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

## Abstract

**:**

## Contents

**1****Introduction****4**- 1.1 The Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
- 1.2 Some Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
- 1.3 The content of the Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

**3****The Theory of Cohomology of KV Algebroids and Their Modules****11**- 3.2 The Theory of KV Cohomology—Version: the Semi-Simplicial Objects . . . . . . . . . . 14
- 3.2.1 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
- 3.2.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
- 3.2.3 Notation-Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
- 3.2.4 The KV Chain Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
- 3.2.5 The V-Valued KV Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
- 3.2.6 Two Cochain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
- 3.2.7 Residual Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

- 3.3 The Theory of KV Cohomology—Version the Anomaly Functions . . . . . . . . . . . . . 20
- 3.3.1 The General Challenge CH${\left(\mathbb{D}\right)}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
- 3.3.2 Challenge CH${\left(\mathcal{D}\right)}$ for KV Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
- 3.3.3 The KV Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
- 3.3.4 The Total Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
- 3.3.5 The Residual Cohomology, Some Exact Sequences, Related Topics, DTO-HEG-IGE-ENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

**5****The Information Geometry, Gauge Homomorphisms and the Differential Topology****35**- 5.1 The Dualistic Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
- 5.1.1 Statistcal Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
- 5.1.2 A Uselful Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
- 5.1.3 The Homological Nature of Gauge Homomorphisms . . . . . . . . . . . . . . . . 41
- 5.1.4 The Homological Nature of the Equation FE
^{∇∇*}. . . . . . . . . . . . . . . . . . . 43 - 5.1.5 Computational Relations. Riemannian Foliations. Symplectic Foliations: Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
- 5.1.6 RiemannianWebs—SymplecticWebs in Statistical Manifolds . . . . . . . . . . . 49

- 5.2 The Hessian Information Geometry, Continued . . . . . . . . . . . . . . . . . . . . . . . . 51
- 5.3 The a-Connetions of Chentsov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
- 5.4 The Exponential Models and the Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . 55

**7****Some Highlighting Conclusions****59**- 7.1 The Total KV Cohomology and the Differential Topology . . . . . . . . . . . . . . . . . . 59
- 7.2 The KV Cohomology and the Geometry of Koszul . . . . . . . . . . . . . . . . . . . . . . 60
- 7.3 The KV Cohomology and the Information Geometry . . . . . . . . . . . . . . . . . . . . 60
- 7.4 The Differential Topology and the Information Geometry . . . . . . . . . . . . . . . . . . 60
- 7.5 The KV Cohomology and the Linearization Problem for Webs . . . . . . . . . . . . . . . 60

**8****B. The Theory of StatisticaL Models****61**- 8.1 The Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
- 8.2 The Category ${\mathcal{FB}(\mathsf{\Gamma},\mathsf{\Xi})}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
- 8.3 The Category ${\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
- 8.3.1 The Objects of ${\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
- 8.3.2 The Global Probability Density of a Statistical Model . . . . . . . . . . . . . . . . 70
- 8.3.3 The Morphisms of ${\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
- 8.3.4 Two Alternative Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
- 8.3.5 Fisher Information in ${\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

- 8.4 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
- 8.4.1 The Entropy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
- 8.4.2 The Fisher Information as the Hessian of the Local Entropy Flow . . . . . . . . . 75
- 8.4.3 The Amari-Chentsov Connections in ${\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})}$ . . . . . . . . . . . . . . . . . . . 75
- 8.4.4 The Homological Nature of the Probability Density . . . . . . . . . . . . . . . . . 76
- 8.4.5 Another Homological Nature of Entropy . . . . . . . . . . . . . . . . . . . . . . . 77

**10****The Homological Statistical Models****83**- 10.1 The Cohomology Mapping of HSM${(\mathsf{\Xi},\mathsf{\Omega})}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
- 10.2 An Interpretation of the Equivariant Class [Q] . . . . . . . . . . . . . . . . . . . . . . . . 85
- 10.3 Local Vanishing Theorems in the Category ${\mathcal{HSM}(\mathsf{\Xi},\mathsf{\Omega})}$ . . . . . . . . . . . . . . . . . . 85

**13****Highlighting Conclusions****91**- 13.1 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
- 13.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
- 13.3 KV Homology and Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
- 13.4 The Homological Nature of the Information Geometry . . . . . . . . . . . . . . . . . . . 91
- 13.5 Homological Models and Hessian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 92

**A****Appendix A****92**- A.1 The Affinely Flat Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
- A.2 The Hessian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
- A.3 The Geometry of Koszul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
- A.4 The Information Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
- A.5 The Differential Topology of a Riemannian Manifold . . . . . . . . . . . . . . . . . . . . 94

