A generalization of Berger's theorem on almost 1/4-pinched manifolds. II
[For part I, cf. Bull. Am. Math. Soc., New Ser. 12, 260-264 (1985; Zbl 0563.53031).] Manifolds with sectional curvature K(M)\(\geq 1\) and diameter \(d(M)>\pi /2- c\), and the geometric and regularity properties of the limit metrics in the sense of Gromov are studied. We prove the existence of c depending on the dimension, the upper bound of the curvature and the injectivity radius (which can be replaced by the volume and if the dimension is odd) such that if M has the curvature and diameter as above then either i) M is homeomorphic to a sphere, ii) M is simply-connected and has the cohomology ring structure of symmetric spaces of rank 1, or iii) M is not simply connected and it admits a smooth metric with 1/4-pinching and diameter \(\pi\) /2. We obtained diffeomorphisms in (ii) under slightly stronger hypotheses. These results generalize Berger's result on the even dimensional manifolds with pinching less than 1/4, and extend the results of Gromoll- Grove for \(d(M)=\pi /2\) in the cohomological sense. The limit metrics obtained by using Gromov's compactness theorem are shown to be differentiable (also obtained by Peters, Greene-Wu) in some sense. Our study further develops some geometric properties of the limit metric which are not valid for arbitrary \(C^{1+a}\) metrics. Some consequences (not the theorems themselves) of Rauch Comparison Theorems are valid on the limit metric even though neither curvature nor Jacobi fields are defined.
scientific article; zbMATH DE number 3967359