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Article

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

by
Michel Nguiffo Boyom
ALEXANDER GROTHENDIECK INSTITUTE, IMAG-UMR CNRS 5149-c.c.051, University of Montpellier, PL. E. Bataillon, F-34095 Montpellier, France
Entropy 2016, 18(12), 433; https://doi.org/10.3390/e18120433
Submission received: 1 June 2016 / Accepted: 16 November 2016 / Published: 2 December 2016
(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)

Abstract

:
Let us begin by considering two book titles: A provocative title, What Is a Statistical Model? McCullagh (2002) and an alternative title, In a Search for Structure. The Fisher Information. Gromov (2012). It is the richness in open problems and the links with other research domains that make a research topic exciting. Information geometry has both properties. Differential information geometry is the differential geometry of statistical models. The topology of information is the topology of statistical models. This highlights the importance of both questions raised by Peter McCullagh and Misha Gromov. The title of this paper looks like a list of key words. However, the aim is to emphasize the links between those topics. The theory of homology of Koszul-Vinberg algebroids and their modules (KV homology in short) is a useful key for exploring those links. In Part A we overview three constructions of the KV homology. The first construction is based on the pioneering brute formula of the coboundary operator. The second construction is based on the theory of semi-simplicial objects. The third construction is based on the anomaly functions of abstract algebras and their abstract modules. We use the KV homology for investigating links between differential information geometry and differential topology. For instance, “dualistic relation of Amari” and “Riemannian or symplectic Foliations”; “Koszul geometry” and “linearization of webs”; “KV homology” and “complexity of models”. Regarding the complexity of a model, the challenge is to measure how far from being an exponential family is a given model. In Part A we deal with the classical theory of models. Part B is devoted to answering both questions raised by McCullagh and B Gromov. A few criticisms and examples are used to support our criticisms and to motivate a new approach. In a given category an outstanding challenge is to find an invariant which encodes the points of a moduli space. In Part B we face four challenges. (1) The introduction of a new theory of statistical models. This re-establishment must answer both questions of McCullagh and Gromov; (2) The search for an characteristic invariant which encodes the points of the moduli space of isomorphism class of models; (3) The introduction of the theory of homological statistical models. This is a pioneering notion. We address its links with Hessian geometry; (4) We emphasize the links between the classical theory of models, the new theory and Vanishing Theorems in the theory of homological statistical models. Subsequently, the differential information geometry has a homological nature. That is another notable feature of our approach. This paper is dedicated to our friend and colleague Alexander Grothendieck.

Contents

  • 1Introduction        4
    • 1.1The Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
    • 1.2Some Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
    • 1.3The content of the Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
  • 2Algebroids, Moduls of Algebroids, Anomaly Functions        8
    • 2.1The Algebroids and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
    • 2.2Anomaly Functions of Algebroids and of Modules . . . . . . . . . . . . . . . . . . . . . . 9
  • 3The Theory of Cohomology of KV Algebroids and Their Modules        11
    • 3.1The Theory of KV Cohomology—Version the Brute Formula of the Coboundary Operator 11
      • 3.1.1The Cochain Complex CKV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
      • 3.1.2The Total Cochain Complex Cτ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
    • 3.2The Theory of KV Cohomology—Version: the Semi-Simplicial Objects . . . . . . . . . . 14
      • 3.2.1Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
      • 3.2.2Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
      • 3.2.3Notation-Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
      • 3.2.4The KV Chain Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
      • 3.2.5The V-Valued KV Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
      • 3.2.6Two Cochain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
      • 3.2.7Residual Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
    • 3.3The Theory of KV Cohomology—Version the Anomaly Functions . . . . . . . . . . . . . 20
      • 3.3.1The General Challenge CH ( D ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
      • 3.3.2Challenge CH ( D ) for KV Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
      • 3.3.3The KV Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
      • 3.3.4The Total Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
      • 3.3.5The Residual Cohomology, Some Exact Sequences, Related Topics, DTO-HEG-IGE-ENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
  • 4The KV Topology of Locally Flat Manifolds        26
    • 4.1The Total Cohomology and Riemannian Foliations . . . . . . . . . . . . . . . . . . . . . . 26
    • 4.2The General Linearization Problem of Webs . . . . . . . . . . . . . . . . . . . . . . . . . . 28
    • 4.3The Total KV Cohomology and the Differential Topology Continued . . . . . . . . . . . 30
    • 4.4The KV Cohomology and Differential Topology Continued . . . . . . . . . . . . . . . . . 34
      • 4.4.1Kernels of 2-Cocycles and Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . 34
  • 5The Information Geometry, Gauge Homomorphisms and the Differential Topology        35
    • 5.1The Dualistic Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
      • 5.1.1Statistcal Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
      • 5.1.2A Uselful Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
      • 5.1.3The Homological Nature of Gauge Homomorphisms . . . . . . . . . . . . . . . . 41
      • 5.1.4The Homological Nature of the Equation FE∇∇* . . . . . . . . . . . . . . . . . . . 43
      • 5.1.5Computational Relations. Riemannian Foliations. Symplectic Foliations: Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
      • 5.1.6RiemannianWebs—SymplecticWebs in Statistical Manifolds . . . . . . . . . . . 49
    • 5.2The Hessian Information Geometry, Continued . . . . . . . . . . . . . . . . . . . . . . . . 51
    • 5.3The a-Connetions of Chentsov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
    • 5.4The Exponential Models and the Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . 55
  • 6The Similarity Structure and the Hyperbolicity        58
  • 7Some Highlighting Conclusions        59
    • 7.1The Total KV Cohomology and the Differential Topology . . . . . . . . . . . . . . . . . . 59
    • 7.2The KV Cohomology and the Geometry of Koszul . . . . . . . . . . . . . . . . . . . . . . 60
    • 7.3The KV Cohomology and the Information Geometry . . . . . . . . . . . . . . . . . . . . 60
    • 7.4The Differential Topology and the Information Geometry . . . . . . . . . . . . . . . . . . 60
    • 7.5The KV Cohomology and the Linearization Problem for Webs . . . . . . . . . . . . . . . 60
  • 8B. The Theory of StatisticaL Models        61
    • 8.1The Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
    • 8.2The Category FB ( Γ , Ξ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
      • 8.2.1The Objects of FB ( Γ , Ξ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
      • 8.2.2The Morphisms of FB ( Γ , Ξ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
    • 8.3The Category GM ( Ξ , Ω ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
      • 8.3.1The Objects of GM ( Ξ , Ω ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
      • 8.3.2The Global Probability Density of a Statistical Model . . . . . . . . . . . . . . . . 70
      • 8.3.3The Morphisms of GM ( Ξ , Ω ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
      • 8.3.4Two Alternative Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
      • 8.3.5Fisher Information in GM ( Ξ , Ω ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
    • 8.4Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
      • 8.4.1The Entropy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
      • 8.4.2The Fisher Information as the Hessian of the Local Entropy Flow . . . . . . . . . 75
      • 8.4.3The Amari-Chentsov Connections in GM ( Ξ , Ω ) . . . . . . . . . . . . . . . . . . . 75
      • 8.4.4The Homological Nature of the Probability Density . . . . . . . . . . . . . . . . . 76
      • 8.4.5Another Homological Nature of Entropy . . . . . . . . . . . . . . . . . . . . . . . 77
  • 9The Moduli Space of the Statistical Models        78
  • 10The Homological Statistical Models        83
    • 10.1The Cohomology Mapping of HSM ( Ξ , Ω ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
    • 10.2An Interpretation of the Equivariant Class [Q] . . . . . . . . . . . . . . . . . . . . . . . . 85
    • 10.3Local Vanishing Theorems in the Category HSM ( Ξ , Ω ) . . . . . . . . . . . . . . . . . . 85
  • 11The Homological Statistical Models and the Geometry of Koszul        88
  • 12Examples        88
  • 13Highlighting Conclusions        91
    • 13.1Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
    • 13.2Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
    • 13.3KV Homology and Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
    • 13.4The Homological Nature of the Information Geometry . . . . . . . . . . . . . . . . . . . 91
    • 13.5Homological Models and Hessian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 92
  • AAppendix A        92
    • A.1The Affinely Flat Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
    • A.2The Hessian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
    • A.3The Geometry of Koszul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
    • A.4The Information Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
    • A.5The Differential Topology of a Riemannian Manifold . . . . . . . . . . . . . . . . . . . . 94

1. Introduction

1.1. The Notation

Throughout the paper we use tha following notation. N is the set of non negative integers, Z is the ring of integers, R is the field of real numbers, C ( M ) is the associative commutative algebra of real valued smooth functions in a smooth manifold M. Let ∇ be a Koszul connection in a manifold M, R is the curvature tensor of ∇. It is defined by
R ( X , Y ) = X Y Y X [ X , Y ] .
T is the torsion tensor of ∇. It is defined by
T ( X , Y ) = X Y Y X [ X , Y ] .
Let X be a smooth vector field in M. L X is the Lie derivative of ∇ in the direction X · ι ( X ) R is the inner product by X. To a pair of Koszul connections ( , * ) we assign three differential operators. They are denoted by D * , D and D .
(A.1)
D * is a first order differential operator. It is defined in the vector bundle H o m ( T M , T M ) . Its values belong to the vector bundle H o m ( T M 2 , T M ) .
(A.2)
D and D are 2nd order differential operators. They are defined in the vector bundle T M . Their values belong to the vector bundle H o m ( T M 2 , T M ) . Let X be a section of T M and let ψ be a section of T * M T M . The differential operators just mentioned are defined by
(1a) D * ( ψ ) = * ψ ψ , (1b) D ( X ) = L X ι ( X ) R , (1c) D ( X ) = 2 ( X ) .
Part A of this paper is partially devoted to the global analysis of the differential equation
F E ( * ) : D * ( ψ ) = O .
The solutions to F E ( * ) are useful for addressing the links between the KV homology, the differential topology and the information geometry.
The purpose of a forthcoming paper is the study of the differential equations
F E * ( ) : D ( X ) = 0 ,
F E * * ( ) : D ( X ) = 0 .
In the Appendix A to this paper we overview the role played by the solutions to F E * * ( ) in some still open problems.

1.2. Some Explicit Formulas

Let x = ( x 1 , . . . , x m ) be a system of local coordinate functions of M. In those coordinates the Christoffel symbols of both ∇ and * are denoted by Γ i j : k and Γ i j : k * respectively. We use those coordinate functions for presenting an element ψ M ( * ) as a matrix [ ψ i j ] . Thus by setting i = x i one has
i j = Γ i j : k k .
We focus on F E ( * ) and of F E * * ( ) . They are equivalent to the following system of partial differential equations
[ S i j : k ] : ψ k j x i 1 m ( Γ i j : ψ k Γ i : k * ψ j ) = 0 ,
[ Θ i j k ( X ) ] : 2 X k x i x j + α [ Γ i α k X α x j + Γ j α k X α x i Γ i j X k x α α ] + α [ Γ j α k x i + β [ Γ j α β Γ i β k Γ i j β Γ β k α ] ] X α = 0 .
In Part A we address the links between the following topics DTO, HGE, IGE and ENT. Those topics are presented as vertices of a square whose centre is denoted by K V H .
(1)
DTO stands for Differential TOpology. In DTO, FWE stands for Foliations and WEbs.
(2)
HGE stands for Hessian GEometry. Its sources are the geometry of bounded domains, the topology of bounded domains, the analysis in bounded domains. Among the notable references are [1,2,3]. Hessian geometry has significant impacts on thermodynamics, see [4,5], About the impacts on other related topics the readers are referred to [6,7,8,9,10,11,12].
(3)
IGE stands for Information GEometry. That is the geometry of statistical models. More generally its concern is the differential geometry of statistical manifolds. The range of the information geometry is large [13]. Currently, the interest in information geometry is increasing. This comes from the links with many major research domains [14,15,16]. We address some significant aspects of those links. Non-specialist readers are referred to some fundamental references such as [17,18]. See also [4,19,20,21,22,23]. The information geometry also provides a unifying approach to many problems in differential geometry, see [21,24,25]. The information geometry has a large scope of applications, e.g., physics, chemistry, biology and finance.
(4)
ENT stands for ENTropy. The notion of entropy appears in many mathematical topics, in Physics, in thermodynamics and in mechanics. Recent interest in the entropy function arises from its topological nature [14]. In Part B we introduce the entropy flow of a pair of vector fields. The Fisher information is then defined as the Hessian of the entropy flow.
(5)
KVH stands for KV Homology. The theory of KV homology was developed in [9]. The motivation was the conjecture of M. Gerstenhaber in the category of locally flat manifolds. In this paper we emphasize other notable roles played by the theory of KV homology. It is also useful for discussing a problem raised by John Milnor in [26].
The conjecture of Gerstenhaber is the following claim.
Every restricted theory of deformation generates its proper cohomology theory [27].
Loosely speaking, in a restricted theory of deformation one has the notion of both infinitesimal deformation and trivial deformation. The challenge is the search for a cochain complex admitting infinitesimal deformations as cocycles. In the present paper, K V H is useful for emphasizing the links between the vertices D T O , H G E , I G E and E N T . That is our reason for devoting a section to K V H .
Warning. 
We propose to overview the structure of this paper. The readers are advised to read this paper as through it were a wander around the vertices of the square “DTO-HGE-IGE-ENT”. Thus, depending on his interests and his concerns a reader could walk several times across the same vertex. For instance the information geometry appears in many sections, depending on the purpose and on the aims.

