6,789 research outputs found
A step towards the Alekseevskii Conjecture
We provide a reduction in the classification problem for non-compact,
homogeneous, Einstein manifolds. Using this work, we verify the (Generalized)
Alekseevskii Conjecture for a large class of homogeneous spaces.Comment: 12 pages, proof of main result simplified, final version to appear in
Math. Annale
Rigidity of gradient Ricci Solitons
We define a gradient Ricci soliton to be rigid if it is a flat bundle where is Einstein. It is known that not all
gradient solitons are rigid. Here we offer several natural conditions on the
curvature that characterize rigid gradient solitons. Other related results on
rigidity of Ricci solitons are also explained in the last section.Comment: 16 page
Warped product Einstein metrics over spaces with constant scalar curvature
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.Comment: 38 pages, 1 appendi
On the classification of warped product Einstein metrics
In this paper we take the perspective introduced by Case-Shu-Wei of studying
warped product Einstein metrics through the equation for the Ricci curvature of
the base space. They call this equation on the base the -Quasi Einstein
equation, but we will also call it the -Einstein equation. In
this paper we extend the work of Case-Shu-Wei and some earlier work of Kim-Kim
to allow the base to have non-empty boundary. This is a natural case to
consider since a manifold without boundary often occurs as a warped product
over a manifold with boundary, and in this case we get some interesting new
canonical examples. We also derive some new formulas involving curvatures which
are analogous to those for the gradient Ricci solitons. As an application, we
characterize warped product Einstein metrics when the base is locally
conformally flat.Comment: 29 pages. Minor changes and references updated. Submitted versio
Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons
In this paper we consider connections between Ricci solitons and Einstein
metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton
admits an Einstein one-dimensional extension if the soliton derivation can be
chosen to be normal. Using our previous work on warped product Einstein
metrics, we show that every normal semi-algebraic Ricci soliton also admits a
-dimensional Einstein extension for any . We also prove converse
theorems for these constructions and some geometric and topological structure
results for homogeneous warped product Einstein metrics. In the appendix we
give an alternative approach to semi-algebraic Ricci solitons which naturally
leads to a definition of semi-algebraic Ricci solitons in the non-homogeneous
setting.Comment: 28 pages. supersedes the earlier version of arXiv:1110.245
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