Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

Abstract

In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold Mn of a complex space form M˜(4ϵ) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of Mn are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold Mn, in which Mn is homeomorphic to a sphere Sn if n≥4 and if n=3, then M3 is homotopic to a sphere S3 under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

The Homology of Warped Product Submanifolds of Spheres and Their Applications

“…In recent years, a significant number of studies have been conducted on the geometric structure and topological characteristics of submanifolds in various ambient spaces. These structures and characteristics have been demonstrated in numerous papers covering numerous applications, such as Euclidean spaces [6], complex projective spaces [7], CR-warped product submanifolds in Sasakian space forms [5], CR-warped product submanifolds in Euclidean spaces [8], CR-warped products in nearly Kaeher manifolds [9,10], CR-warped products in hyperbolic spaces [11], and many others (see [12,13]). There is a closed relationship between the absence of stable currents and the vanishing homology groups of submanifolds in various ambient manifold classes.…”