1.3. The content of the Paper

This paper is divided into Part A and Part B.
Section 1 is the Introduction. Section 2 is devoted to algebroids, modules of algebroids and the theory of KV homology of the Koszul-Vinberg algebroids. To introduce the KV cohomology we have adopted three approaches. Each approach is based on its specific machinery. However, the readers will face three cochain complexes which are pairwise quasi isomorphic. The KV cohomology is present throughout this paper. At the end of Part B the reader will see that the theory of statistical models is but a vanishing theorem in the theory of KV hohomology. The first approach is based on the pioneering fundamental brute formula of the coboundary operator. Historically, the brute formula is the first to have been constructed [9].
This first approach is used in many sections of this paper. Regarding the theory of deformation of the Koszul Geometry, the KV cohomology is the solution to the conjecture of Gerstenhaber. In the theory of modules of KV algebroids the role played by the KV cohomology is practically SIMILAR to the role played by the Hocshild cohomology in the category of associative alggebroids and their modules. This last remark holds for the role played by the Chevalley-Eilenberg cohomology in the category of Lie algebroids and their modules. Nevertheless, our comparison fails in the theory of Extension of modules over algebroids. In both categories of extensions of modules over associative algebroids and Lie algebroids the moduli space of equivalence class is encoded by cohomology classes of degree one. In the category of extensions of KV modules the moduli space is encoded by a spectral sequence. That was a unexpected feature in [9]. The pioneering coboundary operator of Nijenhuis [28] may be derived from the total brute coboundary operator introduced in [29].
The second approach is based on the notion of simplecial objects.
The third approach is based on the theory of anomaly functions for abstract algebras and their abstract modules. The idea has emerged from recent correspondences with one of my former teachers. The KV anomaly function of a Koszul connection ∇ may be expressed in terms of the ∇-Hessian operators 2 , namely
K V ( X , Y , Z ) = < 2 ( Z ) , ( X , Y ) > < 2 ( Z ) , ( Y , X ) > .
This approach is a powerful for addressing the relationships between the global analysis, the differential topology and the information geometry. The approach by the anomaly functions suggests many conjectures. Among those conjectures is the following.
Conjecture. Every anomaly function of algebras and of modules yields a theory of cohomology of algebras and modules.
Section 3. This section is devoted to the theory of KV (co)homology of Koszul-Vinberg algebroids. We focus on cohomological data which are used in the paper.
Section 4. This section is devoted the KV algebroids which are defined by structures of locally flat manifold. The KV cohomology theory is used for highlighting the impacts on the differential topology of the information geometry and its methods. We make the most of some relationships between the KV cohomology and the global analysis of the differential equation F E * ( * ) . We also sketch the global analysis of the differential equation
F E * * ( ) .
This leads to the function
L C r b ( ) Z .
We explain how to interpret r b as a distance. (See the Appendix A to this paper ). For instance, the function r b gives rise to an numerical invariant r b ( M ) which measures how far from being an exponential family is a statistical model M . This result is a significant contribution to the information geometry, see [18,22,24].
Section 5. We are interested in how interact the information geometry, the KV cohomology and the geometry and Koszul. In particular we relate the notion of hyperbolicity and vanishing theorems in the KV cohomology.
Section 6. This section is devoted to the homological version of the geometry of Koszul. Our approach involves the dualistic relation of Amari. The KV cohomology links the dualistic relation with the geometry of Koszul.
Section 7. In this section summarize the highlighting features of Part A.
Section 8. This is the starting section of the second part B. This Part B is devoted to new insights in the theory of statistical models. On 2002 Peter McCullar raised the provocative question.
What Is a Statistical Model
Across the world (Australia, Canada, Europe, US) the M c C u l l a g h s paper became the object of many criticisms and questions by eminent theoretical and applied statisticians [30].
Part B is aimed at supplying some deficiencies in the current theory of statistical models. We address some criticisms which support the need of re-establishing the theory of statistical model for measurable sets. Those criticisms are used for highlight the lack of both Structure and Relations. Those criticisms also highlight the search of M. Gromov [15]. The need for structures and relations was the intuition of Peter McCullagh. Loosely speaking there is a lack of Intrinsic Geometry in the sense of Erlangen. Subsequently the lack of intrinsic geometry yields other things that are lacking. The problem of the moduli space of models is not studied, although this would be crucial for applied information geometry, and for applied statistics. That might be a key in reading some the controversy about [30].
Section 9. In this section we address the problem of moduli space of statistical models. The problem of moduli space in a category is a major question in Mathematic. It is generally a difficult problem that involves finding a characteristic invariant which encodes the point of the moduli space. Such an invariant is a crucial step toward the geometry and the topology of a moduli space. Among other needs, the problem of encoding the moduli space of models has motivated our need of a new approach, that is to say the need of a theory having nice mathematical structure and relations. In this Part B the problem of the moduli space is solved. To summarize the theorem describing the moduli spaces of statistical models we need the following notation.
A gauge structure in a manifold M is a pair ( M , ) where ∇ is a Koszul connection in M. The category of gauge structures in M is denoted by L C ( M ) . We are concerned with the vector bundle T * 2 M of bi-linear forms in the tangent bundle T M . The sheaf of sections of T * 2 M is denoted by B L ( M ) .
The category of m-dimensional statistical models (to be defined) of a measurable ( Ξ , Ω ) is denoted by GM m ( Ξ , Ω ) . The category of random functors
L C ( M ) × Ξ B L ( M )
is denoted by F ( L C , B L ) ( M ) . One of the interesting breakthrough in this Part.B is the following solution to the problem of moduli.
Theorem 1.
There exists a functor
GM m ( Ξ , Ω ) M q M B L ( M )
which determines a model M up to isomorphism.
Let p be the probability density of a model M . The mathematical expectation of q M ( ) is defined by
E ( q M ( ) ) = Ξ p q M ( ) .
The quantity E ( q M ) ( ) does not depend on the Koszul connection ∇. It is called the Fisher information of M .
This theorem emphasizes the Search for structure [16].
Section 10. This section is devoted to introduce the category of homological statistical models. This may be interpreted as a variant of the topology of the information. Another approach is to be found in Baudot-Bennequin [31].
The current theory (as in [17]) is called the classical (or local) theory. This means that a statistical model as in [17,18] is derived from the localization of a homological model. Loosely speaking such a model expresses a local vanishing theorem in the theory of homological statistical models.
Section 11. This section is devoted to discussing the links between the geometry of Koszul and the theory of homological statistical models. Those investigations lead to this notable feature.
The Geometry of Koszul, the homological statistical models and the classical information geometry locally look alike.
Section 12. Through Section 9 the framework is the category of equivariant locally trivial fibration. This assumption is weakened in Section 12. We recall the relationships between the Cech cohomology and the theory of locally trivial fiber bundle. We extend the scope of applications of the methods of the information geometry. Those extensions produce some interesting results. Here is an instance.
Theorem 2.
Let M be an oriented compact real analytic manifold and let C ω ( M 2 ) be the space of real valued analytic functions in M 2 . There exists a non trivial map of C ω ( M 2 ) in the family of (positive) stratified Riemannian foliation in M.
Section 13. This Section 13 is a variant of Section 7.
Section 14 is an appendix we have mentioned. It is devoted to overview a few new significant results. Those results are derived from the global analysis of the differential operators
D , D , L C ( M ) .
The solutions to a few open problems are announced.

2. Algebroids, Moduls of Algebroids, Anomaly Functions

The purpose of this section is to introduce basic notions in the algebraic topology of locally flat manifolds.

2.1. The Algebroids and Modules

Given a smooth fiber bundle
B M
the set of smooth sections of B is denoted by Γ ( B ) .
Definition 1.
An (abstract) real algebra is a real vector space A endowed with a bilinear map
A × A A
Definition 2.
An (abstract) real two-sided module of an (abstract) algebra A is a real vector space W with two bilinear mappings
A × W W ,
W × A W
Warning. 
Here algebra means a multiplication a · b without any rule of calculations. So the product a · b · c is meaningless.
Throughout this paper, the smooth manifolds we deal with are connected and paracompact. In a smooth manifold M all geometrical objects we are interested in are smooth as well.
The vector space of smooth vector fields in a manifold M is denoted by X ( M ) . It is a left module of the associative commutative algebra C ( M ) .
Consider a real vector bundle
E M .
The real vector space of sections of E is denoted by Γ ( E ) .
Definition 3.
A real algebroid over a smooth manifold M is a real vector bundle whose vector space of sections is a real algebra.
So the vector space of sections of a real algebroid E is endowed with a R -bilinear map
Γ ( E ) × Γ ( E ) ( s , s * ) s · s * Γ ( E )
To simplify the multiplication of two sections is denoted s · s * .
Definition 4.
A two-sided module of an algebroid E is a vector bundle
V M
whose vector space of sections is a two-sided module of the algebra Γ ( E ) .
Let s be section E and let v be a section of V . Both left action s on v and the right action of s on v are denoted by s · v and v · s .
Definition 5.
An anchored vector bundle over M is a pair
( E , b )
formed by a real vector bundle E and a vector bundle homomorphism
E e b ( e ) T M .
The homomorphism b is called the anchor map.

2.2. Anomaly Functions of Algebroids and of Modules

Let V be a two-sided module of an algebroid ( E , b ) .
Definition 6.
An anomaly function of an algebroid E is a 3-linear map A E of Γ ( E ) 3 in Γ ( E ) whose values A E ( s 1 , s 2 , s 3 ) belong to s p a n R [ ( s i · s j ) · s k , s i · ( s j · s k ) ; i , j , k [ 1 , 2 , 3 ] ] . An anomaly function of an E -module V is a 3-linear map A EV of Γ ( E ) 2 × Γ ( V ) in Γ ( V ) whose values A EV ( s , s * , v ) belong to s p a n R [ ( s · s * ) · v , s · ( s * · v ) s , s * Γ ( E ) , v Γ ( V ) ] .
In this paper we are interested in some anomaly functions which have strong geometrical impacts. They are defined below.
Definition 7.
Let E be an algebroid and let s , s * , s * * Γ ( E ) .
(1) 
The associator anomaly function of E is defined by
A s s ( s , s * , s * * ) = ( s · s * ) · s * * s · ( s * · s * * ) .
(2) 
The Koszul-Vinberg anomaly function of E is defined by
K V ( s , s * , s * * ) = A s s ( s , s * , s * * ) A s s ( s * , s , s * * ) .
(3) 
The Jacobi anomaly functions of E are defined by
J ( s , s * , s * * ) = ( s · s * ) · s * * + ( s * · s * * ) · s + ( s * * · s ) · s * .
Definition 8.
Let v be a section of a two-sided E -module V .
(1) 
The associator anomaly function of a left module V is defined as
A s s ( s , s * , v ) = ( s · s * ) · v s · ( s * · v ) .
(2) 
The KV anomaly functions of a two sided module V are defined as
K V ( s , s * , v ) = A s s ( s , s * , v ) A s s ( s * , s , v ) ,
K V ( s , v , s * ) = ( s · v ) · s * s · ( v · s * ) ( v · s ) · s * + v · ( s · s * ) .
Definition 9.
We keep the notation used above. Let s , s * be sections of E , let v be a section of V and f C ( M ) .
(1) 
The Leibniz anomaly function of an anchored algebroid E is defined by
L ( s , f , s * ) = s · ( f s * ) d f ( b ( s ) ) s * f s · s * .
(2) 
The Leibniz anomaly function of the E -module V is defined by
L ( s , f , v ) = s · ( f v ) d f ( b ( s ) ) v f s · v .
A category of algebroids and modules of algebroids is defined by its anomaly functions. The anomaly functions are also used for introducing theories of homology of algebroids.
Some categories of anchored algebroids play important roles in the differential geometry.
Definition 10.
(A1): A Lie algebroid is an anchored algebroid ( E , b ) satisfying the identities
s · s * = 0 ,
L ( s , f , s * ) = 0 .
(B1): A KV algebroid is an anchored algebroid ( E , b ) satisfying the identities
K V ( s , s * , s * * ) = 0 ,
L ( s , f , s * ) = 0 .
(B2): A vector bundle V is a module of Lie algebroid ( E , b ) if it satisfies the identities
L ( s , f , v ) = 0 ,
( s · s * ) · v s · ( s * · v ) + s * · ( s · v ) = 0 .
A vector bundle V is a two-sided KV module of a Koszul-Vinberg algebroid ( E , b ) if it satisfies the identities
L ( s , f , v ) = 0 ,
K V ( s , s * , v ) = 0 ,
K V ( s , v , s * ) = 0 .
Warning. 
Consider a vector V space as the trivial vector bundle
V × O 0 .
Then we get
Γ ( V × 0 ) = V .
Therefore an algebra is an anchored algebroid over a point; its anchor map of is the zero map. Therefore, the Leibniz anomaly of an algebra is nothing but the bilinearity of the multiplication. So the notion of KV algebra and KV module is clear.

