Homotopy type of manifolds with partially horoconvex boundary

Let M be an n-dimensional compact connected manifold with boundary, > 0 a constant and 1 ≤ q ≤ n - 1 an integer. We prove that M supports a Riemannian metric with the interior q-curvature Kq and the boundary q-curvature q ≥ 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 ≥ and boundary principal curvature ≥ is diffeomorphic to the standard closed n-ball.