 
Summary: RIGIDITY OF QUASIEINSTEIN METRICS
JEFFREY CASE, YUJEN SHU, AND GUOFANG WEI
Abstract. We call a metric quasiEinstein if the mBakryEmery 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 quasiEinstein metrics and prove several rigidity results. We also give a
splitting theorem for some Kšahler quasiEinstein 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, ef
dvolg), where
M is a complete ndimensional 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
mBakryEmery Ricci tensor
(1.1) Ricm
f = Ric + Hessf 
