Warped product submanifolds of Lorentzian paracosymplectic manifolds

TL;DR

This paper investigates the properties and existence conditions of warped product submanifolds within Lorentzian paracosymplectic manifolds, revealing nonexistence results and characterizing their geometric structure.

Contribution

It provides new nonexistence results and characterizations for warped product semi-invariant, semi-slant, and anti-slant submanifolds in Lorentzian paracosymplectic manifolds.

Findings

Warped product semi-invariant submanifolds with normal characteristic vector are Riemannian products.

Distributions in warped product semi-invariant submanifolds are always integrable.

Conditions for existence and nonexistence of warped product semi-slant and anti-slant submanifolds.

Abstract

In this paper we study the warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold in the form {$M=M_{T}\times_{f}M_{\bot}$} of Lorentzian paracosymplectic manifold such that the characteristic vector field is normal to $M$ is an usual Riemannian product manifold where totally geodesic and totally umbilical submanifolds of warped product are invariant and anti-invariant, respectively. We prove that the distributions involved in the definition of a warped product semi-invariant submanifold are always integrable. A necessary and sufficient condition for a semi-invariant submanifold of a Lorentzian paracosymplectic manifold to be warped product semi-invariant submanifold is obtained. We also investigate the existence and nonexistence of warped product semi-slant and warped product anti-slant submanifolds in a Lorentzian paracosymplectic manifold.