Degeneracy of Koszul Homological Series on Lie Algebroids: Production of All Affine Structures, Production of All Riemannian Foliations and Production of All Fedosov Structures

Abstract

The framework of the research whose part of results are published in this work is the category of real vector bundles over finite dimensional differentiable manifolds. The objects of studies are gauge structures on these vector bundles. We are interested in the dynamical properties of the holonomy groups of Koszul connections as well as on their topological properties, i.e., properties that are of homological nature. For the most part the context is the subcategory of Lie algebroids. In addition to other investigations, three open problems are studied in detail. (P1-Affine Geometry): When is a Koszul connection an affine connection? (P2-Riemannian Geometry): When is a Koszul connection a metric connection? (P3-Fedosov Geometry): When is a Koszul connection a symplectic connection? In the category of tangent Lie algebroids our homological approach leads to deep relations of our homological ingredients with the open problem of how to produce labeled foliations the most studied of which are Riemannian foliations. On a Lie algebroid we define two families of differential equations, the family of differential Hessian equations and the family of differential gauge equations. The solutions of these differential equations are implemented to construct homological ingredients which are key tools for our studies of open problems we are concerned with. We introduce Koszul Homological Series. This notion is a machine for converting obstructions whose nature is vector space into obstructions whose nature is homological class. We define the property of Degeneracy and the property Nondegeneracy of Koszul homological Series. The property of Degeneracy is implemented to solve problems (P1), (P2), and (P3). In the abundant literature on Riemannian foliations, we have only cited references directly related to the open problems which are studied using the tools which are introduced in this work. Thus, the property of nondegeneracy is implemented to give a complete solution of the problem posed by E. Ghys, (P4-Differential Topology): How does one produce Riemannian foliations? See our Theorems 12 and 13, which are fruits of a happy conjunction between gauge geometry and differential topology.

1. Introduction

Une différence entre un mathématicien appliqué et un mathématicien pur: Lorsque le premier ne réussit pas à résoudre un problème il change de méthode. Lorsque le second n’arrive pas à résoudre un problème il change de problème.

Eugenio Calabi, Colloque du Centenaire de la naissance d’Elie Cartan, Institut Fourier, Saint-Martin-d’Hères 1968.

Why did I quote the sentences from Eugenio Calabi that served as the oral introduction to his lecture at the Elie Cartan Colloquium in Saint-Martins-d’Hères in 1968? The initial problem that preceded those studied in this paper was the search for a numerical invariant that could serve as a characteristic obstruction to the existence of symplectic structure on a differentiable manifold M. Exchanges with Alan Weinstein, for whom I have great respect, revealed that my method led to tautologies. This led to shift from the first problem to the problem of special symplectic connections on Lie algebroids, which is the symplectic version of the widely studied problem of metric connection on tangent Lie algebroids [1,2,3].

Thus, inspired by Eugenio Calabi’s adage that I quoted, I abandoned the field of differential geometry for gauge geometry, the study of Koszul connections in vector bundles. Another factor that prompted us to change our focus was the numerous discussions we had with Dmitri Alekseevsky, whom I would like to thank here. The readers of this work will see that all the problems I studied and solved are formulated within the framework of gauge geometry.

Through this paper, differentiable manifolds and mappings are assumed to be smooth, i.e., $C^{\infty }.$ Furthermore, manifolds are always assumed connected and paracompact.

Let M be an m-dimensionnal differentiable manifold and let ∇ be a torsion-free Koszul connection on the tangent vector bundle

$$TM\to M.$$

One of the important open problems is the following:

P1: When is ∇ the Levi Civita connection of a positive Riemannian structure $(M,g)$?

This problem is one of those widely studied, see [1,2], and widely debated, see [3]. Some researchers have raised the subsidiary question of whether ∇ determines the positive Riemannian metrics of which it is the Levi Civita connection.

In this work, an approach that does no require the vanishing torsion of ∇ is presented, still considering pseudo-Riemannian connections. For that purpose, we extend the framework to the category of general Lie algebroids

$$A=(V,M,a,[-,-\left]\right).$$

The couple $(V,M)$ stands for the vector bundle

$$V\to M,$$

and a, called anchor mapping, is a vector bundle morphism

$$a:(V,M)\to (TM,M),$$

The vector space of sections $\Gamma \left(V\right)$ is a real Lie algebra whose bracket is

$$\Gamma \left(X\right)\times \Gamma \left(V\right)\ni (s,s^{\prime })\to [s,s^{\prime }]\in \Gamma \left(V\right).$$

The bracket just defined is subject to the following requirement:

$$[s,f{s}^{\prime }]=df\left(a\left(s\right)\right)s^{\prime }+f[s,s^{\prime }]\forall (s,s^{\prime },f)\in \Gamma \left(V\right)\times \Gamma \left(V\right)\times C^{\infty }\left(M\right).$$

In addition to the problem of metric connection on the category of Lie algebroids, two other open problems emerge as follows:

P2: = Problem of affine connection: The question of whether $(A,\nabla )$ is an affine structure on the Lie algebroid A.

P3: = Problem of symplectic connection: The question of whether ∇ is a symplectic connection on the Lie algebroid A. In the case of a tangent Lie algebroid $A\left(M\right)$, the existence of a symplectic connection is the first step toward the answer of the question of whether a given torsion-free gauge structure $(TM,\nabla )$ arises from a Fedosov structure $(M,\omega ,\nabla )$ [4,5].

On a Lie algebroid A, we define two families of differential equations as follows: the family of gauge equations and the family of Hessian equations. The implementations of the solutions of these differential equations are sufficient to solve the open problems P2 and P3. In particular, the solutions of gauges equations are implemented to achieve the study of the problem of metric connections and to answer the subsidiary problem which is stated above. Alternatively, a remarkable fact is that the existence of solutions to the gauge equation will also solve an open problem posed by E. Ghys in the quantitative differential topology as follows:

P4: How does one produce Riemannian foliations? See [6].

It must be highlighted that the current literature on Riemannian foliations deals with foliations in positive Riemannian manifolds [7,8,9,10]. Our approach to producing Riemannian foliations deals with Riemannian manifolds of arbitrary signature.

Below is a short overview of the contents of this paper.

This work is divided into sections including this introduction.

Section 2 is devoted to notation and to a few notions attached to Vector bundles and to Lie algebroids. In addition to the classical gauge dynamics, i.e., the action of the gauge group $GL(V,M)$ on Koszul connections on a vector bundle $(V,M)$, we introduce the metric dynamics which is the action of the group $GM(V,M)$ which is generated by the symmetries defined by inner products. We define gauge equations and other notions which are derived from the solutions of gauge equations. Among these notions are short exact sequences (28) and (29) which link the solutions of gauge equations with relevant bilinear forms on a vector bundle.

Section 3 is devoted to materials which will be deeply implemented in subsequent sections. These materials are objects and properties linked to Koszul connections and which are of homological nature. The relevant relationships and various bridges between these homological notions and the problems P1, P2, and P3 are clarified. Theorems 1 and 2 are among the key tools in our approach to problems P1, P2, and P3 on Lie algebroids.

Section 4 is devoted to the sketch of the gauge topology of Lie algebroids. Section 4 is the core of the paper. Two efficient homological tools are defined. Two major families of objects are defined. (1) The family of Koszul homological series. (2) The (graded) family of degeneracy of Koszul homological series. These two families are implemented to study problems P1, P2, and P3.

Section 5 serves as a transition. On the one hand, it is a section of reminders of the challenges. On the other hand, it is a section of reminders of the tools which are implemented to face these challenges.

Section 6 is partially a return to topology and to affine geometry on Lie algebroids. Given an affine Lie algebroid $(A,\nabla )$ and a left A-module

$$W\to M.$$

The couple $(\nabla ,W)$ gives rise to three types of cochain complexes.

(1)

The KV complex $C_{KV}(\nabla ,W)$ as in [11,12].

(2)

The total KV complex $C_{\tau }(\nabla ,W)$.

(3)

The Chevalley–Eilenberg $C_{CE}(A,W)$

When $(W,M)$ is the trivial vector bundle

$$M\times R\to M,$$

the complex (3) is the de Rham complex. Section 6 is also devoted to these cochain complexes and to their ramifications, such as the affine hyperbolic structure in the sense of Kaup–Koszul [13,14]. See our Theorem 9 (in this paper) about hyperbolic foliations. Section 6 highlights the passage from gauge geometry toward the differential geometry.

Section 7 is totally devoted to canonical tangent Lie algebroids of differentiable manifolds. The results of Section 4, Section 5 and Section 6 acquire new resonances in tangent Lie algebroids, see Theorems 12 and 13. In particular, Theorem 13 completely solves the problem of producing all Riemannian foliations in any signature. Thus, it goes beyond the scope of the problem posed by E. Ghys.

Section 8 offers a reading of Section 7 from the point of view of Erlangen by Felix Klein. In addition to the gauge dynamics GL(TM,M) and the metric dynamics GM(TM,M), Gau(TM,M) is acted by the group of diffeomorphisms $Dif{f}^{★}\left(M\right)$ the invariants of which are called geometric invariants.

Section 9 is devoted to introducing special Fedosov manifolds. Each odd Betti number of a compact special Fedosov manifold is even. This statement follows from the following property of special Fedosov manifolds, as follows: every special Fedosov manifold possesses a canonical Kaehlerian metric. A notable consequence is an interplay between information geometry and the problem of existence of a metric connection. More precisely, in a special statistical Fedosov manifold $(M,g,\omega ,\nabla )$, the Koszul connection ∇ and its g-dual ${\nabla }^{★}$ are both the Levi Civita connection of a Riemannian metric, cf. Theorem 17.

Section 10 is devoted to a few short comments.

2. Gauge Geometry of General Lie Algebroids

2.1. Notation and Basic Notions

Given a differentiable manifold M, the couple $(V,M)$ stands for a real vector bundle V over a differentiable manifold M, the mapping $\pi$ in the following diagram being implicit:

$$V\stackrel{\pi }{⟶}M$$

(1)

$\Gamma \left(V\right)$ is the $C^{\infty }\left(M\right)$-module of sections of $(V,M)$.

$C^{\infty }\left(M\right)$ is the associative algebra of real valued smooth functions on M.

An inner product on $(V,M)$ is a nondegenerate symmetric $C^{\infty }\left(M\right)$-bilinear mapping

$$\Gamma \left(V\right)\times \Gamma \left(V\right)\ni (s,s^{\prime })\to g(s,s^{\prime })\in C^{\infty }\left(M\right)$$

(2)

The set of all inner products on $(V,M)$ is denoted by $\mathit{Me}\left(V\right)$.

A symplectic product on $(V,M)$ is a nondegenerate skew symmetric $C^{\infty }\left(M\right)$-bilinear mapping

$$\Gamma \left(V\right)\times \Gamma \left(V\right)\ni (s,s^{\prime })\to \omega (s,s^{\prime }).$$

(3)

The set of all symplectic products on $(V,M)$ is denoted by $\mathit{Symp}\left(V\right)$.

Given a non negative integer k a $(V,M)$-valued differential k-form on M is skew-symmetric $C^{\infty }\left(M\right)$-k-multilinear mapping of $\Gamma \left(TM\right)$ in $\Gamma \left(V\right)$,

$${\Lambda }^k(\Gamma \left(TM\right))\ni X_1\wedge \dots \wedge X_k\to \theta (X_1\wedge \dots \wedge X_k)\in \Gamma \left(V\right).$$

(4)

The $C^{\infty }\left(M\right)$-module of $(V,M)$-valued differential k-forms is denoted by ${\Omega }^k(M,V)$.

A Koszul connection on $(V,M)$ is a linear mapping

$$\Gamma \left(V\right)\ni s\to \nabla s\in {\Omega }^1(M,V)$$

(5)

which is subject to the following requirement

$$\nabla (f·s)=f·\nabla s+df·s\forall f\in C^{\infty }\left(M\right).$$

(6)

Thus, by putting

$$\nabla s\left(X\right)={\nabla }_{X}s,$$

one has

$${\nabla }_X(f·s)=f·{\nabla }_{X}s+df\left(X\right)·s.$$

Definition 1.

Let ∇ be a Koszul connection on $(V,M)$, then the couple $(V,\nabla )$ is called a gauge structure on $(V,M).$

The family of all gauge structures on $(V,M)$ is denoted by Gau(V,M).

Let $(V^{★},M)$ be the dual vector bundle of $(V,M)$ and let $\left(End\right(V),M)$ be the vector bundle of vector bundle morphisms of $(V,M)$ in itself. We canonically identify the vector bundle $\left(End\right(V),M)$ with the tensor product of vector bundles $(V^{★}\otimes V,M)$.

The $C^{\infty }\left(M\right)$-module of sections of $End\left(V\right)$ is denoted by gl(V,M).

GL(V,M) is the set of invertible sections in gl(V,M).

The action

$$\mathit{GL}(V,M)\times \mathit{Gau}(V,M)\ni (\Phi ,\nabla )\to {\nabla }^{\varphi }\in \mathit{Gau}(V,M)$$

is defined as follows:

$${\nabla }^{\Phi }=\Phi \circ \nabla \circ {\Phi }^{-1}.$$

(7)

Thus, if s is a section of $(V,M)$, the $(V,M)$-valued differential 1-form ${\nabla }^{\Phi }s$ is defined as follows:

$${\nabla }_X^{\Phi }s=\Phi \left({\nabla }_X{\Phi }^{-1}\left(s\right)\right).$$

(8)

The couple

$$\left[\mathit{GL}\right(V,M),\mathit{Gau}(V,M\left)\right]$$

is called the category of gauge structures on $(V,M).$

2.2. Metric Connections and Symplectic Connections on Vector Bundles

Definition 2.

A Koszul connection ∇ is called a metric connection on $(V,M)$ if there exists an inner product g subject to the following identity

$$X.g(s,s^{\prime })-g({\nabla }_{X}s,s^{\prime })-g(s,{\nabla }_{X}s{s}^{\prime })=0.$$

(9)

Mutatis mutandis, a Koszul connection ∇ is called a symplectic connection on (V,M) if there exists a symplectic product ω which is subject to the following identity,

$$X\omega (s,s^{\prime })-\omega ({\nabla }_{X}s,s^{\prime })-\omega (s,{\nabla }_X{s}^{\prime })=0.$$

(10)

2.3. Problem of Metric Connection and Problem of Symplectic Connection on Vector Bundles

Warning I. From the point of view of the Erlangen program of Felix Klein, we should think of gauge geometry as the study of invariants of the gauge group GL(V,M).

Warning II. In this work we also think of gauge geometry as the study of invariants of a gauge structure $(V,\nabla )$, viz.,. The study of those objects defined on $(V,M)$ which are ∇-parallel.

Thus arise two open problems on the category of gauge structures on a vector bundle

$$V\to M.$$

Problem 1. When is a Koszul connection metric connection?

References [1,2,3] deal with Problem 1 on canonical tangent algebroids $(TM,M)$.

Problem 2. There are many open problems regarding connections on symplectic structures [15,16]. In this work, one of the questions we are interested in is as follows: When is a Koszul connection a symplectic connection?

Of course Problem 2 makes sense only when the rank of $(V,M)$ is even.

In the case $(V,M)$ is the tangent vector bundle $(TM,M)$, the solution to Question 1 is well known and widely discussed when ∇ is torsion-free; see R. Atkins [1] and B. Schmidt [2]. A strategy suggested by Thurston consists of calculating the holonomy group of ∇, see Bill Thurston Stack Exchange [3]. Regarding Question 2, there is no answer up to author knownledge.

2.4. The Metric Dynamic on Gau(V,M)

Every inner product

$$(s,s^{\prime })\to g(s,s^{\prime })$$

gives rise to the following symmetry on Gau(V,M):

$$\nabla \to {\nabla }^g$$

(11)

where ${\nabla }^g$ is defined by the following identity:

$$g({\nabla }_X^{g}s,s^{\prime })=Xg(s,s^{\prime })-g(s,{\nabla }_X{s}^{\prime }).$$

(12)

It is obvious that (11) is a symmetry, that is to say that

$${\left({\nabla }^g\right)}^g=\nabla .$$

The group $\mathit{GM}(V,M)$ which is generated by symmetries as in (12) is called the metric group of Gau(V,M).

The couple $\left[\mathit{GM}\right(V,M),\mathit{Gau}(V,M\left)\right]$ is called the metric dynamics on Gau(V,M).

Remark: It is easy to check that $GL(V,M)\cap GM(V,M)$ is not trivial.

Definition 3.

(1) Objects and properties defined on $(V,M)$ which are invariant under the action of the group GL(V,M) are called gauge invariants.

(2) Objects and properties defined on $(V,M)$ which are invariant under the action of the group GM(V,M) are called metric invariants.

Warning III. In addition to Question 1 and Question 2, this work is also devoted to the study of properties and objects defined on Gau(V,M) which are either metric invariants or gauge invariants. This quest is the gauge geometry following Felix Klein Erlangen.

Let g be an inner product on $(V,M)$, let ∇ be a Koszul connection on $(V,M)$, and let $\varphi$ be a section of GL(V,M). We define the inner $g^{\varphi }$ and Koszul connection ${\nabla }^{\varphi }$ by putting

$$g^{\varphi }(s,s^{\prime })=g({\varphi }^{-1}\left(s\right),{\varphi }^{-1}\left(s^{\prime }\right))$$

(13)

$${\nabla }_X^{\varphi }s=\varphi \left({\nabla }_X{\varphi }^{-1}\left(s\right)\right).$$

(14)

Thus, by incorporating (13) and (14), the metric dynamics and the gauge dynamics are linked with each other as follows:

$${\left({\nabla }^{\varphi }\right)}^{g^{\varphi }}={\left({\nabla }^g\right)}^{\varphi }.$$

(15)

2.5. Gauge Equations on Vector Bundles

To every pair $(\nabla ,{\nabla }^{★})$ of Koszul connections on $(V,M)$, we assign the following first-order differential operator

$$\Gamma \left(End\left(V\right)\right)\ni \varphi \to D^{\nabla {\nabla }^{★}}\left(\varphi \right)\in {\Omega }^1(M,End\left(V\right))$$

which is defined as follows

$$D^{\nabla {\nabla }^{★}}\left(\varphi \right)={\nabla }^{★}\circ \varphi -\varphi \circ \nabla .$$

The End(V)-valued differential 1-form $D^{\nabla {\nabla }^{★}}\left(\varphi \right)$ is defined as follows:

$$\left[D^{\nabla {\nabla }^{★}}\left(\varphi \right)\right]\left(X\right)={\nabla }_X^{★}\circ \varphi -\varphi \circ {\nabla }_X.$$

(16)

Mutatis mutandis, we get the second differential operator

$$\left[D^{{\nabla }^{★}\nabla }\left(\varphi \right)\right]\left(X\right)={\nabla }_X\circ \varphi -\varphi \circ {\nabla }_X^{★}.$$

(17)

Thus, we will concern ourselves with the following two first-order linear differential equations:

$${\nabla }^{★}\circ \varphi -\varphi \circ \nabla =0,$$

(18)

$$\nabla \circ \psi -\psi \circ {\nabla }^{★}=0.$$

(19)

(18) and (19) are called gauge equations of the pair $(\nabla ,{\nabla }^{★})$.