## 1. Introduction

#### 1.1. The Notation

- (A.1)
- ${D}^{\nabla {\nabla}^{*}}$ is a first order differential operator. It is defined in the vector bundle $Hom(TM,TM)$. Its values belong to the vector bundle $Hom(T{M}^{\otimes 2},TM)$.
- (A.2)
- ${D}^{\nabla}$ and ${D}_{\nabla}$ are 2nd order differential operators. They are defined in the vector bundle $TM$. Their values belong to the vector bundle $Hom(T{M}^{\otimes 2},TM)$. Let X be a section of $TM$ and let ψ be a section of ${T}^{*}M\otimes TM$. The differential operators just mentioned are defined by$$\begin{array}{}\mathrm{(1a)}& \hfill {D}^{\nabla {\nabla}^{*}}(\psi )={\nabla}^{*}\circ \psi -\psi \circ \nabla ,\mathrm{(1b)}& \hfill {D}^{\nabla}(X)={L}_{X}\nabla -\iota (X){R}^{\nabla},\mathrm{(1c)}& \hfill {D}_{\nabla}(X)={\nabla}^{2}(X).\end{array}$$Part A of this paper is partially devoted to the global analysis of the differential equation$$FE(\nabla {\nabla}^{*}):\phantom{\rule{1.em}{0ex}}{D}^{\nabla {\nabla}^{*}}(\psi )=O.$$The solutions to $FE(\nabla {\nabla}^{*})$ 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$$F{E}^{*}(\nabla ):\phantom{\rule{1.em}{0ex}}{D}^{\nabla}(X)=0,$$$$F{E}^{**}(\nabla ):\phantom{\rule{1.em}{0ex}}{D}_{\nabla}(X)=0.$$In the Appendix A to this paper we overview the role played by the solutions to $F{E}^{**}(\nabla )$ in some still open problems.

#### 1.2. Some Explicit Formulas

- (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].

**Warning.**

#### 1.3. The content of the Paper

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Algebroids, Moduls of Algebroids, Anomaly Functions

#### 2.1. The Algebroids and Modules

**Definition**

**1.**

**Definition**

**2.**

**Warning.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.2. Anomaly Functions of Algebroids and of Modules

**Definition**

**6.**

**Definition**

**7.**

- (1)
- The associator anomaly function of $\mathcal{E}$ is defined by$$Ass(s,{s}^{*},{s}^{**})=(s\xb7{s}^{*})\xb7{s}^{**}-s\xb7({s}^{*}\xb7{s}^{**}).$$
- (2)
- The Koszul-Vinberg anomaly function of $\mathcal{E}$ is defined by$$KV(s,{s}^{*},{s}^{**})=Ass(s,{s}^{*},{s}^{**})-Ass({s}^{*},s,{s}^{**}).$$
- (3)
- The Jacobi anomaly functions of $\mathcal{E}$ are defined by$$J(s,{s}^{*},{s}^{**})=(s\xb7{s}^{*})\xb7{s}^{**}+({s}^{*}\xb7{s}^{**})\xb7s+({s}^{**}\xb7s)\xb7{s}^{*}.$$

**Definition**

**8.**

- (1)
- The associator anomaly function of a left module $\mathcal{V}$ is defined as$$Ass(s,{s}^{*},v)=(s\xb7{s}^{*})\xb7v-s\xb7({s}^{*}\xb7v).$$
- (2)
- The KV anomaly functions of a two sided module $\mathcal{V}$ are defined as$$KV(s,{s}^{*},v)=Ass(s,{s}^{*},v)-Ass({s}^{*},s,v),$$$$KV(s,v,{s}^{*})=(s\xb7v)\xb7{s}^{*}-s\xb7(v\xb7{s}^{*})-(v\xb7s)\xb7{s}^{*}+v\xb7(s\xb7{s}^{*}).$$

**Definition**

**9.**

- (1)
- The Leibniz anomaly function of an anchored algebroid $\mathcal{E}$ is defined by$$L(s,f,{s}^{*})=s\xb7(f{s}^{*})-df(b(s)){s}^{*}-fs\xb7{s}^{*}.$$
- (2)
- The Leibniz anomaly function of the $\mathcal{E}$-module $\mathcal{V}$ is defined by$$L(s,f,v)=s\xb7(fv)-df(b(s))v-fs\xb7v.$$

**Definition**

**10.**

**Warning.**

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

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

#### 3.1.1. The Cochain Complex ${C}_{KV}$.

**Definition**

**11.**

**The conjecture of Gerstenhaber:**

**Comments.**

**Features.**

#### 3.1.2. The Total Cochain Complex ${C}_{\tau}.$

- (1):
- $\phantom{\rule{1.em}{0ex}}[{\delta}_{\tau}w](a)=-a\xb7w+wa\phantom{\rule{1.em}{0ex}}\forall (a,w)\in \mathcal{A}\times W,$
- (2):
- $\phantom{\rule{1.em}{0ex}}[{\delta}_{\tau}f](\xi )=\sum _{1}^{q+1}{(-1)}^{i}[{a}_{i}\xb7f({\partial}_{i}\xi )-f({a}_{i}\xb7{\partial}_{i}\xi )+(f({\partial}_{i,q+1}^{2}\xi \otimes {a}_{i}))\xb7{a}_{q+1}]\phantom{\rule{1.em}{0ex}}\forall f\in {C}_{\tau}^{q}(\mathcal{A},W).$

