Remarks on $\eta$-Einstein unit tangent bundles

TL;DR

This paper investigates the conditions under which the unit tangent bundle of a 4-dimensional Einstein manifold, with a standard contact metric structure, is $\eta$-Einstein, revealing a characterization related to constant sectional curvature.

Contribution

It establishes a precise criterion linking the $\eta$-Einstein property of the unit tangent bundle to the base manifold's constant sectional curvature values.

Findings

Unit tangent bundle of 4D Einstein manifold is $\eta$-Einstein iff the base has constant curvature 1 or 2.

Characterization of $\eta$-Einstein condition in terms of base manifold's curvature.

Provides geometric conditions for $\eta$-Einstein structures on tangent bundles.

Abstract

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the $\eta$-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is $\eta$-Einstein manifold if and only if base manifold is the space of constant sectional curvature 1 or 2.