3. The Theory of Cohomology of KV Algebroids and Their Modules

This section is devoted to the cohomology of KV algebroids and KV modules of KV algebroids. KV stands for Koszul-Vinberg. We shall introduce three approaches to the theory of KV cohomology. Each approach has its particular advantage. So, depending on the needs or on the concerns one or other approach may be convenient. The three approaches are called “Version brute formula”, “Version semi simplicial objects”, “Version anomaly functions”. The same graded vector space is common to the three constructions. They differ in their coboundary operators. However, three constructions lead to cohomology complexes which are pairwise quasi isomorphic.
Each construction leads to two cochain complexes. Those complexes are called the KV complex and total KV complex. They are denoted by C K V * and C τ * . In final we obtain six cohomological complexes.

3.1. The Theory of KV Cohomology—Version the Brute Formula of the Coboundary Operator

The geometric framework is the category of real KV algebraoids and their two sided modules. However our machineries only make use of R -multi-linear calculations in the vector spaces of sections of vector bundles. Without any damage we replace the categories of KV algebroids and modules of KV algebroids by the categories of KV algebras and abstract modules of KV algebras.

3.1.1. The Cochain Complex C K V .

Let W be a two-sided module of a KV algebra A .
Definition 11.
The vector subspace J ( W ) W is defined by
( a · b ) · w a · ( b · w ) = 0 a , b A
We consider the Z -graded vector space
C K V ( A , W ) = q C K V q ( A , W ) .
The homogeneous vector sub-spaces are defined by
C K V q ( A , W ) = 0 q < 0 ,
C K V 0 ( A , W ) = J ( W ) ,
C K V q ( A , W ) = H o m R ( A q , W ) q > 0 .
Before pursuing we fix the following notation.
Let
ξ = a 1 . . . a q + 1 A q + 1
and let a A ,
i ξ = a 1 . . . a ^ i . . . a q + 1 ,
i , k + 1 2 ξ = i ( k + 1 ξ ) ,
a . ξ = 1 q + 1 a 1 . . . a j 1 a . a j a j + 1 . . . a q + 1 .
We are going to define the coboundary operator
δ K V : C q ( A , W ) C q + 1 ( A , W ) .
The coboundary operator is a linear map. It is defined by
(4a) [ δ K V ( w ) ] ( a ) = a · w + w · a w J ( W ) , [ δ K V f ] ( ξ ) = 1 q ( 1 ) i [ a i · f ( i ξ ) f ( a i · i ξ ) + ( f ( i , q + 1 2 ξ a i ) ) · a q + 1 ] f C K V q ( A , W ) , (4b) ξ A q + 1 .
The operator δ K V satisfies the identity
δ K V 2 f = 0 f C K V ( A , W ) .
Therefore the pair ( C K V * ( A , W ) , δ K V ) is a cochain complex. Its cohomology space is denoted by
H K V ( A , W ) = q H K V q ( A , W ) .
The conjecture of Gerstenhaber: Comments.
A KV algebra A is a two-sided module of itself. An infinitesimal deformations of A is a 1-cocycle of C K V ( A , A ) [9]. By the conjecture of Gerstenhaber the cohomology complex C K V ( A , A ) is generated by the theory of deformations in the category of KV algebras.
The theory of deformation of KV algebras is the algebraic version of the theory of deformation of locally flat manifolds [2]. Therefore, the complex C K V ( A , a ) is the solution to the conjecture of Muray Gerstenhaber in the category of locally flat manifolds [27].
Features. 
(1) The 2nd cohomology space H K V 2 ( A , A ) is the space of non trivial deformations of A .
The definition of KV algebra of a locally flat manifold will be given in the next section.
Following [2] every hyperbolic locally flat manifold has non trivial deformations. Thus, if A is the KV algebra of a hyperbolic locally flat manifold then
H K V 2 ( A , A ) 0 .
(2) Let W be a two-sided module of a KV algebra A . We consider W as a trivial KV algebra, viz
w · w * = 0 w , w * W .
Let E X T K V ( A , W ) be the set of equivalence classes of short exact sequences of KV algebras
0 W B A 0 .
An interpretation of the 2nd cohomology space of C K V ( A , W ) is the identification
H K V 2 ( A , W ) = E X T K V ( A , W ) .
Let W , W * be two-sided modules of A . Let E X T A ( W * , W ) be the set of equivalence classes of exact short sequences of two-sided A -modules
0 W T W * 0 .
In both the category of associative algebras and the category of Lie algebras we have
H H 1 ( A , H o m R ( W * , W ) ) = E X T A ( W * , W ) ,
H C E 1 ( A , H o m R ( W * , W ) ) = E X T A ( W * , W ) .
Here H H ( A , ) stands for Hochschild cohomology of an associative algebra A and H C E ( A , ) stands for cohomology of Chevalley-Eilenberg of a Lie algebra A.
Unfortunately in the category of KV modules of KV algebras this interpretation of the first cohomology space fails. Loosely speaking in the category of KV algebras the set H 1 ( A , H o m ( W * , W ) ) is not canonically isomorphic to set E X T A ( W * , W ) [9].

3.1.2. The Total Cochain Complex C τ .

The purpose is the total complex
C τ ( A , W ) = q C τ q ( A , W ) .
Its homogeneous vector subspaces are defined by
C τ q ( A , W ) = 0 q < 0 ,
C τ 0 ( A , W ) = W ,
C τ q ( A , W ) = H o m R ( A q , W ) q > 0 .
The total coboundary operator is a linear map
C τ q ( A , W ) C τ q + 1 ( A , W ) .
That operator is defined by
(1):
[ δ τ w ] ( a ) = a · w + w a ( a , w ) A × W ,
(2):
[ δ τ f ] ( ξ ) = 1 q + 1 ( 1 ) i [ a i · f ( i ξ ) f ( a i · i ξ ) + ( f ( i , q + 1 2 ξ a i ) ) · a q + 1 ] f C τ q ( A , W ) .
The pair
( C τ * ( A , W ) , δ τ )
is a cochain complex, viz
δ τ 2 = 0 .
The derived cohomology space is denoted by
H τ ( A , W ) = q H τ q ( A , W ) .
It is called the W-valued total KV cohomology of A .

3.2. The Theory of KV Cohomology—Version: the Semi-Simplicial Objects

Let V be a two-sided module of a KV algebra A . Our aim is the construction of semi simplicial A -modules whose derived cochain complex is quasi isomorphic to the KV cochin complex C K V ( A , V ) .

3.2.1. Extension

We start by considering the vector space
B = A R .
Its elements are denoted by ( s + λ ) . We endow B with the multiplication which is defined by
( s + λ ) · ( s * + λ * ) = s · s * + λ s * + λ * s + λ λ * .
With the multiplication we just defined, B is a real KV algebra. In other words we have
K V ( X 1 , X 2 , X 3 ) = 0 .
Here
X j = s j + λ j .
In the A -module V we have a structure of left B -module which is defined by
( s + λ ) · v = s · v + λ v ( s + λ ) B , v V .

3.2.2. Construction

Let B ˜ be the vector space spanned by A × R . Its elements are finite linear combinations of ( s , λ ) , s A × R .
The tensor algebra of B ˜ is denoted by T ( B ˜ ) . It has a Z -grading. its homogeneous vector sub-spaces are defined by
T q ( B ˜ ) = B ˜ q .
A monomial element is denoted by
ξ = x 1 x 2 . . . x q .
Here
x j = ( s j , λ j ) A × R .
The KV algebra A is a two-sided ideal of the KV algebra B . Thereby, the vector space B ˜ is canonically a left module of A .
We define the natural two-sided action of R in B ˜ by setting
λ · ( s * , λ * ) = ( λ s , λ λ * ) ,
( s * , λ * ) · λ = ( λ s * , λ * λ ) .
Thereby every vector subspace T q ( B ˜ ) is a left KV module of B . Here the left action of B in T q ( B ˜ ) is defined
( s + λ ) · ξ = s · ξ + λ ξ .
Before continuing we recall the (extended) action of A in tensor space T q ( B ˜ ) ,
s · ( x 1 x 2 . . . x q ) = j = 1 q x 1 x 2 . . . s · x j . . . x q .
We recall a notation which has been used in the last subsections,
j ξ = x 1 x 2 . . . x j ^ . . . x q .
The symbol x j ^ means that x j is missing. Let 1 R be the unit element, then 1 ˜ stands for ( 0 , 1 ) B ˜ . We are going to construct semi simplicial modules of B .

3.2.3. Notation-Definitions

Implicitly we use set isomorphism
B ˜ x = ( s , λ ) X * = s + λ B .
Then ξ T q ( B ˜ ) one has
1 ˜ * · ξ = ξ ·
We go back to the Z -graded B -module
T * ( B ˜ ) = q T q ( B ˜ ) .
Definition 12.
Let j , q be two positive integers with j < q , let
ξ = x 1 x 2 . . . x q .
The linear maps
d j : T q ( B ˜ ) T q 1 ( B ˜ )
and
S j : T q ( B ˜ ) T q + 1 ( B ˜ )
are defined by
d j ξ = X j * · j ξ ,
S j ξ = e j ( 1 ˜ ) ξ
The right member of the last equality has the following meaning
e j ( 1 ˜ ) ξ = x 1 x 2 . . . x j 1 1 ˜ x j . . . x q
Structure. 
The maps d j and S j satisfy the following identities
(5a) d i d j = d j 1 d i i f i j , (5b) S i S j = S j + 1 S i i f i < j , (5c) ( S j 1 d i d i S j ) ( ξ ) = e j 1 ( x i ) i ξ i f 1 < i < j , (5d) ( d i + 1 S j S j d i ) ( ξ ) = e j ( x i ) i ξ i f j + 1 < i q , (5e) d i ( S i ( ξ ) ) = ξ i f i = j .
Definition 13.
The system T q ( B ) , d i , S i is called the canonical semi simplicial module of B .

3.2.4. The KV Chain Complex

From the canonical simplicial B -module we derive the chain complex C * ( B ) . it has a Z -grading which is defined by
(6a) C q ( B ) = 0 i f q < 0 , (6b) C 0 ( B ) = R , (6c) C q ( B ) = T q ( B ˜ ) i f q > 0 .
Now one defines the ( linear) boundary operator
d : C q ( B ) C q 1 ( B )
by setting
d ( C 0 ( B ) ) = 0 ,
d ( C 1 ( B ) ) = 0 ,
d ξ = 1 q ( 1 ) j d j ξ i f q > 1 .
By the virtue of (5a) we have
d 2 = 0 .

