Torus manifolds and non-negative curvature

Abstract

A torus manifold $M$ is a $2n$-dimensional orientable manifold with an effective action of an $n$-dimensional torus such that $M^T\ne \varnothing$. In this paper we discuss the classification of torus manifolds which admit an invariant metric of non-negative curvature. If $M$ is a simply connected torus manifold which admits such a metric, then $M$ is diffeomorphic to a quotient of a free linear torus action on a product of spheres. We also classify rationally elliptic torus manifolds $M$ with $H^{\text{odd}}(M;\mathbb{Z})=0$ up homeomorphism.

26 pages; There was a gap in the arguments in section 4 therefore Theorem 1.1 is only proved for torus manifolds with vanishing odd degree integer cohomology; results about non-simply connected torus manifolds added; to appear in J. Lond. Math. Soc

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