Interaction of Codazzi Couplings with (Para-)Kähler Geometry

Abstract

We study Codazzi couplings of an affine connection $$\nabla $$ with a pseudo-Riemannian metric g, a nondegenerate 2-form $$\omega $$ , and a tangent bundle isomorphism L on smooth manifolds, as an extension of their parallelism under $$\nabla $$ . In the case that L is an almost complex or an almost para-complex structure and $$(g, \omega , L)$$ form a compatible triple, we show that Codazzi coupling of a torsion-free $$\nabla $$ with any two of the three leads to its coupling with the remainder, which further gives rise to a (para-)Kahler structure on the manifold. This is what we call a Codazzi-(para-)Kahler structure; it is a natural generalization of special (para-)Kahler geometry, without requiring $$\nabla $$ to be flat. In addition, we also prove a general result that g-conjugate, $$\omega $$ -conjugate, and L-gauge transformations of $$\nabla $$ , along with identity, form an involutive Abelian group. Hence a Codazzi-(para-)Kahler manifold admits a pair of torsion-free connections compatible with the $$(g, \omega , L)$$ . Our results imply that any statistical manifold may admit a (para-)Kahler structure as long as one can find an L that is compatible to g and Codazzi coupled with $$\nabla $$ .