On almost Blaschke manifolds. II
In part I [Math. Z. 190, 63-72 (1985; Zbl 0549.53040)] the author studied the topology of a Riemannian manifold (M,g) whose injectivity radius i(M) is close to its diameter d(M) and announced the following result: Let (M,g) be a 3-dimensional compact Riemannian manifold and K denote its sectional curvature. Assume that one of the following holds: (i) There is a positive number \(\delta\) such that we have \(K\geq -\delta^ 2\) and \(\sinh (\delta i(M))>a(\delta d(M))\cdot \sinh (\delta d(M))\) where \[ a(\delta d(M)):=\sin [\frac{\pi}{2}-\frac{1}{10}\sin^{- 1}\{\frac{\sinh(\frac{1}{10}\delta d(M))}{\sinh (2\delta d(M))}\}]. \] (ii) \(K\geq 0\) and \(i(M)>a\cdot d(M)\) hold where \(a:=\sin (\pi /2-(1/10)\sin 1/20)\). Then M is diffeomorphic to the 3-dimensional sphere \(S^ 3\) or real projective space \(P^ 3\). In this paper the author gives a detailed proof of this result.
scientific article; zbMATH DE number 3949224