Summary: Metrics of positive Ricci curvature on bundles
Igor Belegradek Guofang Wei \Lambda
May 11, 2001
Abstract
We construct new examples of manifolds of positive Ricci curvature which, topologi­
cally, are vector bundles over compact manifolds of almost nonnegative Ricci curvature.
In particular, we prove that if E is the total space of a vector bundle over a compact
manifold of nonnegative Ricci curvature, then E \Theta R p admits a complete metric of
positive Ricci curvature for all large p .
1 Introduction
According to the soul theorem of J. Cheeger and D. Gromoll, a complete open manifold of
nonnegative sectional curvature, denoted K – 0 , is the total space of a vector bundle over
a compact manifold with K – 0 . Manifolds of nonnegative Ricci curvature are much more
flexible, and nowadays there are many examples of complete manifolds of Ric – 0 which are
not even homotopy equivalent to complete manifolds of K – 0 . These include manifolds not
homotopy equivalent to closed manifolds [GM85], manifolds not satisfying Gromov's Betti
numbers estimate [SY89], manifolds of infinite topological type (see [Men00] and references
therein), manifolds with not virtually­abelian fundamental group [Wei88], compact spin
Ricci­flat 4­manifolds with nonzero “
A­genus [Bes87, 6.27], [Lot00], and complements to

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

Collections: Mathematics