The unknowns $\varphi$ and $\psi$ are sections of End(V).

The vector spaces of solutions of (18) and of (18) are denoted by

$$J_{{\nabla }^{★}\nabla }\left(V\right)$$

and by

$$J_{\nabla \nabla ★}\left(V\right),$$

respectively.

2.6. Two Fundamental Short Exact Sequences on Vector Bundles

We are concerned with a couple $(g,\nabla )$ where g is a positive inner product on the vector bundle $(V,M)$ and ∇ is a Koszul connection on $(V,M)$. We then deal with the pair $(\nabla ,{\nabla }^g)$.

To make it simpler, we will set

$$J_{\nabla ,g}\left(V\right)=J_{\nabla {\nabla }^g}\left(V\right)$$

$$J_{{\nabla }^g,g}\left(V\right)=J_{{\nabla }^g\nabla }\left(V\right).$$

Given

$$\varphi \in J_{\nabla ,g}$$

we introduce the pair of bilinear products $\left(q\right(g,\varphi ),\omega (g,\varphi \left)\right)$, which are defined as

$$2q(g,\varphi )(s,s^{\prime })=g(\varphi \left(s\right),s^{\prime })+g(s,\varphi \left(s^{\prime }\right)),$$

(20)

$$2\omega (g,\varphi )(s,s^{\prime })=g(\varphi \left(s\right),s^{\prime })-g(s,\varphi \left(s^{\prime }\right)).$$

(21)

We also introduce the following pair

$$(\Phi ,{\Phi }^{★})\subset \Gamma \left(End\left(V\right)\right),$$

which is defined as follows:

$$g(\Phi \left(s\right),s^{\prime })=q(g,\varphi )(s,s^{\prime }),$$

(22)

$$g({\Phi }^{★}\left(s\right),s^{\prime })=\omega (g,\varphi )(s,s^{\prime }).$$

(23)

Definition 4.

If $\varphi \in \Gamma \left(\mathit{End}\right(V\left)\right),$ its transpose ${\varphi }^t$ is defined by the following relation:

$$g\left(\varphi s,s^{\prime }\right)=g(s,{\varphi }^t{s}^{\prime }),$$

for any sections $s,s^{\prime }$ of $(V,M).$

Proposition 1.

If ϕ satisfies the gauge equation ${\nabla }^g\varphi =\varphi \nabla ,$ then ${\nabla }^g{\varphi }^t={\varphi }^t\nabla .$

Proof. 

Let X be a vector field, and let $s,s^{\prime }$ be any sections of $(V,M).$ Then,

$$Xg\left({\varphi }^{t}s,s^{\prime }\right)=Xg\left({\varphi }^{t}s,s^{\prime }\right),$$

hence,

$$g\left({\nabla }_X^g{\varphi }^{t}s,s^{\prime }\right)+g\left({\varphi }^{t}s,{\nabla }_X{s}^{\prime }\right)=g\left({\nabla }_{X}s,\varphi s^{\prime }\right)+g\left(s,{\nabla }_X^g\varphi s^{\prime }\right)$$

Using the gauge equation, it comes to the following:

$$\begin{array}{cc}\hfill g\left({\nabla }_{X}s,\varphi s^{\prime }\right)+g\left(s,{\nabla }_X^g{s}^{\prime }\right)& =g\left({\nabla }_{X}s,\varphi s^{\prime }\right)+g\left(s,\varphi {\nabla }_X{s}^{\prime }\right)=\hfill \\ & g\left({\varphi }^t{\nabla }_{X}s,s^{\prime }\right)+g\left({\varphi }^{t}s,{\nabla }_X{s}^{\prime }\right)\hfill \end{array}.$$

Finally,

$$g\left({\nabla }_X^g{\varphi }^{t}s,s^{\prime }\right)-g\left({\varphi }^t{\nabla }_{X}s,s^{\prime }\right)=0,$$

and the claim is proved. □

Proposition 2.

Let $\varphi \in \Gamma \left(\mathit{End}\right(V\left)\right).$ Then the bilinear product $T_{\varphi }$ defined by the following:

$$T_{\varphi }(s,s^{\prime })=g\left(\varphi s,s^{\prime }\right)$$

is ∇-parallel.

Proof. 

Let X be a vector field, and let $s,s^{\prime }$ be any sections of $(V,M).$ By definition,

$$\left({\nabla }_X{T}_{\varphi }\right)(s,s^{\prime })=X{T}_{\varphi }(s,s^{\prime })-T_{\varphi }\left({\nabla }_{X}s,s^{\prime }\right)-T_{\varphi }\left(s,{\nabla }_X{s}^{\prime }\right),$$

so:

$$\left({\nabla }_X{T}_{\varphi }\right)(s,s^{\prime })=g\left({\nabla }_X^g\varphi s,s^{\prime }\right)+g\left(\varphi s,{\nabla }_X{s}^{\prime }\right)-g\left(\varphi {\nabla }_{X}s,s^{\prime }\right)-g\left(\varphi s,{\nabla }_X{s}^{\prime }\right).$$

Since ${\nabla }_X^g\varphi =\varphi {\nabla }_X,$

$$\left({\nabla }_X{T}_{\varphi }\right)(s,s^{\prime })=0.$$

□

Lemma 1.

The bilinear products $q(g,\varphi )$ and $\omega (g,\varphi )$ are ∇-parallel.

Proof. 

$q(g,\varphi )$ (resp., $\omega (g,\varphi )$) is just the previous bilinear product for the symmetric (resp., skew-symmetric) part of $\varphi$, and both satisfy the gauge equation. □

Here are two corollaries of Lemma 1.

Corollary 1.

The sections Φ and ${\Phi }^{★}$ are solutions of the differential Equation (18).

Proof. 

$$\begin{gathered} \left(A\right):\left({\nabla }_{X}q(g,\varphi )\right)(s,s^{\prime })=Xg(\Phi \left(s\right),s^{\prime })-g(\Phi \left({\nabla }_{X}s\right),s^{\prime })-g(\Phi \left(s\right),{\nabla }_X{s}^{\prime }) \\ =g(\Phi \left({\nabla }_{X}s\right)-{\nabla }_X^g\Phi \left(s\right),s^{\prime }) \\ =0\forall (X,s,s^{\prime }). \end{gathered}$$

Therefore,

$$\Phi \left({\nabla }_{X}s\right)-{\nabla }_X^q\Phi \left(s\right)=0\forall (X,s).$$

Similarly, computing

$$\left(B\right):\left({\nabla }_X\omega (g,\varphi )\right)(s,s^{\prime })=0\forall (X,s,s^{\prime })$$

yields the following identity:

$${\nabla }_X^g{\Phi }^{★}\left(s\right)-{\Phi }^{★}\left({\nabla }_{X}s\right)=0.$$

□

Corollary 2.

Lemma 1 yields the following g-orthogonal decompositions:

$$V=Ker(\Phi )\oplus Im(\Phi ),$$

(24)

$$V=Ker\left({\Phi }^{★}\right)\oplus Im\left({\Phi }^{★}\right)$$

(25)

Proof. 

Let us take

$$\Phi \left(s\right)\in Ker(\Phi ),$$

then

$$\begin{gathered} \left(A\right):g(\Phi (\Phi \left(s\right)),s^{\prime })=g(\Phi \left(s\right),\Phi \left(s^{\prime }\right)) \\ =0\forall s^{\prime }. \end{gathered}$$

Therefore,

$$g(\Phi (s),\Phi (s\left)\right)=0.$$

Since g is positive definite, we get

$$\Phi \left(s\right)=0.$$

The same argument yields the following fact,

$$\left(B\right):Ker\left({\Phi }^{★}\right)\cap Im\left({\Phi }^{★}\right)=0.$$

□

Let $H(\nabla )$ and $H\left({\nabla }^g\right)$ be the holonomy groups of ∇ and ${\nabla }^g$, respectively.

The following proposition is a straight corollary of Lemma 1.

Proposition 3.

When we take into account (16) and (17), then

(1) 

$Ker(\Phi )$ and $Ker\left({\Phi }^{★}\right)$ are stable under the action of the holonomy group $H(\nabla )$.

(2) 

$Im(\Phi )$ and $Im\left({\Phi }^{★}\right)$ are stable under the action of the holonomy group $H\left({\nabla }^g\right)$.

The vector space of ∇-parallel symmetric bilinear products on $(V,M)$ and the vector space of ∇-parallel skew symmetric bilinear products on $(V,M)$ are denoted by ${\mathbf{S}}_2^{\nabla }\left(V\right)$ and by ${\Omega }_2^{\nabla }\left(V\right)$, respectively.

Then we introduce the following mapping:

$$J_{\nabla ,g}\ni \varphi \to (\Phi ,{\Phi }^{★})\subset J_{\nabla ,g}$$

(26)

and we perform it to set the following identification:

$$({\Phi }^{★},\Phi )=(\omega (g,\varphi ),q(g,\varphi )).$$

(27)

By virtue of Lemma 1 we know that

$$(\omega (g,\varphi ),q(g,\varphi ))\in {\Omega }_2^{\nabla }\left(V\right)\times {\mathbf{S}}_2^{\nabla }\left(V\right)$$

Therefore we obtain the following fundamental short exact sequences of vector spaces:

$$0\to {\Omega }_2^{\nabla }\left(V\right)\to J_{\nabla ,g}\left(V\right)\to {\mathbf{S}}_2^{\nabla }\left(V\right)\to 0.$$

(28)

$$0\to {\Omega }_2^{{\nabla }^g}\left(V\right)\to J_{{\nabla }^g,g}\left(V\right)\to {\mathbf{S}}_2^{{\nabla }^g}\left(V\right)\to 0.$$

(29)

Exact sequences (28) and (29) will be involved to discuss some important open problems in differential topology. Among these open problems is the following question, cf. [6],

EGQ: How does one produce Riemannian foliations?

For details, see P. Molino [8], BL Reinhart [7], and Moedijk-MCul [10]. It is to be noted that many of these references deal with Riemannian foliations on positive definite Riemannian manifolds. In contrast, our approach to producing Riemannian foliations is based on fundamental exact sequences (28) and (29). We will prove that our method used to produce Riemannian foliations works in pseudo-Riemannian geometry. Thus we will use sequences (28) and (29) to completely solve that open problem of production of all Riemannian foliations.

3. Gauge Geometry and Gauge Topology on Lie Algebroids

In this section we focus on Koszul connections on algebroids.

3.1. Basic Algebraic Tools

Reminder: Henceforth we remind the reader that Gauge invariants are either objects or properties which are invariant under the action of the gauge group. Metric invariants are either properties or objects that are invariant under the action of the metric group. By gauge topology on a Lie algebroid A we mean the study of gauge concerns on A that are linked with objects or with properties which are of homologcal nature on A.

We remind the reader that the structure of a Lie algebroid on a vector $(V,M)$ is a couple $(a,[-,-\left]\right)$ where a is vector bundle morphism

$$a:(V,M)\to (TM,M)$$

and the following arrow

$$\Gamma \left(V\right)\times \Gamma \left(V\right)\ni (s,s^{\prime })\to [s,s^{\prime }]\in \Gamma \left(V\right)$$

provides a structure of real Lie algebra on $\Gamma \left(V\right)$ which is related with a as follows

$$[s,f.s^{\prime }]=\left(a\left(s\right)\right)f.s+f.[s,s^{\prime }]\forall (s,s^{\prime })\subset \Gamma \left(V\right),\forall f\in C^{\infty }\left(M\right).$$

(30)

The requirement (30) implies that

$$a\left([s,s^{\prime }]\right)=[a\left(s\right),a\left(s^{\prime }\right)],$$

(31)

where the right side member is the Poisson bracket of vector fields.

We also recall that a left module of Lie algebroid $(V,M,a,[-,-\left]\right)$ is a vector bundle $(W,M)$ whose space of sections $\Gamma \left(W\right)$ is a real left module of the Lie algebra $\Gamma \left(V\right)$. Moreover, the left action

$$\Gamma \left(V\right)\times \Gamma \left(W\right)\ni (s,w)\to s.w\in \Gamma \left(W\right)$$

satisfies the following identity:

$$s.fw=df\left(a\right(s\left)\right)w+fs.w\forall f\in C\infty \left(M\right).$$

Before pursuing, we put

$$A=(V,M,a,[-,-\left]\right).$$

(32)

Both $(V,M)$ and $(TM,M)$ are examples of left modules of A under the following left actions:

$$L_s\left(s^{\prime }\right)=[s,s^{\prime }]\forall (s,s^{\prime })\subset \Gamma \left(V\right);$$

(33)

$$L_s\left(X\right)=[a\left(s\right),X]\forall (s,X)\in \Gamma \left(V\right)\times \Gamma \left(TM\right).$$

(34)

Let us put

$$V^{p,q}=V^{★\otimes p}\otimes V^{\otimes q},$$

(35)

$$T^{p,q}\left(M\right)=T{M}^{★\otimes p}\otimes T{M}^{\otimes q}.$$

(36)

By extending (33) and (34) to tensor spaces, the vector bundles $V^{p,q}$ and $T^{p,q}\left(M\right)$ are made into left modules of the Lie algebroid A.

Occasionally, if there is no risk of confusion, we use the following notation:

${\Omega }_{p,q}^k(M,V)$ represents the vector space $V^{p,q}$-valued differential k-forms on M. In other words,

$${\Omega }_{p,q}^k(M,V)={\Omega }^k(M,V^{p,q}).$$

$H_{p,q}^k(M,V)$ is the Chevalley–Eilenberg cohomology space of M with coefficients in $V^{p,q}$.

$H_{p,q}^k(A,V)$ is the Chevalley–Eilenberg cohomology space of A with coefficients in $V^{p,q}$.

Notation: On a Lie algebroid

$$A=(V,M,a,[-,-\left]\right),$$

the k-th exterior power of the mapping a is denoted by $a^{\Lambda k}$, thus

$$a^{\Lambda k}(s_1\wedge \dots \wedge s_k)=a\left(s_1\right)\wedge \dots \wedge a\left(s_k\right).$$

(37)

Therefore, given an A-module $(W,M)$ we put

$${\Omega }^k(A,W^{p,q})={\Omega }^k(M,W^{p,q})\circ a^{\Lambda k}.$$

(38)

Without the statement of the contrary, the basic definition we are dealing with is given below.

Definition 5.

A gauge structure on a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

is a first-order differential operator

$$\Gamma \left(V\right)\ni s\to \nabla s\in {\Omega }^1(A,V^{1,1})$$

which is subject to the following requirement,

$$\nabla f.s=df.s+f.\nabla s\forall f\in C^{\infty }\left(M\right).$$

Warning.

According to (37) and to (38), here is the meaning of the requirement ∇ subject to

$$(df.s+f.\nabla s)\left(s^{\prime }\right)=df\left(a\left(s^{\prime }\right)\right)s+f{\nabla }_{a\left(s^{\prime }\right)}s.$$

(39)

Let us consider a Lie algebroid

$$A=(V,M,a,[-,-\left]\right);$$

let us endow A with a gauge structure as follows:

$$(A,\nabla )=(V,M,a,[-,-],\nabla ).$$

The curvature $R^{\nabla }$ and the torsion $T^{\nabla }$ are defined as follows:

$$R^{\nabla }(s,s^{\prime };s^{″})={\nabla }_{a\left(s\right)}{\nabla }_{a\left(s^{\prime }\right)}{s}^{″}-{\nabla }_{a\left(s^{\prime }\right)}{\nabla }_{a\left(s\right)}{s}^{″}-{\nabla }_{a\left([s,s^{\prime }]\right)}{s}^{″},$$

(40)

$$T^{\nabla }(s,s^{\prime })={\nabla }_{a\left(s\right)}{s}^{\prime }-{\nabla }_{a\left(s^{\prime }\right)}s-[s,s^{\prime }].$$

(41)

On a Lie algebroid A, a gauge structure $(A,\nabla )$ whose tensor $T^{\nabla }$ vanishes identically is called symmetric (or torsion-free).

Of course there exist symmetric gauge structures. Indeed, every gauge structure $(A,\nabla )$ is associated with the symmetric gauge structure $(A,{\nabla }^{\prime })$, which is defined as follows:

$${\nabla }_{a\left(s\right)}^{\prime }{s}^{\prime }={\nabla }_{a\left(s\right)}{s}^{\prime }-{\displaystyle \frac{1}{2}}{T}^{\nabla }(s,s^{\prime }).$$

The family of all symmetric gauge structures on A is denoted by SGau(A).

Warnings: We recall that on a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

every inner product g admits a unique symmetric metric connection ∇ which is defined by the following well known Koszul formula:

$$\begin{array}{c}g({\nabla }_{a\left(s^{″}\right)}s,s^{\prime })={\displaystyle \frac{1}{2}}[a\left(s\right)g(s^{\prime },s^{″})-a\left(s^{\prime }\right)g(s,s^{″})+a\left(s^{″}\right)g(s,s^{\prime })\\ -g([s,s^{\prime }],s^{″})-g([s^{\prime },s^{″}],s)-g([s,s^{″}],s^{\prime })]\forall (s,s^{\prime },s^{″})\subset \Gamma \left(V\right).\end{array}$$

(42)

Mutatis mutandis, on A, every symplectic product ω admits symmetric symplectic connections. Those symmetric symplectic may be constructed following a formalism introduced in Biliavski–Cahen–Gutt–Rawnsky [16].

Indeed, starting from any symmetric connection ${\nabla }^0$, one defines

$$\Gamma \left(V\right)\times \Gamma \left(V\right)\ni (s,s^{\prime })\to N(s,s^{\prime })\in \Gamma \left(V\right),$$

as follows:

$$\left({\nabla }_{a\left(s\right)}^0\omega \right)(s^{\prime },s^{″})=\omega (N(s,s^{\prime }),s^{″}).$$

(43)

Then the following connection is symplectic,

$${\nabla }_{a\left(s\right)}{s}^{\prime }={\nabla }_{a\left(s\right)}^0{s}^{\prime }+{\displaystyle \frac{1}{3}}(N(s,s^{\prime })+N(s^{\prime },s)).$$

(44)

Otherwise, it satisfies the following identity,

$$a\left(s\right)\omega (s^{\prime },s^{″})-\omega ({\nabla }_{a\left(s\right)}{s}^{\prime },s^{″})-\omega (s^{\prime },{\nabla }_{a\left(s\right)}{s}^{″})=0\forall (s,s^{\prime },s^{″})\subset \Gamma \left(V\right).$$

(45)

Definition 6.

An affine structure on a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

is a gauge structure $(A,\nabla )$ whose both curvature $R^{\nabla }$ and torsion $T^{\nabla }$ vanish identically.

