Topological properties on warped product submanifolds of space forms

The current article aspires to formulate the sufficient condition for the Laplacian and the gradient of the warping function of a compact warped product submaniold F^{{α1+α2}} in a space form F c^α1+α2+k1 (c) that provides the trivial homology and fundamental groups. Also, we validate the instability of current flows in π1( F^α1+α2). The constraints also apply to the warped function eigenvalues, integral Ricci curvature, and Hessian tensor.