(Para-)Holomorphic Connections for Information Geometry

Abstract

On a statistical manifold $(M,g,\nabla )$, the Riemannian metric g is coupled to an (torsion-free) affine connection $\nabla$, such that $\nabla g$ is totally symmetric; $\{\nabla ,g\}$ is said to form “Codazzi coupling”. This leads $\nabla ^{*}$, the g -conjugate of $\nabla$, to have same torsion as that of $\nabla$. In this paper, we investigate how statistical structure interacts with L in an almost Hermitian and almost para-Hermitian manifold (M, g, L), where L denotes, respectively, an almost complex structure J with $J^{2}=- \mathrm{id}$ or an almost para-complex structure K with $K^{2}=\mathrm{id}$. Starting with $\nabla ^{L}$, the L -conjugate of $\nabla$, we investigate the interaction of (generally torsion-admitting) $\nabla$ with L, and derive a necessary and sufficient condition (called “Torsion Balancing” condition) for L to be integrable, hence making (M, g, L) (para-)Hermitian, and for $\nabla$ to be (para-)holomorphic. We further derive that $\nabla ^L$ is (para-)holomorphic if and only if $\nabla$ is, and that $\nabla ^*$ is (para-)holomorphic if and only if $\nabla$ is (para-)holomorphic and Codazzi coupled to g. Our investigations provide concise conditions to extend statistical manifolds to (para-)Hermitian manifolds.