Transformations and Coupling Relations for Affine Connections

The statistical structure on a manifold \(\mathfrak {M}\) is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection \(\nabla \) on the \(T\mathfrak {M}\), such that \(\nabla g\) is totally symmetric, forming, by definition, a “Codazzi pair” \(\{\nabla , g\}\). In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on \(T\mathfrak {M}\)), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of \(\nabla \) with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.

[1]  U. Simon Transformation Techniques for Partial Differential Equations on Projectively Flat Manifolds , 1995 .

[2]  Keiko Uohashi,et al.  On \alpha-conformal equivalence of statistical submanifolds , 2002 .

[3]  U. Simon,et al.  Codazzi-Equivalent Affine Connections , 2009 .

[4]  Takashi Kurose ON THE DIVERGENCES OF 1-CONFORMALLY FLAT STATISTICAL MANIFOLDS , 1994 .

[5]  U. Simon,et al.  Codazzi Tensors and the Topology of Surfaces , 1998 .

[6]  Su Buqing,et al.  Affine differential geometry , 1983 .

[7]  Geometric methods for solving Codazzi and Monge-Ampère equations , 1994 .

[8]  On dual-projectively flat affine connections , 1995 .

[9]  Jun Zhang,et al.  Nonparametric Information Geometry: From Divergence Function to Referential-Representational Biduality on Statistical Manifolds , 2013, Entropy.

[10]  野水 克己,et al.  Affine differential geometry : geometry of affine immersions , 1994 .

[11]  Udo Simon,et al.  Chapter 9 - Affine Differential Geometry , 2000 .

[12]  Michel Nguiffo Boyom,et al.  KV Cohomology in Information Geometry , 2013 .

[13]  H. Matsuzoe On realization of conformally-projectively flat statistical manifolds and the divergences , 1998 .

[14]  U. Simon,et al.  Generating higher order Codazzi tensors by functions , 1999 .

[15]  Hiroshi Matsuzoe,et al.  GENERALIZATIONS OF CONJUGATE CONNECTIONS , 2009 .

[16]  K. Nomizu Affine Differential Geometry , 1994 .

[17]  Jun Zhang,et al.  Transformations and coupling relations for affine connections , 2016 .

[18]  Udo Simon,et al.  Introduction to the affine differential geometry of hypersurfaces , 1991 .

[19]  Calyampudi R. Rao,et al.  Chapter 4: Statistical Manifolds , 1987 .

[20]  M. C. Chaki ON STATISTICAL MANIFOLDS , 2000 .

[21]  Hirohiko Shima,et al.  Geometry of Hessian Structures , 2013, GSI.

[22]  Takashi Kurose Conformal-Projective Geometry of Statistical Manifolds , 2002 .

[23]  Jun Zhang,et al.  Divergence Function, Duality, and Convex Analysis , 2004, Neural Computation.