Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">S</mi><mi>m</mi></msup><mo>(</mo><mi>c</mi><mo>)</mo><mo>×</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>

Abstract

Let Sm(c) denote a sphere with a positive constant curvature c and Mn(n≥3) be an n-dimensional compact pseudo-umbilical submanifold in a Riemannian product space Sm(c)×R with a nonzero parallel mean curvature vector (PMC), where R is a Euclidean line. In this paper, we prove a sequence of pinching theorems with respect to the Ricci, sectional and scalar curvatures of Mn, which allow us to generalize some classical curvature pinching results in spheres.

Keywords

• sectional curvature
• Ricci curvature
• parallel mean curvature vector
• pseudo-umbilical
• product space
• pinching theorems