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

Abstract

<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 ^{[<xref ref-type="bibr" rid="b17">17</xref>]}. </abstract>