In view of this definition, the question of whether every Lie algebroid admits affine structures arises naturally. One of our aims is to point out a characteristic obstruction to the existence of affine structure on a Lie algebroid. That obstruction is of homological nature.

3.2. The Hessian Differential Operators of Gauge Structures on Lie Algebroids

The gauge structure $(A,\nabla )$ is associated with the $V^{2,1}$-valued differential operator ${\nabla }^2$ which is defined on $(V,M)$,

$$\Gamma \left(V\right)\ni s^{″}\to {\nabla }^2{s}^{″}\in \Gamma \left(V^{2,1}\right).$$

The $C^{\infty }\left(M\right)$-bilinear ${\nabla }^2{s}^{″}$ is defined as follows:

$$\left({\nabla }^2{s}^{″}\right)(s,s^{\prime })={\nabla }_{a\left(s\right)}\left({\nabla }_{a\left(s^{\prime }\right)}{s}^{″}\right)-{\nabla }_{a\left({\nabla }_{a\left(s\right)}{s}^{\prime }\right)}{s}^{″}.$$

(46)

Now we introduce the Hessian equation of $(A,\nabla )$,

$${\nabla }^2{s}^{″}=0.$$

(47)

The unknown $s^{″}$ is an element of $\Gamma \left(V\right)$. The differential Equation (47) is of order 2 and type 0.

Let $(x_1,\dots ,x_m)$ be local coordinates. We put

$$Z={\sum }_p{Z}^p{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_p}}$$

The geometric symbol of the Hessian equation above is as follows:

$${\Theta }_{ij}^p:{\displaystyle \frac{{\mathsf{\partial }}^2{Z}^p}{\mathsf{\partial }{x}_i\mathsf{\partial }{x}_j}}.$$

Consequently, the vector space of its solutions is finite dimensional and

$$dim\left(J_{\nabla }\right)\le m(m+2).$$

If the Koszul connection ∇ is torsion-free, then the corresponding Hessian equation is a Lie equation. The vector space of solutions of the Hessian Equation (47) is denoted by

$$J_{\nabla }\left(A\right)=\left\{s\in \Gamma \left(V\right)s.t.{\nabla }^{2}s=0\right\}$$

(48)

We endow $\Gamma \left(V\right)$ with the product defined by $(V,\nabla )$, viz.,

$$s.s^{\prime }={\nabla }_{a\left(s\right)}{s}^{\prime }.$$

(49)

Proposition 4.

Under the product in (49), $J_{\nabla }\left(A\right)$ is an associative algebra.

Proof. 

In fact, let $\xi$ be a germ of a section of $(V,M)$ defined on an open set

$$U\subset M.$$

$$\xi \in J_{\nabla }\left(A\right)$$

if and only if

$$s.(s^{\prime }.\xi )=(s.s).\xi \forall (s^{\prime },s^{″})\subset J_{\nabla }\left(A^{\circ }\right).$$

Thus, if

$$(\xi ,{\xi }^{\prime })\subset J_{\nabla }\left(A\right),$$

then

$$\begin{gathered} s.(s^{\prime }.(\xi .{\xi }^{\prime }))=s.((s^{\prime }.\xi ).{\xi }^{\prime }) \\ =(s.(s^{\prime }.\xi )).{\xi }^{\prime } \\ =((s.s^{\prime }).\xi ).{\xi }^{\prime } \\ =(s.s^{\prime })(\xi .{\xi }^{\prime }) \end{gathered}$$

□

Proposition 5.

If $(A,\nabla )$ is an affine structure on a Lie algebroid A, then the associative algebra $J_{\nabla }\left(A\right)$ is $(A,\nabla )$-preserving.

Proof. 

We recall the action of $J_{\nabla }\left(A\right)$ on $(A,\nabla )$ as follows:

$${(L_s\nabla )}_s^{\prime }{s}^{″}=[s,{\nabla }_{a\left(s^{\prime }\right)}{s}^{″}]-{\nabla }_{a\left([s,s^{\prime }]\right)}{s}^{″}-{\nabla }_{a^{\prime }\left(s^{\prime }\right)}[s,s^{\prime }]$$

Since both $T^{\nabla }$ and $R^{\nabla }$ vanish identically, we rewrite the right side of the equality above as follows:

$$\begin{gathered} {\nabla }_{a\left(s\right)}\left({\nabla }_{a\left(s^{\prime }\right)}{s}^{″}\right)-{\nabla }_{a\left({\nabla }_{a\left(s^{\prime }\right)}{s}^{″}\right)}s-{\nabla }_{a\left([s,s^{\prime }]\right)}{s}^{″}-{\nabla }_{a\left(s^{\prime }\right)}\left({\nabla }_{a\left(s\right)}{s}^{\prime }\right)+{\nabla }_{a\left(s^{\prime }\right)}\left({\nabla }_{a\left(s^{″}\right)}s\right) \\ =R^{\nabla }(s,s^{\prime };s^{″})+\left({\nabla }^{2}s\right)(s^{\prime },s^{″}) \\ =0\forall s\in J_{\nabla }\left(A\right),\forall (s^{\prime },s^{″})\subset \Gamma \left(V\right). \end{gathered}$$

□

Definition 7.

A gauge structure $(A,\nabla )$ is called regular if $dim\left(J_{\nabla }\left(A\right)\left(p\right)\right)$ does not depend on the point $p\in M$.

The family of all regular symmetric gauge structures is denoted by RSGau(A).

On a regular Lie algebroid A, we also fix g, a positive definite inner product on $(V,M)$. Then we denote by $J_{\nabla }^g\left(A\right)$ the Lie subalgebra of $(\Gamma (V),[-,-\left]\right)$ which is generated by all the sections of $(V,M)$ which are g-orthogonal to $J_{\nabla }\left(A\right)$.

Of course we already noticed that according to (33) and (34), all of the vector spaces $V^{p,q}$ and $T{M}^{p,q}$ are left modules of the Lie algebra $J^g\nabla \left(A\right)$. We are interested in the Chevalley–Eilenberg cohomology of $J^g\nabla \left(A\right)$ with coefficients in those left modules.

The present context is the Lie algebroid

$$A=(V,M,a,[-,-\left]\right),$$

and $(A,\nabla )$ is a regular torsion-free gauge structure on A. We recall the formula we are concerned with,

$${(L_s\nabla )}_{a\left(s^{\prime }\right)}{s}^{″}=[s,{\nabla }_{a\left(s^{\prime }\right)}{s}^{″}]-{\nabla }_{a\left([s,s^{\prime }]\right)}{s}^{″}-{\nabla }_{a\left(s^{\prime }\right)}[s,s^{″}],\forall (s,s^{\prime },s^{″})\subset \Gamma \left(V\right).$$

We put

$$A_{\nabla }\left(V\right)=\left\{s\in \Gamma \left(V\right)s.t{(L_s\nabla )}_{a\left(s^{\prime }\right)}{s}^{″}=0\forall (s^{\prime },s^{″})\subset \Gamma \left(V\right)\right\}.$$

(50)

One of the key tools that we will frequently use in the following is Theorem 1, which we prove (below).

Theorem 1.

The vector space $A_{\nabla }$ (V) as in (50) cannot contain any non-null left module of the associative algebra $C^{\infty }\left(M\right).$

Demonstration. We consider

$$s\in A_{\nabla }\left(V\right),(f,h)\subset C^{\infty }\left(M\right)$$

with

$$s\ne 0$$

and we put

$$\xi =hs.$$

If one assumes that

$$spa{n}_{C^{\infty }\left(M\right)}\left(s\right)\subset A_{\nabla }\left(V\right),$$

then

$$(\xi ,f\xi )\subset A_{\nabla }\left(V\right)\forall (\xi ,f)\in \Gamma \left(V\right)\times C^{\infty }\left(M\right).$$

Therefore, one has the following identity:

$$(care.1)-[{\nabla }_{a\left(s^{\prime }\right)}{s}^{″},f\xi ]+{\nabla }_{a\left([s^{\prime },f\xi ]\right)}{s}^{″}+{\nabla }_{a\left(s^{\prime }\right)}[s^{″},f\xi ]=0\forall (f,s^{\prime },s^{″}).$$

We take into account the following identity:

$$L_{\xi }\nabla =0;$$

then the left hand member of (care.1) yields the following identity:

$$\begin{gathered} -df\left(a\left({\nabla }_{a\left(s^{\prime }\right)}{s}^{″}\right)\right)\xi +df\left(a\left(s^{\prime }\right)\right){\nabla }_{\xi }{s}^{″} \\ +\left(a\left(s^{\prime }\right)(a\left(s^{″}\right).f)\right)\xi +df\left(a\left(s^{″}\right)\right){\nabla }_{a\left(s^{\prime }\right)}\xi df\left(a\left(s^{\prime }\right)\right)[s^{″},\xi ]=0\forall (f,s^{\prime },s^{″}). \end{gathered}$$

Since ∇ is torsion-free, the identity above yields the following one:

$$(care.2)(df\left(a\left({\nabla }_{a\left(s^{\prime }\right)}{s}^{″}\right)\right)-a\left(s^{\prime }\right)\left(a\left(s^{″}\right)f\right))\xi =df\left(a\left(s^{″}\right)\right)({\nabla }_{a\left(s^{\prime }\right)}\xi +df\left(a\left(s^{\prime }\right)\right){\nabla }_{a\left(s^{″}\right)}\xi ).$$

Now remember that

$$\xi =hs,$$

then (care.2) becomes

$$\begin{gathered} (df\left(a\left({\nabla }_{a\left(s^{\prime }\right)}{s}^{″}\right)\right)-a\left(s^{\prime }\right)\left(a\left(s^{″}\right)f\right))hs=df\left(a\left(s^{″}\right)\right){\nabla }_{a\left(s^{\prime }\right)}hs+df(a\left(s^{″}\right){\nabla }_{a\left(s^{″}\right)}hs \\ =h[df\left(a\left(s^{″}\right)\right){\nabla }_{a\left(s^{\prime }\right)}s+df\left(a\left(s^{\prime }\right)\right){\nabla }_{a\left(s^{″}\right)s}] \\ +[df\left(a\left(s^{\prime }\right)\right)dh\left(a\left(s^{″}\right)\right)+df\left(a\left(s^{″}\right)\right)deh\left(a\left(s^{\prime }\right)\right)]s. \end{gathered}$$

One takes into account (care.2) to deduce the following identity

$$(care.3)[df\left(a\left(s^{\prime }\right)\right)dh\left(a\left(s^{″}\right)\right)+df\left(a\left(s^{″}\right)\right)dh\left(a\left(s^{\prime }\right)\right)]s=O\forall (f,h)\subset C^{\infty }\left(M\right).$$

The conclusion of identity (care.3) is

$$s=0.$$

There is contradiction; consequently, the theorem is demonstrated.

Comments:

Comm. 1. In global analysis on differentiable manifolds, it is well known that given a gauge structure $(TM,M,D)$ the subgroup of D-preserving diffeomorphisms $Aff(M,D)$ is a finite dimensional Lie group, see Theorem 23 in [17]. In fact, one has the following inequality:

$$dim\left(Aff(M,D)\right)\le m^2+m$$

here,

$$m=dim\left(M\right).$$

The demonstration of Theorem 23 in [17] involves both the fundamental form of the principal $Gl(m,R)$ bundle of linear frames $R^1\left(M\right)$ and the horizontal distribution $H_D\subset T{R}^1\left(M\right)$ which is attached to D. These data are used to provide $R^1\left(M\right)$ with a parallelism which is invariant under the action of $Aff(M,D)$ on $R^1\left(M\right)$. Since this action of $Aff(M,D)$ is effective, one easily deduces that $Aff(M,D)$ is a subgroup of isometries of positive inner product on the tangent vector bundle

$$T{R}^1\left(M\right)\to R^1\left(M\right).$$

For details, readers are referred to Kobayashi-Nomizu [17] (p. 232, Theorem 23). In contrast to our strategy of direct proof, the strategy based on Riemannian geometry is not efficient for general Lie algebroids.

Comm. 2. Below we will state and prove Theorem 2, which illustrates that Theorem 1 does not work on general left modules of the Lie algebroid

$$A=(V,M,a,[-,-\left]\right).$$

We consider a real vector bundle $(W,M)$ which is a left module of A whose left action

$$\Gamma \left(V\right)\times \Gamma \left(W\right)\ni (s,\xi )\to s.\xi \in \Gamma \left(W\right)$$

(51)

satisfies the next two properties

$$(fs.\xi )\left(x\right)=f\left(x\right)(s.\xi )\left(x\right),$$

(52)

$$(s.f\xi )\left(x\right)=df\left(a\right(s\left)\right(x\left)\xi \right(x)+f(x\left)\right(s.\xi \left)\right(x\left)\right).$$

(53)

Let us assume that ∇ is a connection on $(W,M)$. Then we put

$$A_{\nabla }\left(W\right)=\left\{s\in \Gamma \left(V\right)s.t{L}_s\nabla =0\right\}.$$

(54)

Problem: Let $(W,M)$ be a left module of the Lie algebroid A, and let $(W,M,\nabla )$ be a gauge structure on $(W,M)$. The problem is knowing under what conditions $A_{\nabla }\left(W\right)$ contains a non-null module of $C^{\infty }\left(M\right)$.

Otherwise said, there exists

$$s\in A_{\nabla }\left(W\right)$$

such that

$$C^{\infty }\left(M\right)s\subset A_{\nabla }\left(W\right),$$

which means that

$$fs.{\nabla }_{a\left(s^{\prime }\right)}w-{\nabla }_{a\left([fs,s^{\prime }]\right)}w-{\nabla }_{a\left(s^{\prime }\right)}fs.w=0\forall f\in C^{\infty }\left(M\right),\forall (w,s^{\prime })\in \Gamma \left(W\right)\times \Gamma \left(V\right).$$

(55)

If we take into account (53), then (55) is reduced to the following identity,

$$df\left(a\left(s^{\prime }\right)\right)[{\nabla }_{a\left(s\right)}w-s.w]\forall \Gamma \left(V\right)\times \Gamma \left(V\right)\times (\Gamma \left(W\right)\times C^{\infty }\left(M\right)$$

(56)

Consequently, we are led to the following three alternatives:

Alternative 1: f is first integral of the distribution $a\left(V\right)\subset TM$.

Alternative 2: ${\nabla }_{a\left(s\right)}w=s.w\forall w\in \Gamma \left(W\right)$. Thus, this Alternative 2 implies the following inclusion

$$C^{\infty }\left(M\right)\subset A_{\nabla }\left(W\right).$$

Alternative 3: $s=0$. The expression (55) jointed to the injectivity of a excludes Alternative 1.

It remains to discuss Alternatives 2 and 3.

We recall that, according to Formulas (52) and (53), the left action of A on W looks like the restriction connections on W along the leaves of the following foliation

$$a\left(V\right)\subset TM.$$

Thus, a connection (on $(W,M)$) whose restriction along the foliation $a\left(V\right)$ coincides with the left action as in (51) is called a canonical connection on W and is denoted by ${\nabla }^{can}$.

Thus (53)–(55) yield the following statement:

Theorem 2.

Let us consider an injective Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

and let $(W,M)$ be a left module of A according to (52) and (53). Let $(W,M,\nabla )$ be a gauge structure on $(W,M)$.

(I) 

If ∇ is not a canonical connection on $(W,M)$, then the situation is Alternative 3; therefore, $A_{\nabla }$(W) does not contain any non-null module of the associative algebra $C^{\infty }\left(M\right)$.

(II) 

If ∇ is a canonical connection on $(W,M)$, then the situation is Alternative 2. Consequently, $A_{\nabla }\left(W\right)$ is module of the associative algebra $C^{\infty }\left(M\right).$

Let us briefly recall the following arrow

$$p_k:{\Omega }^k(M,W)\to {\Omega }^k(A,W)$$

(57)

where

$${\Omega }^K(A,W)={\Omega }^k(M,W)\circ a^k.$$

Let ${\Omega }_b^1(A,W)$ be the kernel of $p_1$. Thus, one has the following exact sequence:

$$0\to {\Omega }_b^1(A,W)\to {\Omega }^1(M,W)\to {\Omega }^1(A,W).$$

(58)

The two theorems above will be repeatedly used in the proofs and are considered central in our work. It is clear that $Hom(\Gamma \left(W\right),{\Omega }_b^1(A,W))$ is a left module of the Lie algebra $\Gamma \left(V\right)$ whose k-th space of Chevalley–Eilenberg cohomology is denoted by

$$H_{CE}^k(\Gamma \left(V\right),Hom(\Gamma \left(W\right),{\Omega }_B^1(A,W))).$$

In regard to Theorem 2, the family of all canonical gauge structures on $(W,M)$ is denoted by $\mathit{CGau}(\mathit{W},\mathit{M})$. Therefore, it is easy to check the following statement,

Proposition 6.

There exists a canonical one-to-one correspondence between $\mathit{CGau}(\mathit{W},\mathit{M})$ and the cohomology space $H_{CE}^0(\Gamma \left(V\right),Hom(\Gamma \left(W\right),{\Omega }_b^1(A,W)))$.

Warnings. From the viewpoint of the differential topology, ${\Omega }_b^k(A,W)$ is the space of $(W,M)$-valued basic differential k-forms on the foliated space $(M,a(V\left)\right)$, see [7,8,18].

4. Tools from the Differential Gauge Operators on Vector Bundles

Here we are referring to decompositions (24) and (25).

The Lie subalgebras of $(\Gamma (V),[-,-\left]\right)$ which are generated by $\Gamma \left(Ker\right(\Phi \left)\right)$ and by $\Gamma \left(Ker\left({\Phi }^{★}\right)\right)$, are denoted by $L(\nabla ,\Phi )$ and by $L(\nabla ,{\Phi }^{★})$, respectively. Thus,

$$\begin{gathered} \left(L\right(\nabla ,\Phi ),[-,-\left]\right)\subset (\Gamma (V),[-,-\left]\right), \\ (L(\nabla ,{\Phi }^{★}),[-,-])\subset (\Gamma \left(V\right),[-,-]). \end{gathered}$$

4.1. Three Canonical Arrows

We aim to introduce three arrows whose targets are Lie subalgebras of the Lie algebra $(\Gamma (V),[-,-\left]\right)$.

The affine arrow

$$\mathit{RSGau}\left(\mathit{A}\right)\ni \nabla \to J_{\nabla }^g\left(A\right).$$

The metric arrow

$$J_{\nabla ,g}\ni \varphi \to L(\nabla ,\Phi ).$$

The symplectic arrow

$$J_{\nabla ,g}\ni \varphi \to L(\nabla ,{\Phi }^{★}).$$

On the one hand, we depend on Lie algebroids and on the modules of the Lie algebroid in view; on the other hand, one takes into account (34). Then it is clear that all of the vector spaces $\Gamma \left(V^{p,q}\right)$ and $\Gamma \left(T^{p,q}\left(M\right)\right)$ are left modules of each of the following three three targets: (i) the target of the affine arrow, (ii) the target of the metric arrow, (iii) the target of the symplectic arrow.

