Isotropic quasi-Einstein manifolds

Abstract

We investigate the local structure of four-dimensional Lorentzian quasi-Einstein manifolds under conditions on the Weyl tensor. We show that if the Weyl tensor is harmonic and the potential function preserves this harmonicity then, in the isotropic case, the manifold is necessarily a $pp$-wave. Using the quasi-Einstein equation, further conclusions are obtained for $pp$-waves.

References in corpus (14)

• Generalized quasi-Einstein manifolds with harmonic Weyl tensor
• A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time
• Singularity theorems and the Lorentzian splitting theorem for the Bakry-Emery-Ricci tensor
• Gravity and Scalar Fields
• Ricci Solitons and Einstein-Scalar Field Theory
• Some Applications of Ricci Flow in Physics
• Ambient metrics for $n$-dimensional $pp$-waves
• Cosmological singularities in Bakry-Émery spacetimes
• The Weyl Tensor of Gradient Ricci Solitons
• Cosmological singularity theorems and splitting theorems for N-Bakry-Emery spacetimes
• Scalar-tensor gravitation and the Bakry-Emery-Ricci tensor
• New Restrictions on the Topology of Extreme Black Holes
• Bakry-Émery black holes
• Conformal pure radiation with parallel rays

Cited by in corpus (1)

• Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case