On the differentiable sphere theorem for manifolds with Ricci curvatures bounded from above
Abstract
In the present paper, we prove that if is an -dimensional compact Riemannian manifold and if where , and are the sectional and Ricci curvatures of respectively, then is diffeomorphic to a spherical space form where is a finite group of isometries acting freely. In particular, if is simply connected, then it is diffeomorphic to the Euclidian sphere
Reference
1. Berger, M.: Sur quelques varieties riemaniennes suffisamment pincées. Bull. Soc. Math. France, 88, 57—71 (1960).
2. Brendle, S., Schoen, R. M.: Classification of manifolds with weakly 1/4-pinched curvatures. Acta Math., 200, 1—13 (2008).
3. Xu, H.-W., Gu, J.-Ru.: The differentiable sphere theorem for manifolds with positive Ricci curvature. Proc. AMS, 140:3, 1011—1021 (2012).
4. Cao, X., Gursky, M. J., Tran, H.: Curvature of the second kind and a conjecture of Nishikawa. Commentarii Mathematici Helvetici, 98:1, 195—216 (2023).
5. Rovenski, V., Stepanov, S., Tsyganok, I.: On the Betti and Tachibana numbers of compact Einstein manifolds. Mathematics, 7, 1210 (2019).