A Pinching Theorem for Compact Minimal Submanifolds in Warped Products I×fSm(c)

Abstract

Let Sm(c) be a Euclidean sphere of curvature c>0 and R be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type I×fSm(c), where f:I→R+ is a smooth positive function on an open interval I of R. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products Sm(c)×R to Riemannian warped products I×fSm(c).