We are particularly concerned with the Chevalley–Eilenberg cohomology of Lie algebras $J_{\nabla }^g\left(A\right)$, $L(\nabla ,\Phi )$, and $L(\nabla ,{\Phi }^{★})$ with coefficients in $\Gamma \left(T^{p,q}\left(M\right)\right)$ or in $\Gamma \left(V^{p,q}\right)$. We are also interested in interpretations of some particular cohomology classes which are called canonical Koszul classes.

Our construction of these objects of homological nature was inspired by the works of Koszul in [19]. These tools are used to convert obstructions whose nature is vector space into obstructions whose nature is cohomological class.

4.2. Koszul Homological Series

Here is the context of this subsection. We are dealing with a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

which is endowed with a torsion-free gauge structure

$$(A,\nabla )$$

and with a positive inner product

$$(V,M,g).$$

We refer to Formulas (24)–(26) to remind the three Lie algebras: $J_{\nabla }^g$, $L(\nabla ,\Phi )$, $L(\nabla ,{\Phi }^{★})$. Let us fix a gauge structure $(V^{p,q},D)$. The Koszul connection D gives rise to the following three ${\Omega }^1(A,V^{p,q})$-valued Chevalley–Eilenberg 1-cocycles: $k_{\infty }^a$, $k_{\infty }^m$, and $k_{\infty }^s$, which are defined as follows:

$$J_{\nabla }^g\left(A\right)\ni s\to k_{\infty }^{a,p,q}\left(s\right)=L_{s}D\in {\Omega }^1(A,V^{p,q}),$$

(59)

$$L(\nabla ,\Phi )\ni s\to k_{\infty }^{m,p,q}\left(s\right)=L_{s}D\in {\Omega }^1(A,V^{p,q}),$$

(60)

$$L(\nabla ,{\Phi }^{★})\ni s\to k_{\infty }^{s,p,q}\left(s\right)=L_{s}D\in {\Omega }^1(A,V^{p,q}).$$

(61)

We recall that $L_s$ is the extension to the tensor spaces of the action of the Lie algebroid A on its canonical A-module $(V,M)$.

For instance, if

$$D\in Gau(V^{1,1},M),$$

then one has

$${\left(L_{s}D\right)}_{s^{\prime }}{s}^{″}=[s,D_{a\left(s^{\prime }\right)}{s}^{″}]-D_{[a\left(s\right),a\left(s^{\prime }\right)]}{s}^{″}-D_{a\left(s^{\prime }\right)}[s,s^{″}]).$$

(62)

It is easy to see that the cohomology classes of these cocycles are independent of the choice of the Koszul connection D.

Based on the four Formulas (59)–(62), we introduce the following three Koszul homological series.

The Koszul affine homological series are the following family:

$$\left\{F^{a,p,q}(g,\nabla ):=N^2\ni (p,q)\to \left[k_{\infty }^{a,p,q}\right]\in H_{CE}^1(J_{\nabla }^g\left(A\right),{\Omega }^1(A,V^{p,q}))\right\}.$$

(63)

The Koszul metric homological series are the following family:

$$\left\{F^{m,p,q}(g,\nabla ,\varphi ):=N^2\ni (p,q)\to \left[k_{\infty }^{m,p,q}\right]\in H_{CE}^1(L(\nabla ,\Phi ),{\Omega }^1(A,V^{p,q}))\right\}.$$

(64)

The Koszul symplectic homological series are the following family:

$$\left\{F^{s,p,q}(g,\nabla ,\varphi ):=N^2\ni (p,q)\to \left[k_{\infty }^{s,p,q}\right]\in H_{CE}^1(L(\nabla ,{\Phi }^{★}),{\Omega }^1(A,V^{p,q}))\right\}.$$

(65)

Definition 8.

Given a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

(1) the arrow

$$\mathit{RGau}\left(\mathit{V}\right)\ni \nabla \to \left\{F^{a,p,q}(g,\nabla ),(p,q)\subset N\right\}$$

is called a Koszul affine homological functor of A.

(2) The term $F^{a,p,q}(\nabla ,g)$ is called an affine (p,q)-homology class of A.

(3) The Lie algebroid A is called affinely nondegenerate if its Koszul homological functor does not contain any affine vanishing homology class; otherwise, it is affinely degenerate.

(4) The Lie algebroid A is affinely (p,q)-degnerate if its Koszul affine homological functor contains an affine (p,q)-vanishing homology class, viz., there exists some $\nabla \in \mathit{RGau}\left(\mathit{V}\right)$ such that

$$F^{a,p,q}(g,\nabla )=0.$$

Nota bene: It must be noted that this property of nondegeneracy is independent of the choice of the positive inner product g.

Nota bene: Mutatis mutandis, the following notions are clearly unambiguous:

(5) The Lie algebroid A is metrically (p,q)-nondegenerate.

Otherwise said, for any fixed couple $(p,q)$ of positive integers, the set $\left\{F^{m,p,q}(g,\nabla ,\varphi )\right\}$ does not contain a vanishing class.

(6) The Lie algebroid A is symplectically (p,q)-nondegenerate.

Otherwise said, for any fixed couple $(p,q)$ of positive integers, the set $\left\{F^{s,p,q}(g,\nabla ,\varphi )\right\}$ does not contain a vanishing class.

Comments: The notion of nondegeneracy may be restricted to a fixed gauge structure $(A,\nabla )$. This means that depending on the issue and on the concern, the parameter ∇ is fixed in (63)–(65). Consequently, the following statements are unambiguous:

(7) $(A,\nabla )$ is affinely (p,q)-nondegenerate, (respectively, affinely (p,q)-degenerate).

(8) $(A,\nabla )$ is metrically nondegenerate, (respectively, metrically (p,q)-degenerate.

(9) $(A,\nabla )$ is symplectically (p,q)- nondegenerate, (respectively, symplectically (p,q)-degenerate.

Restrictions {(7),(8),(9)} are useful and effective for examining many interesting questions such as the problem of metric connection and the problem of symplectic connection.

Without going into details, some examples are as follows:

Example 1.

Restriction (7) may be implemented to address the question of whether $(A,\nabla )$ is an affine structure on A. Regarding the case of canonical tangent algebroids, see Goldman W. M. [20] and Smilie J. [21].

Example 2.

Restriction (7) may be implemented to study the question of whether a statistical structure in A is a Hessian structure.

Example 3.

Restriction (7) may be used to discuss the question of whether a statistical model of a measurable set is an exponential model. The examples just mentioned are of great interests in both pure information geometry and applied information geometry.

To make it simpler, one must retain that information geometry is the study of geometric properties of statistical models [22,23,24,25,26].

Example 4.

Both Restriction (8) and Restriction (9) may be involved to study the question of whether a statistical structure on $(A,\nabla )$; namely, $(A,\nabla ,g)$, admits a compatible symplectic structure $(A,\nabla ,\omega )$. This means that the following requirements are satisfied,

$${\delta }^{\nabla }g=0,$$

$$\nabla \omega =0.$$

Example 4 can be implemented to construct Hamiltonian systems on statistical manifolds.

The set of all vanishing (p,q)-classes is noted by $F_{\delta }^{X,p,q}$; here, X stands for one of the three letters a: = affine, m: = metric, and s: = symplectic.

Henceforth, the family of all affinely vanishing (p,q)-classes of A is denoted by

$$F_{\delta }^{a,p,q}\left(A\right).$$

(66)

The family of all metrically vanishing (p,q)-classes of A is denoted by

$$F_{\delta }^{m,p,q}\left(A\right).$$

(67)

The family of all symplectically vanishing (p,q)-classes of A is denoted by

$$F_{\delta }^{s^{,}p,q}\left(A\right).$$

(68)

4.3. Nondegeneracy as Homological Characteristic Obstruction

The goal of this subsection is to point out Gauge Topology interpretations of Koszul homological functors of Lie algebroids.

In the spirit of Theorem 1, we aim to point out that the notion of homological nondegeneracy is a machine to convert an obstruction by a $C^{\infty }\left(M\right)$-module into an obstruction by a cohomology class.

The generic picture is the following: the framework is the category of injective Lie algebroids over the same base manifolds and isomorphisms of Lie algebroids.

We fix a Lie algebroid

$$A=(V,M,a,[-,-\left]\right).$$

Let $(W,M)$ be a left module of the Lie algebroid A. We recall the meaning of the ingredients as follows:

(1)

The letter a stands for a vector bundle homomorphism

$$a:(V,M)\to (TM,M),$$

(2)

$\Gamma \left(W\right)$ is a left module of the Lie algebra $(\Gamma (V),[-,-\left]\right)$ and the following left action

$$\Gamma \left(V\right)\times \Gamma \left(W\right)\ni (s,w)\to s.w\in \Gamma \left(W\right)$$

is subject to the following requirements:

$$(s.fw)\left(p\right)=df\left(a\right(s\left)\right(p\left)\right)w\left(p\right)+f\left(p\right)(s.w)\left(p\right);$$

(69)

$$(\left(fs\right).w)\left(p\right)=f\left(p\right)\left(sw\right)\left(p\right)\forall (p,f,s,w)\in M\times C^{\infty }\left(M\right)\times \Gamma \left(V\right)\times \Gamma \left(W\right).$$

(70)

Of course all of the vector bundles $W^{p,q}$ are left modules of the Lie algebroid A.

We recall that ${\Omega }^k(M,W^{p,q})$ is the space of $W^{p,q}$-valued differential k-forms on M and

$${\Omega }^k(A,W^{p,q})={\Omega }^k(M,W^{p,q})\circ a^{\Lambda k}$$

(71)

We observe that a Koszul connection D on $W^{p,q}$ is a ${\Omega }^1(M,W^{p,q})$-valued first-order differential operator

$$\Gamma \left(W^{p,q}\right)\ni \xi \to D\xi \in {\Omega }^1(M,W^{p,q}).$$

The canonical Koszul 1-cocycle which is associated with connection D is the following mapping:

$$\Gamma \left(V\right)\ni s\to k_{\infty }\left(s\right)=L_{s}D.$$

(72)

The action of A on ${\Omega }^1(M,W^{p,q})$ is connection-preserving if, on $(W^{p,q},M)$, there exists a Koszul connection $D^{★}$ that satisfies the following identity:

$$L_s{D}^{★}=O\forall s\in \Gamma \left(V\right).$$

(73)

Miscellaneous 1: The space of first-order differential operators which are defined from $(W,M)$ to ${\Omega }^1(M,W)$ is denoted by $D{O}^1(W,{\Omega }^1(M,W))$.

Let

$$(\Delta ,s)\in D{O}^1(W,{\Omega }^1(M,W)\times \Gamma (V\left)\right)$$

the differential operator

$$L_s\Delta \in D{O}^1(W,{\Omega }^1(M,W))$$

is defined as follows:

$${\Omega }^1(M,W)\ni (L_s\Delta )\left(w\right)=L_s(\Delta \left(w\right))-\Delta (s.w).$$

Given a vector field

$$X\in \Gamma \left(TM\right)$$

the vector differential 1-form $L_s(\Delta \left(w\right))$ is defined as follows:

$$L_s(\Delta \left(w\right))\left(X\right)=s.(\Delta \left(w\right)\left(X\right))-(\Delta \left(w\right)\left([a\left(s\right),X]\right)).$$

Miscellaneous 2:

Remember that we implicitly use the following identification of vector bundles,

$$T{M}^{★}\otimes W^{★}\otimes W=Hom(TM\otimes W,W).$$

Therefore, put

$$Bil(TM\times W;W)=\Gamma (T{M}^{★}\otimes W^{★}\otimes W).$$

Under the notation as in (72) it is easy to check that

$$L_{s}D\in Bil(TM\times W,W)$$

which means the following identity,

$$\begin{gathered} \left[\left(L_{s}D\right)(fX,w)\right]\left(p\right)=\left[L_{s}D(X,fw)\right]\left(p\right) \\ =f\left(p\right)\left[\left(L_{s}D\right)(X,w)\right]\left(p\right)\forall (p,f,s,X,w)\in M\times C^{\infty }\left(M\right)\times \Gamma \left(V\right)\times \Gamma \left(TM\right)\times \Gamma \left(W\right). \end{gathered}$$

Here is an interpretation of the degeneracy of the Koszul homological functor.

Proposition 7.

Given a left module $(W,M)$ of a Lie algebroid A, the associated Koszul homological functor is (p,q)-degenerate if and only if the action of A on $(W^{p,q},M)$ is connection-preserving.

Proof. 

The (p,q)-degeneracy is sufficient.

The (p,q)-degeneracy means that, if D is a Koszul connection on $W^{p,q}$, then the cocycle

$$\Gamma \left(V\right)\ni s\to L_{s}D\in {\Omega }^1(A,W^{p,q})$$

is exact. Therefore, there exists a $W^{p,q}$-valued differential 1-form

$$\theta \in {\Omega }^1(M,W^{p,q})$$

such that

$$L_{s}D=\left(d_{CE}\theta \right)\left(s\right)$$

$$=L_s\theta .$$

More precisely,

$$\begin{gathered} s.\left(D_Y\xi \right)-D_{\left[a\right(s),Y]}\xi -D_Y(s.\xi )=s.\left(\theta (Y;\xi )\right)-\theta ([a\left(s\right),Y];\xi )-\theta (Y;s.\xi ), \\ \forall (s,Y,\xi )\subset \Gamma \left(V\right)\times \Gamma \left(TM\right)\times \Gamma \left(W^{p,q}\right). \end{gathered}$$

The right side member of the equality above is $C^{\infty }$(M)-bilinear with respect to $(Y,\xi )$. Therefore,

$$D^{★}=D-\theta$$

is a Koszul connection on $W^{p,q}$ and

$$L_s{D}^{★}=0\forall s\in \Gamma \left(V\right).$$

Then, the action of A on $W^{p,q}$ is connection preserving.

The (p,q)-degeneracy is necessary.

Let us assume that the action of A on $(W,M)$ is connection-preserving. Then there exists a Koszul connection ∇ such that

$$L_s\nabla =0\forall s\in \Gamma \left(V\right).$$

According the notation in (63), the following cocycle vanishes identically,

$$\begin{gathered} \Gamma \left(V\right)\ni s\to k_{\infty }^{p,q}\left(s\right) \\ =L_s\nabla \in {\Omega }^1(M,W^{p,q}) \end{gathered}$$

Therefore, the Koszul homological class $F^{p,q}(g,\nabla )$ vanishes.

The proposition is proved. □

We recall that a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

is injective if a is injective.

Without the explicit statement of the contrary we will be dealing with injective Lie algebroids.

Definition 9.

A Koszul–Vinberg structure (KV structure in short) on a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

is a product

$$\Gamma \left(V\right)\times \Gamma \left(V\right)\ni (s,s^{\prime })\to s.s^{\prime }\in \Gamma \left(V\right)$$

which is subject to the following requirements:

$$\begin{gathered} (r.1):(s.s^{\prime }).s^{″}-s.(s^{\prime }.s^{″})=(s^{\prime }.s).s^{″}-s^{\prime }.(s.s^{″})\forall (s,s^{\prime },s^{″})\subset \Gamma \left(V\right), \\ (r.2):s.s^{\prime }-s^{\prime }.s=[s,s^{\prime }]\forall (s,s^{\prime })\subset \Gamma \left(V\right). \end{gathered}$$

Warning. As a straight consequence of requirement (r.2), we have the following identity:

$$s.f{s}^{\prime }=df\left(a\left(s\right)\right)s^{\prime }+fs.s^{\prime }\forall f\in C\infty \left(M\right).$$

We are now in a position to state and prove the following three theorems.

Theorem 3.

For a Lie algebroid A to admit a Koszul–Vinberg structure it is sufficient and necessary that it is affinely (1,1)-degenerate.

Proof. 

The affinely (1,1)-degeneracy is sufficient.

On a Lie algebroid

$$A=(V,M,a,[-,-\left]\right),$$

we assume that A is affinely (1,1)-degenerate. Then it carries a regular symmetric gauge structure $(A,\nabla )$ and a positive definite inner product g such that

$$F^{a,1,1}(g,\nabla )=0.$$

Otherwise said, the cocycle

$$J_{\nabla }^g\left(A\right)\ni s\to L_{s}D$$

is exact for any Koszul connection D on the tangent vector bundle $(TM,M)$. Therefore, given a gauge structure $(TM,M,D)$, there exists

$$\theta \in \Gamma \left(T{M}^{1,1}\right)$$

such that the $J_{\nabla }^g\left(A\right)$-action on $(TM,M)$ is $D^{★}$-preserving, where

$$D^{★}=D-\theta .$$

Consequently, according to (54), one has the following inclusion

$$J_{\nabla }^g\left(A\right)\subset A_D^{★}\left(TM\right).$$

Since $J_{\nabla }^g\left(A\right)$ is a $C^{\infty }\left(M\right)$-module, by virtue of Theorem 1, one has

$$J_{\nabla }^g\left(A\right)=0.$$

So, on the one hand, we have the following equality,

$$rank\left(J_{\nabla }\left(A\right)\right)=rank\left(V\right).$$

which yields the following consequence:

$$Spa{n}_{C^{\infty }\left(M\right)}\left(J_{\nabla }\left(A\right)\right)=\Gamma \left(V\right).$$

On the other hand, we have

$$R^{\nabla }(s,s^{\prime };s^{″})=0\forall (s,s^{\prime },s^{″})\subset J_{\nabla }\left(A\right).$$

Since the following arrow

$$(s,s^{\prime },s^{″})\to R^{\nabla }(s,s^{\prime };s^{″})$$

is $C^{\infty }\left(M\right)$-three-multilinear, we conclude that

$$R^{\nabla }(s,s^{\prime };s^{″})=0\forall (s,s^{\prime },s^{″})\subset \Gamma \left(V\right).$$

To finish, we define the Koszul–Vinberg product $s★s^{\prime }$ by putting

$$s★s^{\prime }={\nabla }_s{s}^{\prime }\forall (s,s^{\prime })\subset \Gamma \left(V\right).$$

The affinely (1,1)-degeneracy is necessary.

Indeed, let us assume that the Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

carries a Koszul–Vinberg structure

$$\Gamma \left(V\right)\times \Gamma \left(V\right)\ni (s,s^{\prime })\to s★s^{\prime }\in \Gamma \left(V\right).$$

We recall that the Koszul–Vinberg product $s★s^{\prime }$ satisfies the following two identities,

$$\begin{gathered} (s,s^{\prime };s^{″})=(s^{\prime },s;s^{″}), \\ s★s^{\prime }-s^{\prime }★s=[s,s^{\prime }], \end{gathered}$$

where $(s,s^{\prime };s^{″})$ stands for the associator of the product $s★s^{\prime }$ and $[s,s^{\prime }]$ is the bracket of the Lie algebroid A.

