A New Class of Contact Pseudo Framed Manifolds with Applications

Abstract

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifolds $\left(M,g,f,\lambda ,\xi \right)$ by a real tensor field $f$ of type $\left(\mathrm{1,1}\right)$ , a real function $\lambda$ such that $f^3={\lambda }^{2}f$ where $\xi$ is its characteristic vector field. We prove in our main Theorem 2 that $M$ admits a closed 2-form $\mathrm{\Omega }$ if $\lambda$ is constant. In 1976, Blair proved that the vector field $\xi$ of a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general, $\xi$ of 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.