A compactness theorem in Finsler geometry

Let (M, F) be a complete Finsler manifold and P be a minimal and compact submanifold of M. The k-Ricci curvature Ric(k) (x), x is an element of M is a differential invariant that interpolates between the flag curvature and the Ricci scalar. We prove that if the k-Ricci curvature satisfies the condition integral(infinity)(0) Ric(k) (t) > 0 along any geodesic gamma : [0, infinity) -> M, t -> gamma(t) emanating orthogonally from P or f(-infinity)(0) Ric(k)(t) > 0 along any geodesic gamma : (-infinity, 0] -> M, t -> gamma(t) arriving orthogonally to P, then M is compact.