Communications in Analysis and Geometry

Volume 30 (2022)

Number 5

A softer connectivity principle

Pages: 1093 – 1119



Luis Guijarro (Department of Mathematics, Universidad Autónoma de Madrid, and ICMAT CSIC-UAM-UCM-UC3M, Madrid, Spain)

Frederick Wilhelm (Department of Mathematics, University of California, Riverside, Calif., U.S.A.)


We give soft, quantitatively optimal extensions of the classical Sphere Theorem, Wilking’s connectivity principle and Frankel’s Theorem to the context of Rick curvature. The hypotheses are soft in the sense that they are satisfied on sets of metrics that are open in the $C^2$-topology.

The first author was supported by research grants MTM2017-85934-C3-2-P from the MINECO and PID2021-124195NB-C32 from the AEI, and by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO).

This work was supported by a grant from the Simons Foundation (#358068, Frederick Wilhelm).

Received 11 December 2018

Accepted 27 December 2019

Published 17 March 2023