Stable currents and homology groups in a compact CR-warped product submanifold with negative constant sectional curvature

Abstract

The main purpose of this paper is to highlight some striking features of the relationship between homology groups and compact warped product submanifolds geometry with negative constant sectional curvature that follows from Lawson and Simons (1973). Using the result of Fu and Xu (2008), we prove that there does not exist stable integral p-currents and their homology groups are zero in a compact CR-warped product submanifold Mn from a complex hyperbolic space ℂHm(−4) under extrinsic conditions involving the Laplacian of warped function and the squared norm of gradient of the warping function. Moreover, under the same extrinsic conditions and using the generalized Poincaré’s conjecture, we derive new topological sphere theorem on a compact CR-warped product submanifold Mn, we prove that Mn is homeomorphic to the sphere Sn if n=4. Also, if n=3 then M3 is homotopic to the sphere S3 and this result follows from the work of Sjerve (1973). Finally, the same types of theorems are constructed in terms of some mechanical tools such as Dirichlet energy and Hamiltonian as well.