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

#### 3.2.1. Extension

#### 3.2.2. Construction

#### 3.2.3. Notation-Definitions

**Definition**

**12.**

**Structure.**

**Definition**

**13.**

#### 3.2.4. The KV Chain Complex

#### 3.2.5. The V-Valued KV Homology

**Definition**

**14.**

#### 3.2.6. Two Cochain Complexes

**Comments.**

#### 3.2.7. Residual Cohomology

**Definition**

**15.**

**Definition**

**16.**

- (1)
- The vector subspace of residual KV cocycles of degree q is denoted by ${Z}_{KVres}^{q}$.
- (2)
- The vector subspace of residual coboundaries of degree q is defined by ${B}_{KVres}^{q}={H}_{e}^{q}(\mathcal{B},V)+{d}_{KV}({C}_{KV}^{q-1}(\mathcal{B},V)).$ The residual KV cohomology space of degree q is the quotient vector space.
- (3)
- ${H}_{KVres}^{q}(\mathcal{B},V)=\frac{{Z}_{KVres}^{q}}{{B}_{KVres}^{q}}.$
- (4)
- By replacing the KV complex by the total KV complex one defines the vector space of residual total cocycles ${Z}_{\tau res}^{q}$ and the space of residual total coboundaries ${B}_{\tau res}^{q}$. Therefore we get the residual total KV cohomology space$${H}_{\tau res}^{q}(\mathcal{A},V)=\frac{{Z}_{\tau res}^{q}}{{B}_{\tau ,res}^{q}}$$

**Some**

**Comments.**

- $(c.1)$:
- We replace the KV $\mathcal{B}$ by $\mathcal{A}$. Then we obtain the exact sequences$$\to {H}_{KVres}^{q-1}(\mathcal{A},V)\to {H}_{e}^{q}(\mathcal{A},V)\to {H}_{KV}^{q}(\mathcal{A},V)\to {H}_{KVres}^{q}(\mathcal{A},V)\to $$$$\to {H}_{\tau res}^{q-1}(\mathcal{A},V)\to {H}_{e}^{q}(\mathcal{A},V)\to {H}_{\tau}^{q}(\mathcal{A},V)\to {H}_{\tau res}^{q}(\mathcal{A},V)\to $$
- $(c.2)$:
- The KV cohomology difers from the total cohomology. Loosely speaking their intersecttion is the equivariant cohomology ${H}_{e}^{*}(\mathcal{B},V)$ their difference is the residual cohomology. The domain of their efficiency are different as well. Here are two illustrations.

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

#### 3.3.1. The General Challenge $CH(\mathbb{D})$

- (1)
- V is an (abstract) two sided module of an (abstract) algebra $\mathcal{A}$.
- (2)
- ${A}_{\mathcal{A}}$ and ${A}_{\mathcal{AV}}$ are fixed anomaly functions of $\mathcal{A}$ and of V respectively.
- (3)
- $Hom(T(\mathcal{A}),V)$ stands for the $\mathbb{Z}$-graded vector space$$Hom(T(\mathcal{A}),V)={\oplus}_{q}Ho{m}_{\mathbb{R}}({\mathcal{A}}^{\otimes q},V).$$

**Property**

**Δ**

#### 3.3.2. Challenge $CH(\mathcal{D})$ for KV Algebras

**Lemma**

**1.**

**Definition**

**17.**

**Lemma**

**2.**

#### 3.3.3. The KV Cohomology

**Theorem**

**3.**

#### 3.3.4. The Total Cohomology

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

## 4. The KV Topology of Locally Flat Manifolds

#### 4.1. The Total Cohomology and Riemannian Foliations

**Definition**

**18.**

**Warning.**

**Remark 1**(Important Remarks)

**Theorem**

**4.**

**Warning.**

**Definition**

**19.**

- (1.1)
- $rank(g)=constant,$
- (1.2)
- ${L}_{X}g=0\phantom{\rule{1.em}{0ex}}\forall X\in \mathsf{\Gamma}(Ker(g)).$ A symplectic foliation is a ( de Rham) closed differential 2-form ω which satisfies
- (2.1)
- $rank(\omega )=constant$,
- (2.2)
- ${L}_{X}\omega =0\phantom{\rule{1.em}{0ex}}\forall X\in \mathsf{\Gamma}(Ker(\omega ))$.

**Warning.**

**Definition**

**20.**

**Definition**

**21.**

**Theorem**

**5.**

#### 4.2. The General Linearization Problem of Webs

**Definition**

**22.**

**Definition**

**23.**

**A**

**Comment.**

**Definition**

**24.**

**Definition**

**25.**

#### 4.3. The Total KV Cohomology and the Differential Topology Continued

**Definition**

**26.**

**Definition**

**27.**

**Definition**

**28.**

- (1)
- $(M,\mathcal{D})$ is transversally Riemannian if there exists a $g\in {\mathcal{S}}_{2}(M)$ such that$$\mathcal{D}=\mathcal{K}er(g).$$
- (2)
- $(M,\mathcal{D})$ is transversally symplectic if there exists a (de Rham) closed differential 2-form ω such that$$\mathcal{D}=Ker(\omega )$$

**Definition**

**29.**

**Proposition**

**1.**

**Proof.**

- (1)
- Suppose that$$0<rank(\mathcal{D})<dim(M)$$Therefore, $(M,B)$ is a D-geodesic Riemannian foliation.
- (2)
- Suppose that$$rank(\mathcal{D})=O.$$Then $(M,B)$ is a Riemannian manifold the Levi-Civita connection of which is D. Therefore, the proposition holds. ☐

**Corollary**

**1.**

**Theorem**

**6.**

**Proof.**

**Conclusion.**

**Definition**

**30.**

**Proposition**

**2.**

#### 4.4. The KV Cohomology and Differential Topology Continued

#### 4.4.1. Kernels of 2-Cocycles and Foliations

- (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 $\mathcal{A}$.

