Homotopy type of manifolds with partially horoconvex boundary

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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.