We set

$${\nabla }_{a\left(s\right)}{s}^{\prime }=s★s^{\prime }\forall (s,s^{\prime })\subset \Gamma \left(V\right).$$

It is easy to check that $(A,\nabla )$ is an affine structure on $A.$

Since A is injective, we then focus on the regular involutive distribution

$$a\left(V\right)\subset TM.$$

Let

$$F_x\subset M$$

be a leaf of $a\left(V\right)$ through

$$x\in M.$$

Then $F_x$ is endowed with locally flat

$$(F_x,D),$$

where

$$D_{a\left(s\right)}a\left(s^{\prime }\right)=a\left({\nabla }_{a\left(s\right)}{s}^{\prime }\right).$$

Thus, the couple

$$(a\left(V\right),D)=\left\{(F_x,D),x\in M\right\}$$

is a k-dimensional locally flat foliation of M.

By the machinery of the Haefliger pseudogroup, there exists a foliated atlas of M whose local charts have the following form

$$U\ni x\to (X\left(x\right),Y\left(x\right))\in R^k\times R^{m-k},$$

and a local chart change has the following form

$$(X,Y)\to (X^{\prime }(X,Y),Y^{\prime }\left(Y\right))$$

where $X^{\prime }(X,Y)$ is the affine function of X.

We already pointed out that elements of $J_D$ are infinitesimal transformations of $(F_x,D)$. Therefore, one has

$$rank\left(J_D\right)=rank\left(a\left(V\right)\right).$$

It is easy to check the following identity:

$$D_{a\left(s\right)}{D}_{a\left(s^{\prime }\right)}a\left(s^{″}\right)-D_{\left[D_{a\left(s\right)}a\left(s^{\prime }\right)\right]}a\left(s^{″}\right)=a({\nabla }_{a\left(s\right)}{\nabla }_{a\left(s^{\prime }\right)}{s}^{″}-{\nabla }_{\left[a\left({\nabla }_{a\left(s\right)}{s}^{\prime }\right)\right]}{s}^{″}).$$

Based on this identity, we conclude that

$$\begin{gathered} rank\left(J_{\nabla }\right)=rank\left(J_D\right) \\ =rank\left(V\right) \end{gathered}$$

Therefore,

$$F^{a,1,1}(\nabla )=O.$$

Thus A is affinely (1,1)-degenerate. □

Theorem 4.

A Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

admits a Koszul–Vinberg structure if and only it admits an affine structure.

Proof. 

(i)

Let us assume that A admits Koszul–Vinberg structure

$$\Gamma \left(V\right)\times \Gamma \left(V\right)\ni (s,s^{\prime })\to s.s^{\prime }\in \Gamma \left(V\right).$$

Then we define the symmetric gauge structure $(A,\nabla )$ by putting

$${\nabla }_{a\left(s\right)}{s}^{\prime }=s.s^{\prime }.$$

Direct calculations yield

$$R^{\nabla }=0$$

and

$$T^{\nabla }=0.$$

Thus, $(A,\nabla )$ is an affine structure on A.

(ii)

Conversely, given an affine structure $(A,\nabla )$, the Koszul–Vinberg product

$$(s,s^{\prime })\to s.s^{\prime }$$

is defined as follows

$$s.s^{\prime }={\nabla }_{a\left(s\right)}{s}^{\prime }$$

All of the axioms of the Koszul–Vinberg structure are satisfied by this definition. □

Theorem 5.

For a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

to admit a symplectic structure, it is sufficient and necessary for it to be symplectically (1,1)-degenerate.

Proof of Theorem 5. 

Symplectically (1,1)-degeneracy is sufficient.

Let us assume that A is symplectically (1,1)-degenerate. Then A admits a gauge structure $(A,\nabla )$ and a positive definite inner product g and a gauge structure $(TM,M,D)$ which are subject to the following requirements:

The gauge equation

$${\nabla }^g\otimes \varphi -\varphi \circ \nabla =0$$

has a solution $\varphi$ such that the ${\Omega }^1(M,T{M}^{1,1})$-valued cocycle

$$L(\nabla ,{\Phi }^{★})\ni s\to L_{s}D\in {\Omega }^1(M,T{M}^{1,1})$$

is exact. Then there exists

$$\theta \in {\Omega }^1(M,T{M}^{1,1})$$

such that

$$\begin{gathered} L_{s}D=\left(d_{CE}\theta \right)\left(s\right) \\ =L_s\theta \forall s\in L(\nabla ,{\Phi }^{★}). \end{gathered}$$

Therefore, if we put

$$D^{★}=D-\theta ,$$

the action of $L(\nabla ,{\Phi }^{★})$ on ${\Omega }^1(M,T{M}^{1,1})$ is $D^{★}$-preserving. So, on the one hand, one has the following inclusion

$$L(\nabla ,{\Phi }^{★})\subset A_{D^{★}}\left(TM\right)$$

and, on the other hand, we know that the vector space $L(\nabla ,{\Phi }^{★})$ is the left module of the associative algebra $C^{\infty }\left(M\right)$. Then, by virtue of Theorem 1, we deduce that

$$L(\nabla ,{\Phi }^{★})=0.$$

Finally, ${\Phi }^{★}$ is an invertible solution of the gauge equation

$${\nabla }^g\circ \varphi -\varphi \circ \nabla =0.$$

This situation yields two symplectic products on the vector bundle

$$V\to M,$$

$$\begin{gathered} \omega (s,s^{\prime })=g({\Phi }^{★}\left(s\right),s^{\prime }) \\ {\omega }^{★}(s,s^{\prime })=g({\Phi }^{★-1}\left(s\right),s^{\prime }) \end{gathered}$$

It is easy to check that

$$\begin{gathered} \nabla \omega =O, \\ {\nabla }^g{\omega }^{★}=0. \end{gathered}$$

The sufficiency is proved.

Symplectically (1,1)-degeneracy is necessary.

Let us assume that A has a symplectic product

$$\Gamma {\left(V\right)}^2\ni (s,s^{\prime })\to \omega (s,s^{\prime })\in C^{\infty }\left(M\right).$$

Let $(A,{\nabla }^0)$ be a symmetric gauge structure on A. Using the same formalism as in [16], we define

$$\mathit{N}\in \Gamma \left(V^{2,1}\right)$$

by the following formula:

$${\nabla }_{a\left(s\right)}^0\omega (s^{\prime },s^{″})=\omega (\mathit{N}(s,s^{\prime }),s^{″}).$$

We set

$${\nabla }_{a\left(s\right)}{s}^{\prime }={\nabla }_{a\left(s\right)}^0{s}^{\prime }+{\displaystyle \frac{1}{3}}(\mathit{N}(s,s^{\prime })+\mathit{N}(s^{\prime },s)).$$

Then the gauge structure $(A,\nabla )$ is torsion-free as well, and it satisfies the following identity,

$${\nabla }_a\left(s\right)\omega =0\forall s\in \Gamma \left(V\right).$$

Using an auxiliary positive inner product g, we pose the gauge equation o

$${\nabla }^g\otimes \varphi -\varphi \circ \nabla =0.$$

There exists

$$\varphi \in J_{\nabla ,g}$$

such that

$$g({\Phi }^{★}\left(s\right),s^{\prime })=\omega (s,s^{\prime })\forall (s,s^{\prime })\subset \Gamma \left(V\right).$$

Therefore, because the algebra $L(\nabla ,{\Phi }^{★})$ is spanned by the kernel of ${\Phi }^{★}$, one deduces that

$$L(\nabla ,{\Phi }^{★})=0$$

Consequently, we get

$$F^{s,1,1}(\nabla )=0.$$

Finally, A is symplectically (1,1)-degenerate. □

Remark 1.

We have pointed out that the following symplectic product

$$\omega (s,s^{\prime })=g({\Phi }^{★}\left(s\right),s^{\prime })$$

is ∇-parallel, viz.,

$$a\left(s\right)\left(\omega (s^{\prime },s^{″})\right)-\omega ({\nabla }_{a\left(s\right)}{s}^{\prime },s^{″})-\omega (s,{\nabla }_{a\left(s\right)}{s}^{″})=0\forall (s,s^{\prime },s^{″})\subset \Gamma \left(V\right).$$

Remark 2.

The mapping ${\Phi }^{★-1}$ satisfies the following requirement:

$$\nabla \otimes {\Phi }^{★-1}-{\Phi }^{★-1}\circ {\nabla }^g=0.$$

As we already pointed that ${\Phi }^{★-1}$ gives another symplectic product

$${\omega }^{★}(s,s^{\prime })=g({\Phi }^{★-1}\left(s\right),s^{\prime })$$

which verifies the following identity:

$$\left({\nabla }_{a\left(s\right)}^g{\omega }^{★}\right)(s^{\prime },s^{″})=0.$$

Remark 3.

The triplet $(\omega ,{\omega }^{\prime },{\Phi }^{★})$ satisfies the following identity:

$$\omega (s,s^{\prime })={\omega }^{\prime }({\Phi }^{★2}\left(s\right),s^{\prime }).$$

Thus, ${\Phi }^{★2}$ is the recursion operator for the couple $(\omega ,{\omega }^{\prime })$ [27].

Remark 4.

Let us put

$$\left(L_S\omega \right)(s^{\prime },s^{″})=a\left(s\right)\left(\omega (s^{\prime },s^{″})\right)-\omega ([s,s^{\prime }],s^{″})-\omega (s^{\prime },[s,s^{″}])$$

If $(A,\nabla )$ is a Koszul–Vinberg structure on A, then we get the following identity:

$$\left(L_s\omega \right)(s^{\prime },s^{″})+\left(L_{s^{\prime }}\omega \right)(s^{″},s)+\left(L_{s^{″}}\omega \right)(s,s^{\prime })=0.$$

Theorem 6.

On a Lie algebroid

$$A=(V,M,a,[-,-\left]\right),$$

for a Koszul connection ∇ to be a metric connection, it is sufficient and necessary for it to be metrically (1,1)-degenerate.

Hint:

On $(V,M)$, we fix a positive definite inner product g. Under the metrically (1,1)-degeneracy of $(A,\nabla )$, there exists

$$\varphi \in J_{\nabla ,g}\left(A\right)$$

such that the following $T{M}^{1,1}$-valued Chevalley–Eilenberg 1-cocycle

$$L(\nabla ,{\Phi }^{★})\ni s\to L_{s}D\in {\Omega }^1(M,T{M}^{1,1})$$

is exact. Thus, there exists

$${\theta }^{″}\in {\Omega }^1(M,T{M}^{1,1})$$

such that the action of $L(\nabla ,\Phi )$ on ${\Omega }^1(M,T{M}^{1,1})$ is $D^{″}$-preserving where

$$D^{″}=D-{\theta }^{″}.$$

Therefore, one has the following inclusion

$$L(\nabla ,\Phi )\subset A_{D^{″}}\left(TM\right).$$

Since $L(\nabla ,\Phi )$ is a module of $C^{\infty }\left(M\right)$, Theorem 1 tells one that

$$L(\nabla ,\Phi )=0.$$

To pursue this, one defines the following inner product on $(V,M)$:

$$\left(q(g,\varphi )\right)(s,s^{\prime })=g(\Phi \left(s\right),s^{\prime })\forall (s,s^{\prime })\subset \Gamma \left(V\right).$$

This inner product satisfies the following identity:

$$a\left(s\right)\left(q(g,\varphi )\right)(s^{\prime },s^{″})-q(g,\varphi )({\nabla }_{a\left(s\right)}{s}^{\prime },s^{″})-\left(q(g,\varphi )\right)(s^{\prime },{\nabla }_{a\left(s\right)}{s}^{″})=0.$$

Thus ∇ is a metric connection on $(V,M,q(g,\varphi \left)\right).$

Theorem 6 is proven

Below is the symplectic analogous of Theorem 6.

Theorem 7.

On a Lie algebroid

$$A=(V,M,a,[-,-\left]\right),$$

For a Koszul connection ∇ to be symplectic connection, it is sufficient and necessary to it to be symplectically (1,1)-degenerate.

Hint: The demonstration is exactly similar to Proposition 6.

4.4. Some Comments on Gauge Geometry

In the sections devoted to gauge geometry we have been concerned with the following questions:

Question (4.4.1): On a vector bundle $(V,M)$, when is the Koszul connection ∇ metric connection? This is nothing other than Problem 1.

Question (4.4.2): On an even rank vector bundle $(V,M)$, when is the Koszul connection a ∇ symplectic connection? This is nothing other than Problem 2.

Question (4.4.3): When does the Lie algebroid $(V,M,a,[-,-\left]\right)$ admit a Koszul–Vinberg structure?

Warnings: Regarding Question (4.4.3) on tangent algebroids of differentiable manifolds, the readers are referred to Goldman’s survey [20] and to the bibliography therein. We also refer the readers to Smilie’s search for obstructions [21].

$(A★)$: We have introduced gauge equations of a pair of gauge structures and we have involved solutions of gauge equations to discuss question (4.4.1) and question (4.4.2) in the context of Lie Algebroids.

$(A★★)$: We have introduced the Hessian equation of a Lie Algebroid and we have involved its solutions to discuss question (4.4.3).

$(A★★★):$ Our homological approach to a question cannot be implemented on general vector bundles. In those cases, what is actually available is the direct calculation of the holonomy groups of gauge structures.

We have recalled a construction of Koszul to observe that every inner product on a Lie algebroid admits a unique symmetric metric connection. Consequently, every Lie algebroid is metrically (1,1)-degenerate.

A well known subsidiary question.

We have already recalled that the problem of metric connection has been widely discussed in the literature, see Richard Atkins [1] and and B. Schmidt [2]. Many authors raised the subsidiary question of whether the connection determines the metrics.

Analogously, a question concerning the problem symplectic connection is relevant. We are in the position to answer that affirmatively on Lie algebroids.

On a Lie algebroid

$$A=(V,M,a,[-,-\left]\right),$$

every gauge structure $(A,\nabla )$ determines the vector space ${\mathit{S}}_2^{\nabla }\left(V\right)$ of all inner products g such that

$$\nabla g=0.$$

The solution is our fundamental short exact sequence (28), viz.,.

Theorem 8.

All ∇-parallel either singular inner products or definite inner products are provided by the following splitting short exact sequence which is linked with gauge equations

$$0\to {\Omega }_2^{\nabla }\left(A\right)\to J_{\nabla ,g}\left(A\right)\to {\mathit{S}}_2^{\nabla }\left(A\right)\to 0.$$

Warnings.

Before pursuing this, the author underlines and invites the readers to keep in mind that the approach of differential geometry (or of the global analysis on manifolds) and that of gauge geometry are $[source<>target]$ of each other.

[DG≫GG]:= Approach from the differential geometry realm to the gauge geometry realm.

In differential geometry (or in global analysis ) on a manifold M, we start from G-structures

$$P\to M,$$

(74)

and then we are interested in connections ∇ which are adapted to (74). This means that the holonomy group $H(\nabla )$ is a subgroup of G.

[GG≫DG]:= Approach from the gauge geometry realm to the differential geometry realm.

In gauge geometry, as practiced in this work, we start from a gauge structure $(V,M,\nabla )$ on a vector bundle

$$V\to M,$$

(75)

and then we are interested in G-structures to which ∇ is adapted.

Those are the meanings of both the problem of metric connection and the problem of symplectic connection.

Our methods to study the approach [GG≫DG] are based on ideas which consist of looking for obstructions of a Homological nature.

To make our methods successful, open problems we are interested in are associated with Lie subalgebras of sections of Lie algebroids. We aim to make those Lie algebras characteristic obstructions to solve the open problems they are associated with.

Those Lie algebras serve us in extending some canonical homological constructions introduced by J.-L. Koszul [19].

That is why we have introduced Koszul homological series and their degeneracy.

4.5. Koszul Homology Series and Their Degeneracy

The notion of homological degeneracy produces sufficient and necessary conditions for question (4.4.1), question (4.4.2), and question (4.4.3) on the category of injective Lie algebroids and isomorphisms of Lie algebroids over the same base manifold. In forthcoming sections, we will implement those machineries to study questions (4.4.1), (4.4.2), and (4.4.3) in the context of tangent Lie algebroids.

5. Supplements to Affine Structures of Lie Algebroids

This short section is devoted to motivating interests in some basic homological objects. There is an abundance of literature on topics involving the homology of Lie algebroids, e.g., [28,29]. Here, we focus on the aspects of the homology on Lie algebroids that are implemented in this works [19].

5.1. A Conjecture of Muray Gerstenhaber

In [30], M. Gerstenhaber posed the following conjecture:

Every restrict theory of Deformation generates its proper theory of cohomology.

This conjecture has been solved for the category of Koszul–Vinberg algebras and for their modules [11]. This category is the algebraic versus the hyperbolic structures in the sense of Koszul in [11,19].

The next sections are devoted to recalling aspects which are of interest to the questions which are studied in this work.

5.2. Operational Tools

By operational tools we mean notions that can be technically manipulated to obtain concrete solutions to problems such as the effective numerical resolutions of algebraic or differential equations. For instance, how does one produce all Riemannian metric tensors which have the same Levi Civita connection ∇? All the special symplectic connections on a fixed Lie algebroid are parameterized by the first prolongation of the linear symplectic algebra sp(n) [16,31]. Conversely, how does one effectively product all the symplectic products which admit the same special symplectic connection ∇? In this regard, we emphasize the operational tool nature of Gauge Equations (18) and (19), which are of linear first order, as well as their fruits that are the short exact sequences (29) and (29). These tools can be numerically manipulated by the computer.

Seen from this angle, this work can be regarded as a provision of operational tools for applied gauge geometry. This wink will be renewed in Section 7. as well as Theorem 13.

5.3. Some Major Structures on Lie Algebroids

We devote this short subsection to recall the major challenges. The framework of challenges we face is the category of real Lie algebroids

$$A=(V,M,a,[-,-\left]\right).$$

In a Lie algebroid A, a gauge structure $(A,\nabla )$ may bear one of the following three labels:

L1: The Koszul connection ∇ is an Affine connection.

L2: The Koszul connection ∇ is a Metric connection.

L3: The Koszul connection ∇ is a Symplectic connection.

The challenge is to find a characteristic obstruction to each of these three labels.

6. Cohomology of Affine Algebroids and Their Modules: Some Examples of Applications

Let $(A,\nabla )$ be an affine structure on a Lie algebroid

$$A=(V,M,a,[-,-\left)\right].$$

We recall that ∇ is an ${\Omega }^1(A,V^{1,1})$-valued first-order differential operator

$$\Gamma \left(V\right)\ni s\to \nabla s\in {\Omega }^1(A,V^{1,1})$$

which is subject to the following requirements:

$$\begin{gathered} \nabla f.s=df.s+f\nabla s, \\ {\nabla }_{a\left(s\right)}{s}^{\prime }-{\nabla }_{a\left(s^{\prime }\right)}s=[s,s^{\prime }]\forall (f,s,s^{\prime })\in C^{\infty }\left(M\right)\times \Gamma \left(V\right)\times \Gamma \left(V\right). \end{gathered}$$

On the Lie algebroid A, the affine structure $(A,\nabla )$ yields the structure of Koszul–Vinberg algebra whose product $s★s^{\prime }$ is defined as follows

$$s.s^{\prime }={\nabla }_{a\left(s\right)}{s}^{\prime }\forall (s,s^{\prime })\subset \Gamma \left(V\right).$$

6.1. Two-Sided Modules of $(A,\nabla )$

Definition 10.

