Mathematics

An almost torus manifold M is a closed (2n+1)-dimensional orientable Riemannian manifold with an effective, isometric n-torus action such that the fixed point set is non-empty. Almost torus manifolds are analogues of torus manifolds in odd dimension and share many of the characteristics of torus manifolds. Recently Wiemeler classified simply connected torus manifolds with non-negative sectional curvature. In this talk we describe a structure theorem for the orbit spaces of simply connected almost torus manifolds with non-negative sectional curvature. The analysis of the structure of almost torus manifolds also lets us combine this structure theorem with results about extending the n-torus action to a smooth (n+1)-torus action to conclude that the orbit space is a face-acyclic combinatorial polytope as in the case of torus manifolds. As a consequence, a classification of non-negatively curved, simply connected almost torus manifolds can be obtained.