Topological properties on warped product sub- manifolds of space forms

Abstract

The current article aspires to formulate the sufficient condition for the Laplacian and the gradient of the warping function of a compact warped product submanifold $\mathbf{F}^{\alpha_1+\alpha_2}$ in a space form $\mathbb{F}_c^{\alpha_1+\alpha_2+k_1}(c)$ that provide the trivial homology and fundamental groups. Also, we validate the instability of current flows in $\pi_1(\mathbf{F}^{\alpha_1+\alpha_2})$. The constraints also apply to the warped function eigenvalues, integral Ricci curvature, and Hessian tensor.