3.2.5. The V-Valued KV Homology

We keep the notation used in the preceding sub-subsection. So the vector spaces A , B and V are the same as in the preceding subsubsection.
We consider the Z -graded vector space
C * ( B , V ) = q C q ( B , V ) .
Its homogeneous sub-spaces are defined by
C q ( B , V ) = 0 i f q < 0 ,
C 0 ( B , V ) = V ,
C q ( B , V ) = T q ( B ˜ ) V i f q > 0 .
Every homogeneous vector subspace C q ( B , V ) is a left module of the KV algebra B . The left action is defined by
s · ( ξ v ) = s · ξ v + ξ s · v .
Let j and q be two positive integers such that j < q .
Let ξ = x 1 x 2 . . . x q . To define the linear map
d j : C q ( B , V ) C q 1 ( B , V )
we put
d j ( ξ v ) = X j * · ( j ξ v ) .
Henceforth one defines the boundary operator
d : C q ( B , V ) C q 1 ( B , V )
by setting
d = 1 q ( 1 ) j d j .
So we obtain a chain complex whose homology space of degree q is denoted by H q ( B , V ) .
Definition 14.
The graded vector space
H * ( B , V ) = q H q ( B , V )
is called the total homology of B with coefficients in V.

3.2.6. Two Cochain Complexes

We are going to define two cochain complexes. They are denoted by C K V ( B , V ) and by C τ ( B , V ) respectively.
We recall that the vector subspace J ( V ) V is defined by
( s · s * ) · v s · ( s * · v ) = 0 s s * B .
Let us set
C K V 0 ( B , V ) = J ( V ) ,
C τ 0 ( B , V ) = V ,
C q ( B , V ) = H o m R ( T q ( B ˜ ) q 1 .
Let ( j , q ) be a pair of non negative integers such that j < q . We are going to define the linear map
d j : C q ( B , V ) C q + 1 ( B , V ) .
Given f C q ( B , V ) and
ξ = x 1 . . . x q + 1
we put
d j f ( ξ ) = X j * · f ( j ξ ) f ( d j ξ ) .
The family of linear mappings d j has property S · 1 , viz
d j d i = d i d j 1 i , j w i t h i < j .
We use these data for constructing two cochain complexes. They are denoted by ( C K V * , d K V ) and by ( C τ * , d τ ) respectively. The underlying graded vector spaces are defined by
C K V = J ( V ) q > 0 C q ( B , V ) ,
C τ = V q > 0 C q ( B , V ) .
Their coboundary operators are defined by
( d K V v ) ( s ) = s · v ,
( d τ w ) ( s ) = s w ,
d K V ( f ) = 1 q ( 1 ) j d j ( f ) i f q > 0 ,
d τ ( f ) = 1 q + 1 ( 1 ) j d j ( f ) i f q > 0 .
The simplicial formula (5a) yields the identities
d K V 2 = 0 ,
d τ 2 = 0 .
The cohomology space
H K V ( B , V ) = q H K V q ( B , V )
is called the V-valued KV cohomology of B .
The cohomology space
H τ ( B , V ) = q H τ q ( B , V )
is called the V-valued total KV cohomology of B .
The algebra A is a two-sided ideal of the KV algebra B . Mutatis mutandis our construction gives the cohomology spaces H K V ( A , V ) and H τ ( A , V ) . They are called the V-valued KV cohomology and the V-valued total KV cohomology of A .
Comments. 
Though the spectral sequences are not the purpose of this paper we recall that the pair ( A B ) gives rise to a spectral sequences E r i j [32,33,34]. The term E 0 i j is nothing other than H K V ( A , V ) [29]. In other words one has
H K V q ( A , V ) = 0 j q E 0 j , q j .

3.2.7. Residual Cohomology

Before pursuing we introduce the notion of residual cohomology. It will be used in the section be devoted the homological statistical models.
The machinery we are going to introduce is similar to the machinery of Eilenberg [35]. In particular we introduce the residual cohomology. Our construction leads to an exact cohomology sequence which links the residual cohomology with the equivariant cohomology. We restrict the attention to the category of left modules of KV algebroids. We keep our previous notation.
We recall that for every positive integer q > 0 the vector space C q ( B , V ) is a left module of B . The left action of s B is defined by
( s · f ) ( ξ ) = s · f ( ξ ) f ( s · ξ ) .
Definition 15.
A cochain f C q ( B , V ) is called a left invariant cochain if
s · f = 0 s B s .
A straightforward consequence of this definition is that a left invariant cochain is a cocycle of both C K V * and C τ * . The vector subspace of left invariant q-cochains of B is denoted by H e q ( B , V ) . It is easy to see that
Z τ q ( B , V ) Z K V q ( B , V ) = H e q ( B , V ) ,
Z τ q ( A , V ) Z K V q ( A , V ) = H e q ( A , V ) .
Definition 16.
A KV cochain of degree q whose coboundary is left invariant invariant is called a residual KV cocycles.
(1) 
The vector subspace of residual KV cocycles of degree q is denoted by Z K V r e s q .
(2) 
The vector subspace of residual coboundaries of degree q is defined by B K V r e s q = H e q ( B , V ) + d K V ( C K V q 1 ( B , V ) ) . The residual KV cohomology space of degree q is the quotient vector space.
(3) 
H K V r e s q ( B , V ) = Z K V r e s q B K V r e s q .
(4) 
By replacing the KV complex by the total KV complex one defines the vector space of residual total cocycles Z τ r e s q and the space of residual total coboundaries B τ r e s q . Therefore we get the residual total KV cohomology space
H τ r e s q ( A , V ) = Z τ r e s q B τ , r e s q
The definitions above lead to the cohomological exact sequences which is similar to those constructed by Eilenberg machinery [35]. We are going to pay a special attention to two cohomology exact sequences.
(1) At one side the operator d K V yields a canonical linear map
H K V r e s q ( B , V ) H e q + 1 ( B , V ) .
(2) At another side every KV cocycle is a residual cocycle and every KV coboundary is a residual coboundary as well. Then one has a canonical linear map
H K V q ( B , V ) H K V r e s q ( A , V ) .
Those canonical linear mappings yield the following exact sequences
H K V r e s q 1 ( B , V ) H e q ( B , V ) H K V q ( B , V ) H K V r e s q ( B , V )
H τ r e s q 1 ( B , V ) H e q ( B , V ) H τ q ( B , V ) H τ r e s q ( B , V )
Some Comments.
( c . 1 )
We replace the KV B by A . Then we obtain the exact sequences
H K V r e s q 1 ( A , V ) H e q ( A , V ) H K V q ( A , V ) H K V r e s q ( A , V )
H τ r e s q 1 ( A , V ) H e q ( A , V ) H τ q ( A , V ) H τ r e s q ( A , V )
( c . 2 )
The KV cohomology difers from the total cohomology. Loosely speaking their intersecttion is the equivariant cohomology H e * ( B , V ) their difference is the residual cohomology. The domain of their efficiency are different as well. Here are two illustrations.
Example 1.
In the introduction we have stated a conjecture of M. Gerstenhaber, namely Every Restricted Theory of Deformation Generates Its Proper Theory of Cohomology.
From the viewpoint of this conjecture, the KV cohomology is the completion a long history [2,9,28]. Besides Koszul and Nijenhuis, other pioneering authors are Vinberg, Richardson, Gerstenhaber, Matsushima, Vey.
The challenge was the search for a theory of cohomology which might be generated by the theory of deformation of locally flat manifolds [8]. The expected theory is the now known KV theory of KV cohomolgy [9].
Example 2.
The total cohomology is close to both the pioneering Nijenhuis work [28,36]. In [29] we have constructed a spectral sequence which relates to [28,36].
From another viewpoint, the total KV cohomology is useful for exploring the relationships between the information geometry and the theory of Riemannian foliations. This purpose will be addressed in the next sections.

3.3. The Theory of KV Cohomology—Version the Anomaly Functions

This subsection is devoted to use the KV anomaly functions for introducing the theory of cohomology of KV algebroids and their modules.
This viewpoint leads to an unifying framework for introducing the theory of cohomology of abstract algebras and their abstract two-sided modules. Here are a few examples of cohomology theory which are based on the anomaly functions.
Example 1.
The theory of Hochschild cohomology of associative algebras is based on the associator anomaly function.
Example 2.
The theory of Chevalley-Eilenberg-Koszul cohomology of Lie algebras is based on the Jacobi anomaly function.
Example 3.
The theory of cohomology of Leibniz algebras is based on the Jacobi anomaly function as well.

3.3.1. The General Challenge C H ( D )

We consider data
D = [ ( A , A A ) , ( V , A A V ) , H o m ( T ( A ) , V ) ] .
Here
(1)
V is an (abstract) two sided module of an (abstract) algebra A .
(2)
A A and A AV are fixed anomaly functions of A and of V respectively.
(3)
H o m ( T ( A ) , V ) stands for the Z -graded vector space
H o m ( T ( A ) , V ) = q H o m R ( A q , V ) .
Let A D be the category of (abstract) algebras and (abstract) modules whose structures are defined by the pair ( A A , A AV ) . So the rules of calculations in the category A are defined by the identities
A A ( a , b , c ) = 0 ,
A AV ( a , b , v ) = 0 .
The challenge is the search of a particular family of linear maps
H o m ( A q , V ) f d q ( f ) H o m ( A q + 1 , V ) .
Such a particular family d q must satisfy a condition that we call the property Δ.
Property Δ
ξ = a 1 a 2 . . . a q + 2 A q + 2 , f H o m ( A q , V ) the quantity [ d q + 1 ( d q ( f ) ) ] ( ξ ) depends linearly on the values of the anomaly functions
A A ( a i , a j , a k ) , A A V ( a i , a j , v )
Let us assume that a family d q is a solution to C H ( D ) . Then the category A D admits a theory of cohomology with coefficients in modules.
The next is devoted to this challenge in the category of KV algebras and KV modules. The geometry version is the category of KV algebroids and KV modules of KV algebroids.