A two-sided module of $(A,\nabla )$ is a vector bundle $(W,M)$ endowed with two real bilinear mapping

$$\Gamma \left(V\right)\times \Gamma \left(W\right)\ni (s,\xi )\to s.\xi \in \Gamma \left(W\right),$$

(76)

$$\Gamma \left(W\right)\times \Gamma \left(V\right)\ni (\xi ,s)\to s.\xi \in \Gamma \left(W\right);$$

(77)

these left and right actions are subject to the following three requirements:

$$(s,s^{\prime };\xi )=(s^{\prime },s;\xi ),$$

(78)

$$(s,\xi ;s^{\prime })=(\xi ;s;s^{\prime }).$$

(79)

We recall that both the left side member and the right side member of the identities above are associator-like.

We then introduce the following vector subspace

$$\begin{gathered} J\left(W\right)\subset \Gamma \left(W\right): \\ \xi \in J\left(W\right) \end{gathered}$$

if and only if

$$(s,s^{\prime };\xi )=0\forall (s,s^{\prime })\subset \Gamma \left(V\right).$$

(80)

6.2. The W-Valued KV Cohomology of $(A,\nabla )$

The W-valued cochain complex of affine structure $(A,\nabla )$ is the following $\mathbb{Z}$-graded vector space

$$\begin{gathered} C_{KV}(\nabla ,W)={\oplus }_q{C}^q(\nabla ,W), \\ {C}^q(\nabla ,W)=Hom(\Gamma {\left(V\right)}^{\otimes q},\Gamma \left(W\right))\forall q>0. \end{gathered}$$

We use the operator $\delta$ that Albert Nijenhuis named the brut formula as follows:

$$\begin{gathered} C^q(\nabla ,W)\ni F\to \delta F\in C^{q+1}(\nabla ,W); \\ \delta \xi \left(s\right)=s.\xi -\xi .s\forall (\xi ,s)\subset \Gamma \left(W\right)\times \Gamma \left(V\right); \end{gathered}$$

(81)

$$\begin{array}{c}\delta F(s_0\otimes \dots \otimes s_q)={\sum }_{i<q}{(-1)}^i\left[\right[s_i.F(\dots \otimes \widehat{s_i}\otimes \dots )\\ +F(\dots \otimes \widehat{s_i}\otimes \dots \widehat{s_q}\otimes s_i).s_q\\ -{\sum }_{j\ne i}F(\dots \otimes \widehat{s_i}\otimes \dots \otimes s_i.s_j\otimes \dots )\left]\right]\end{array}$$

(82)

The q-th cohomology space of $C_{KV}(\nabla ,W)$ is denoted by

$$H_{KV}^q(\nabla ,W)={\displaystyle \frac{ker(\delta :C^q(\nabla ,W)\to C^{q+1}(\nabla ,W))}{\delta C^{q-1}}}$$

The trivial vector bundle

$$R\times M\to M$$

is a left module of $(A,\nabla )$ under the following left action:

$$s.f=df\left(a\left(s\right)\right)\forall (s,f)\subset \Gamma \left(V\right)\times C^{\infty }\left(M\right).$$

Thus, $J\left(C^{\infty }\left(M\right)\right)$ is the vector space of affine functions, viz.,

$$(s,s,s^{\prime };f)=0\forall (s,s^{\prime },f)\subset \Gamma {\left(V\right)}^2\times C^{\infty }\left(M\right).$$

The q-th real KV cohomolgy of the affine structure $(A,\nabla )$ is denoted by

$$H_{KV}^q(\nabla ,R)={\displaystyle \frac{ker(\delta :C^q(\nabla ,C^{\infty }\left(M\right))\to C^{q+1}(\nabla ,C^{\infty }\left(M\right)))}{\delta C^{q-1}(\nabla ,C^{\infty }\left(M\right))}}$$

(83)

6.3. Left Module-Valued Total Cohomology of an Affine Structure on a Lie Algebroid

Let W be a left module of an affine structure $(A,\nabla )$ on a Lie algebroid

$$A=(V,M,a,[-,-\left]\right).$$

Besides W-valued KV cohomology which recalled in the precedent subsection, there is another cohomology of $(A,\nabla )$ with coefficients in W.

We consider the vector space $C_{\tau }(\nabla ,W)$ whose q-th homogeneous space is the following vector space:

$$C_{\tau }^q=Hom(\Gamma {\left(V\right)}^{\otimes q},\Gamma \left(W\right)).$$

The operator

$$C_{\tau }^q\ni f\to d_{\tau }f\in C_{\tau }^{q+1}$$

is defined as follows:

$$\left(d_{\tau }w\right)\left(s\right)=s.w\forall w\in W.$$

(84)

$$\left(d_{\tau }f\right)(s_0\otimes \dots \otimes s_q)={\sum }_{i\le q}{(-1)}^i[s_i(f(\dots \otimes \widehat{s_i}\otimes \dots )-{\sum }_{j\ne i}f(\dots \widehat{s_i}\otimes \dots \otimes s_i.s_j\otimes \dots )]$$

(85)

Remember that in the right-hand member of (85)

$$s_i.s_j={\nabla }_{a\left(s_i\right)}{s}_j.$$

The total q-th total cohomology space is denoted by

$$H_{\tau }^q(\nabla ,W)={\displaystyle \frac{ker(d_{\tau }:C_{\tau }^q\to C^{q+1}\tau )}{d_{\tau }\left(C_{\tau }^{q-1}\right)}}$$

(86)

When W is the trivial vector bundle

$$M\times R\to M$$

the corresponding total KV cohomology is denoted by

$$H_{\tau }^q(\nabla ,R)$$

In the next subsection, we are interested in the second cohomology spaces $H_{KV}^2(\nabla ,V)$ and $H_{KV}^2(\nabla ,R).$

6.4. Links with Classical Chevalley–Eilenberg Cohomology and with the De Rham Scalar Cohomology of the Lie Algebroid A

Every left $(A,\nabla )$-module W is a left module of the Lie algebroid $A=(V,M,a,[-,-\left]\right)$. We define its space of q-cochains as follows

$$C_{CE}^q(A,W)=Hom(\Gamma {\left(V\right)}^{\Lambda q},\Gamma \left(W\right)).$$

We recall the operator

$$C_{CE}^q(A,W)\ni \Theta \to d_{CE}\Theta \in C_{CE}^{q+1}(A,W):$$

$$\left(d_{CE}w\right)\left(s\right)=s.w\forall w\in W.$$

(87)

$$\begin{array}{c}(d_{CE}\Theta (s_0\wedge \dots \wedge s_q)={\sum }_i{(-1)}^i[s_i.\Theta (\dots \wedge \widehat{s_i}\wedge \dots )]\\ +{\sum }_{i<j}{(-1)}^{i+j}\Theta ([s_i,s_j]\wedge \dots \wedge \widehat{s_i}\wedge \dots \wedge \widehat{s_j}\wedge \dots ).\end{array}$$

(88)

When W is the trivial vector bundle

$$R\times M\to M$$

Formula (88) is nothing but the De Rham operator, viz.,

$$\begin{array}{c}\left(d_{dR}F\right)(s_0\wedge \dots \wedge s_q)={\sum }_i{(-1)}^{i}a\left(s_i\right).(F(\dots \wedge \widehat{s_i}\wedge \dots )\\ +{\sum }_{i<j}F([s_i,s_j]\wedge \dots \wedge \widehat{s_i}\wedge \dots \wedge \widehat{s_j}\wedge \dots )\end{array}$$

(89)

It is easy to check the following claims:

The Chevalley–Eilenberg complex $(C_{CE}(A,W),d_{CE})$ is a subcomplex of the total KV complex $\left(C_{\tau }(\nabla ,W)\right)$, viz.,

$$(C_{CE}(A,W),d_{CE})\subset (C_{\tau }(\nabla ,W),d_{\tau }).$$

(90)

Let one set be the following quotient complex:

$$(Q(A,W),d_Q)=({\displaystyle \frac{(C_{\tau }(A,W),d_{CE})}{(C_{CE}(A,W),d_{\tau })}},d_Q)$$

then one has the following exact sequence of cochain complexes:

$$0\to \left(C_{CE}\right((A,W),d_{CE}\left)\right)\to (C_{\tau }(A,W),d_{\tau })\to (Q\left(A\right),d_Q)\to 0.$$

Consequently, the Chevalley–Eilenberg cochomology is linked with the total cohomology via the following long exact sequence:

$$\to H^p\left(Q(A,W)\right)\to H_{CE}^{p+1}(A,W)\to H_{\tau }^{p+1}(A,W)\to H^{p+1}\left(Q(A,W)\right)\to$$

We consider the case of the de Rham complex $(\Omega (A,R),d_{dR})$, viz.,

$$(\Omega (A,R),d_{dR})\subset (C_{\tau }(\nabla ,R),d_{\tau }).$$

(91)

By setting the following quotient

$$(Q(A,R),d_Q)=({\displaystyle \frac{(C_{\tau }(\nabla ,R),d_{\tau })}{(\Omega (A,R),d_{dR})}},d_Q),$$

one deduces the following link between real de Rham cohomology and total real KV cohomology of $(A,\nabla )$,

$$\to H^p\left(Q(A,R)\right)\to H_{dR}^{p+1}\left((A,R)\right)\to H_{\tau }^{p+1}(\nabla ,R)\to H^{p+1}(Q(A,R)\to$$

(92)

Miscellaneous. Before pursuing we consider the following tensor products of cochain complexes:

$$C(A,\nabla )=(\Omega (A,R),d_R)\otimes (C_{\tau }(\nabla ,R),d_{\tau }).$$

It is bigraded

$$C^{i,j}(A,\nabla )={\Omega }^i(A,R)\otimes C_{\tau }^j(\nabla ,R).$$

Thus, given

$$\omega \otimes \theta \in C^{i,j}(A,\nabla )$$

the coboundary operator of $C(A,\nabla )$, namely, d, is defined as follows:

$$d(\omega \otimes \theta )=d_R\omega \otimes \theta +{(-1)}^i\omega \otimes d_{\tau }\theta .$$

The bigrading $C^{i,j}(A,\nabla )$ is bounded in i. Thus, we obtain an example double complex, viz., $\left(C\right(A,\nabla ),d)$, whose total cohomology is easily calculable by using spectral sequences, see useful details in Moore C. C. and Schocher C. [18].

Some among these links we just pointed out will be implemented to discuss a problem in differential topology which has been raised by E. Ghys, cf. [8]. We recall that this problem is how to produce all Riemannian foliations.

We introduce the following canonical relationship:

$$H_{KV}^2(\nabla ,R)\to H_{dR}^2(A,R).$$

(93)

Relation (93) is produced by the following correspondence:

$$C_{KV}^2(\nabla ,R)\ni \Theta \to {\Lambda }_{\Theta }\in {\Omega }^2(A,R)$$

where

$${\Lambda }_{\Theta }(s,s^{\prime })=\Theta (s,s^{\prime })-\Theta (s^{\prime },s).$$

(94)

Warning: (94) may be implemented to investigate the symplectic structure in a Lie algebroid.

The idea is based on the following claim:

Lemma 2.

Assume that

$$\delta \Theta =O,$$

then the vector subspace

$$L(\Theta )=ker\left({\Lambda }_{\Theta }\right)$$

is a Lie subalgebra of the Lie algebra $(\Gamma (V),[-,-\left]\right)$.

Furthermore, the de Rham cohomology class

$$[\Lambda ]\in H_{dR}^2(A,R)$$

depends only on the KV cohomology class

$$[\Theta ]\in H_{KV}^2(\nabla ,R).$$

Hint

Direct calculations yield the following result,

$${\sum }_{cyclic}\delta \Theta (s,s^{\prime },s^{″})=2d_{dR}\Lambda (s,s^{\prime },s^{″}).$$

Then, $\Lambda$ is de Rham closed. Thus, $ker(\Lambda )$ is in involution.

The vector space of two cocycles of a cochain complex $\mathit{C}$ is denoted by $Z^2\left(\mathit{C}\right)$.

Then the Lemma 2 yields the following linear mapping:

$$Z_{KV}^2(A,\nabla )\ni \Theta \to {\Lambda }_{\Theta }\in Z_{dR}^2(A,R).$$

(95)

The arrow that will be used for our purpose is the following:

$$Z_{KV}^2(A,\nabla )\ni \Theta \to L(\Theta )\subset \Gamma \left(V\right).$$

(96)

Therefore, under the notation already used, we introduce the homological series

$$Z_{KV}^2(A,\nabla )\ni \Theta \to \left\{\left[k_{\infty }^{p,q}\left(L(\Theta )\right)\right]\in H_{CE}^1(L(\Theta ),T{M}^{p,q})\right\}$$

(97)

Functor (97) is called (p,q)-degenerate if there exists a two-cocycle $\Theta$ such that

$$\left[k_{\infty }^{p,q}\left(L(\Theta )\right)\right]=0.$$

Without going into details, an interesting claim is presented as follows:

Proposition 8.

If (97) is (1,1)-degenerate, then there exists a two-cocycle Θ whose skew symmetric part ${\Lambda }_{\Theta }$ is a symplectic product on $(A,\nabla )$.

Hint: One applies Theorem 1 to $L(\Theta )$.

6.5. Hessian Structure on Affine Structure

Definition 11.

A Hessian structure on $(A,\nabla )$ is a definite symmetric two-cocycle

$$g\in Z_{KV}^2(A,\nabla )$$

On an affine structure $(A,\nabla )$, it is easy to check the proposition.

Proposition 9.

On an affine structure $(A,\nabla )$, the following assertions are equivalent:

(i) 

The inner product g is a Hessian structure on $(A,\nabla )$;

(ii) 

$(A,{\nabla }^g)$ is an affine structure as well and g is a Hessian structure on $(A,{\nabla }^g).$

Warnings:

The proposition above has another interesting rephrasing.

Indeed, we already mentioned two dynamics on the category of gauge structures Gau(V,M): the gauge group GL(V,M) and the metric group GM(V,M). On a Lie algebroid $(V,M,a,[-,-\left]\right)$, the curvature of connection is both a gauge invariant and a metric invariant in the meanings that we proceed to make precise.

Let $(A,\nabla )$ be a gauge structure on an algebroid A. For

$$\begin{gathered} \Phi \in \mathit{GL}(\mathit{V},\mathit{M}), \\ \Phi .\nabla =\Phi \circ \nabla \circ {\Phi }^{-1}, \end{gathered}$$

Then,

$$R^{\Phi .\nabla }=\Phi \circ R^{\nabla }\circ {\Phi }^{-1}.$$

(98)

Let g be an inner product on $(V,M)$; $R^{\nabla }$ and $R^{{\nabla }^g}$ are subject to the following identity:

$$g(R^{\nabla }(s,s^{\prime };s^{″}),s^{‴})+g(s^{″},R^{{\nabla }^g}(s,s^{\prime }:s^{‴}))=0.$$

(99)

Lemma 3.

The following three assumptions are equivalent:

$$\begin{gathered} (a.1)R^{\nabla }=0, \\ (a.2)R^{\Phi .\nabla }=0, \\ (a.3)R^{{\nabla }^g}=0. \end{gathered}$$

In contrast with the curvature, the torsion of the connection is neither a gauge invariant nor a metric invariant in the sense of the Lemma above.

Thus arises the following question:

Given a gauge structure $(A,\nabla )$ and an inner product $(V,M,g)$ on a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

under what condition on g is the g-dual of ∇ torsion-free?

Further, in this work, we will underline a link between the question just mentioned and the theorem of M. Gromov, which states that any open differentiable manifold M which admits an almost complex structure admits a symplectic structure.

On an Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

Let one be given an inner product $(V,M,g)$, a gauge structure $(V,M,\nabla )$, and its g-dual $(V,M,{\nabla }^g)$.

On $(V,M,g,\nabla )$, we write the expression that looks like Formula (4.1.2), viz.,

$$\begin{gathered} {\delta }^{\nabla }g(s,s^{\prime };s^{″})=a\left(s\right).g(s^{\prime },s^{″})-g({\nabla }_{a\left(s\right)},s^{″}) \\ -g(s^{\prime },{\nabla }_{a\left(s\right)}{s}^{″})-a\left(s^{\prime }\right).g(s,s^{″})+g({\nabla }_{a\left(s^{\prime }\right)}s,s^{″})+g(s,{\nabla }_{a\left(s^{\prime }\right)}{s}^{″}). \end{gathered}$$

By direct calculations, it is easy to check the following identity:

$${\delta }^{\nabla }g(s,s^{\prime };s^{″})=g(T^{{\nabla }^g}(s,s^{\prime })-T^{\nabla }(s,s^{\prime }),s^{″}).$$

Thus the conclusion we will be interested in is the following (small) lemma.

Lemma 4.

Under the notation above, the following assumptions are equivalent:

$$\begin{gathered} (b.1)T^{{\nabla }^g}-T^{\nabla }=0, \\ (b.2){\delta }^{\nabla }g=0. \end{gathered}$$

A remark. Let one consider the following triplet,

$$T(g,\nabla )=\left\{{\delta }^{\nabla }g,T^{\nabla },T^{{\nabla }^g}\right\}$$

By virtue of Lemma 4, the nullity of two components of $T(g,\nabla )$ leads to the nullity of the third component. We are therefore faced with the following two relevant configurations determined by the nullity or the non-nullity of the curvature $R^{\nabla }$.

If

$$\begin{gathered} (cf.1.1)({\delta }^{\nabla }g,T^{\nabla })=(0,0) \\ (cf.1.2)R^{\nabla }\ne 0, \end{gathered}$$

then the pair $(g,\nabla )$ is (called) a Statistical structure.

If

$$\begin{gathered} (cf.2.1)({\delta }^{\nabla }g,T^{\nabla })=(0,0), \\ (cf.2.2)R^{\nabla }=0, \end{gathered}$$

then the pair $(g,\nabla )$ is (called) a Hessian structure.

Notation: The set of all Hessian structures on affine structure $(A,\nabla )$ is denoted by $Hess(A,\nabla )$.

Definition 12.

A Hessian structure

$$(g,\nabla )\in Hess(A,\nabla )$$

is called almost hyperbolic if

$$\left[g\right]=O\in H_{KV}^2(\nabla ,R).$$

Remember that

$$\left[g\right]=g+{\delta }^{\nabla }{C}_{KV}^1(\nabla ,R).$$

Thereby, it is easy to check the following statement:

Proposition 10.

The set of all almost hyperbolic Hessian structures $AHH(A,\nabla )$ is an open cone in the vector space $B_{KV}^2(A,\nabla )$.

Here,

$$B_{KV}^2(A,\nabla )={\delta }^{\nabla }{C}_{KV}^1(\nabla ,R).$$

The proposition is analogous to the theorem of J.-L. Koszul, see [32].

