Almost Hermitian structures on tangent bundles

TL;DR

This work characterizes when the natural almost Hermitian structure $(J^D,\tilde{g}^D)$ on the tangent bundle $TM$, induced by a Riemannian metric $g$ and an affine connection $D$ on $M$, yields almost Kähler, Kähler, or Einstein geometries. It establishes that $TM$ is almost Kähler iff the dual connection $D^*$ is torsion-free, and that $TM$ is Kähler iff $(M,g,D)$ is a Hessian manifold (i.e., $D$ and $D^*$ are flat); furthermore, if the Sasaki metric is Einstein then $R^D=0$. The authors provide explicit constructions of Kähler–Einstein structures on $TM$, including a 1-parameter family of almost Kähler structures and Hessian-models from the manifold of multivariate normal distributions, thereby linking tangent-bundle complex geometry with Hessian and information-theoretic structures. These results illuminate how base-manifold geometric data lift to $TM$ and offer non-compact, non-Kähler Einstein examples with potential implications for the Goldberg conjecture in non-compact settings.

Abstract

In this article, we consider the almost Hermitian structure on $TM$ induced by a pair of a metric and an affine connection on $M$. We find the conditions under which $TM$ admits almost Kähler structures, Kähler structures and Einstein metrics, respectively. Moreover, we give two examples of Kähler-Einstein structures on $TM$.

Theorems & Definitions (18)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Lemma 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Theorem 3.1
• proof
• Theorem 3.2