3.3.2. Challenge C H ( D ) for KV Algebras

Let W be a two-sided module of an abstract algebra A . We assume that the following bilinear mappings are non trivial applications
A × W ( X , w ) X · w W ,
W × A ( w , X ) w · X W .
Let f H o m ( A q , W ) . We consider a monomial ξ A q + 1 , so
ξ = X 1 . . . X q + 1 A q + 1 .
Our construction is divided into many STEPS.
Step 1.
Let ( i < j ) be a pair of positive integers with 1 i < j q . The linear the map
S [ i , j ] ( f ) H o m ( A q + 1 , V ) .
S [ i j ] is defined by
S [ i , j ] ( f ) ( X 1 . . . X q + 1 ) = ( 1 ) j [ X j · f ( X 1 . . . X i . . . X j ^ X j + 1 . . . X q + 1 )
+ ( f ( X 1 . . . X i . . . X j ^ . . . X q + 1 ^ X j ) · X q + 1
ω ( f ) f ( X 1 . . . X j · X i . . . X j ^ . . . X q + 1 ) ]
+ ( 1 ) i [ X i · f ( X 1 . . . X i ^ . . . X j . X q + 1 )
+ ( f ( X 1 . . . X i ^ . . . X j . . . X q + 1 ^ X i ) ) · X q + 1
ω ( f ) f ( X 1 . . . X i ^ . . . X i · X j . . . X q + 1 ) ] .
In the right side member of S [ i , j ] ( f ) ( ξ ) the coefficient ω ( f ) is the degree of f, viz ω ( f ) = q for all f H o m ( A q , W ) .
Step 2.
For every pair ( i , q + 1 ) with 1 i q we define the map S [ i , q + 1 ] ( f ) by
S [ i , q + 1 ] ( f ) ( X 1 . . . X q + 1 ) = ( 1 ) i [ X i · f ( X 1 . . . X i ^ . . . X q + 1 )
+ ( f ( X 1 . . . X i ^ . . . X q + 1 ^ X i ) ) · X q + 1
ω ( f ) f ( X 1 . . . X i ^ . . . X i · X q + 1 ) ] .
Step 3.
Let g H o m ( A q + 1 , W ) and let
ξ = X 1 . . . X q + 2 A q + 2 .
Let i , j , k be three positive integers such that i < j < q + 2 ; k q + 2 . We have already introduced the notation
k ξ = X 1 . . . X ^ k . . . X q + 2 ,
k , q + 2 2 ξ = X 1 . . . X ^ k . . . . . . X ^ q + 2 .
We define S [ i , j ] k ( g ) H o m ( A q + 2 , W ) by setting
S [ i , j ] k ( g ) ( ξ ) = ( 1 ) i + k [ X k · g ( k ξ ) + ( g ( k , q + 2 2 ξ X k ) ) · X q + 2
+ ω ( g ) g ( X 1 . . . X k · X i . . . X k ^ . . . X q + 2 ) ]
+ ( 1 ) j + k [ X k · g ( k ξ ) + ( g ( k , q + 2 2 ξ ) X k ) · X q + 2
+ ω ( g ) g ( X 1 . . . X k · X j . . . X k ^ . . . X q + 2 ) ] .
Given a triple ( i , j , k ) with i < j < k < q + 2 we put
S [ i , j , k ] ( g ) ( ξ ) = S [ i , j ] k ( g ) ( ξ ) + S [ i , k ] j ( g ) ( ξ ) + S [ j , k ] i ( g ) ( ξ ) .
The proof of the following statement is based on direct calculations.
Lemma 1.
( * * * * ) : [ i < j ] S [ i , j ] ( g ) ( ξ ) = [ i < j < k ] S [ i , j , k ] ( g ) ( ξ )
Let f H o m ( A q , W ) . In both the left side and the right side of the equality ( * * * * ) we replace g by i < j S [ i , j ] ( f ) . Then we obtain a linear mapping
H o m ( A q , W ) f E * * * * ( f ) H o m ( A q + 2 , W ) .
Our aim is to evaluate ξ of E * * * * ( f ) at ξ A q + 2 . Here
ξ = X 1 . . . X q + 2 .
To calculate [ E * * * * ( f ) ] ( ξ ) we take into account both STEP1 and STEP2. Then we obtain
[ E * * * * ( f ) ] ( ξ ) = [ i < j < q + 2 ; 1 k q + 2 ] [ E [ i j k ] * * * * ( f ) ] ( ξ ) .
At the right side member
[ E [ i j k ] * * * * ( f ) ] ( ξ ) = ( 1 ) i + j [ K V ( X i , X j , f ( X 1 . . . X i ^ . . X j ^ . . . X k . . . X q + 2 ) )
+ K V ( X i , f ( X 1 . . . X i ^ . . . X j ^ . . . X q + 1 X j ) , X q + 2 )
+ K V ( X j , f ( X 1 . . . X i ^ . . . X j ^ . . . X q + 1 X i ) , X q + 2 )
+ ω ( f ) ( ω ( f ) + 1 ) f ( X 1 . . . X i ^ . . . X j ^ . . . K V ( X i , X j , X k ) . . . X q + 2 ) ] .
Step 4.
We are in position to face C H ( D ) .
Definition 17.
Let f H o m ( A q , W ) and ξ = X 1 . . . X q + 1 A q + 1 . We take into account Step 1, Step 2 and Step 3. Therefore, we define the linear map
H o m ( A q , W ) f f H o m ( A q + 1 , W )
by putting
[ f ] ( ξ ) = 1 i < j q + 1 S [ i , j ] ( f ) ( ξ )
The following lemma is a straightforward consequence of the machinery in STEP3.
Lemma 2.
2 f ( ξ ) = [ i < j < q + 2 ] ; 1 k q + 2 [ E [ i j k ] * * * * ( f ) ] ( ξ )
Lemma 2 tells us that 2 f ( ξ ) depends linearly on the values of the KV anomaly functions.
The challenge C H ( D ) is won in the category of KV algebras and their two-sided KV modules.
We replace the category of KV algebras and their two-sided modules by the category of KV algebroids and their bi-modules. Then we win the geometry version of C H ( D .
We use Lemma 2 for introducing a theory of KV homology of KV algebras and their two-sided modules.

3.3.3. The KV Cohomology

Let W be a two sided KV module of a KV algebra A . We consider the graded vector space
C K V = q C K V q .
The homogeneous subspaces are defined by C K V q = 0 if q is a negative integer, C K V 0 = J ( W ) , C K V q = H o m ( A q , W ) if q is a positive integer.
We define the linear map
C K V q f K V f C q + 1 K V
by setting
(7a) K V ( w ) ( X ) = X · w + w · X i f w J ( W ) , (7b) K V f = [ i < j ] S [ i , j ] ( f ) i f q > 0 .
By Lemma 2 we obtain the following statement
Theorem 3.
For every two sided KV module W of a KV algebra A the pair ( C K V * , K V ) is a cochain complex.

3.3.4. The Total Cohomology

Let W be a two-sided module of a KV algebra A . Our concern is the Z -graded vector space
C τ = W + q > 0 C q ( A , W ) .
For our present purpose the maps S i j are not subject the requirement as in Step 2.
We define the coboundary operator τ by setting
τ w ( X ) = X · w + w · X w i n W ,
τ f ( ξ ) = 1 i < j q + 1 S [ i , j ] ( f ) ( ξ ) q > 0 .
The quantity ( τ 2 f ( ξ ) depends linearly on the KV anomaly functions of the pair ( A , W ) . Thus the pair ( C τ * , τ ) is a cochain complex. Its cohomology is called the W-valued total KV cohomology of A . We denote it by H τ * ( A , W ) .

3.3.5. The Residual Cohomology, Some Exact Sequences, Related Topics, DTO-HEG-IGE-ENT

In the next sections we will see that the links between the information geometry and the differential topology involve the real valued total KV cohomology of KV algebroids. Many relevant relationships are based on the exact sequences
H K V r e s q 1 ( A , R ) H e q ( A , R ) H K V q ( A , R ) H K V r e s q ( A , R )
H τ r e s q 1 ( A , R ) H e q ( A , R ) H τ q ( A , R ) H τ r e s q ( A , R )
Now we are provided with cohomological tools which will be used in the next sections.
We plan to perform KV cohomological methods for studying some links between the vertices of the square “DTO, IGE, ENT HGE” as in Figure 1. We recall basic notions.
DTO stands for Differential TOpology.
The purposes: Riemannian foliations and Riemannian webs. Symplectic foliations and symplectic webs. Linearization of webs.
Our aims: We use cohomological methods for constructing Riemannian foliations, Riemannian webs, linearizable webs.
Nowadays, there does not exist any criterion for deciding whether a manifold supports those differential topological objects. Our aim is to discuss sufficient conditions for a manifold admitting those structures. Our approach leads to notable results. The key tools are the KV cohomology and the dualistic relation of Amari. Both the KV cohomology and the dualistic relation product remarkable split exact sequences. Notable results are based on those exact sequences. HGE stands for Hessian GEometry.
The purposes: Hessian structures, geometry of Koszul, hyperbolicity, cohomological vanishing theorems. Our aims: The geometry of Koszul is a cohomological vanishing theorem. Statistical geometry and vanishing theorem, the solution to a hold question of Alexander K Guts (announced).
Theorem 3 as in [2] may be rephrased in the framework of the theory of KV homology. For a compact locally flat manifold ( M , ) being hyperbolic it is necessary and sufficient that C K V 2 ( A , C ( M ) ) contains a positive definite EXACT cocycle. To be hyperbolic is a geometrical-topological property of the developing map of locally flat manifolds. To be hyperbolicitic means that the image of the developing is a convex domain not containing any straight line. This formulation is far from being a homological statement. So the Hessian GEOmetry is a link between the theory of KV homology and the Riemannian Riemannian geometry.
The geometry of Koszul, the geometry of homogeneous bounded domains and related topics have been studied by Vinberg, Piatecci-Shapiro and many other mathematicians [3]. The geometry of Siegel domains belongs to that galaxy [7,12]. Almost all of those studies are closely related to the Hessian geometry.
Among the open problems in the Hessian geometry are two questions we are concerned with. The first is to know whether the metric tensor g of a Riemannian manifold is a Hessian metric. Alexander K. Guts raised this question in a mail (to me) forty years ago. The second question is to know whether a locally flat manifold admits a Hessian tensor metric. The solutions to those two problems are announced in the Appendix A to this paper.
IGE stands for Information GEometry.
The purposes: The differential geometry of statistical models, the complexity of statistical models, ramifications of the information geometry.
Our aims: We revisit the classical theory of statistical models, requests of McCullagh and Gromov. A search of a characteristic invariant. The moduli space of models. The homological nature of the information geometry.
The information geometry is the differential geometry in statistical models for measurable sets. In both the theoretical statistics and the applied statistics the exponential families and their generalizations are optimal statistical models. There are many references, e.g., [17,18,22,37]. Here Murray-Rice 1.15 means Murray-Rice Chapter 1, Section 15. A major problem is to know whether a given statistical model is isomorphic to an exponential model. That is what we call the complexity problem of statistical models. This challenge is a still open problem. It explicitly arises from the purposes which are discussed in [22] here, see also [30]. In the appendix to this paper we present a recently discovered invariant which measures how far from being an exponential family is a given model. That invariant is useful for exploring the differential topology of statistical models. That is particularly important when models are singular, viz models whose Fisher information is not inversible.
ENT stands for ENTropy.
Pierre Baudot and Daniel Bennequin recently discovered that the entropy function has a homological nature [31]. We recall that in 2002 Peter McCullagh raised a fundamental geometric-topological question in the theory of information: What Is a Statistical Model? [30] A few years after Misha Gromov raised a similar request: The Search of Structure. Fisher Information [15,16].
Those two titles are two formulations of the same need.
The paper of McCullagh became the subject of controversy. It gave rise to questions, discussions, criticisms, see [30].
In Part B of this paper we will be addressing this fundamental problem. A reading of the McCullagh paper would be useful for drawing a comparison between our approach and [15,16,30].

4. The KV Topology of Locally Flat Manifolds

4.1. The Total Cohomology and Riemannian Foliations

In this section we focus on the KV algebroids which are defined by structures of locally flat manifolds. To facilitate a continuous reading of this paper we recall fundamental notions which are needed.
Definition 18.
A locally flat manifold is a pair ( M , D ) . Here D is a torsion free Koszul connection whose curvature tensor R D vanishes identically.
The pair ( M , D ) defines a Koszul-Vinberg algebroid
A = ( T M , D , 1 )
The anchor map is the identity map of T M . The multiplication of sections is defined by D, viz
X · Y = D X Y
forall X , Y X ( M ) .
The KV algebra of ( M , D ) is the algebra
A : = ( X ( M ) , D ) .
The cotangent bundle T * M is a left module of the KV algebroid ( T M , D , 1 ) . For every ( X , Y , θ ) X ( M ) × X ( M ) × Γ ( T * M ) the differential 1-form X · θ is defined by
[ X · θ ] ( Y ) = [ d ( θ ( Y ) ) ] ( X ) θ ( X · Y ) .
In the right hand member of the equality above d ( θ ( Y ) ) is the exterior derivative of the real valued function θ ( Y ) .
Let S 2 ( T * M ) be the vector bundle of symmetric bi-linear forms in M.
The vector space of sections of S 2 ( T * M ) is denoted by S 2 ( M ) , viz
S 2 ( M ) = Γ ( S 2 ( T * M ) ) .
The vector space S 2 ( M ) is a left module of the KV algebra A . The left action of A in S 2 ( M ) is defined by
( X · g ) ( Y , Z ) = [ d g ( Y , Z ) ] ( X ) g ( X · Y , Z ) g ( Y , X · Z ) .
We put
Ω 1 ( M ) = Γ ( T * M ) .
The T * M -valued cohomology of the KV algebroid ( T M , D , 1 ) is but the cohomology of A with coefficients in Ω 1 ( M ) . The KV cohomology and the total cohomology are denoted by
H K V * ( A , Ω 1 ( M ) ) ,
H τ * ( A , Ω 1 ( M ) ) .
Warning. 
We observe that elements of S 2 ( M ) may be regarded as 1-cochains of A with coefficients in its left module Ω 1 ( M ) . By [29] we have
Z τ 2 ( A , C ( M ) ) = S 2 A ( M ) .
At another side we have the cohomolgy exact sequence
H K V r e s 1 ( A , V ) H K V e 2 ( A , V ) H K V 2 ( A , V ) H K V r e s 2 ( A , V )
By Equations (8) and (9) we obtain the inclusion maps
S 2 A ( M ) Z K V 1 ( A , Ω 1 ( M ) ) Z K V 2 ( A , R ) .
Mutatis mutandis one also has
S 2 A ( M ) Z τ 1 ( A , Ω 1 ( M ) Z τ 2 ( A , R ) .
Remark 1 (Important Remarks)
We give some subtle consequences of (1).
(R.1) Every exact total 2-cocycle ω C τ 2 ( A , R ) is a skew symmetric bilinear form. Viz one has the identity
ω ( X , X ) = 0 X A .
(R.2) Every symmetric KV 2-cocycle g Z K V 2 ( A , R ) is locally an exact KV cocycle, viz in a neighbourhood of every point there exists a local section θ Ω 1 ( M ) such that
g = δ K V θ .
(R.3) Every symmetric total 2-cocycle is a left invariant cochain, viz
Z τ 2 ( A , R ) S 2 ( M ) = S 2 A ( M ) .
By (R.1) and (R.3) we obtain the inclusion map
S 2 A ( M ) H τ 2 ( A , R ) .
Let H d R 2 ( M ) be the second cohomology space of the de Rham complex of M. The following theorem is useful for relating the total KV cohomology and the differential topology.
Theorem 4.
[29] There exists a canonical linear injection of H d R 2 ( M ) in H τ 2 ( A , R ) such that
H τ 2 ( A , R ) = H d R 2 ( M ) S 2 A ( M )
The theorem above highlights a fruitful link between the total KV cohomology and the differential topology. We are particularly interested in D-geodesic Riemannian foliations in a locally flat manifold ( M , D ) .
Warning. 
Throughout this paper a Riemannian metric tensor in a manifold M is a non-degenerate symmetric bilinear form in M.
A positive metric tensor is a positive definite metric tensor.
In the next we use the following definition of Riemannian foliation and symplectic foliation.
Definition 19.
A Riemannian foliation is an element g S 2 ( M ) which has the following properties
(1.1) 
r a n k ( g ) = c o n s t a n t ,
(1.2) 
L X g = 0 X Γ ( K e r ( g ) ) . A symplectic foliation is a ( de Rham) closed differential 2-form ω which satisfies
(2.1) 
r a n k ( ω ) = c o n s t a n t ,
(2.2) 
L X ω = 0 X Γ ( K e r ( ω ) ) .
Warning. 
When g is positive semi-definite our definition is equivalent to the classical definition of Riemannian foliation [38,39,40].
The complete integrability of K e r ( g ) and the conditions to be satisfied by the holonomy of leaves are equivalent to the Property (2.2).
The set of Riemannian foliations in a manifold M is denoted by RF ( M ) . The last theorem above yields the inclusion map
H τ 2 ( A , R ) H d R 2 ( M ) RF ( M ) .
We often use the notion of affine coordinates functions in a locally flat manifold. For non specialists we recall two definitions and the link between them.
Definition 20.
An m-dimensional affinely flat manifold is an m-dimensional smooth manifold M admitting a complete atlas ( U j , ϕ j ) whose local coordinate changes coincide with affine transformations of the affine space R m .
We denoted an affine atlas by
A = ( U j , ϕ j ) .
Definition 21.
An affinely flat structure ( M , A ) and a locally flat structure ( M , ) are compatible if local coordinate functions of ( M , A ) are solutions to the Hessian equation
2 x j = O
Theorem 5.
For every positive integer m the relation to be compatible with a locally flat manifold is an equivalence between the category of m-dimensional affinely flat manifolds and the category of m-dimensional locally flat manifolds.

