  Let $M$ be an $n$-dimensional compact connected manifold with boundary,
$\kappa>0$ a constant and $1\leq q\leq n-1$ an integer. We prove that $M$
supports a Riemannian metric with the interior $q$-curvature $K_q\geq
-q\kappa^2$ and the boundary $q$-curvature $\Lambda_q\geq q\kappa$, if and only
if $M$ has the homotopy type of a CW complex with a finite number of cells with
dimension $\leq (q-1)$. Moreover, any Riemannian manifold $M$ with sectional
curvature $K\geq -\kappa^2$ and boundary principal curvature $\Lambda\geq
\kappa$ is diffeomorphic to the standard closed $n$-ball.
