The Diffeomorphism Type of Manifolds with Almost Maximal Volume

TL;DR

This paper proves that Riemannian manifolds with sectional curvature bounded below and volume close to the maximum are diffeomorphic to spheres or real projective spaces, revealing their topological structure under near-maximal volume conditions.

Contribution

It establishes a diffeomorphism classification for manifolds with almost maximal volume relative to curvature and radius bounds, extending understanding of their topological types.

Findings

Manifolds with volume close to the upper bound are diffeomorphic to spheres or real projective spaces.

The volume bound is sharp for the classification.

The result links geometric bounds to topological classification.

Abstract

The smallest $r$ so that a metric $r$-ball covers a metric space $M$ is called the radius of $M$. The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.