Homology of warped product submanifolds in the unit sphere and its applications

Abstract

In this work, several pinched conditions on the Laplacian and gradient of the warping function are found in consideration of warped product submanifolds structure that force to homology groups vanish with no stable currents. Also, it is proved that a warped product pointwise semi-slant submanifold [Formula: see text] that is compact and oriented in an odd-dimensional spheres [Formula: see text] and [Formula: see text], has no stable integral [Formula: see text]-currents and [Formula: see text]-currents, respectively, and their homology groups are null, provided squared norm of the gradient for warping function satisfies some extrinsic restrictions including the Laplacian of the warping function, pointwise slant functions in addition to dimension of fiber of warped product immersions. Moreover, under assumption of extrinsic condition on the warping function, it is show [Formula: see text] being homeomorphic to a standard sphere [Formula: see text] with [Formula: see text] and homotopic to a standard sphere [Formula: see text] with [Formula: see text]. Further, the same results are generalized for contact CR-warped product submanifolds of same ambient spaces.