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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
- 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
- 4 The KV Topology of Locally Flat Manifolds 26
- 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
- 6 The Similarity Structure and the Hyperbolicity 58
- 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
- 8.3 The Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
- 8.3.1 The Objects of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
- 8.3.2 The Global Probability Density of a Statistical Model . . . . . . . . . . . . . . . . 70
- 8.3.3 The Morphisms of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
- 8.3.4 Two Alternative Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
- 8.3.5 Fisher Information in . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . 75
- 8.4.4 The Homological Nature of the Probability Density . . . . . . . . . . . . . . . . . 76
- 8.4.5 Another Homological Nature of Entropy . . . . . . . . . . . . . . . . . . . . . . . 77
- 9 The Moduli Space of the Statistical Models 78
- 10 The Homological Statistical Models 83
- 11 The Homological Statistical Models and the Geometry of Koszul 88
- 12 Examples 88
- 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)
- is a first order differential operator. It is defined in the vector bundle . Its values belong to the vector bundle .
- (A.2)
- and are 2nd order differential operators. They are defined in the vector bundle . Their values belong to the vector bundle . Let X be a section of and let ψ be a section of . The differential operators just mentioned are defined byPart A of this paper is partially devoted to the global analysis of the differential equationThe solutions to 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 equationsIn the Appendix A to this paper we overview the role played by the solutions to 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].
1.3. The content of the Paper
2. Algebroids, Moduls of Algebroids, Anomaly Functions
2.1. The Algebroids and Modules
2.2. Anomaly Functions of Algebroids and of Modules
- (1)
- The associator anomaly function of is defined by
- (2)
- The Koszul-Vinberg anomaly function of is defined by
- (3)
- The Jacobi anomaly functions of are defined by
- (1)
- The associator anomaly function of a left module is defined as
- (2)
- The KV anomaly functions of a two sided module are defined as
- (1)
- The Leibniz anomaly function of an anchored algebroid is defined by
- (2)
- The Leibniz anomaly function of the -module is defined by
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 .
3.1.2. The Total Cochain Complex
- (1):
- (2):
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
3.2.4. The KV Chain Complex
3.2.5. The V-Valued KV Homology
3.2.6. Two Cochain Complexes
3.2.7. Residual Cohomology
- (1)
- The vector subspace of residual KV cocycles of degree q is denoted by .
- (2)
- The vector subspace of residual coboundaries of degree q is defined by The residual KV cohomology space of degree q is the quotient vector space.
- (3)
- (4)
- By replacing the KV complex by the total KV complex one defines the vector space of residual total cocycles and the space of residual total coboundaries . Therefore we get the residual total KV cohomology space
- :
- We replace the KV by . Then we obtain the exact sequences
- :
- The KV cohomology difers from the total cohomology. Loosely speaking their intersecttion is the equivariant cohomology 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
- (1)
- V is an (abstract) two sided module of an (abstract) algebra .
- (2)
- and are fixed anomaly functions of and of V respectively.
- (3)
- stands for the -graded vector space
3.3.2. Challenge for KV Algebras
3.3.3. The KV Cohomology
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
- (1.1)
- (1.2)
- A symplectic foliation is a ( de Rham) closed differential 2-form ω which satisfies
- (2.1)
- ,
- (2.2)
- .
4.2. The General Linearization Problem of Webs
4.3. The Total KV Cohomology and the Differential Topology Continued
- (1)
- is transversally Riemannian if there exists a such that
- (2)
- is transversally symplectic if there exists a (de Rham) closed differential 2-form ω such that
- (1)
- Suppose thatTherefore, is a D-geodesic Riemannian foliation.
- (2)
- Suppose thatThen is a Riemannian manifold the Levi-Civita connection of which is D. Therefore, the proposition holds. ☐
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 .
- (1)
- The arrow
- (2)
- The arrow
- (3)
- If a 2-cocycle g is a symmetric form then 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
- (1)
- A dual pair is called a flat pair if the connection D is flat, viz
- (2)
- A flat pair is called a dually flat pair if both and are locally flat manifolds.
- (1)
- Both D and are torsion free.
- (2)
- D is torsion free and A is symmetric, viz
- (3)
- is torsion free and the metric tensor g a is KV cocycle of the KV algebra of the locally flat manifold .
- (4)
- The flat pair is a dually flat pair.
- (1)
- is a dually flat pair.
- (2)
- (1)
- .
- (2)
- is a dually flat pair.
- (1)
- (2)
- .
- (1)
- (2)
- (1)
- ,
- (2)
5.1.1. Statistcal Reductions
5.1.2. A Uselful Complex
5.1.3. The Homological Nature of Gauge Homomorphisms
- (1)
- (2)