On almost Blaschke manifolds II

Abstract

where Br(Om) denotes the metric r-ball centered at the origin o m in the tangent space TraM and ExPm denotes the exponential mapping from T~M onto M. In general the diameter d(M) is equal to or greater than the injectivity radius i(M). In special case when the diameter is equal to the injectivity radius, we say that such (M, g) is a Blaschke manifold. For example compact symmetric spaces of rank one so called CROSSes have such a property. The famous Blaschke conjecture asks; Is any Blaschke manifold isometric to a CROSS? It is known that Blaschke manifolds have the cohomology types of the CROSSes, although in I-2, 11] R.Bott and H.Nakagawa have shown that this conclusion holds under much weaker assumption. Moreover M. Berger proved that Blaschke manifolds which are diffeomorphic to spheres or real projective spaces are isometric to standard ones ([1]). Recently it was shown that any Blaschke manifold is homeomorphic to one of the CROSSes in low dimensions by H. Gluck, F. Warner and C.T. Yang [7]. See also H. Sato and T. Mizutani [14, 15]. In this paper we study the following problem from a different view point. What is the topology of a riemannian manifold whose injectivity radius is close to its diameter? We get the following results in the lower dimensional case.