Warped product Einstein metrics over spaces with constant scalar curvature

Chenxu He; Peter Petersen; William Wylie

Home/Archive/arXiv

arXiv · 1012.3446

Warped product Einstein metrics over spaces with constant scalar curvature

Chenxu He·Peter Petersen·William Wylie

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Chenxu He, Peter Petersen, William Wylie. 2010-12-15. Warped product Einstein metrics over spaces with constant scalar curvature. https://arxiv.org/abs/1012.3446

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Futaki invariant on Hopf manifolds

The Futaki invariant is a fundamental tool in Kähler geometry representing an obstruction to the existence of Kähler-Einstein metrics. Recently, it was generalized to compact complex manifolds. In this paper, we prove that it vanishes on Hopf manifolds.

math.DG↗

arXiv2026

Remarks on potential functions of noncompact quasi-Einstein manifolds

In this article, we study the set of potential functions on noncompact quasi-Einstein manifolds. We show that the space of all positive potential functions on a three-dimensional noncompact quasi-Einstein manifold has dimension at most two, and that equality holds if and only if the manifold is isometric to a product $B\times\mathbb{R}$, where $B$ is a $λ$-Einstein surface or one of the examples obtained by L. Berard Bergery and described in Besse's book. Moreover, we prove that any asymptotically flat $n$-dimensional quasi-Einstein manifold with $λ=0$ is necessarily Ricci-flat.

math.DG↗

arXiv2026

Adjusted connections on non-abelian bundle gerbes

Higher gauge theory for non-abelian structure 2-groups faces significant challenges when extending beyond the fake-flat sector, which suffers from limited applicability in physical models. A promising resolution involves equipping 2-groups with additional structure, known as adjustments. We present a comprehensive theory of adjusted connections on non-abelian bundle gerbes, classified by Saemann's adjusted version of non-abelian differential cohomology. This theory enables, in particular, a new coordinate-independent formulation of Tellez-Dominguez' lifting theorem, establishing a correspondence between adjusted connections on non-abelian bundle gerbes and connections on abelian bundle 2-gerbes.

math.DG↗