The Lawson-Simons' theorem on warped product submanifolds with geometric information

Abstract

The main objective of this paper is to investigate topological properties from the view point of compact warped product submanifolds of a space form with the vanishing constant sectional curvature. That is, we prove the non-existence of stable integral $p$-currents in a compact oriented warped product pointwise semi-slant submanifold $M^n$ in the Euclidean space $\mathbb{R}^{p+2q}$ which satisfies an operative condition involving the Laplacian of a warped function and a pointwise slant function, and show that their homology groups are zero on this operative condition. Moreover, under the assumption of extrinsic conditions, we derive new topological sphere theorems on a warped product submanifold $M^n$, and prove that $M^n$ is homeomorphic to $\mathbb{S}^n$ if $n=4$, and $M^n$ is homotopic to $\mathbb{S}^n$ if $n=3$. Furthermore, the same results are generalized for CR-warped products and our results recovered \cite{S4}.

Keywords

• warped product manifolds
• integral p-currents
• sphere thoerems
• cr-warped products
• ricci-flat