© Printed in India Convexity of spheres in a manifold without conjugate points

Abstract. For a non-compact, complete and simply connected manifold M without conjugate points, we prove that if the determinant of the second fundamental form of the geodesic spheres in M is a radial function, then the geodesic spheres are convex. We also show that if M is two or three dimensional and without conjugate points, then, at every point there exists a ray with no focal points on it relative to the initial point of the ray. The proofs use a result from the theory of vector bundles combined with the index lemma