On the topology of CR-warped product submanifolds

Abstract

We obtain a necessary condition for homology group to be zero on CR-warped product submanifold in Euclidean spaces in terms of second fundamental form of the submanifold and warping function. By using this condition, we show that such CR-warped product submanifold is a homotopy sphere.

On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature

“…More recently, geometric, topological, and differentiable rigidity theorems of the Riemannian submanifold connecting to parallel mean curvature in space forms such that c + H 2 > 0 have been obtained in terms of Ricci curvature in [5]. In some articles such as [3,[6][7][8][9][10][11][12][13][14][15][16][17], several results have been derived on topological and differentiable structures of singular submanifolds and submanifolds with specific effective conditions for the second fundamental form, sectional curvatures, and Ricci curvatures.…”

“…However, very few topological obstructions to warped product submanifolds with positive sectional curvature are known; for example, Sahin et al [13] verified some outcomes for the non-existence of the stable current and vanishing homology groups into a contact CR-warped product which immersed in a sphere with an odd dimension, by putting suitable restrictions on the Laplacian and the gradient of the warping function. Taking the benefits of the constant section curvature which could be zero or one, Sahin [13,14] extended this study on a class of CR-warped product in an Euclidean space and in the nearly Kaehler six-sphere. By assuming negative constant section curvature, Ali et al [18][19][20] obtained various results on CR-warped product, especially on the complex hyperbolic spaces, and many structures about this subject remain open.…”

Applications of stable currents and homology groups in CR-warped products of complex space forms

“…Some new and interesting results for trivial homology groups and stable currents on submanifolds have been gotten by putting some restrictions on on the second fundamental form (see [15,16,22,23,23,24,26,27,29]).…”

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

“…where H is the mean curvature of M n and c is a constant sectional curvature. Motivated by the non-existence of stable currents or stable submanifolds, a number of topological properties have been studied in [2,5,[7][8][9][10][11][12][13][14] by using Theorem 1.…”