**Theorem**

**7.**

- (1)
- The arrow$${Z}_{KV}^{2}(\mathcal{A},{C}^{\infty}(M))\ni g\to Ker(g)$$
- (2)
- The arrow$${Z}_{KV}^{2}(\mathcal{A},{C}^{\infty}(M))\ni g\to {K}^{0}er(g)$$
- (3)
- If a 2-cocycle g is a symmetric form then $Ker(g)$ is a stratified locally flat transversally Riemannian foliation.

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

- (i)
- The first is the existence problem for Riemannian foliations.
- (ii)
- The second is the linearization of webs.

#### 5.1. The Dualistic Relation

**Definition**

**31.**

**Definition**

**32.**

- (1)
- A dual pair $(M,g,D,{D}^{*})$ is called a flat pair if the connection D is flat, viz ${R}^{\nabla}=0.$
- (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.

**Proposition**

**3.**

- (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 ${\mathcal{A}}^{*}$ of the locally flat manifold $(M,{D}^{*})$.
- (4)
- The flat pair $(M,g,D,{D}^{*})$ is a dually flat pair.

**Proof.**

**A**

**Comment.**

**Proposition**

**4.**

- (1)
- $(M,g,{D}^{0},{D}^{g})$ is a dually flat pair.
- (2)
- ${\delta}_{KV}^{0}(g)=0$

- (1)
- $\varphi \in Sym(g)$.
- (2)
- $(M,{g}_{\varphi},{D}^{0},{D}^{\varphi})$ is a dually flat pair.

- (1)
- $\varphi \in {G}^{0},$
- (2)
- ${g}_{\varphi}\in \mathcal{H}es(M,{D}^{0})$.

**Lemma**

**3.**

- (1)
- ${g}_{\varphi}\in \mathcal{H}es(M,{D}^{0}),$
- (2)
- $\varphi \in {Z}_{\tau}^{1}({\mathcal{A}}^{*},{\mathcal{A}}^{*}).$

**Hint.**

**Corollary**

**2.**

- (1)
- ${g}_{\varphi}\in \mathcal{H}yp(M,{D}^{0})$,
- (2)
- $\varphi \in {B}^{1}\tau ({\mathcal{A}}^{*},{\mathcal{A}}^{*})$

**Proof of**

**Corollary.**

**Reminder.**

#### 5.1.1. Statistcal Reductions

**Theorem**

**8**

**Definition**

**33.**

#### 5.1.2. A Uselful Complex

**Remark**

**2.**

#### 5.1.3. The Homological Nature of Gauge Homomorphisms

**Theorem**

**9.**

- (1)
- $\psi \in \mathcal{M}(D,{D}^{*}),$
- (2)
- ${\delta}_{1,2}(\psi \otimes {q}_{\psi})=0$

**Proof.**

**Lemma**

**4**

**A**

**Comment.**

#### 5.1.4. The Homological Nature of the Equation $F{E}^{\nabla {\nabla}^{*}}$

**Corollary**

**3.**

- (1)
- $\psi \otimes {q}_{\psi}$ is an exact (1,2)-cocyle,
- (2)
- $\psi \in {B}_{\tau}^{1}({\mathcal{A}}^{*},{\mathcal{A}}^{*})$.

**Proof.**

**Proposition**

**5.**

#### 5.1.5. Computational Relations. Riemannian Foliations. Symplectic Foliations: Continued

**Lemma**

**5.**

**An**

**Idea.**

**Definition**

**34.**

**Proposition**

**6.**

**Reminder.**

**Digressions.**

- (a)
- A ∇-geodesic symplectic foliation $\omega \in {\mathsf{\Omega}}^{\nabla}$ might carry richer structures such as Kahlerian structures.
- (b)
- Suppose that the manifold M is compact and suppose that $g\in {\mathcal{S}}_{2}^{\nabla}(M)$ is a positive Riemannian foliation, viz$$g(X,X)\ge 0\phantom{\rule{1.em}{0ex}}\forall X.$$
- (c)
- In the principal bundle of first order linear frames of M the analog of a Koszul connection ∇ is a principal connection 1-form ω whose curvature form is denoted by Ω. The curvature form is involved in constructing characteristic classes of M, (the formalism of Chern-Weill.)

**Theorem**

**10.**

**Proof.**

**A Useful**

**Comment.**

**Proposition**

**7.**

**Definition**

**35.**

- (1)
- The g-symmetric part of ψ, ${\psi}^{+}$ is defined by$$g({\psi}^{+}(X),Y)=\frac{1}{2}[g(\psi (X),Y)+g(X,\psi (Y))].$$
- (2)
- The g-skew symmetric part of ψ, ${\psi}^{-}$ is defined by$$g({\psi}^{-}(X),Y)=\frac{1}{2}[g(\psi (X),Y)-g(X,\psi (Y))].$$

**Theorem**

**11.**

- (1)
- The g-symmetric part ${\psi}^{+}$ is an element $\mathcal{M}(\nabla ,{\nabla}^{*})$ whose rank is constant.
- (2)
- We have the g-orthogonal decomposition$$TM=Ker({\psi}^{+})\oplus im({\psi}^{+}).$$
- (3)
- If both ∇ and ${\nabla}^{*}$ are torsion free then $Ker({\psi}^{+})$ and $Im({\psi}^{+})$ are completely integrable.

