Homotopy type of manifolds with partially horoconvex boundary

Abstract

Let [Formula: see text] be an [Formula: see text]-dimensional compact connected manifold with boundary, [Formula: see text] a constant and [Formula: see text] an integer. We prove that [Formula: see text] supports a Riemannian metric with the interior [Formula: see text]-curvature [Formula: see text] and the boundary [Formula: see text]-curvature [Formula: see text], if and only if [Formula: see text] has the homotopy type of a CW complex with a finite number of cells with dimension [Formula: see text]. Moreover, any Riemannian manifold [Formula: see text] with sectional curvature [Formula: see text] and boundary principal curvature [Formula: see text] is diffeomorphic to the standard closed [Formula: see text]-ball.