On almost Blaschke manifolds. I
Let \(i(M)\) be the injectivity radius of a Riemannian manifold \((M,g)\) and \(d(M)\) be its diameter. The purpose of this paper is to determine the topology of a Riemannian manifold \((M,g)\) when \(i(M)\) is close to \(d(M)\) in lower \((2,3)\) dimensional cases. In the 2-dimensional case under the assumption that the Gaussian curvature of \((M,g) \geq -\delta^ 2\) (resp. \(\geq 0)\) and \(\sinh (\delta i(M))>(\sqrt{3}/2)\cdot\sinh (\delta d(M))\) (resp. \(i(M)>(\sqrt{3}/2)d(M))\), it follows that \(M\) is diffeomorphic to \(S^ 2\) or \(RP^ 2\). In the 3-dimensional case under the assumption that the sectional curvature of (M,g) \(\geq\delta^ 2\) and \(\sin (\delta i(M))>(\sqrt{3}/2)\cdot\sin (\delta d(M)),\) it follows that M is diffeomorphic to \(S^ 3\) or \(RP^ 3\). These results are best possible. Under the hypothesis that i(M) is sufficiently close to d(M) the same result follows without any positive curvature assumption. To prove these results the author uses the structure of the cut locus.
scientific article; zbMATH DE number 3876127