**A**

**Digression.**

#### 5.1.6. Riemannian Webs—Symplectic Webs in Statistical Manifolds

- (a)
- $rank(g)=constant,$
- (b)
- ${L}_{X}g=0\forall X\in \mathsf{\Gamma}(Ker(g)).$

**Definition**

**36.**

**Definition**

**37.**

**Theorem**

**12.**

**Corollary**

**4.**

#### 5.2. The Hessian Information Geometry, Continued

**Definition**

**38**

**Definition**

**39.**

**Theorem**

**13**

**Theorem**

**14.**

#### 5.3. The α-Connetions of Chentsov

- (1)
- $\forall \xi \in \mathsf{\Xi}$ the function$$\mathsf{\Theta}\ni \theta \to P(\theta ,\xi )$$
- (2)
- $\forall \theta \in \mathsf{\Theta}$ the triple$$(\mathsf{\Xi},\mathsf{\Omega},P(\theta ,-))$$
- (3)
- $\forall \theta ,{\theta}^{*}\in \mathsf{\Theta}$ with $\theta \ne {\theta}^{*}$ there exists $\xi \in \mathsf{\Xi}$ such that$$P(\theta ,\xi )\ne P({\theta}^{\prime},\xi ).$$
- (4)
- The differentiation ${d}_{\theta}$ commutes with the integration ${\int}_{\mathsf{\Xi}}.$ The Fisher information of a model $(\mathsf{\Theta},P)$ is the symmetric bi-linear form g which is defined by$$g(X,Y)(\theta )={\int}_{\mathsf{\Xi}}P(\theta ,\xi )[{[{d}_{\theta}log(P)]}^{\otimes 2}(X,Y)](\theta ,\xi )d\xi .$$Here ${d}_{\theta}$ stands for the differentiation with respect to the argument $\theta \in \mathsf{\Theta}$.
- (5)
- The Fisher information is positive definite.

**Remark**

**3.**

**Proposition**

**8.**

- (1)
- Every non zero singular section$$\mathbb{R}\ni \alpha \to {B}_{\alpha}\in {\mathcal{S}}^{\alpha}(\mathsf{\Theta})$$$$T\mathsf{\Theta}=Ker({\psi}^{+\alpha})\oplus im({\psi}^{+\alpha}).$$
- (2)
- By replacing ${\mathcal{S}}^{\alpha}(\mathsf{\Theta})$ by ${\mathsf{\Omega}}_{2}^{\nabla}(\mathsf{\Theta})$ every non zero singular section$$\mathbb{R}\ni \alpha \to {\omega}_{\alpha}\in {\mathsf{\Omega}}_{2}^{\nabla}(\mathsf{\Theta})$$

**Reminder.**

- (i)
- $\alpha \to {B}_{\alpha}$ is called a singular section if each ${B}_{\alpha}$ is non inversible.
- (ii)
- $\alpha \to {\omega}_{\alpha}$ is called a simple section if each ${\omega}_{\alpha}$ is simple.

**Theorem**

**15.**

**Proposition**

**9.**

**The Sketch of**

**Proof**

- (1)
- If $Ker(B)=0$, then both ${\nabla}^{\alpha}$ and ${\nabla}^{-\alpha}$ coincide with the Levi-Civita connection of B. This implies $\alpha =0$, this contradicts our choice of α.
- (2)
- If $Ker(B)\ne 0$ then $Ker(B)$ and $Ker{(B)}^{\perp}$ are geodesic for both ${\nabla}^{\alpha}$ and ${\nabla}^{-\alpha}$. Thus the pair $(Ker(B),Ker{(B)}^{\perp})$ defines a g-orthogonal 2-web.

#### 5.4. The Exponential Models and the Hyperbolicity

**Definition**

**40.**

**Theorem**

**16.**

- (1)
- There exists $\nabla \in \mathcal{LF}(\mathsf{\Theta})$ such that$${\delta}_{KV}g=0,$$
- (2)
- The model $(\mathsf{\Theta},P)$ is an exponential model.

**Demonstration.**

**Some**

**Comments.**

- (i)
- It must be noticed that the demonstration above is independent of the rank of the Fisher information g. Therefore, the theorem holds in singular statistical models.
- (ii)
- In regular statistical models the theorem above leads to the notion of e-m-flatness as in [18].
- (iii)
- When the Fisher information g is semi-definite the dualistic relation is meaningless. However data $(\mathsf{\Theta},g,\nabla ,{\nabla}^{*})$ may be regarded as data depending on the transversal structure of the distribution $Ker(g)$.
- (iv)
- In the analytic category the Fisher information is a Riemannian foliation. Therefore, both the information geometry and the topology of information are transversal concepts. This may be called the transversal geometry and the transversal topology of Fisher-Riemannian foliations.
- (v)
- The theorem above does not solve the question as how far from being an exponential family is a given model. It only tells us that exponential families are objects of the Hessian geometry.

## 6. The Similarity Structure and the Hyperbolicity

**Theorem**

**17.**

- (1)
- The locally flat manifold $(M,\nabla )$ is hyperbolic,
- (2)
- the locally flat manifold $(M,{\nabla}^{*})$ admits a global similarity vector field ${H}^{*}$.

