A New Class of Contact Pseudo Framed Manifolds with Applications

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

Anti-invariant Riemannian submersions from Sasakian manifolds with totally umbilical fibers

The purpose of this paper is to study anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds such that characteristic vector field is vertical or horizontal vector field. We first show that any anti-invariant Riemannian submersions from Sasakian manifold is not a Riemannian submersion with totally umbilical fiber. Then we introduce anti-invariant Riemannian submersions from Sasakian manifolds with totally contact umbilical fibers. We investigate the totally contact geodesicity of fibers of such submersions. Moreover, under this condition, we investigate Ricci curvature of anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds.

Existence of some classes of N(k)-quasi Einstein manifolds

The object of the present paper is to study some classes of N(k)-quasi Einstein manifolds. The existence of such manifolds are proved by giving non-trivial physical and geometrical examples. It is also proved that the characteristic vector field of the manifold is killing as well as parallel unit vector fields under certain curvaturerestrictions.

Some Results on D-Homothetic Deformation On Almost Paracontact Metric Manifolds

In this work, we apply the notion of D-homothetic deformation on an almost paracontact metric manifolds and show that the structure after the deformation is also almost paracontact metric structure. Also, we state the classes of almost paracontact metric structures having parallel characteristic vector field and get some results about D- homothetic deformations on these classes.

An Overview of the History of Projective Representations of Groups

In this work we investigate the possible classes of seven-dimensional almost paracontact metric structures induced by the three-forms of $G_2^*$ structures. We write the projections that determine to which class the almost paracontact structure belongs, by using the properties of the $G_2^*$ structures. Then we study the properties that the characteristic vector field of the almost paracontact metric structure should have such that the structure belongs to a specific subclass of almost paracontact metric structures.

∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds

In this paper, we consider *-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if (M, g) is a Kenmotsu manifold and g is a *-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a *-Ricci soliton, then we show that M is of constant sectional curvature –1. Next, we show that if M admits a *-Ricci soliton whose potential vector field is collinear with the characteristic vector field ξ, then M is Einstein and soliton vector field is equal to ξ. Finally, we prove that if g is a gradient almost *-Ricci soliton, then either M is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of M. We verify our result by constructing examples for both *-Ricci soliton and gradient almost *-Ricci soliton.

Lightlike hypersurfaces of an indefinite Kaehler manifold of a quasi-constant curvature

We study lightlike hypersurfaces M of an indefinite Kaehler manifold M̅ of quasi-constant curvature subject to the condition that the characteristic vector field ζ of M̅ is tangent to M. First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface M of M̅ such that(1) the screen distribution S(TM) is totally umbilical or(2) M is screen conformal.

Some results on null hypersurfaces in $(LCS)$-manifolds

We prove that a Lorentzian concircular structure $ (LCS)$-manifold does not admit any null hypersurface which is tangential or transversal to its characteristic vector field. Due to the above, we focus on its ascreen null hypersurfaces, and show that such hypersurfaces admit a symmetric Ricci tensor. Furthermore, we prove that there is no any totally geodesic ascreen null hypersurfaces of a conformally flat $(LCS)$-manifold.

Generating Curves of Minimal Ruled Real Hypersurfaces in a Nonflat Complex Space Form

We first provide a necessary and sufficient condition for a ruled real hypersurface in a nonflat complex space form to have constant mean curvature in terms of integral curves of the characteristic vector field on it. This yields a characterization of minimal ruled real hypersurfaces by circles. We next characterize the homogeneous minimal ruled real hypersurface in a complex hyperbolic space by using the notion of strong congruency of curves.