Theorem 9.

On the tangent Lie algebroid $(TM,M,1_{TM},[-,-])$ every hyperbolic structure admits non-trivial deformations.

Reminder.

In our context, $(V,M,a,[-,-\left]\right)$ is injective. Then the distribution

$$a\left(V\right)\subset TM$$

is a regular foliation whose leaves are equipped with the Koszul connection D which defined by the following formula:

$$D\circ a=a\circ \nabla .$$

(100)

Formula (100) has the following meaning:

$$D_{a\left(s\right)}a\left(s^{\prime }\right)=a\left({\nabla }_{a\left(s\right)}{s}^{\prime }\right)\forall (s,s^{\prime })\in \Gamma \left(V\right).$$

Along every leaf of a(V), the Koszul connection D defines a locally flat structure in the sense of Jean-Louis Koszul [32].

One also puts

$$g\ast (a\left(s\right),a\left(s^{\prime }\right))=g(s,s^{\prime }).$$

(101)

Remember that the mapping a is the vector bundle isomorphism of $(V,M)$ on $\left(a\right(V),M)$. Thus, given

$$(X,Y)\subset \Gamma \left(a\right(V\left)\right)$$

the Formula (100) tells us that

$$D_{X}Y=a\left({\nabla }_X{a}^{-1}\left(Y\right)\right)$$

and

$$g\ast (X,Y)=g(a^{-1}\left(X\right),a^{-1}\left(Y\right)).$$

Therefore, $\left(a\right(V),D)$ is an affine structure on the following Lie algebroid

$$A\ast =(a\left(V\right),M,1_{a\left(V\right)},{[-,-]}_P)$$

Furthermore, $g\ast$ is a Hessian structure on $(A\ast ,D)$ and ${[-,-]}_P$ stands for the Poisson bracket of vector fields. Further, it is easy to check that

$$[g\ast ]=0\in H_{KV}^2(\nabla ,R)$$

if and only if

$$[g\ast ]=0\in H_{KV}^2(D,R).$$

6.6. $H_{KV}^2(\nabla ,R)$ and Hessian Structures on Lie Algebroids

Reminders. We recall that given an n-dimensional locally flat manifold $(M,\nabla )$ and a fixed point $p\in M$, the universal covering of M is the following quotient

$$\tilde{M}={\displaystyle \frac{C^0((0,[0,1]),(p,M))}{H}}$$

where H stands for the fixed ends homotopy relation. Let ${\tau }_{\nabla }$ be the parallel transport along the paths

$$[0,1]\ni t\to c\left(t\right)\in M,$$

then we put

$$D\left(\left[c\right]\right)={\int }_0^1{\tau }_{\nabla }^{-1}\left({\displaystyle \frac{dc\left(s\right)}{ds}}\right)ds\in T_{p}M.$$

Therefore, the covering mapping is

$$\tilde{M}\ni \left[c\right]\to c\left(1\right)\in M,$$

and the developing mapping is

$$\tilde{M}\ni \left[c\right]\to D\left(\left[c\right]\right)\in T_{p}M.$$

We put

$$\Omega =D\left(\tilde{M}\right)\subset T_{p}M.$$

Then $(M,\nabla )$ is called hyperbolic if Ω is convex and does not contain any straight line. To learn more, the readers are referred to Koszul [32] Warnings. Before continuing, we recall that, in the context of differential geometry, Hessian structures play significant roles in information geometry, see [12,22,24,33]. They also play important roles in the Lie Group Theory of Heath, following Jean Marie Souriau [26].

The following implementation of $H_{KV}^2(\nabla ,R)$ is a foliation versus of a theorem of Koszul, see [32] (Theorem 3).

Proposition 11.

Under the notation used above, we assume that the following assertions hold:

(i) 

The Hessian structure g is positive.

(ii) 

$\left[g\right]=0\in H_{KV}^2(\nabla ,R).$

Then every compact leaf of $a\left(V\right)$ is hyperbolic in the sense we just recalled above.

Hint.

(a)

We focus on a compact leaf F whose fundamental group is noted $\pi \left(F\right)$. The couple $(F,D)$ is a locally flat structure whose universal covering is noted $(F^{★},D^{★}).$

(b)

The developing mapping is noted $\Delta$. The affine holomony representation of $(F,D)$ is denoted h. Thus, we put

$$\Gamma =h\left(\pi \right(F\left)\right)\subset Aff\left(k\right).$$

Here, F is k-dimensional.

(c)

To conclude, one takes into account that the data below satisfy the following identity,

$$\Delta (u.x)=\gamma \left(u\right).\Delta \left(x\right)\forall (u,x)\subset \pi \left(F\right)\times F^{★}.$$

6.7. $H_{KV}^2(\nabla ,V)$ and Deformations of $(A,\nabla )$

One considers the series of symmetric two-cochains

$$\theta \left(t\right)={\sum }_{q\in \mathbb{N}}{t}^q{\theta }_q$$

with

$${\theta }_q\in C^2(\nabla ,V)and{\theta }_q(s,s^{\prime })={\theta }_q(s^{\prime },s).$$

Let us put

$$\nabla \left(t\right)=\nabla +\theta \left(t\right)$$

The question is under what condition do the couples $(A,\nabla (t\left)\right)$ form series of affine structures on A.

It is easy to check the following statement.

Proposition 12.

A necessary condition for $(A,\nabla (t\left)\right)$ to be a series of affine structures on A is

$${\delta }_{KV}{\theta }_1=0.$$

The proposition above is a nod to $H_{KV}^2(\nabla ,V)$. Its analogues are well known in the theory of deformations of algebraic structures, see Nijenhis-Richardson [34] and S. Piper [35] for the theory of deformations of algebraic structures, and see also M. Kontsevich [36] for applications of the theory of deformations of associative algebras to the quantization of Poisson structures.

6.8. $H_{KV}^2(\nabla ,W)$ and Extensions of Affine Structures on Lie Algebroids

We consider an affine structure $(A,\nabla )$ on a Lie algebroid

$$A=(V,M,a,[-,-\left]\right).$$

Let $(W,M)$ be a vector bundle. We assume that $(W,M)$ is a left module of the affine structure $(A,\nabla )$, the left action of which is denoted as follows:

$$\Gamma \left(V\right)\times \Gamma \left(W\right)\ni (s,w)\to s.w\in \Gamma \left(W\right).$$

We are using notation (76) and the KV complex $C_{KV}(\nabla ,W)$ as in (82). Thus the following requirement is satisfied:

$$(s,s^{\prime };w)=(s^{\prime },s;w)$$

where

$$(s,s^{\prime };w)=\left({\nabla }_{a\left(s\right)}{s}^{\prime }\right).w-s.(s^{\prime }.w)$$

We consider the following exact sequence of vector bundles,

$$0\to (W,M)\to (W\oplus V,M)\to (V,M)\to 0.$$

(102)

Using a W-valued two-cochain

$$\theta \in C_{KV}^2(\nabla ,W)$$

we define the following product on $\Gamma (W\oplus V)$:

$$(w,s).(w^{\prime },s^{\prime })=(s.w^{\prime }+\theta (s,s^{\prime }),{\nabla }_{a\left(s\right)}{s}^{\prime }).$$

(103)

Here, the question is under what requirements the product (103) is a Koszul–Vinberg product on the vector bundle $W\oplus V.$

Without going into details, we state the following results; see [11] to learn more.

Theorem 10.

(a) Formula (103) defines a Koszul–Vinberg structure on $W\oplus V$ if and only if θ is KV twp-cocycle of the cochain complex $C_{KV}(\nabla ,W)$ as in Section 6.2.

(b) $Ext\left(\right(A,\nabla ),W)$ being the set of all equivalent classes of extensions of $(A,\nabla )$ by W, there is one-to-one correspondence between $Ext\left(\right(A,\nabla ),W)$ and $H_{KV}^2(\nabla ,W).$

Here is a useful overview.

The first six sections of which this is the sixth are devoted to gauge geometry on general Lie algebroids. To enlighten non-specialist readers, it is useful for us to recall our four fundamentals.

(F1): Our first fundamental is to state the main open problems which are studied on general Lie algebroids. Our approach to these open problems is based on the search for characteristic obstructions; that is to say notions which provide necessary and sufficient criteria.

(F2): Our second fundamental is to clear obstructions which are either Lie subalgebroids or Lie subalgebras of sections of Lie algebroids. The active arms of gauge equations are the two exact sequences (28) and (29).

(F3): Our third fundamental is Hessian Equation (47) and the object which is defined as in (48).

These fundamentals are used to associate an obstruction of vector nature with each of the open problems. Then, with each obstruction of vector nature, we associate an object of homological nature which is called a Koszul homological series.

(F4): Our fourth fundamental is the introduction of the notion of degeneracy of Koszul homological series. The notion of degeneracy converts the nullity of vector spaces into degeneracy of Koszul homological series.

(F5): Our fifth fundamental. Finally, from the differential geometry point of view, geometric properties of Koszul connections emerge from the dynamical properties of their holonomy groups.

Under the gauge geometry viewpoint, geometric properties of Koszul connections emerge from their homological properties via the notion of degeneracy of their Koszul homological series.

6.9. Some Major Gauge Structures on Lie Algebroids

Now we are in a position to introduce some relevant gauge structures on Lie algebroids. These notions are based on the notion of Koszul homology series and on their (p,q)-degeneracy.

We are in position to state the main definitions.

Definition 13.

On a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

(105) 

$F_{\delta }^{a,1.1}\left(A\right)$ is called an affine structure on A;

(106) 

$F_{\delta }^{m,1,1}\left(A\right)$ is called a metric structure on A;

(107) 

$F_{\delta }^{s,1,1}\left(A\right)$ is called a Fedosov structure on A.

Without going into details, we state the following theorem:

Theorem 11.

On a Lie algebroid

$$A=(V,M,a,[-,-\left]\right)$$

$\left\{F_{\delta }^{a,1,1}\left(A\right)\right\}$ is gauge invariant,

$\left\{F_{\delta }^{m,1,1}\left(A\right)\right\}and\left\{F_{\delta }^{s,1,1}\left(A\right)\right\}$ are both gauge invariants and metric invariants.

Hint: The tools which are used to prove this theorem are Formulas (7) and (15) and the exact sequences (28) and (29).

Warnings:

(a) 

It is to be noted that every Lie algebroid is metrically (1,1)-degenerate.

(b) 

In contrast with (a), every non-orientable Lie algebroid is symplectically (1,1)-nondegenerate.

(c) 

Whatever the differential manifold M, the tangent Lie algebroid $(T{T}^{★}M,T^{★}M)$ is symplectically (1,1)-degenerate.

7. Tangent Lie Algebroids: (1,1)-Nondegeneracy and Production of Labeled Foliations

Apart from the introduction, the central subject in the six precedent sections is the geometry of gauge structures on the category of general vector bundles over the same base manifold.

Section 7 is mainly devoted to the study of the tangent vector bundles which are canonically Lie algebroids under the Poisson bracket of vector fields. The central issue is to realize and to complete the machineries which are developed in the precedent first six sections that are devoted to general Lie algebroids.

Miscellaneous:

Among the major facts we pointed out are relationships between (1,1)-nondegeneracy and productions of labeled foliations, the most studied of which are Riemannian foliations, as follows: [6,7,8,9].

In fact, in addition to gauge dynamics GL(TM,M) and metric dynamics GM(TM,M), the category gauge structures on a tangent bundle $(TM,M)$ is acted by the group of diffeomorphisms of M that we note $Dif{f}^{★}(TM,M)$. The invariants of $Dif{f}^{★}(TM,M)$ are called geometric invariants.

For instance, while torsion $T^{\nabla }$ is neither gauge-invariant nor metric-invariant on general Lie algebroids, it is geometric invariant on tangent Lie algebroids. So on general Lie algebroids, we encounter situations where the homological degeneracy provides sufficient and necessary criteria to answer affirmatively to certain questions.

We aim to apply those materials to regular Lie subalgebroids of $(TM,M,1_{TM},[-,-])$. Those regular Lie subgroupoids are nothing but involutive regular distributions, viz., regular foliations.

7.1. Applications to Tangent Lie Algebroids $A\left(M\right)$

The context of this Section 7 is the category tangent Lie algebroids

$$A\left(M\right)=(TM,,M,1_{TM},[-,-]).$$

The bracket $[X,Y]$ is the Poisson bracket of vector fields.

Without explicit statement of the contrary, we will be dealing with SGau(TM,M). We proceed to implement the machineries which are developed in the first six sections above.

We start from the following data.

The gauge equations and the splitting exact sequences

$$0\to {\Omega }_2^{\nabla }\left(A\left(M\right)\right)\to J_{\nabla ,g}\left(A\left(M\right)\right)\to {\mathit{S}}_2^{\nabla }\left(A\left(M\right)\right)\to 0,\nabla \in \mathit{SGau}(\mathit{TM},\mathit{M}).$$

(104)

We focus on the following data:

The family of Koszul affinely vanishing-(1,1)-classes

$$F_{\delta }^{a,1,1}\left(A\left(M\right)\right),$$

(105)

the family of Koszul metrically vanishing-(1,1)-classes

$$F_{\delta }^{m,1,1}\left(A\left(M\right)\right),$$

(106)

the family of Koszul symplectically vanishing-(1,1)-classes

$$F_{\delta }^{s,1,1}\left(A\left(M\right)\right).$$

(107)

Definition 14.

On the Lie algebroid $A\left(M\right)$,

(105) 

is called an affine structure on $A\left(M\right)$.

(106) 

is called a metric structure on $A\left(M\right)$.

(107) 

is called a Fedosov structure on $A\left(M\right)$.

Regarding (105), concerning the search for obstructions to the existence of affine structures, see Smilie J. [21]. For a survey on the conjecture of Markus and related items, see the survey of W.M. Goldman [20], and see also Y. Carrière [37] Before further implementing sequence (104), we recall two definitions.

Definition 15.

A Riemannian foliation on the manifold M is a singular smooth inner product on $A\left(M\right)$, and g is subject to the following two requirements:

(r.1) rank(g) = constant,

(r.2) $i_{X}g=0$ implies $L_{X}g=0$.

Warning: In the classic definition of Riemannian foliation as in Moerdijk-MrCul [10], in Molino [8], in Reinhart [7], in Haefliger [9], and Ghys [6], the singular inner product g is assumed to be positive, viz.,

$$0\le g(X,X)\forall X\in \Gamma \left(TM\right).$$

We omitted this restriction in order to extend the theory of Riemannian foliations in Riemannian manifolds with positive signature.

Definition 16.

A symplectic foliation on M is a singular smooth symplectic product on $A\left(M\right)$, and ω which subject to the following requirements:

(r.1) $rank\left(\omega \right)=constant$,

(r.2) ω is de Rham-closed.

E. Ghys raised the problem of how to construct Riemannian foliations, see [6], Appendix E of [8].

The analogous problem versus symplectic foliations is how to produce all symplectic foliations.

One of our main aims is to completely solve the problem of producing all Riemannian foliations [7,8,9]. Exact sequences as in (104) take us in a position to completely solve the problem of production of all Riemannian foliations. The notation is that we used in Formulas (22) and (23). Regarding Riemannian foliations, here is our key statement,

Theorem 12.

Given a gauge structure

$$\left(A\right(M),\nabla )\in \mathit{SGau}(\mathit{TM},\mathit{M})$$

and an auxiliary positive inner product g, we take into account the exact sequence (104), viz.,

$$O\to {\Omega }_2^{\nabla }\left(A\left(M\right)\right)\to J_{\nabla ,g}\left(A\left(M\right)\right)\to {\mathit{S}}_2^{\nabla }\left(A\left(M\right)\right)\to 0.$$

Letting

$$q(g,\varphi )(s,s^{\prime })={\displaystyle \frac{1}{2}}\left(g(\varphi \left(s\right),s^{\prime })+g(s,\varphi \left(s^{\prime }\right))\right):$$

(1) 

The correspondence

$$J_{\nabla ,g}\ni \varphi \to q(g,\varphi )$$

as in our Formula (22) sends $J_{\nabla ,g}\left(A\left(M\right)\right)$ onto the set of all totally ∇-geodesic Riemannian foliations ${\mathit{S}}_2^{\nabla }\left(A\left(M\right)\right)$.

(2) 

Conversely, all Riemannian foliations are obtained by this process.

Demonstration.

(d.1) Since $\left(A\right(M),\nabla )$ is symmetric, by virtue of Lemma 1 and Proposition 2, the rank of $q(g,\varphi )$ is constant and

$$\begin{gathered} \left(L_{s}q(g,\varphi )\right)(s^{\prime },s^{″})=q(g,\varphi )({\nabla }_s^{\prime }s,s^{″})+q(g,\varphi )(s^{\prime },{\nabla }_s^{″}s) \\ =0\forall s\in Ker\left(q\right(g,\varphi \left)\right). \end{gathered}$$

Then $q(g,\varphi )$ is a totally ∇-geodesic Riemannian foliation.

(d.2) The proof of the converse is based on works of S. E. Kozlov [38] and those of S. N. Kupeli [39] on singular metric tensors. The results of Kozlov and those of Kupeli ensure that if a symmetric bilinear form q satisfies conditions (r.1) and (r.2) as in Definition 15, then there exists a symmetric Koszul connection ∇ such that

$${\nabla }_{s}q=O\forall s\in \Gamma \left(TM\right).$$

In [38], what is called the acyclicity condition is the property (r.2) as in Definition 15.

Thus, given an auxiliary positive inner product g, there exists

$$\Phi \in \Gamma \left(End\right(TM,M\left)\right)$$

such that

$$q(s,s^{\prime })=g(\Phi \left(s\right),s^{\prime })\forall (s,s^{\prime })\subset \Gamma (TM,M).$$

Further, one has the following identity,

$$\begin{gathered} s.g(\Phi \left(s^{\prime }\right),s^{″})=g(\Phi ({\nabla }_s{s}^{\prime },s^{″})+g(\Phi \left(s^{\prime }\right),{\nabla }_s{s}^{″}) \\ =g({\nabla }_s^g\Phi \left(s^{\prime }\right),s^{″})+g(\Phi \left(s^{\prime }\right),{\nabla }_s{s}^{″}). \end{gathered}$$

Thus we get

$${\nabla }_s^g\Phi \left(s^{\prime }\right)-\Phi \left({\nabla }_s{s}^{\prime }\right)=0.$$

Then, finally, one obtains the following identity:

$$\Phi \in J_{\nabla ,g}\left(A\left(M\right)\right).$$

The theorem is demonstrated.

