A New Class of Contact Pseudo Framed Manifolds with Applications

K L Duggal,Luca Vitagliano

doi:10.1155/2021/6141587

Abstract

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifoldsM,g,f,λ,ξby a real tensor fieldfof type1,1, a real functionλsuch thatf3=λ2fwhereξis its characteristic vector field. We prove in our main Theorem 2 thatMadmits a closed 2-formΩifλis constant. In 1976, Blair proved that the vector fieldξof a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general,ξof a normal CPF-manifold is non-Killing. We also have established a link of CPF-hypersurfaces with curvature, affine, conformal collineations symmetries, and almost Ricci soliton manifolds, supported by three applications. Contrary to the odd-dimensional contact manifolds, we construct several examples of even- and odd-dimensional semi-Riemannian and lightlike CPF-manifolds and propose two problems for further consideration.

Highlights

Let M be an m-dimensional real differential manifold.Recently, we introduced in [1] a new class of differentiable structure called pseudo Cauchy Riemann structure on M defined by a real tensor field J of type (1, 1) at every point of M satisfying J2 λ2I, (1)where λ is a nonzero real function on M
In same paper [1], we introduced a revised version of a contact manifold, called contact pseudo framed (CPF)-manifold (M, g, f, λ) by a real tensor field f of type (1, 1) and a real function λ such that f3 λ2f, and T(M) splits into a direct sum of two subbundles, International Journal of Mathematics and Mathematical Sciences namely, im(f) with a pseudo Cauchy Riemann (PCR)structure and 1-dimensional ker(f)
Contrary to the odddimensional contact manifolds, for the first time in the literature, we presented some examples of a class of evendimensional CPF-manifolds and shown that the metric of PCR and CPF-manifolds is not severely restricted

Summary

Introduction

Let M be an m-dimensional real differential manifold.Recently, we introduced in [1] a new class of differentiable structure called pseudo Cauchy Riemann structure on M defined by a real tensor field J of type (1, 1) at every point of M satisfying J2 λ2I, (1)where λ is a nonzero real function on M. Us, in support of eorem 2, we have an example of an odd-dimensional semi-Riemannian ACPF-hypersurface (M5, f, g, λ, η, ξ), of a metric PCR-manifold (M6, g), which admits a 2-form Ω and spacelike or timelike ξ as ε is 1 or −1.

Results

Conclusion