Odd-dimensional wiedersehen manifolds are spheres

Abstract

Let M be a connected, simply connected, compact Riemannian Λ-manifold without boundary, n > 2, such that for any m E Λf, the cut locus of m in M is a single point.It is known that M is diffeomorphic to the ^-sphere S n .(This fact is not used in the present paper.)Moreover, every geodesic returns to its beginning point and is smoothly closed.Following Green [2], we call M a wiedersehen n-manifold.It is easily seen that in M, all closed geodesies are of the same length, say 2πr, r > 0. Whether M is isometric to a euclidean w-sphere S" of radius r is usually referred to as the Blaschke problem (for spheres).Recently, Berger [1] made use of an inequality given by Kazdan [3] to prove that vol M > vol S r n , and that the equality holds iff M is isometric to S".On the other hand, Weinstein [4] has proved the following result.If M is a connected compact Riemannian n-manifold in which all geodesies are smoothly closed and have the same length, say 2πr, if UM is the space of unit tangent vectors of A/, CM is the space of (oriented) closed geodesies in M, α is the Euler class of the natural circle fibration π: UM -» CM, and CM is so oriented that the value (α"" 1 , [CM]) of α"" 1 at the fundamental class [CM] is positive, then 2 vol M = (a n ~\ [CM]} vol S r n .Therefore the evaluation of vol M depends only on that of <α π ~\ [CM]}.It is remarked in [4] that, when n is even, (α"" 1 , [CM]} = 2. Hence for any even n > 2, vol M -vol S" 9 and thus M and S" are isometric.The purpose of this paper is to show that for any odd n > 1, (a n ~\ [CM]} = 2 remains valid and hence M and S" are isometric.The Blaschke problem (for spheres) is thus completely solved.