Characterizations Of Weakly Conformally Flat And Quasi Einstein Manifolds

TL;DR

This paper characterizes and classifies weakly conformally flat and quasi-Einstein manifolds, including warped products, contact manifolds, and certain semi-Riemannian spaces, revealing their geometric structures and properties.

Contribution

It provides new characterizations and classifications of weakly conformally flat and quasi-Einstein manifolds, especially in contact and semi-Riemannian contexts.

Findings

Warped product of a line and fiber is weakly conformally flat and quasi Einstein iff the fiber is Einstein.

Contact and K-contact manifolds satisfying these conditions are characterized and classified.

Semi-Riemannian manifolds with harmonic Weyl tensor and specific vector fields are classified by Petrov types and Bach tensor.

Abstract

First, we show that a warped product of a line and a fiber manifold is weakly conformally flat and quasi Einstein if and only if the fiber is Einstein. Next, we characterize and classify contact (in particular, $K$-contact) Riemannian manifold satisfying weakly (and doubly weakly) conformally flat and quasi-Einstein ($\eta$-Einstein) conditions. Finally, we provide local classification and characterization of a semi-Riemannian (including the 4-dimensional spacetime) with harmonic Weyl tensor and a non-homothetic conformal (including closed) vector field, in terms of Petrov types and Bach tensor.