Compact Einstein warped product spaces with nonpositive scalar curvature

Abstract

Abstract. We study Einstein warped product spaces. As a result, we prove the following: if M is an Einstein warped product space with nonpositive scalar curvature and compact base, then M is simply a Riemannian product space.

On Bach flat warped product Einstein manifolds

“…Theorem 1.1 is a direct consequence of the following more general result combined with some early results in [He et al 2012].…”

“…In this section, we set up our notation and recall some well-known facts on warped product Einstein manifolds. For more detail, see for example [He et al 2012] and references therein.…”

“…Next we recall some properties of warped product Einstein manifolds; the proofs can be found in [He et al 2012]. The function ρ is defined by…”

“…If m is positive, then from [Kim and Kim 2003] and the comparison theorem of m-Bakry-Emery tensors in [Qian 1997, Theorem 5], a (λ, n+m)-Einstein manifold is compact if and only if λ > 0. Using [He et al 2012, Theorem 1.5], the global classification of warped product Einstein manifolds with harmonic Weyl tensor, and W (∇ f, •, •, ∇ f ) = 0, Theorem 1.2 has the following corollary: Corollary 1.7.…”

A non-existence result for compact Einstein warped products

“…The question of existence of compact Einstein manifolds has also been addressed recently in [7], where it is shown necessarily for these manifolds to have positive scalar curvature.…”