Geometric Mechanics on Warped Product Semi-Slant Submanifold of Generalized Complex Space Forms

In this study, we develop a general inequality for warped product semi-slant submanifolds of type M n = N T n 1 × f N ϑ n 2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.

A General Inequality for CR-Warped Products in Generalized Sasakian Space Form and Its Applications

In the present paper, by considering the Gauss equation in place of the Codazzi equation, we derive new optimal inequality for the second fundamental form of CR-warped product submanifolds into a generalized Sasakian space form. Moreover, the inequality generalizes some inequalities for various ambient space forms.

Theory of Hypersurfaces yn–1 in yn Space and Geodesics on this Hypersurfaces yn–1

The hypersurface yn–1 in yn space is studied for this piece of work. We established the correlation between tensors of hypersurface yn–1 and tensors of embedding space yn . The second non-symmetrical tensor of hypersurface has been introduced, which have been obtained from the analog of Peterson-Codazzi equation in nonsymmetrical case.Also we have introduced the tensor that is associated with square of angle between normal and adjacent normal and it is represented in terms of metric and second tensors of hypersurface. The geodesics on hypersurface have been studied, and nontrivial example of geodesics on hypersurface with torsion and Euclid metric was constructed.

Curvatures and discrete Gauss–Codazzi equation in (2 + 1)-dimensional loop quantum gravity

We derive the Gauss–Codazzi equation in the holonomy and plane-angle representations and we use the result to write a Gauss–Codazzi equation for a discrete (2 + 1)-dimensional manifold, triangulated by isosceles tetrahedra. This allows us to write operators acting on spin network states in (2 + 1)-dimensional loop quantum gravity, representing the 3-dimensional intrinsic, 2-dimensional intrinsic, and 2-dimensional extrinsic curvatures.