The 28th Statistical Machine Learning Seminar (2016.2.4)
Time: February 4 (Thu), 2016. 16:00-17:00
Place: Seminar Room 2
We study the interaction of a torsion-free affine connection nabla with three objects on a manifold M: a pseudo-Riemannian metric g, a skew-symmetric symplectic form omega, and a tangent-bundle isomorphism L, two special cases being L= J (almost complex structure, J^2 = -id) and L=K (almost para-complex structure, K^2 = id). It is well known that Codazzi coupling of nabla with g gives rise to the statistical structure. It is shown here that Codazzi coupling of nabla with any two of the compatible triple (g, omega, L) will lead to its coupling with the remainder, which further gives rise to a (para-)Kahler structure on the manifold. We call this Codazzi-(para-)Kahler structure, which is a natural generalization of special (para-)Kahler geometry, without requiring nabla to be flat. In fact, we prove a general result that g-conjugate, omega-conjugate, and L-gauge transformations of nabla, along with the identity transform, form a 4-element Klein group. This leads a Codazzi-(para-)Kahler manifold to admit a pair of torsion-free connections compatible with the (g, omega, L). Finally, we give an example of Codazzi-(para-)K\"ahler manifold, namely, the alpha-Hessian structure studied in information geometry. (Joint work with Teng Fei, MIT).