In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold in the hyperbolic space satisfy various extrinsic restrictions, then has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.

E. Study map is one of the most basic and powerful mathematical tools to study lines in line geometry, it has symmetry property. In this paper, based on the E. Study map, clear expressions were developed for the differential properties of one-parameter Lorentzian dual spherical movements that are coordinate systems independent. This eliminates the requirement of demanding coordinates transformations necessary in the determination of the canonical systems. With the proposed technique, new proofs for Euler–Savary, and Disteli’s formulae were derived.

The prime objective of the approach is to give geometric classifications of k -almost Ricci solitons associated with paracontact manifolds. Let M 2 n + 1 ( φ , ξ , η , g ) be a paracontact metric manifold, and if a K -paracontact metric g represents a k -almost Ricci soliton ( g , V , k , λ ) and the potential vector field V is Jacobi field along the Reeb vector field ξ , then either k = λ − 2 n , or g is a k -Ricci soliton. Next, we consider K -paracontact manifold as a k -almost Ricci soliton with the potential vector field V is infinitesimal paracontact transformation or collinear with ξ . We have proved that if a paracontact metric as a k -almost Ricci soliton associated with the non-zero potential vector field V is collinear with ξ and the Ricci operator Q commutes with paracontact structure φ , then it is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) . Finally, we have deduced that a para-Sasakian manifold admitting a gradient k -almost Ricci soliton is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) .

In this paper, we give an estimate of the first eigenvalue of the Laplace operator on minimally immersed Legendrian submanifold $N^n$ in Sasakian space forms ${\tilde{N}}^{2n+1}\left(ϵ\right)$ . We prove that a minimal Legendrian submanifolds in a Sasakian space form is isometric to a standard sphere ${\mathbb{S}}^n$ if the Ricci curvature satisfies an extrinsic condition which includes a gradient of a function, the constant holomorphic sectional curvature of the ambient space and a dimension of $N^n$ . We also obtain a Simons-type inequality for the same ambient space forms ${\tilde{N}}^{2n+1}\left(ϵ\right)$ .

This work examines some classical results of Bertrand curves for timelike ruled and developable surfaces using the E. Study map. This provides the ability to define two timelike ruled surfaces which are offset in the sense of Bertrand. It is shown that every timelike ruled surface has a Bertrand offset if and only if an equation should be satisfied between the dual geodesic curvatures. Some new results and theorems related to the developability of the Bertrand offsets of timelike ruled surfaces are also obtained.