Homotopy type of manifolds with partially horoconvex boundary
Abstract
Let MM be an nn-dimensional compact connected manifold with boundary, κ>0κ>0 a constant and 1≤q≤n−11≤q≤n−1 an integer. We prove that MM supports a Riemannian metric with the interior qq-curvature Kq≥−qκ2Kq≥−qκ2 and the boundary qq-curvature Λq≥qκΛq≥qκ, if and only if MM has the homotopy type of a CW complex with a finite number of cells with dimension ≤(q−1)≤(q−1). Moreover, any Riemannian manifold MM with sectional curvature K≥−κ2K≥−κ2 and boundary principal curvature Λ≥κΛ≥κ is diffeomorphic to the standard closed nn-ball.