Small Covers, infra-nilmanifolds and positive curvature

TL;DR

This paper characterizes small covers that are infra-nilmanifolds as real Bott manifolds and explores how positive or nonnegative curvature conditions impose topological and combinatorial restrictions on these manifolds.

Contribution

It establishes a precise classification of certain small covers as real Bott manifolds and analyzes the impact of curvature conditions on their topology and combinatorics.

Findings

All infra-nilmanifold small covers are real Bott manifolds

Curvature conditions restrict the topology of small covers

Results extend to real moment-angle complexes

Abstract

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.