CR-submanifolds of Some Lorentzian Manifolds and K-manifolds

TL;DR

This paper investigates various types of CR-submanifolds within Lorentzian manifolds, providing new theoretical results on their geometric properties, structure, and examples, expanding understanding of submanifold theory in these complex settings.

Contribution

It introduces new results on the properties and classifications of CR-submanifolds in Lorentzian Concircular Structure, Para-Sasakian, S-, and Kenmotsu manifolds, including examples.

Findings

Results on totally umbilical CR-submanifolds with specific properties.

Characterization of D-totally geodesic and D-umbilic CR-submanifolds.

Examples of CR-submanifolds in (LCS)n-manifolds and GKM manifolds.

Abstract

In this paper I have studied about CR(Cauchy-Riemann)-submanifolds of Lorentzian Concircular Structure manifold ((LCS)n-manifold), Lorentzian Para-Sasakian(LP)-cosymplectic manifold, S-manifold and Generalized Kenmotsu (GKM) manifold. I have discussed some results regarding distribution, structure vector field, totally geodesic submanifold, leaf etc.. I have obtained results on totally umbilical contact CR-submanifold where the anti-invariant distribution has some properties. Next, I have studied some results about D-totally geodesic CR-submanifold (D is the distribution), a contact CR-submanifold, D(perp)-totally geodesic CR-submanifold, xi-horizontal CR-submanifold where the distribution is integrable (here xi is the structure vector field). Also I have proved some results on D-umbilic CR-submanifold, mixed totally geodesic CR-submanifold, foliate xi-horizontal mixed totally geodesic CR-submanifold, leaf of the distribution, totally geodesic leaf, CR-product etc.. I have given an example of a CR-submanifold of an (LCS)n-manifold and at last, I have given an example of a GKM manifold.