Null Homology Groups and Stable Currents in Warped Product Submanifolds of Euclidean Spaces

Abstract

In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin’s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.