Summary: Metrics of positive Ricci curvature on vector
bundles over nilmanifolds
Igor Belegradek Guofang Wei \Lambda
We construct metrics of positive Ricci curvature on some vector bundles over tori
(or more generally, over nilmanifolds). This gives rise to the first examples of manifolds
with positive Ricci curvature which are homotopy equivalent but not homeomorphic to
manifolds of nonnegative sectional curvature.
According to the soul theorem of J. Cheeger and D. Gromoll, a complete open manifold
of nonnegative sectional curvature is diffeomorphic to the total space of the normal bundle
of a compact totally geodesic submanifold. Furthermore, a finite cover of any compact
manifold with sec – 0 is diffeomorphic to the product C \Theta T where T is a torus, and C is
simplyconnected with sec – 0 . (The same is actually true when sec is replaced by Ric.)
It was shown in [ ¨
OW94, BKb, BKa] that a majority of vector bundles over C \Theta T admit no
metric with sec – 0 provided dim(T ) is sufficiently large. For example, any vector bundle
over a torus whose total space admits a metric with sec – 0 becomes trivial in a finite
cover, and there are only finitely many such bundles in each rank.
It is then a natural question which of the bundles can carry metrics with Ric – 0 or Ric ? 0 .