Proof of Blaschke's Sphere Conjecture

For a brief history of the problem and a proof under stringent additional conditions see [2]. Let us first recall some of the known facts about such surfaces which we shall use. Proofs may be found in [l, §102]. Normalize the conjugate distance so it equals T. Then every geodesic is closed and has length 2ir. The mapping of any point into its conjugate point (determined uniquely, independently of the geodesic ray used) is an isometric involution of M onto itself. Any two geodesies intersect in a pair of mutually conjugate points. Denote the unit tangent bundle of M by T, and set dK = dAd, where dA is the element of area on M and d is the differential of angle between unit vectors based at the same point. An element of T will be denoted by e, or by the pair (x, ), where 0 is a fiber coordinate in some local product representation of T; set p(e) = #. The geodesic flow in T takes e after time / into the element et, the end point of the lift into the bundle of the geodesic segment of length t whose initial element is e. I t is well known that dK, the kinematic density, is invariant under this flow.