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-)Kähler structure on the manifold. This is what we call a Codazzi-(para-)Kähler structure; it is a natural generalization of special (para-)Kähler 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-)Kähler 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-)Kähler structure as long as one can find an L that is compatible to g and Codazzi coupled with $\nabla$.