**Definition**

**41.**

- (1)
- The gauge structure $(M,\nabla )$ is called a similarity structure if ∇ admits a global similarity vector field $H\in \mathcal{X}(M)$.
- (2)
- A dual pair $(M,g,\nabla ,{\nabla}^{*})$ is a similarity dual pair if either $(M,\nabla )$ or $(M,{\nabla}^{*})$ is a similarity structure.

**Proposition**

**10.**

## 7. Some Highlighting Conclusions

#### 7.1. The Total KV Cohomology and the Differential Topology

#### 7.2. The KV Cohomology and the Geometry of Koszul

#### 7.3. The KV Cohomology and the Information Geometry

#### 7.4. The Differential Topology and the Information Geometry

#### 7.5. The KV Cohomology and the Linearization Problem for Webs

- (i)
- Elements of ${\mathbb{G}}^{p}[{\mathsf{\Omega}}_{2}^{\nabla}(M)]$ are LINEARIZABLE symplectic k-webs.
- (ii)
- Elements of ${\mathbb{G}}^{p}[{\mathcal{S}}_{2}^{\nabla}(M)]$ are LINEARIZABLE Riemannian k-webs.

## 8. B. The Theory of StatisticaL Models

**Definition**

**42.**

**Definition**

**43.**

- (1)
- The manifold Θ is an open subset of the m-dimensional Euclidean space ${\mathbb{R}}^{m}$.
- (2)
- P is a positive real valued function$$\mathsf{\Theta}\times \mathsf{\Xi}\ni (\theta ,\xi )\to P(\theta ,\xi )\in \mathbb{R}$$
- (3)
- The function $P(\theta ,\xi )$ is differentiable with respect to $\theta \in \mathsf{\Theta}$.
- (4)
- For every fixed $\theta \in \mathsf{\Theta}$ one set$${P}_{\theta}=P(\theta ,-)$$$$(\mathsf{\Xi},\mathsf{\Omega},{P}_{\theta})$$$${\int}_{\mathsf{\Xi}}{P}_{\theta}(\xi )d\xi =1$$Furthermore the operation of differentiation$${d}_{\theta}=\frac{d}{d\theta}$$
- (5)
- $(\mathsf{\Theta},P)$ is identifiable, viz for $\theta ,{\theta}^{*}\in \mathsf{\Theta}$$${P}_{\theta}={P}_{{\theta}^{*}}$$$$\theta ={\theta}^{*}$$
- (6)
- The Fisher information$${g}_{\theta}(X,Y)={\int}_{\mathsf{\Xi}}P(\theta ,\xi ){[{d}_{\theta}log(P(\theta ,\xi ))]}^{\otimes 2}(X,Y)d\xi $$

**Some Criticisms.**

- (i)
- $(i):$ $P(\theta ,t)$ is smooth,
- (ii)
- $P(0,t)=P(2\pi ,t)\phantom{\rule{1.em}{0ex}}\forall t\in \mathbb{R}$,
- (iii)
- the $\frac{d}{d\theta}$ commutes with ${\int}_{\mathbb{R}}$,
- (iv)
- $P(\theta ,t)\le 1\phantom{\rule{1.em}{0ex}}\forall (\theta ,t)\in {S}^{1}\times \mathbb{R}$,
- (v)
- if $0<\theta ,{\theta}^{*}<2\pi $ then ${P}_{\theta}={P}_{\theta}^{*}$ if and only if $\theta ={\theta}^{*}$,
- (vi)
- ${\int}_{-\infty}^{+\infty}P(\theta ,t)dt=1.$

**A Digression.**

**Another Construction.**

- (1)
- If $\theta \ne {\theta}^{*}$ there exists ${t}^{**}\in {\mathbb{R}}^{m}$ such that$$P(\theta ,{t}^{**})\ne P({\theta}^{*},{t}^{**}),$$
- (2)
- $P(\theta ,t)\le 1\forall (\theta ,t)\in T{\mathbb{T}}^{m},$
- (3)
- ${\int}_{{\mathbb{R}}^{m}}P(\theta ,t)dt=1.$

#### 8.1. The Preliminaries

**Definition**

**44.**

- (1)
- for every vector field X the real number $Q(x,\xi )[X,X]$ is non negative, furthermore $\forall v\in {T}_{x}M\setminus \left\{0\right\}\exists \xi \in \mathsf{\Xi}$ such that$$Q(x,\xi )(v,v)>0,$$
- (2)
- for every $\xi \in \mathsf{\Xi}$, the random KV cochain$$(X,Y)\to {Q}_{\xi}(X,Y)(x)$$$${Q}_{\xi}(X,Y)(x)=Q(x,\xi )({X}_{x},{Y}_{x})$$

