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