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RIGIDITY OF QUASI-EINSTEIN METRICS JEFFREY CASE, YU-JEN SHU, AND GUOFANG WEI
 

Summary: RIGIDITY OF QUASI-EINSTEIN METRICS
JEFFREY CASE, YU-JEN SHU, AND GUOFANG WEI
Abstract. We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor
is a constant multiple of the metric tensor. This is a generalization of Einstein
metrics, which contains gradient Ricci solitons and is also closely related to
the construction of the warped product Einstein metrics. We study properties
of quasi-Einstein metrics and prove several rigidity results. We also give a
splitting theorem for some Kšahler quasi-Einstein metrics.
1. Introduction
Einstein metrics and their generalizations are important both in mathematics
and physics. A particular example is from the study of smooth metric measure
spaces. Recall a smooth metric measure space is a triple (Mn
, g, e-f
dvolg), where
M is a complete n-dimensional Riemannian manifold with metric g, f is a smooth
real valued function on M, and dvolg is the Riemannian volume density on M.
A natural extension of the Ricci tensor to smooth metric measure spaces is the
m-Bakry-Emery Ricci tensor
(1.1) Ricm
f = Ric + Hessf -

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics