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

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 vanishing, and M3 is homotopic to the sphere S3 under various extrinsic restrictions, involving eigenvalue of function, integral Ricci curvature, Hessian tensor. The results paper can be considered extension Xin’s work framework a submanifold, when base manifold minimal ambient manifolds.