Almost Hermitian structures on tangent bundles

TL;DR

This paper investigates conditions for tangent bundles to admit almost Hermitian, Kähler, and Einstein structures, providing specific criteria and examples of Kähler-Einstein structures.

Contribution

It establishes new conditions for tangent bundles to possess various Hermitian structures and presents explicit examples of Kähler-Einstein metrics.

Findings

Conditions for almost Kähler, Kähler, and Einstein structures on tangent bundles

Criteria for tangent bundles to admit Kähler-Einstein structures

Two explicit examples of Kähler-Einstein tangent bundle structures

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 K\"ahler structures, K\"ahler structures and Einstein metrics, respectively. Moreover, we give two examples of K\"ahler-Einstein structures on $TM$.