4.2. The General Linearization Problem of Webs

In the framework RF ( M ) the inclusion
H τ 2 ( A , R ) H d R 2 ( M ) RF ( M )
may be rewritten as the exact sequence
O H d R 2 ( M ) H τ 2 ( A , R ) RF ( M ) .
Let ( M , ) be a locally flat manifold whose KV algebra is denoted by A . Every finite family in H τ 2 ( A , R ) is a family of ∇-geodesic Riemannian foliations.
There does not exist any criterion to know whether a manifold supports Riemannian foliations. The exact cohomology sequences we have been performing provide us with a cohomological method for constructing Riemannian foliations in the category of locally flat manifolds. This is an impact of the theory of KV homology on DTO.
In the next section we will introduce other new ingredients which highlight the impacts on DTO of the information geometry.
Further we will see that those new machineries from the information geometry have a homological nature.
Another major problems in the differential topology is the linearization of webs. Among references are [41,42,43].
Definition 22.
Consider a finite family of distributions D j T M , j : = 1 , 2 , . . . , k . Those distributions are in general position at a point x M if for every subset J 1 , 2 , . . . , k one has
d i m ( j J D j ( x ) ) = min d i m ( M ) , j J d i m ( D j ( x ) ) .
Definition 23.
A k-web in M is a family of completely integrable distributions which are in general position everywhere in M.
A Comment.
The distributions belonging to a web may have different dimensions. An example of problem is the symplectic linearization of lagrangian 2-webs.
Let ( D j , j : = 1 , 2 ) be a lagrangian 2-web in a 2 n -dimensional symplectic manifold ( M , ω ) . The challenge is the search of special local Darboux coordinate functions
( x , y ) = ( x 1 , . . . , x n , y 1 , . . . , y n ) .
Those functions must have three properties
( 1 ) : ω ( x , y ) = Σ j d x j d y j ; ( 2 ) : The leaves of D 1 are defined by x = c o n s t a n t ; ( 3 ) : The leaves of D 2 are defined by y = c o n s t a n t .
Definition 24.
An affine web in an affine space is a web whose leaves are affine subspaces.
Definition 25.
A web in a m-dimensional manifold is linearizable if it is locally diffeomorphic to an affine web in a m-dimensional affine space.
Example 1.
In the symplectic manifold ( R 2 , e x y d x d y ) one considers the lagrangian 2-web which is defined by
L 1 = ( x , y ) | x = c o n s t a n t ,
L 2 = ( x , y ) | y = c o n s t a n t .
This lagrangian 2-web is not symplectic linearizable.
Example 2.
We keep ( L 1 , L 2 ) as in example.1. It is symplectic linearizable in ( R 2 , ( e x + e y ) d x d y ) . The linearization problem for lagrangian 2-webs is closely related to the locally flat geometry [10,44,45].
Example 3.
What about the linearization of the 3-web defined by L 1 : = ( x = c o n s t a n t , y ) , L 2 : = ( x , y = c o n s t a n t ) , L 3 : = e x ( x + y ) = c o n s t a n t , ( x , y ) R 2 .
Up to today the question as to whether it is linearizable is subject to controversies, see [42] and references therein.

4.3. The Total KV Cohomology and the Differential Topology Continued

We implement the KV cohomology to address some open problems in the differential topology. For our purpose we recall a few classical notions which are needed.
Definition 26.
A metric vector bundle over a manifold M is a vector bundle V endowed with a non-degenerate inner product < v , v * > .
A Koszul connection in a vector bundle V is a bilinear map
Γ ( T M ) × Γ ( V ) ( X , v ) X v Γ ( V )
which has the properties
(10a) f X v = f X v v , f C ( M ) , (10b) X f v = d f ( X ) v + f X v v , f C ( M ) .
Definition 27.
A metric connection in ( V , < , > ) is a Koszul connection ∇ which satisfies
d ( < v , v * > ) ( X ) < X v , v * > < v , X v * > = 0 .
Definition 28.
Let ( M , D ) be a foliation in the usual sense, viz D has constant rank and is in involution.
(1) 
( M , D ) is transversally Riemannian if there exists a g S 2 ( M ) such that
D = K e r ( g ) .
(2) 
( M , D ) is transversally symplectic if there exists a (de Rham) closed differential 2-form ω such that
D = K e r ( ω )
A transversally Riemannian foliation and a transversally symplectic foliation are denoted by
( D , g ) , ( D , ω ) .
Definition 29.
Given a Koszul connection ∇, a transversally Riemannian foliation ( D , g ) (respectively a transversally symplectic foliation ( D , ω ) ) is called ∇-geodesic if
g = 0 ,
ω = 0
The notions of transversally Riemannian foliation and transversally symplectic foliation are weaker than the notion of Riemannian foliation and symplectic foliation. However if ∇ a torsion free Koszul connection every ∇geodesic transversally Riemannian foliation is a Riemannian foliation. Every ∇-geodesic transversally symplectic foliation is a symplectic foliation.
For the general theory of Riemannian foliations the readers are referred to [39,40,46], see also the monograph [38] and the references therein.
We have pointed out that criterions for deciding whether a smooth manifold admits Riemannian foliations (respectively symplectic foliations) are missing. Our purpose is to address this existence problem in the category SLC whose objects are symmetric gauge structures. Such an object is a pair ( M , ) where ∇ is a torsion free Koszul connection in M. The category of locally flat structure LF is a subcategory of SLC . The theory of KV homology is useful for discussing geodesic Riemannian foliations in the category LF . In a locally flat manifold ( M , D ) we have been dealing with the decomposition
H τ 2 ( A , R ) = H d R 2 ( M ) S 2 D ( M ) .
Here A is the KV algebra of ( M , D ) .
Let b 2 ( M ) be the second Betti number of M. We define the numerical geometric invariant r ( D ) by
r ( D ) = d i m ( H τ 2 ( A , R ) ) b 2 ( M ) .
Formally r ( D ) is the codimension of H d R 2 ( M ) H τ 2 ( A , R ) , viz
r ( D ) = d i m ( H τ 2 ( A , R ) H d R 2 ( M ) ) .
We consider the exact sequences
O H d R 2 ( M ) H τ 2 ( A , R ) S 2 A ( M ) 0
and
H τ , e 2 ( A , R ) H τ 2 ( A , R ) H τ , r e s 2 ( A , R ) H τ , e 3 ( R , R )
From those exact sequences, one deduces the equality
H τ 2 ( A , R ) H d R 2 ( M ) = H τ , e 2 ( A , R ) H d R 2 ( M ) .
Thus r ( D ) is formally the dimension of S 2 A ( M ) .
The present approach leads to the following statement
Proposition 1.
If r ( D ) > 0 then M admits non trivial D-geodesic Riemannian foliations.
Proof. 
Let B be a non zero element of S 2 A ( M ) and let D be the kernel of B.
(1)
Suppose that
0 < r a n k ( D ) < d i m ( M )
Therefore, ( M , B ) is a D-geodesic Riemannian foliation.
(2)
Suppose that
r a n k ( D ) = O .
Then ( M , B ) is a Riemannian manifold the Levi-Civita connection of which is D. Therefore, the proposition holds. ☐
Before proceeding we define three numerical invariants
r ( M ) = max r ( D ) | D L C ( M ) ,
s ( M , A ) = max r a n k ( B ) | B S 2 A ( M ) ,
s ( M ) = max s ( M , A ) | D LF ( M ) .
The non negative integers r ( M ) and s ( M ) are global geometric invariants. They connect the total KV cohomology to geodesic Riemannian foliations. By this viewpoint the proposition has an interesting corollary.
Corollary 1.
In an m-dimensional manifold M suppose that the following inequalities are satisfied
0 < s ( M ) < m .
Then the manifold M admits a locally flat structure ( M , D * ) which supports a non trivial D * -geodesic Riemannian foliation.
The integer s ( M ) is a local characteristic invariant of some class of 2-webs in Hessian manifolds.
Let ( M , D ) be a locally flat manifold whose KV algebra is denoted by A . we recall that a Hessian metric tensor in ( M , D ) is a inversible cocycle g Z K V 2 ( A , R ) .
Theorem 6.
Let ( M , D , g ) and ( M * , D * , g * ) be m-dimensional Hessian manifolds. We assume that the following inequalities hold
0 < s ( M , D ) = s ( M * , D * ) = s < m .
Then M and M * admit linearizable 2-webs which are locally isomorphic.
Proof. 
The proof is based on methods of the information geometry.
Let A and A * be the KV algebras of ( M , D ) and of ( M * , D * ) respectively. By the hypothesis there exists a pair of geosic Riemannian foliations
( B , B * ) S 2 A × S 2 A *
such that
r a n k ( B ) = r a n k ( B * ) = s .
By the dualistic relation both M and M * admit locally flat structures ( M , D ˜ ) and ( M * , D ˜ * ) defined by
g ( Y , D ˜ X Z ) = X g ( Y , Z ) g ( D X Y , Z ) ,
g * ( Y , D ˜ X * Z ) = X g * ( y , Z ) g * ( D X * Y , Z ) .
Their KV algebras are denoted by A ˜ and A ˜ * .
Step a
There exists a 1-cocycle
ψ Z τ 1 ( A ˜ , A ˜ )
such that
B ( X , Y ) = g ( ψ ( X ) , Y ) ,
K e r ( B ) = K e r ( ψ ) .
By the definition of D ˜ we have
T M = K e r ( ψ ) i m ( ψ ) .
Further i m ( ψ ) is D ˜ -geodesic and K e r ( B ) is D-geodesic. Therefore, the pair
( K e r ( ψ ) , i m ( ψ ) )
is a 2-web in M.
In ( M * , D * , g * ) we obtain similar 2-web
( K e r ( ψ * ) , i m ( ψ * ) ) .
By the choice of B and B * we have
r a n k ( K e r ( ψ ) ) = r a n k ( K e r ( ψ * ) ) = m s .
Now we perform the following arguments.
( a ) : The foliation B is D-geodesic. In a neighbourhood of every point p 0 ( M , D ) we linearize B by choosing appropriate local affine coordinate functions
( x , y ) = ( x 1 , . . . , x m s , y 1 , . . . , y s ) .
The leaves of K e r ( ψ ) are defined by
y = c o n s t a n t .
Thereby those leaves are locally isomorphic to affine sub-spaces.
Step b
The distribution i m ( ψ ) is D ˜ -geodesic. Therefore, near the same point p 0 ( M , D ˜ ) we linearize i m ( ψ by choosing appropriate local affine coordiante functions
( x * , y * ) = ( x 1 * , . . . , y 1 * , . . . ) .
The leaves of i m ( ψ ) are defined by
x * = c o n s t a n t .
Thus near p 0 the foliation defined by î m ( ψ ) is isomorphic to an linear foliation.
Step c
By both step a and step b we choose a neighbourhood of p 0 which is the domain of systems of appropriate local coordinate functions ( x , y ) and ( x * , y * ) . From those data we pick the local coordinate functions
( x , y * ) = ( x 1 , . . . , x m s , y 1 * , . . . , y s * ) .
So we linearize the 2-web ( K e r ( ψ ) , i m ( ψ ) ) with the local coordinate functions ( x , y * ) .
( K e r ( ψ ) , i m ( ψ ) ) .
Thus near the p 0 the 2-web ( K e r ( ψ ) , i m ( ψ ) ) is isomorphic to the linear 2-web ( L 1 , L 2 ) which is defined in R m by
R m = R m s × R s .
Step d
At a point p 0 * in M * we perform the construction as in step a and in steps b and c, then we linearize ( K e r ( ψ * ) , i m ( ψ * ) ) by choosing appropriate local coordinate functions
( x 0 , y 0 * ) = ( x 1 0 , . . . , x m s 0 , y 1 0 * , . . . , y s 0 * ) .
In final, near the p O * M * the web ( K e r ( ψ * ) , i m ( ψ * ) ) is diffeomorphic to the affine web whose leaves are parallel to a decomposition
R m = V m s × V s .
Here V m s and V s are vector subspaces of R m . Their dimensions are m s and s.
Conclusion.
There exists a unique linear transformation ϕ of R m such that
ϕ ( R m s × 0 ) = V m s ,
ϕ ( 0 × R s ) = V s .
Thereby there is a local diffeomorphism Φ of M in M * subject to the requirements
Φ ( p 0 ) = p 0 * ,
( x 0 , y 0 * ) Φ = ( x , y * ) .
The differential of Φ is denoted by Φ * . We express the properties above by
Φ ( p 0 ) = p 0 * ,
Φ * [ K e r ( ψ ) , i m ( ψ ) ] = [ K e r ( ψ * ) , i m ( ψ * ) ] .
This ends the sketch of proof of Theorem. ☐
In the next we use the following definitions.
Definition 30.
A finite family
B J , J Z S 2 A ( M )
is in general position if the distributions K e r ( B j ) , j J are in general position.
The following statement is a straight corollary of the theorem we just demonstrated.
Proposition 2.
In a locally flat manifold ( M , D ) with r ( D ) > 0 every finite family in general position define a linearizable Riemannian web.

