Small Covers, infra-nilmanifolds and positive curvature

Details Small Covers, infra-nilmanifolds and positive curvature Small Covers, infra-nilmanifolds and positive curvature

2011-08-18

arxiv.org/abs/1108.3782v4

Mathematics

Geometric Topology

The main results in this paper have been superseded by arXiv:1111.2174

Scientific paper

We show that all the small covers which are infra-nilmanifolds are exactly real Bott manifolds. This implies that any small cover which admits a flat Riemannian metric must be a real Bott manifold. In addition, we will study small covers which admit Riemannian metrics with positive or nonnegative Ricci curvature or sectional curvature. We will see that these geometric conditions put very strong restrictions on the topology of the small covers and the combinatorial structure of the underlying simple polytopes. Similar geometric problems are also studied for the real moment-angle complex of an arbitrary simple polytope.

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