**Warning.**

**Definition**

**45.**

#### 8.2. The Category $\mathcal{FB}(\mathsf{\Gamma},\mathsf{\Xi})$

#### 8.2.1. The Objects of $\mathcal{FB}(\mathsf{\Gamma},\mathsf{\Xi})$

**Definition**

**46.**

- (1)
- M is a connected m-dimensional smooth manifold. The map$$\pi :\mathcal{E}\to M$$
- (2)
- The pair $(M,D)$ is an m-dimensional locally flat manifold.
- (3)
- There is a group action$$\mathsf{\Gamma}\times [\mathcal{E}\times M]\times {\mathbb{R}}^{m}\ni (\gamma ,[e,x,\theta ])\to [[(\gamma \xb7e),\gamma \xb7x],\tilde{\gamma}\xb7\theta ]\in [\mathcal{E}\times M]\times {\mathbb{R}}^{m}.$$That action is subject to the compatibility requirement$$\pi (\gamma \xb7e)=\gamma \xb7\pi (e)\phantom{\rule{1.em}{0ex}}\forall e\in \mathcal{E}.$$
- (4)
- Every point $x\in M$ has an open neighborhood U which is the domain of a local fiber chart$${\mathsf{\Phi}}_{U}\times {\varphi}_{U}:[{\mathcal{E}}_{U}\times U]\ni ({e}_{x},x)\to [{\mathsf{\Phi}}_{U}({e}_{x}),{\varphi}_{U}(x)]\in [{\mathbb{R}}^{m}\times \mathsf{\Xi}]\times {\mathbb{R}}^{m}.$$The local charts are subject to the following compatibility relation
- $(U,{\varphi}_{U})$ is an affine local chart of the locally flat manifold $(M,D),$
- ${\varphi}_{U}(\pi (e))={p}_{1}({\mathsf{\Phi}}_{U}(e)).$

- (5)
- We set$${\mathsf{\Phi}}_{U}(e)=({\theta}_{U}(e),{\xi}_{U}(e))\in {\mathbb{R}}^{m}\times \mathsf{\Xi}.$$Let $(U,\mathsf{\Phi}\times \varphi )$ and $({U}^{*},{\mathsf{\Phi}}^{*}\times {\varphi}^{*})$ be two local charts with$$U\cap {U}^{*}\ne \varnothing ,$$$$[{\gamma}_{U{U}^{*}}\xb7\mathsf{\Phi}](e)={\mathsf{\Phi}}^{*}(e)\phantom{\rule{1.em}{0ex}}\forall e\in {\mathcal{E}}_{U\cap {U}^{*}}.$$

**Comments.**

#### 8.2.2. The Morphisms of $\mathcal{FB}(\mathsf{\Gamma},\mathsf{\Xi})$

**Definition**

**47.**

#### 8.3. The Category $\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})$

#### 8.3.1. The Objects of $\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})$

**Definition**

**48.**

**A**

**Comment.**

**Definition**

**49.**

#### 8.3.2. The Global Probability Density of a Statistical Model

**Definition**

**50.**

**Comments.**

- (i)
- We take into account the global probability density p. Then an object of the category $\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})$ is denoted by$$[\mathcal{E},\pi ,M,D,p].$$
- (ii)
- The function p is Γ-equivariant. THIS IS THE GEOMETRY in the sense of Erlangen program.
- (iii)
- We have not used any argument depending the dimension of manifolds.

#### 8.3.3. The Morphisms of $\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})$

**Definition**

**51.**

**A**

**Comment.**

**Definition**

**52.**

#### 8.3.4. Two Alternative Definitions

**Definition**

**53.**

**Remark**

**4.**

**Definition**

**54.**

- (1)
- A statistical model is a locally trivial fiber bundle over a locally flat manifold$$\pi :\mathcal{E}\to M.$$The fibers of π are probability spaces.
- (2)
- The functor$$[\mathcal{E},p]\to [M,D]$$

**Definition**

**55.**

#### 8.3.5. Fisher Information in $\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})$

**Definition**

**56.**

- (1)
- g is positive semi-definite,
- (2)
- g is an invariant of the Γ-geometry in $[\mathcal{E},\pi ,M,D,p]$.

#### 8.4. Exponential Models

**Definition**

**57.**

- (1)
- The base manifold M supports a locally flat structure $(M,\nabla )$ and a real valued function $\psi \in {C}^{\infty}(M)$.
- (2)
- The total space $\mathcal{E}$ supports a real valued random function a.
- (3)
- The triple $[a,\nabla ,\psi ]$ is subject to the following requirement
- (4)
- ${\nabla}^{2}{\int}_{F}(a)=O,$
- (5)
- $p(e)=exp[a(e)-\psi (\pi (e))].$

**Remark**

**5.**

**Reminder.**

#### 8.4.1. The Entropy Flow

**Definition**

**58.**

**Theorem**

**18.**

#### 8.4.2. The Fisher Information as the Hessian of the Local Entropy Flow

**Theorem**

**19.**

#### 8.4.3. The Amari-Chentsov Connections in $\mathcal{GM}(\mathsf{\Xi},\mathsf{\Omega})$

#### 8.4.4. The Homological Nature of the Probability Density

#### 8.4.5. Another Homological Nature of Entropy

**Definition**

**59.**

**A**

**Comment.**

## 9. The Moduli Space of the Statistical Models

#### The Hessian Functor

- (1)
- The category $\mathcal{L}$ whose objects are gauge structures $(M,\nabla )$,
- (2)
- The category $\mathcal{GM}$ whose objects are statistical models for measurable sets,
- (3)
- the category $\mathcal{BF}$ whose objects are manifolds equipped bilinear forms,
- (4)
- the category $\mathcal{F}(\mathcal{L}\mathcal{C},\mathcal{BF})$ whose objects are functors$$\mathcal{GM}\to \mathcal{BF}.$$