7.2. Gauge Equations and Productions of All Riemannian Foliations

Let us go back to the exact sequence (104), viz.,

$$O\to {\Omega }_2^{\nabla }\left(A\left(M\right)\right)\to J_{\nabla ,g}\left(A\left(M\right)\right)\to S_2^{\nabla }\left(A\left(M\right)\right)\to O.$$

Let us set

$$RF\left(A\left(M\right)\right)={\cup }_{\left\{\nabla \in SGau\left(A\right(M\left)\right)\right\}}{S}_2^{\nabla }\left(A\left(M\right)\right),$$

(108)

$$J_g\left(A\left(M\right)\right)={\cup }_{\left\{\nabla \in SGau\left(A\right(M\left)\right)\right\}{J}_{\nabla ,g}\left(A\left(M\right)\right)},$$

(109)

$$SF\left(A\left(M\right)\right)={\cup }_{\left\{\nabla \in SGau\left(A\right(M\left)\right)\right\}}{\Omega }_2^{\nabla }\left(A\left(M\right)\right).$$

(110)

We take into account the arrow (26) and the identification as in (27) to define the following arrows:

$$J_g\left(A\left(M\right)\right)\ni \varphi \to \Phi \in RF\left(A\left(M\right)\right).$$

(111)

$$J_g\left(A\left(M\right)\right)\ni \varphi \to {\Phi }^{★}\in SF\left(A\left(M\right)\right).$$

(112)

$$S{F}_{\delta }^{s,1,1}\left(M\right)=F_{\delta }^{s,1,1}\left(A\left(M\right)\right)\cap SF\left(A\left(M\right)\right).$$

(113)

We involve works including that of Richard Atkins [1] and that of N. Kupeli [17] to state a corollary of Theorem 12 which answers the following question: How does one produce Riemannian foliations? see Appendix E in [8]. Using Theorem 12 on (26) and on (27), we can state the following result:

Theorem 13.

(1) The arrow (111) describes all Riemannian foliations on the manifold $M.$

(2) The arrow (112) describes symplectic foliations on the manifold M.

Please note that this result still assumes that a gauge equation exists, whose solutions can be obtained. It is thus not fully constructive. Below we are interested in a few supplements to Definition 14.

Definition 17.

$S{F}_{\delta }^{s,1,1}\left(M\right)$ as in (113) consists of the trivial symplectic foliation on M (i.e., the foliation whose leaves are single points of M) and it is called a symplectic structure on the manifold M.

Thus, quantitatively, the arrow (112) is the production of all symplectic structures on M.

Warnings. Theorem 13 deals with the quantitative study of Riemannian foliations. Our approach aims to produce all Riemannian foliations in (pseudo-)Riemannian geometry while the classical literature deals with Riemannian foliations on positive definite Riemannian Geometry only.

Regarding the Qualitative Study of Riemannian foliations on positive definite Riemannian Geometry, the readers are referred to the survey of André Haefliger in the Bourbaki seminar [9] as well as Gys’s Appendix E in [8]. Regarding the question of how to produce all symplectic foliations, the exact sequence (104) only gives a partial answer which is as follows:

Proposition 13.

Taking into account (23) and splitting the exact (104), the following correspondence

$$J_{\nabla ,g}\left(A\left(M\right)\right)\ni \varphi \to \omega (g,\varphi )\in {\Omega }_2^{\nabla }\left(A\left(M\right)\right)$$

sends $J_{\nabla ,g}\left(A\left(M\right)\right)$ onto the space of all totally ∇-totally geodesic symplectic foliations.

Hint: The proof is similar in part (d.1) as in the demonstration of Theorem 12.

8. Geometric Invariants on Gau(TM,M)

Three Dynamics and Their Invariants

Remember the following dynamics on Gau(TM,M)

(8.2.1) The action of the gauge group GL(TM,M) yielding gauge invariants.

(8.2.2) The action of the metric group GM(TM,M) yielding metric invariants.

(8.2.3) The action of the group of diffeomorphisms $Dif{f}^{★}(TM,M)$ yielding geometric invariants.

Every gauge structure $(TM,M,\nabla )$ is associated with two objects which are

(8.2.4) Torsion $T^{\nabla }$.

(8.2.5) Curvature $R^{\nabla }$.

Given a torsion-free gauge structure $(TM,M,\nabla )$, we are particularly interested in the following possible properties:

(8.2.6) Koszul affinely (1,1)-degeneracy,

(8.2.7) Koszul metrically (1,1)-degeneracy,

(8.2.8) Koszul symplectically (1,1)-degeneracy.

From the dynamics in

$$\left\{(8.2.1),(8.2.2),(8.2.3)\right\}$$

and from the data in

$$\left\{(8.2.4),(8.2.5),(8.2.6),(8.2.7),(7.2.8)\right\},$$

arises the question of which data are invariants of which dynamics.

Without going into details, we summarize some examples.

Are gauge invariants the same data as in {(8.2.5), (8.2.7), (8.2.8)}?

This follows from (22)–(25).

Are metric invariants the same data as in {(8.2.5), (8.2.7), (8.2.8)}?

This follows form both Theorem 6 and (98).

Are geometric invariants the data {(8.2.4), (8.2.5), (8.2.6), (8.2.7), (8.2.8)}.

That is based on (98).

9. Special Fedosov Manifolds and Kaehler Structures: Connections with Information Geometry

This section is devoted to a few topological properties of special Fedosov structures. These topological properties arise from the global analysis of gauge equations, viz., the study of analytic properties of the $J_{\nabla ,g}\left(M\right)$.

Before defining them and to motivate some interest in special Fedosov manifolds, we state one of their topological properties.

Theorem 14.

Every odd Betti number of a compact special Fedosov manifold $(M,\omega ,\nabla )$, $b_{2i+1}\left(M\right)$ is even.

9.1. Statistical Fedosov Manifolds

To start, we introduce the following definition.

Definition 18.

A statistical Fedosov manifold is a quadruplet $(M,g,\omega ,\nabla )$ subject to the following requirements:

(1) $(M,g,\nabla )$ is a statistical manifold,

(2) $(M,\omega ,\nabla )$ is a Fedosov manifold.

Let $(M,{\nabla }^{★})$ be the g-dual of $(M,\nabla )$. We recall that the Koszul connection ${\nabla }^{★}$ is defined by the following identity:

$$g({\nabla }_X^{★}Y,Z)=Xg(Y,Z)-g(Y,{\nabla }_{X}Z).$$

According to our previous notation, $J_{\nabla ,g}\left(M\right)$ and $J_{{\nabla }^{★},g}\left(M\right)$ are the vector spaces of solutions of the gauge equations

$${\nabla }^{★}\otimes \varphi -\varphi \circ \nabla =0,$$

and

$$\nabla \otimes \psi -\psi \circ {\nabla }^{★}=0,$$

respectively.

We consider the unique

$${\Phi }^{★}\in J_{\nabla ,g}\left(M\right)$$

such that

$$\omega (X,Y)=g({\Phi }^{★}\left(X\right),Y)\forall (X,Y).$$

(114)

It is to recall that

$$g({\Phi }^{★2}\left(X\right),X)\le 0\forall X.$$

Therefore, the set of the square root of $-{\Phi }^{★2}$ is denoted by

$$\sqrt{-{\Phi }^{★2}}=\left\{\Psi \in \Gamma \left(Hom(TM,TM)\right):{\Psi }^2=-{\Phi }^{★2}\right\}$$

(115)

We take into account (115); then the positive square root of $-{\Phi }^{★2}$ is denoted by

$${\Phi }_{+}^{★}\in \sqrt{-{\Phi }^{★}2}.$$

Henceforth, we keep the following identification:

$$(M,g,\omega ,\nabla )=(M,g,{\Phi }^{★},\nabla )$$

(116)

9.2. Special Statistical Fedosov Structures

Here is our main object

Definition 19.

A statistical Fedosov manifold $(M,g,{\Phi }^{★},\nabla )$ is called special iff

$${\Phi }_{+}^{★}\in J_{\nabla ,g}\left(M\right).$$

The following claim is obvious:

Proposition 14.

A statistical Fedosov structure $(M,g,{\Phi }^{★},\nabla )$ is special iff its g-dual $(M,g,{\Phi }^{★-1},{\nabla }^{★})$ is special.

We recall that, due to a counterexample from Thurston, the question of whether a compact symplectic manifold admits a Kahler structure was answered negatively. Hence, an interesting result in this section is the following statement.

Theorem 15.

By taking into account the identification (116), one has the following results:

(1) 

Every special statistical Fedosov manifold $(M,g,{\Phi }^{★},\nabla )$ admits a canonical pair of Kaehlerian structures $(M,{\Phi }^{★},J)$ and $(M,{\Phi }^{★-1},J)$.

(2) 

Moreover the Koszul connection ∇ and its g-dual connection ${\nabla }^{★}$ are complex analytic connections on the complex analytic manifold $(M,J).$

Proof. 

We keep the notation just used and assume that

$${\Phi }_{+}^{★}\in \sqrt{-{\Phi }^{★2}}\cap J_{\nabla ,g}\left(M\right),$$

Then the inverse ${\Phi }_{+}^{★-1}$ is a solution of the gauge equation

$$\nabla \otimes \psi -\psi \circ {\nabla }^{★}=0,$$

viz.,

$${\Phi }_{+}^{★-1}\in J_{{\nabla }^{★},g}\left(M\right).$$

We define the almost complex tensor

$$J={\Phi }^{★}\circ {\Phi }_{+}^{★-1}.$$

Therefore, we obtain the following identities:

$$\begin{array}{c}{\nabla }_X^{★}{\Phi }^{★}\left({\Phi }_{+}^{★-1}\left(Y\right)\right)={\Phi }^{★}\left({\nabla }_X{\Phi }_{+}^{★-1}\left(Y\right)\right)\\ ={\Phi }^{★}\left({\Phi }_{+}^{★-1}\left({\nabla }_X^{★}Y\right)\right),\end{array}$$

(117)

$$\begin{array}{c}{\nabla }_X{\Phi }_{+}^{★-1}(\Phi ★\left(Y\right))={\Phi }_{+}^{★-1}\left({\nabla }_X^{★}{\Phi }^{★}\left(Y\right)\right)\\ ={\Phi }_{+}^{★-1}\left({\Phi }^{★}{\nabla }_{X}Y\right)\left)\right)\end{array}$$

(118)

Then, we get the following identities:

$$J\left({\nabla }_X^{★}Y\right)={\nabla }_X^{★}J\left(Y\right)\forall (X,Y),$$

(119)

$$J{\nabla }_{X}Y={\nabla }_{X}JY\forall (X,Y).$$

(120)

Therefore, the Nijenhuis tensor

$$N_J(X,Y)=[X,Y]+J[J\left(X\right),Y]+J[X,J\left(Y\right)]-[J\left(X\right),J\left(Y\right)]$$

vanishes identically. Thus the couple $(M,J)$ is a complex analytic structure on M.

Moreover, both $(TM,\nabla )$ and $(TM,{\nabla }^{★})$ are complex analytic gauge structures.

Consequently, the couple $(TM,{\nabla }^{LC})$ is a complex analytic gauge structure as well.

Here, ${\nabla }^{LC}$ is the Levi Civita connection of $(M,g)$ and is defined as follows:

$$2{\nabla }_X^{LC}Y={\nabla }_X^{★}Y+{\nabla }_{X}Y.$$

Before pursuing, we recall the symplectic forms,

$$\omega (X,Y)=g({\Phi }^{★}\left(X\right),Y),$$

(121)

$${\omega }^{\prime }(X,Y)=g({\Phi }^{★-1}\left(X\right),Y).$$

(122)

It is easy to check the following identities:

$$\omega (JX,JY)=\omega (X,Y)\forall (X,Y),$$

(123)

$${\omega }^{\prime }(JX,JY)={\omega }^{\prime }(X,Y)\forall (X,Y).$$

(124)

The following symmetric bilinear forms, h and $h^{\prime }$, are positive definite:

$$h(X,Y)=\omega (X,JY),$$

(125)

$$h^{\prime }(X,Y)={\omega }^{\prime }(X,JY).$$

(126)

Finally, $(M,\omega ,J)$ and $(M,{\omega }^{\prime },J)$ are Kaehlerian manifolds. □

9.3. Special Fedosov Structure

In the framework of general Fedosov structures $(M,\omega ,\nabla )$, part (1) in Theorem 15 still deserves attention [5].

Indeed, let $Ri{e}^{+}\left(M\right)$ be the category of positive Riemannian structures on the manifold M and isometries.

According to the identification (116) every Fedosov structure $(M,\omega ,\nabla )$ gives rise to the mapping

$$Ri{e}^{+}\left(M\right)\ni g\to {\Phi }_g^{★}\in J_{\nabla g}\left(M\right)$$

(127)

which is defined as follows:

$$\omega (X,Y)=g({\Phi }_g^{★}\left(X\right),Y),$$

(128)

$${\nabla }^g\otimes {\Phi }_g^{★}-{\Phi }_g^{★}\circ \nabla =0.$$

(129)

The square of ${\Phi }_g^{★}$, namely ${\Phi }_g^{★2}$ is symmetric and is negative definite w.r.t. the Riemannian structure $(M,g)$. The positive square root of $-{\Phi }_g^{★2}$ is is denoted by ${\Phi }_{g+}^{★}.$ Therefore, we introduce the almost complex tensor,

$$J={\Phi }_g^{★}\circ {\Phi }_{g+}^{★-1}.$$

Since ∇ is torsion-free, the differential two-form defined as in (128) is de Rham-closed. At another side, the identity (120) implies the integrability of the almost complex structure $(M,J)$.

Finally, if the Fedosov structure $(M,{\Phi }_g^{★},\nabla )$ is special, then

$$(M,\omega ,J)=(M,{\Phi }_g^{★},J)$$

is Kahlerian manifold.

Because the g-dual ${\nabla }^{★}$ may not be torsion-free the following differential two-form

$${\omega }^{\prime }(X,Y)=g({\Phi }_g^{★-1}\left(X\right),Y)$$

may be not de Rham-closed.

Before pursuing, we introduce the following notion:

Definition 20.

A Fedosov structure $(M,\omega ,\nabla )$ is called special if there exists a positive Riemannian structure $(M,g)$ such that

$$\begin{gathered} \omega (X,Y)=g({\Phi }_g^{★}\left(X\right),Y), \\ {\Phi }_{g+}^{★}\in J_{\nabla ,g}\left(M\right). \end{gathered}$$

We keep the notation ${\Phi }_g^{★}$ and ${\Phi }_{g+}^{★}$ which is used above. Our last discussions above yields the following statement.

Theorem 16.

Let $(M,\omega ,\nabla )$ be a special Fedosov manifold. Then there exists $g\in R^{ie+}\left(M\right)$, yielding the following data:

(1) 

$(M,\omega ,\nabla )$ carries the canonical Kahlerian structure

$$(M,\omega ,J)=(M,{\Phi }_g^{★},J)$$

(2) 

∇ is the Levi Civita connection of $(M,\omega ,J)$.

(3) 

The datum $(TM,{\omega }^{\prime },J)$ is Hermitian Lie algebroid.

(4) 

${\nabla }^{★}$ is Hermitian Koszul connection on the Hermitian Lie algebroid $(TM,{\omega }^{\prime },J).$

A straightforward corollary of Theorem 16 is that there exist symplectically (1,1)-degenerate differentiable manifolds M whose $F_{\delta }^{s,1,1}\left(A\left(M\right)\right)$ does not contain any special Fedosov structure. The first such example of $F_{\delta }^{s,1,1}\left(M\right)$ not containing any special Fedosov structure is due to W. Thurston [40,41].

Regarding the open problem of metric connection, statement (2) in Theorem 16 yields a partial answer.

We use the identification as in (116). Thus we consider a statistical Fedosov manifold

$$(M,g,\omega ,\nabla )=(M,g,{\Phi }^{★},\nabla ).$$

We also involve the Riemannian structures $(M,h)$ and $(M,h^{\prime })$ as in (125) and (126), respectively.

Theorem 17.

Under the notation above, if a statistical Fedosov manifold $(M,g,{\Phi }^{★},\nabla )$ is special then the connection ∇ and its g-dual ${\nabla }^{★}$ are the Levi Civita connection of $(M,h)$ and $(M,h^{\prime })$, respectively.

Finally, a Fedosov manifold $(M,\omega ,\nabla )$ admits a Kahler structure $(M,\omega ,J)$ iff it is special.

10. Comments/Conclusions

The purpose of this short section is to frame for the reader the central problems that are studied in this work and to emphasize the results which are obtained.

The subject is the study of properties of Koszul connections on vector bundles. Gauge geometry must be understood as the search of data that are invariant under the action of holonomy groups of Koszul connections.

In addition to inner products and symplectic products on general vector bundles, we have recalled the notion of affine structure on Lie algebroids and their deformations. Among central concerns, we aim to characterize those Koszul connections whose holonomy preserves a structure among those three types of structures just mentioned, viz., metric structure, symplectic structure, and affine structure.

The tools we have involved are of homological nature. We have generalized a canonical homological class which has been introduced by J.-L. Koszul in 1974, see [19]. We have introduced Koszul homological series and their degeneracy. We have successfully involved the notion of Koszul homological degeneracy to Introduce Other Approaches to the central problems which are mentioned above, viz., the problem of metric connection on Lie algebroids, the problem of symplectic connection on Lie algebroids, and the problem of affine structure on Lie algebroids. Thus, on the category of Lie algebroids, the use of Koszul homological degeneracy avoids recourse to calculations of holonomy groups of Koszul connections.

Other tools of excellent efficiency are Theorem 1, Theorem 2, and Proposition 6. These three statements illustrate that, depending on the framework and context, the group of inversible mappings which preserve a connection can be of zero dimension, of finite positive dimension, or of infinite dimension. Theorems 1, 2 and Proposition 6 are the machines for converting obstructions of vector nature into obstructions of homological nature.

An analogous theorem of Theorem 1 is known in global analysis as the theorem of the finiteness of the dimension of Lie group of diffeomorphisms which preserve alinear connection, see Kobayashi-Nomizu [17], Theorem 23. The classical main ingredient used to demonstrate this statement on tangent Lie algebroids is the existence the Fundamental 1-form of principal bundles of linear frames [17,25,42]. This ingredient is absent from principal frame bundles of general Lie algebroids. In this work we involve Theorem 1 only. We have given a direct demonstration of it. However, it is important to note the subtle difference between our Theorem 1 and Theorem 23 as in Kobayashi-Nomizu [17]. The latter shows the finiteness of dimension while Theorem 1 shows that the only $C^{\infty }\left(M\right)$-module is zero.

Section 7 is devoted to implementing those machineries in the canonical tangent Lie algebroids over differentiable manifolds. This is an illustration of approaching the differential geometry from the viewpoint of gauge geometry. For a more in-depth discussion and highly significant examples, readers are referred to K.S. Donaldson [43,44] and to Petrie-Randal [45].

To achieve the complete production of all Riemannian foliations, see our Theorem 13, we have introduced the family of gauge equations and implemented the analysis of their solutions. This approach was inspired by methods of both information geometry and the topology of information, see Amari-Nagaoka [46], Gromov M. [30], Baudot-Bennequin [47], and Nguiffo Boyom [24].

Although based on conceptual approaches, both Theorem 12 and Theorem 13 enjoy the virtue of being constructive theorems.