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.
Ali et al. (Wed,) studied this question.