**Definition**

**60.**

**Reminder.**

**Lemma**

**6.**

**Proof.**

**A Comment.**

**Lemma**

**7.**

- (1)
- ${q}_{{\mathbb{M}}_{1}}={q}_{{\mathbb{M}}_{2}}$,
- (2)
- ${p}_{1}={p}_{2}.$

**Proof.**

- (1)
- Let ${\psi}_{*}$ be the differential of ψ. For $\nabla \in \mathcal{LC}({M}_{1})$ the image ${\psi}_{*}(\nabla )\in \mathcal{LC}({M}_{2})$ is defined by$${[{\psi}_{*}(\nabla )]}_{{X}^{*}}{Y}^{*}={\psi}_{*}[{\nabla}_{{\psi}_{*}^{-1}({X}^{*})}{\psi}_{*}^{-1}({Y}^{*})]$$
- (2)
- It is clear that the datum $[{\mathcal{E}}_{1},\pi ,{M}_{1},{D}_{1},{p}_{2}\circ \mathsf{\Psi}]$ is an object of the category $\mathcal{GM}(X,\mathsf{\Omega})$. Then for vector fields $X,Y$ in ${M}_{1}$ we calculate (at $X,Y$) the right hand member of the following equality$$[{q}_{{p}_{2}\circ \mathsf{\Psi}}(\nabla )]={\nabla}^{2}[log({p}_{2}\circ \mathsf{\Psi})].$$Direct calculations yield$${\nabla}^{2}[log({p}_{2}\circ \mathsf{\Psi})](X,Y)=X\xb7[Y\xb7log({p}_{2}\circ \mathsf{\Psi})]-{\nabla}_{X}Y\xb7log({p}_{2}\circ \mathsf{\Psi})$$$$=X\xb7[Y\xb7log({p}_{2})\circ \mathsf{\Psi}]-{\nabla}_{X}Y\xb7[log({p}_{2})\circ \mathsf{\Psi}]$$$$={\psi}_{*}(X)\xb7[{\psi}_{*}(Y)\xb7log({p}_{2})]-{\psi}_{*}({\nabla}_{X}Y)\xb7log({p}_{2})$$$$=[{\psi}_{*}{(\nabla )}^{2}log({p}_{2})]({\psi}_{*}(X),{\psi}_{*}(Y)).$$Thus for all $\nabla \in \mathcal{LC}({M}_{1})$ we have$${q}_{[{p}_{2}\circ \mathsf{\Psi}]}(\nabla )={q}_{{p}_{2}}({\psi}_{*}(\nabla )).$$

**Lemma**

**8.**

**Theorem**

**20.**

**Demonstration.**

**Reminder.**

- (i)
- Objects of $\mathcal{GM}(\mathsf{\Gamma},\mathsf{\Xi})$ are quintuplets$$\mathbb{M}=[\mathcal{E},\pi ,M,D,p].$$They are called statistical models for the measurable set $(\mathsf{\Xi},\mathsf{\Omega}).$
- (ii)
- Objects of $\mathcal{FB}(\mathcal{MSE})$ are functors$$[\mathcal{E},p]\to [M,D].$$They are called $\mathcal{MSE}$-fibrations.

**Theorem**

**21.**

## 10. The Homological Statistical Models

**Definition**

**61.**

**Comments.**

**Definition**

**62.**

#### 10.1. The Cohomology Mapping of $HSM(\mathsf{\Xi},\mathsf{\Omega})$

**Definition**

**63.**

#### 10.2. An Interpretation of the Equivariant Class [Q]

**Definition**

**64.**

**Proposition**

**11.**

#### 10.3. Local Vanishing Theorems in the Category $\mathcal{HSM}(\mathsf{\Xi},\mathsf{\Omega})$

**Reminder.**

**Definition**

**65.**

- (1)
- A random function f has the property ${p}^{*}-EXP$ if$$exp(f(x,\xi ))\le {\int}_{\mathsf{\Xi}}exp(f(x,\xi ))d{p}^{*}(\xi )\phantom{\rule{1.em}{0ex}}\forall x\in {\mathbb{R}}^{m}.$$
- (2)
- A random closed differential 1-form θ has the property ${p}^{*}-EXP$ if every $x\in {\mathbb{R}}^{m}$ has an open neighbourhood U satisfying the following conditions, $U\times \mathsf{\Xi}$ support a random function f subject to two requirements:
- $\theta =df,$
- f has the property ${p}^{*}-Exp.$

- (3)
- An exact homological statistical model $[\mathcal{E},\pi ,M,D,Q]$ has property ${p}^{*}-EXP$ if there exists a random differential 1-form θ satisfying the following conditions
- θ has the property ${p}^{*}-EXP$,
- $Q={\delta}_{KV}\theta $.

**Theorem**

**22.**

- (1)
- $[\mathcal{E},\pi ,M,D,Q]$ is locally exact.
- (2)
- If the $[\mathcal{E},\pi ,M,D,Q]$ has the property ${p}^{*}-EXP$ then $[\mathcal{E},\pi ,M,D,Q]$ is locally isomorphic to a classical statistical model $(\mathsf{\Theta},P)$ as in [18].

**The Sketch of Proof of**

**(1).**

**The proof of**

**(2).**