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 Rp+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 S-n if n = 4, and M-n is homotopic to Sn if n = 3. Furthermore, the same results are generalized for CR-warped products and our results recovered [17].