The aim of the current article is to formulate sufficient conditions for the Laplacian and a gradient of the warping function of a compact warped product submanifold Σβ1+β2 in a unit sphere Sd that provides trivial homology and fundamental groups. We also validate the instability of current flows in π1(Σβ1+β2). The constraints are also applied to the warped function eigenvalues and integral Ricci curvatures.