4.4. The KV Cohomology and Differential Topology Continued

We have seen how the total cohomology and linearizable Riemannian webs are related. More precisely the theory of KV cohomology provides sufficient conditions for a locally flat manifold admitting linearizable Riemannian webs. That approach is based on the split exact sequence
0 H d R 2 ( M ) H τ 2 ( A , C ( M ) ) S 2 A ( M ) 0 .

4.4.1. Kernels of 2-Cocycles and Foliations

Not all locally flat manifolds admit locally flat foliations. The existence of locally flat foliations is related to the linear holomnomy representation, viz the linear component of the affine holonomy representation of the fundamental group. Via the developing map the affine holonomy representation is conjugate to the natural action of the fundamental group in the universal covering. The KV homology is useful for investigating the existence of locally flat foliations. To simplify we work in the analytic category. So our purposes include singular foliations.
For those purposes we focus on an elementary item which has a notable impacts on our request.
Let ( M , D ) be a locally flat manifold whose KV algebra is denoted by A . Let g C 2 ( A , C ( M ) ) . The left kernel and the right kernel of g are denoted by K e r ( g ) and K 0 e r ( g ) respectively.
K e r ( g ) is defined by
g ( X , Y ) = 0 Y A .
K 0 e r ( g ) is defined by
g ( Y , X ) = 0 Y A .
The scalar KV 2-cocycles have elementary relevant properties
(1)
The left kernel of every KV 2-cocycle is closed under the Poisson bracket of vector fields.
(2)
The right kernel of every KV 2-cocycle is a KV subalgebra of the KV algebra A .
We translate those elementary properties in term of the differential topology
Theorem 7.
In an analytic locally flat manifold ( M , D )
(1) 
The arrow
Z K V 2 ( A , C ( M ) ) g K e r ( g )
maps the set of analytic 2-cocycles in the category of analytic stratified foliations M,
(2) 
The arrow
Z K V 2 ( A , C ( M ) ) g K 0 e r ( g )
maps the set of analytic 2-cocycles in the category of stratified locally flat foliations,
(3) 
If a 2-cocycle g is a symmetric form then K e r ( g ) is a stratified locally flat transversally Riemannian foliation.
The vector subspace of symmetric 2-cocycles the kernels of which are D-geodesic is denoted by Z ˜ K V 2 ( A ) . The corresponding cohomology vector subspace is denoted by
H ˜ K V 2 ( A ) H K V 2 ( A , C ( M ) ) .
By the exact sequence
O H d R 2 ( M ) H τ 2 ( A , C ( M ) ) S 2 A ( M ) 0
we have the inclusion map
H τ 2 ( A , C ( M ) ) H d R 2 ( M ) H ˜ K V 2 ( A ) RF ( M ) .

5. The Information Geometry, Gauge Homomorphisms and the Differential Topology

We combine the dualistic relation with gauge homomorphisms to relate the total cohomology and two problems.
(i)
The first is the existence problem for Riemannian foliations.
(ii)
The second is the linearization of webs.
Those relationships highlight other roles played by the total KV cohomology. Through this section we use the brute coboundary operator.

5.1. The Dualistic Relation

We are interest in the foliation counterpart of the reduction in statistical models. The statistical reduction theorem is Theorem 3.5 as in [18]. We recall the notions which are needed.
Definition 31.
A dual pair is a quadruple ( M , g , D , D * ) where ( M , g ) is a Riemannian manifold, D and D * are Koszul connections in M which are related to the metric tensor g by
X g ( Y , Z ) = g ( D X Y , Z ) + g ( Y , D X * Z ) X , Y , Z .
We recall that a Riemannian tensor is a non degenerate symmetric bilinear 2-form.
The dualistic relation between linear connections plays a central role in the information geometry [17,18,47,48].
Definition 32.
Let ( M , g ) be a Riemannian manifold.
(1) 
A dual pair ( M , g , D , D * ) is called a flat pair if the connection D is flat, viz R = 0 .
(2) 
A flat pair ( M , g , D , D * ) is called a dually flat pair if both ( M , D ) and ( M , D * ) are locally flat manifolds.
Given a dual pair ( M , D , D * ) let us set A = D D * . Here are the relationships between the torsion tensors T D and T D * (respectively the relationship between the curvature tensors R D and R D * )
g ( R D ( X , Y ) · Z , T ) + g ( Z , R D ( X , Y ) · T ) = 0 ,
g ( T D ( X , Y ) , Z ) g ( T D * ( X , Y ) , Z ) = g ( Y , A ( X , Z ) ) g ( X , A ( Y , Z ) ) .
Proposition 3.
Given a flat pair ( M , g , D , D * ) , the following assertions are equivalent.
(1) 
Both D and D * are torsion free.
(2) 
D is torsion free and A is symmetric, viz
A ( X , Y ) = A ( Y , X ) .
(3) 
D * is torsion free and the metric tensor g a is KV cocycle of the KV algebra A * of the locally flat manifold ( M , D * ) .
(4) 
The flat pair ( M , g , D , D * ) is a dually flat pair.
Proof. 
Let us prove that 1 implies ( 2 )
If both T D and T D * vanish identically then A is symmetric, viz A ( X , Y ) = A ( Y , X ) .
Let us prove that ( 2 ) implies ( 3 ) .
Since D is a flat connection, ( 2 ) implies that both the torsion tensor and the curvature tensor of D vanish identically. Then ( M , D ) is a locally flat manifold whose KV complex is denoted by ( C * ( A , R ) , δ K V ) . Using the dualistic relation of the pair ( M , g , D , D * ) one obtains the identity
δ K V g ( X , Y , Z ) = g ( A ( X , Y ) A ( Y , X ) , Z ) = g ( T D * ( X , Y ) , Z ) ,
therefore ( 2 ) implies ( 3 ) .
Let us prove that ( 3 ) implies ( 4 ) .
The assertion ( 3 ) implies that ( M , D * ) is a locally flat manifold. Since g is δ K V -closed D is torsion free. Thereby ( M , g , D , D * ) is a dually flat pair.
Let us prove that ( 4 ) implies ( 1 ) .
This implication derives directly from the definition of dually flat pair. ☐
A Comment.
From the proposition just proved arises a relationship between the dually flatness and the KV cohomology.
Indeed let ( M , D 0 ) be a fixed locally flat manifold whose KV algebra is denoted by A 0 . Let C K V * ( A 0 , R ) be the KV complex of R ˜ -valued cochains of the KV algebroid ( T M , D 0 , 1 ) . We know that every g R i e ( M ) yields a flat pair ( M , g , D 0 , D g ) .
Here D g is the flat Koszul connection defined by
g ( D X g Y , Z ) = X g ( Y , Z ) g ( Y , D X 0 Z ) .
Proposition 4.
The following assertions are equivalent.
(1) 
( M , g , D 0 , D g ) is a dually flat pair.
(2) 
δ K V 0 ( g ) = 0
The scalar KV cohomology of a fixed locally flat manifold ( M , D 0 ) provides a way of constructing new locally flat structures in M. Indeed let us set
H e s ( M , D 0 ) = Z K V 2 ( A 0 , R ) R i e ( M ) .
For every g H e s ( M , D 0 ) there is a unique D g LF ( M ) such that ( M , g , D 0 , D g ) is a dually flat pair.
So the dualistic relation leads to the map
H e s ( M , D 0 ) g D g LF ( M ) .
We recall that a gauge map in T M is a vector bundle morphism of T M in T M which projects on the identity map of M. The readers interested in others topological studies involving connections and gauge transformations are referred to [49].
Given two symmetric cocycles g , g * H e s ( M , D 0 ) there is a unique gauge transformation
ϕ * : T M T M
such that
g * ( X , Y ) = g ( ϕ * ( X ) , Y ) .
The following properties are equivalent
(11a) ϕ ( D X 0 Y ) = D X 0 ϕ ( Y ) , (11b) D g = D g * .
We fix a metric tensor g * H e s ( M , D 0 ) . A gauge transformation ϕ is called g-symmetric if we have
g ( ϕ ( X ) , Y ) = g ( X , ϕ ( Y ) ) ( X , Y ) .
Every g-symmetric gauge transformation ϕ defines the metric tensor
g ϕ ( X , Y ) = g ( ϕ ( X ) , Y ) .
This gives rise to the flat pair
( M , g ϕ , D 0 , D g ϕ ) .
To simplify we set
D ϕ = D g ϕ .
We note S y m ( g ) the subset of g-symmetric gauge transformations ϕ such that the following assertions are equivalent
(1)
ϕ S y m ( g ) .
(2)
( M , g ϕ , D 0 , D ϕ ) is a dually flat pair.
The Lie group of D 0 -preserving gauge transformations of T M is denoted by G 0 . It is easy to see that for every ϕ S y m ( g ) the following assertions are equivalent
(1)
ϕ G 0 ,
(2)
g ϕ H e s ( M , D 0 ) .
Henceforth we deal with a fixed g * H e s ( M , D 0 . The triple ( M , g * , D 0 ) leads to the dually flat pair ( M , g , D 0 , D g * ) . We set
D * = D g * .
The tangent bundle T M is regarded as a left KV module of the KV algebroid ( T M , D * , 1 ) .
The KV algebras of ( M , D 0 ) and of ( M , D * ) are denoted by A 0 and by A * respectively. Their coboundary operators are noted δ 0 and δ * respectively.
We focus on the role played by the total KV cohomology of the algebroid ( M , D * , 1 ) .
Let ϕ be a g * -symmetric gauge transformation. Then ϕ gives rise to the metric tensor g ϕ which is defined by
g ϕ ( X , Y ) = g * ( ϕ ( X ) , Y ) .
Lemma 3.
The following assertions are equivalent,
(1) 
g ϕ H e s ( M , D 0 ) ,
(2) 
ϕ Z τ 1 ( A * , A * ) .
Hint. 
Use the following formula
δ K V 0 g ϕ ( X , Y , Z ) = g * ( δ τ * ϕ ( X , Y ) , Z ) .
Following the pioneering definition as in [2] a hyperbolic locally flat manifold is a positive exact Hessian manifold ( M , D , δ K V θ ) . We extend the notion of hyperbolicity by deleting the condition that δ K V θ is positive. Now denote by H y p ( M , D 0 ) the set of exact Hessian structures in ( M , D 0 ) .
A hyperbolic structure is defined by a triple ( M , D , θ ) where ( M , D ) is a locally flat manifold and θ is a de Rham closed differential 1-form such that the symmetric bilinear δ K V θ is definite.
The following statement is a straightforward consequence of Lemma 3.
Corollary 2.
The following statements are equivalent.
(1) 
g ϕ H y p ( M , D 0 ) ,
(2) 
ϕ B 1 τ ( A * , A * )
Proof of Corollary.
By (1) there exists a (de Rham) closed differential 1-form θ such that
g ϕ ( X , Y ) = X θ ( Y ) θ ( D X 0 Y ) .
Let ξ be the unique vector field such that
θ = ι ξ g * .
Therefore one has
g * ( ϕ ( X ) , Y ) = X g * ( ξ , Y ) g * ( ξ , D X 0 Y ) .
Since the quadruple
( M , g * , D 0 , D * )
is a dually flat pair one has the identity
g * ( ϕ ( X ) , Y ) = g * ( D X * ξ , Y ) .
Thus we get the expected conclusion, viz
ϕ ( X ) = D X * ξ .
Conversely let us assume that there exists a vector ξ satisfying the identity
ϕ ( X ) = D X * ξ .
That leads to the identity
g * ( D X * ξ , Y ) = X g * ( ξ , Y ) g ( ξ , D X 0 Y ) .
In other words one has
g ϕ H y p ( M , D 0 ) .
This ends the proof of Corollary 2. ☐
The set of g * -symmetric gauge transformation is denoted by Σ ( g * ) .
We have the canonical isomorphism
Σ ( g * ) ϕ g ϕ R i e ( M ) .
Now we define the sets
Z ˜ τ 1 ( A * , A * ) = Σ ( g * ) Z τ 1 ( A * A * ) ,
B ˜ τ 1 ( A * , A * ) = Σ ( g * ) B τ 1 ( A * , A * ) .
Combining Lemma 3 and its corollary with the isomorphism Equation (12). Then we obtain the identifications
Z ˜ τ 1 ( A * , A * ) = H e s ( M , D * ) ,
B ˜ τ 1 ( A * , A * ) = H y p ( M , D * ) .
Reminder. 
We recall that a hyperbolic manifold (or a Koszul manifold) is δ K V -exact Hessian manifold ( M , g , D ) .
It is easily seen that the set of positive hyperbolic structures in a locally flat manifold ( M , D ) is a convex subset of H e s ( M , D ) .
So show the Koszul geometry is a vanishing theorem in the theory of KV homology of KV algebroids. The theory of homological statistical model (to be introduced in Part B) is another impact on the information geometry of the KV cohomology.
At the present step we have the relations
H e s ( M , D * ) H y p ( M , D * ) H K V 2 ( A * , R ) ,
Z ˜ τ 1 ( A * , A * ) B ˜ τ 1 ( A * , A * ) = H e s ( M , D * ) H y p ( M , D * ) .
Another outstanding result of Koszul is the non rigidity of compact positive hyperbolic manifolds [2]. The non rigidity means that every open neighborhood of a positive Hyperbolic locally flat manifold ( M , D , δ K V θ ) contain another positive hyperbolic locally flat structure which is not isomorphic to ( M , D ) . This non rigidity property may be expressed with the Maurer–Cartan polynomial function P M C A of ( M , D ) (see the local convexity theorem in [29]. In the next sub-subsection we revisit the notion of dual pair of foliations as in [18].

