Small Covers, infra-solvmanifolds and curvature

TL;DR

This paper characterizes small covers and real moment-angle manifolds that are infra-solvmanifolds or flat, linking their topology to real Bott manifolds and flat tori, and explores curvature conditions affecting their topology and combinatorics.

Contribution

It provides necessary and sufficient conditions for small covers to be infra-solvmanifolds or flat, and analyzes how curvature constraints influence their topology and polytope structure.

Findings

Small covers are infra-solvmanifolds iff diffeomorphic to real Bott manifolds.

Curvature conditions impose strong topological restrictions.

Equivalent conditions for small covers to be homeomorphic to real Bott manifolds.

Abstract

It is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover being homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structure of the underlying simple polytopes.