Numerical properties of Koszul connections

5 1. Prologue 5 2. INTRODUCTION 8 2.1. The general concerns 8 2.2. The geometry of finite dimensional Cartan-Lie groups and abstract Lie groups 9 2.3. The information geometry of complex systems 9 2.4. The gauge group 9 2.5. The overview of the main results: Solutions to some problems EX(S) 10 2.6. Geometry of Lie groups 13 2.7. A comment on Figues 17 3. BASIC NOTIONS. 18 3.1. Cartan-Lie groups and abstract Lie groups 19 4. THE DIFFERENTIAL EQUATIONS 20 4.1. Notation and definitions 21 4.2. The Kuranishi-Spencer formalism 22 4.3. Some categories of geometric structures 24 5. THE FUNDAMENTAL EQUATIONS 27 5.1. Notation 27 5.2. The fundamental gauge equation and its link with foliations 27 5.3. Two numerical fundamental equations 30 5.4. A sketch of the global analysis in symmetric gauge structures 31 6. THE FIRST FUNDAMENTAL EQUATION AND NEW INVARIANTS OF THE LOCALLY FLAT GEOMETRY 34 6.1. Inductive-projective systems of bi-invariant affine Lie groups 35 6.2. Initial objects and Finala objects: the structural theorem of the locally flat geometry 36 7. THE FIRST FUNDAMENATAL EQUATION OF A GAUGE STRUCTURE AND THE FOLIATIONS, continued 37 7.1. The special gauge structures 37 7.2. The topological nature of the semi-inductive systems of bi-invariant affine Lie groups 38 8. NEW NUMERICAL INVARIANTS OF GAUGE STRUCTURES 38 8.1. Numerical invariants of the first fundamental equation 39 8.2. New insights 41 8.3. Some relative gauge invariants 41 8.4. The notion of Hessian defect 41 8.5. Te functor of Amari 42 9. THE FIRST FUNDAMENTAL EQUATION AND THE INFORMATION GEOMETRY 43 9.1. The Fisher information 44 9.2. The connections of Chentsov 44 9.3. The moduli space of Cartan-Lie groups of simply connected locally flat manifolds 45 10. THE FIRST FUNDAMENTAL EQUATION AND SOME OPEN PROBLEMS. SOME NEW THEOREMS EXTH 46 10.1. The first fundamental equation and the locally flat geometry 47 NUMERICAL PROPERTIES OF KOSZUL CONNECTIONS 3 10.2. The first fundamental equation and the Hessian geoemtry in Riemanian manifolds: answer an old question of S-I Amari and A.K. Guts 47 10.3. The first fundamental equation and the Hesian geometry in locally flat manifolds 47 10.4. The first fundamenatal equation and the information geometry 48 10.5. The hyperbolicity after Kaup, Koszul and Vey 49 11. THE MODULI SPACES OF LOCALLY FLAT STRUCTURES, CANONICAL LINEAR REPRESENTATIONS OF FUNDAMENTAL GROUPS 50 11.1. A long digression 50 11.2. The first fundamental equation and the canonical linear representations of the fundamental groups 50 11.3. The moduli space of geometrically complete locally flat manifolds 52 11.4. An algebraic model for complete locally flat geometry 55 12. A SKETCH OF THE KV ALGEBRAIC TOPOLOGY OF LOCALLY FLAT MANIFOLDS 56 12.1. The vector valued cohomology 57 12.2. The scalar valued cohomology 57 13. NAIVE HOMOLOGICAL ALGEBRA, continued 58 13.1. Some remarkable homological functors in gauge structures 58 14. NEW (co)HOMOLOGICAL FUNCTORS, continued 59 14.1. A sketch of the algebraic topology 59 14.2. More sophisticated 60 14.3. The first fundamental equation and deformations of the Poisson bracket of vectors fields 60 15. INTRINSIC MEANINGS OF THE FIRST FUNDAMENTAL EQUATION 61 15.1. Definition-Notation: the special dynamical systems 62 15.2. The existence of special dynamical systems 63 15.3. The differential topological nature of the first fundamental equation, continued 64 15.4. The fundamental gauge equation and regular geodesic foliations 65 16. THE FIRST FUNDAMENTAL EQUATION AND OPEN PROBLEMS : DEMONSTRATIONS OF THEOREMS EXTH 66 16.1. DEMONSTRATION OF THEOREM 10.1. 