Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

Yanlin Li,Fatemah Mofarreh,Akram Ali,Nadia Alluhaibi,Bibhas Ranjan Majhi

doi:10.1155/2021/8554738

Abstract

In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ω p + q in the hyperbolic space ℍ m − 1 satisfy various extrinsic restrictions, then Ω p + q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π 1 Ω p + q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.

Highlights

For any Riemannian manifold Ωn, it is well known that any integral homology class in Hq(Ωn, Z) which is nontrivial is correlated to integral stable currents. is result was initially proved by Federer and Fleming [1]
Utilizing the method of variational calculus for the geometric measure concept combining with the method of Federer and Fleming, Lawson and Simons [2] obtained the optimization for the second fundamental form, which leads to the vanishing homology in a range of intermediate dimensions and the nonexistence of stable currents in the submanifold in the connected space form, and obtained the key theorem of that paper
< p(n − sgn(c)p)c is satisfied, Ωn has no stable p-currents with the vanished pth homology group, i.e., Hp Ωn, Z􏼁 Hq Ωn, Z􏼁 0, (2)

Summary

Introduction and Main Results

For any Riemannian manifold Ωn, it is well known that any integral homology class in Hq(Ωn, Z) which is nontrivial is correlated to integral stable currents. is result was initially proved by Federer and Fleming [1]. N − k}, and if the fundamental group π1(Ωn) is finite and connected, Ωn is homeomorphic to Sn. Using eorem 1, Xu and Gu [16] extended the pinching condition in terms of the Ricci curvature and showed that Ωn in. Motivated by the geometric rigidity ( eorem 2), the second goal of this approach is to prove a new vanishing theorem for compact warped product submanifolds in terms of the Ricci curvature and using the eigenvalue of Laplacian of the warping function. Is an isometric embedding from a compact warped product submanifold Ωp+q into an m-dimensional hyperbolic space Hm(− 1) satisfying the following inequality:. ∇h ∈ KerΠ with the following holds, theorem for a compact warped product submanifold with no need for Ωp+q to be connected. Our result is of significance due to involving the new pinching conditions in terms of the warping function, the integral of the squared norm of the Hessian tensor, the integral Ricci curvature, and the first nontrivial eigenvalue of the warped function

Preliminaries

Proof of Main Results