Mixed curvature almost flat manifolds

Abstract

We prove a mixed curvature analogue of Gromov's almost flat manifolds theorem for upper sectional and lower Bakry-Emery Ricci curvature bounds.

Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

“…Other types of weaker curvature assumptions include certain mixed curvature conditions considered by Kapovitch: in [55], the sectional curvature lower bound in Gromov's almost flat manifold theorem is weakened to a Bakry-Émery Ricci tensor lower bound.…”

“…The other theme of generalization focuses on weakening the curvature assumption (1.1) to lower (Bakry-Émery) Ricci curvature bounds, as examplified by the Colding-Gromov gap theorem: if a Ricci almost non-negatively curved manifold of unit diameter has its first Betti number equal to its dimension, then the manifold is diffeomorphic to a flat torus (see Theorem 2.5). Obviously, the weaker curvature assumption alone is insufficient to conclude the infranil manifold structure, and certain extra assumptions are necessary -just as the case of the Colding-Gromov gap theorem; see also [30,70,49,55] and §2.2 for a brief overview.…”