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Homotopy type of manifolds with partially horoconvex boundary

Mathematical Sciences Institute, Australian National University, Canberra ACT 2601, Australia

E-mail Address: changwei.xiong@anu.edu.au

https://doi.org/10.1142/S0129167X1850074XCited by:0 (Source: Crossref)

Abstract

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

Keywords:

AMSC: 53C20, 53C21

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