Geometry of warped product immersions of Kenmotsu space forms and its applications to slant immersions

In this paper, some relations among the second fundamental form which is an extrinsic invariant, Laplacian of the warping function and constant sectional curvature of a warped product semi-slant submanifold of a Kenmotsu space form and its totally geodesic and totally umbilical submanifolds are described from the exploitation of the Gauss equation instead of the Codazzi equation in the sense of Chen's studies in (2003). These relations provide us an approach to the classifications of equalities by the following case studied of Hasegawa and Mihai (2003). These are exemplified by the classifications of the totally geodesic and totally umbilical submanifolds. Moreover, we provide some applications of the inequality case by using the harmonicity of the smooth warping functions. In particular, we prove the triviality of connected, compact warped product semi-slant manifolds isometrically immersed into a Kenmotsu space form using Hamiltonian, Hessian, and the Kinetic energy of the warped function. Further, we generalize some results for contact CR-warped products in a Kenmotsu space form.