Characterizations and integral formulae for generalized quasi-Einstein metrics

TL;DR

This paper explores the structure of generalized quasi-Einstein metrics, showing that Einstein manifolds in this class are space forms with potential functions, and derives integral formulas leading to rigidity results.

Contribution

It provides new structural equations for generalized quasi-Einstein metrics and establishes conditions under which Einstein manifolds are space forms, along with integral formulas for compact cases.

Findings

Einstein generalized quasi-Einstein manifolds are space forms with potential functions.

Derived integral formulas enable rigidity results for compact manifolds.

Structural equations characterize the geometry of these metrics.

Abstract

The aim of this paper is to present some structural equations for generalized quasi-Einstein metrics which was defined recently by Catino in [12]. In addition, supposing that the Riemannian manifold is Einstein we shall show that it is a space form with a well defined potential function f. Finally, we shall derive some integral formulae for such a class of compact manifolds which permit to obtain some rigidity results.