Homotopy type of manifolds with partially horoconvex boundary

TL;DR

This paper characterizes the homotopy type of manifolds with partially horoconvex boundary based on curvature conditions, showing they are homotopy equivalent to finite CW complexes and identifying conditions for diffeomorphism to a standard ball.

Contribution

It establishes a precise relationship between curvature bounds and the homotopy type of manifolds with boundary, extending understanding of geometric-topological properties under curvature constraints.

Findings

Manifolds with specified curvature bounds have homotopy types of finite CW complexes.

Certain curvature conditions imply the manifold is diffeomorphic to a standard ball.

The results connect curvature conditions with topological and differential structure.

Abstract

Let $M$ be an $n$-dimensional compact connected manifold with boundary, $\kappa>0$ a constant and $1\leq q\leq n-1$ an integer. We prove that $M$ supports a Riemannian metric with the interior $q$-curvature $K_q\geq -q\kappa^2$ and the boundary $q$-curvature $\Lambda_q\geq q\kappa$, if and only if $M$ has the homotopy type of a CW complex with a finite number of cells with dimension $\leq (q-1)$. Moreover, any Riemannian manifold $M$ with sectional curvature $K\geq -\kappa^2$ and boundary principal curvature $\Lambda\geq \kappa$ is diffeomorphic to the standard closed $n$-ball.