66 16.2. DEMONSTRATION OF THEOREM 10.2 68 16.3. DEMONSTRATION OF THEOREM 10.3. 69 16.4. DEMONSTRATION OF THEOREM 10.4. 70 16.5. DEMONSTRATION OF THEOREM 10.5. 70 16.6. DEMONSTRATION OF THEOREM 10.6 71 17. THE GEOMETRIC COMPLETENESS OF LOCALLY FLAT MANIFOLDS 72 17.1. The completeness of affine Lie groups and affine representations 73 18. THE INTRINSIC NATURE OF THE FUNCTION rb, continued. 75 18.1. Cominatorial distancelike 75 18.2. The fundamental equation FE∗(∇) and characteristic obstructions 76 19. THE FUNDAMENTAL EQUATION FE∗(∇) AND THE RIEMANNIAON GEOMETRY 76 19.1. New geometric invariants of Riemannian manifolds 77 19.2. Deformation of metric connections 78 4 MICHEL NGUIFFO BOYOM 19.3. The differential topology of geodesically complete special Riemannian manifolds, continued 78 19.4. Special complete positive Riemannian manifolds, continued 79 19.5. GFHSDS-foliation of statistical models 80 19.6. Special statistical manifolds 82 20. THE FUNDAMENTAL EQUATION FE∗(∇) AND HOMOGENEOUS KAEHLERIAN GEOMETRY 82 20.1. The fundamental equation FE∗(∇) and the problem EXF (S) in homogeneous Kaehlerian manifolds, canonical affine representations 85 20.2. The first fundamental equation FE∗(∇) and flat foliations in geodesically complete Kaehlerian Geometry 86 20.3. Additional numerical invariants of gauge structures 86 21. LINKS OF THE FIRST FUNDAMENTAL EQUATION FE∗(∇) WITH Cartan-Lie GROUPS AND WITH ABSTRACT Lie GROUPS 87 21.1. The fundamental equations FE(∇∇∗) and FE∗(∇) in abstract Lie groups 87 21.2. The problem DL(S : br(G)) 91 22. THE FUNDAMENTAL EQUATION FE(∇∇∗) AND THE SYMPLECTIC GEOMETRY: THE SYMPLECTIC GAPS 92 22.1. The symplectic gap of differentiable manifolds 93 22.2. PROBLEM EX (S : Symp) 93 22.3. The problem DL(S) in the symplectic geometry 95 23. THE FUNDAMENTAL EQUATION FE(∇∇∗) AND THE PROBLEM EXF (S) CONTINUED 96 24. SOME HIGHLIGHTING CONCLUSIONS 99 24.1. The pair (D∇∇ ∗ , D∇) and the pair of functions (rb, sb) 99 24.2. The fundamental equation FE∗(∇) in finite dimensional Lie groups 99 24.3. Hessian defects in abstract Lie groups 99 24.4. The fundamental equation FE∗(∇∇∗) and bi-invariant Riemannian geometry in finite dimensional Lie groups 100 24.5. The fundamental equation FE∗(∇), the homogeneous Kaehlerian geometry and some canonical affine representations 100 24.6. The fundamental equation FE(∇∇∗) and the symplectic geometry 100 24.7. Canonical representations of fundamental groups 100 24.8. In final 101 25. NEW PERSPECTIVES OF APPLICATIONS 101 25.1. New invariants of statistical models 101 25.2. Complex systems and digital science 102 25.3. Needs of geometric structures in complex systems: new challenges 102 25.4. BigData-Mathematic-StatisticalModels-HighPerformanceComputing 102 25.5. The fundamental equations << FE(∇∇∗), FE∗(∇) >> and Complex systems 103 26. APPENDIX. 104 26.1. THE CATEGORY FB(Γ,Ξ) 104 26.2. THE CATEGORY GM(Ξ,Ω). 107 References 112