5.1.1. Statistcal Reductions

The statistical reduction theorem is the following statement.
Theorem 8
([18]).
Let ( M , g , D , D * ) be a dually flat pair and let N be a submanifold of M. Assume that N is either D-geodesic or D * -geodesic. Then N inherits a structure of dually flat pair which is either ( N , g N , D , D N * ) or ( N , g N , D N , D * ) ).
The foliation counterpart of the reduction theorem is of great interest in the differential topology of statistical models see [18]. In the preceding sections we have addressed a cohomological aspect of this purpose. The matter will be more extensively studied in a forthcoming paper (See the Appendix A).
In mathematical physics a principal connection 1-form is called a gauge field.
In the differential geometry a principal connection 1-form in a bundle of linear frames is called a linear connection.
In the category of vector bundle Koszul connections are algebroid counterpart of principal connection 1-forms.
In a tangent bundle T M , depending on concerns and needs Koszul connections may called linear connections or linear gauges.
Definition 33.
Let D , D * L C ( M ) . A vector bundle homomorphism
ψ : T M T M
is called a gauge homomorphism of ( M , D ) in ( M , D * ) if for all pairs of vector fields ( X , Y ) one has
D X * ψ ( Y ) = ψ ( D X Y ) .
The vector space of gauge homomorphisms of ( M , D ) in ( M , D * ) is denoted by M ( D , D * ) . The vector space M ( D , D * ) is not a C ( M ) -module.

5.1.2. A Uselful Complex

In this subsubsection we fix a dually flat pair ( M , g , D , D * ) whose KV algebras are denoted by A and by A * . The tangent bundle T M is endowed the structure left module of the anchored KV algebroids ( T M , D , 1 ) and ( T M , D * , 1 ) . This means that each of the KV algebras A or A * is regarded as a left module of itself.
We consider the tensor product
C = C τ * ( A * , A * ) C τ * ( A , R ) .
We endow C with the Z bi-grading.
C i , 0 = C τ i ( A * , A * ) C ( M ) ,
C 0 , j = A * C τ j ( A , R ) ,
C i , j = C τ i ( A * , A * ) C K V j ( A , R ) .
We recall that C * ( A , R ) stands for C * ( A , C ( M ) ) .
For every non negative integer q we set
C q = Σ i + j = q C i , j .
We defines the linear map
δ i , j : C i , j C i + 1 , j C i , j + 1
by
δ i , j = δ τ 1 + ( 1 ) i δ τ .
So we obtain a linear map
C q C q + 1
Therefore, we consider the bi-graded differential vector space
C : = ( C * * , δ * * ) .
That is a bi-graded cochain complex whose q t h c o h o m o l o g y is denoted by H q ( C ) . The cohomology inherits the bi-grading
H q ( C ) = [ i + j = q ] H i , j ( C ) .
Here
H i , j ( C ) = C i , j [ Z τ i ( A * , A * ) Z τ j ( A , R ) ] i m ( δ i 1 , j ) + i m ( δ i , j 1 )
In the next subsubsection we shall discuss the impacts of this cohomology.
Remark 2.
The pair ( C * * , δ * * ) generates a spectral sequence [34]. That spectral sequence is a useful tool for simultaneously computing both the KV cohomology and the total KV cohomology of KV algebroids. Those matters are not the purpose of this paper.

5.1.3. The Homological Nature of Gauge Homomorphisms

Giving a dually flat pair ( M , g , D , D * ) one considers the linear map
C τ 1 , 0 ( A * , A * ) ψ ψ q ψ C 1 , 2 .
Here the symmetric 2-form q ψ is defined by
q ψ ( X , Y ) = 1 2 [ g ( ψ ( X ) , Y ) + g ( X , ψ ( Y ) ) ] .
To relate the bi-complex ( C * * , δ * * ) and the space of gauge homomorphisms we use the following statement.
Theorem 9.
Given a gauge morphism
ψ : T M T M
the following statements are equivalent
(1) 
ψ M ( D , D * ) ,
(2) 
δ 1 , 2 ( ψ q ψ ) = 0
Proof. 
( 1 ) implies ( 2 ) .
Suppose that ψ M ( D , D * ) . Then we have
D X * ψ ( Y ) = ψ ( D X Y ) ( X , Y ) .
Since both D and D * are torsion free one has the identity
D X * . ψ ( Y ) ψ ( D X * Y ) D Y * ψ ( X ) + ψ ( D Y * X ) = 0 .
Thus ψ is a (1,0)-cocycle of the total KV complex ( C * * , δ * * ) .
At another side the relation D X * ψ = ψ D X leads to the identity
D X q ψ = 0 .
So q ψ is a (0,2)-cocycle of complex ( C * * , δ * * ) . We conclude that
δ 1 , 2 ( ψ q ψ ) = 0 , Q E D .
( 2 ) implies ( 1 ) .
We recall the formula
δ 1 , 2 ( ψ q ψ ) = ( δ τ ψ ) q ψ ψ δ τ q ψ .
By this formula
δ 1 , 2 ( ψ q ψ ) C 2 , 2 C 1 , 3
Thus the statement (2) is equivalent to the system
δ τ ψ = 0 ,
δ τ q ψ = 0 .
To continue the proof we perform the following lemma.
Lemma 4
([29]). For every symmetric cochain B C 0 , 2 , viz
B ( X , Y ) = B ( Y , X )
the following identities are equivalent
δ τ B = 0 ,
B = 0 ,
By Lemma 4 the bilinear form q ψ is D-parallel. Thereby we get the identity
X q ψ ( Y , Z ) q ψ ( D X Y , Z ) q ψ ( Y , D X Z ) = 0 .
To usefully interpret this identity we involve the dualistic relation
X g ( Y , Z ) = g ( D X Y , Z ) + g ( Y , D X * Z ) .
This expression leads to the identity
g ( D X * ψ ( Y ) ψ ( D X Y ) , Z ) + g ( Y , D X * ψ ( Z ) ψ ( D X Z ) ) = 0 .
A highlighting consequence is the identity
D X * ψ ( Y ) ψ ( D X Y ) = D Y * ψ ( X ) ψ ( D Y X ) .
To every vector field X we assign the linear map
Y S X ( Y ) = D X * Y ψ ( D X Y ) .
Then we rewrite Equations (14) and (15) as
g ( S X ( Y ) , Z ) + g ( Y , S X ( Z ) ) = 0 ,
S X ( Y ) = S Y ( X ) .
We consider the last identities in the framework of the Sternberg geometry [50,51].
Since the application
( X , Y ) S X ( Y )
is C ( M ) -bi-linear it belongs to the first Kuranishi-Spencer prolongation of the orthogonal Lie algebra s o ( g ) . Thereby S X ( Y ) vanishes identically. In other words we have
ψ M ( D , D * ) .
This ends the proof of Theorem
A Comment.
The Sternberg geometry is the algebraic counterpart of the global analysis on manifolds. It has been introduced by Shlomo Sternberg and Victor Guillemin. It is an algebraic model for transitive differential geometry [50]. In that approach the Riemannian geometry is a geometry of type one. All of its Kuranishi-Spencer prolongations are trivial. The unique relevant geometrical invariant of the Riemnnian geometry is the curvature tensor of the Levi-Civita connection. Except the connection of Levi-Civita the other metric connections have been of few interest. Really other metric connections may have outstanding impacts on the differential topology. I shall address this purpose in a forthcoming paper.

5.1.4. The Homological Nature of the Equation F E *

Before proceeding we plan to discuss some homological ingredients which are connected to the differential equation
F E * : D * ( ψ ) = 0 .
Let us consider a dually flat pair ( M , g * , D , D * ) and the KV complex
ψ