Characterizing inequalities for existence of contact CR-warped product submanifolds of generalized Sasakian space forms admitting a nearly cosymplectic structure

This paper studies the contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly cosymplectic structure. Some inequalities for the existence of these types of warped product submanifolds are established, the obtained inequalities generalize the results that have acquired in \cite{atceken14}. Moreover, we also estimate another inequality for the second fundamental form in the expressions of the warping function, this inequality also generalizes the inequalities that have obtained in \cite{ghefari19}. In addition, we also explore the equality cases.

Remarks on Almost Cosymplectic 3-Manifolds with RICCI Operators

Let M be an almost cosymplectic 3-h-a-manifold. In this paper, we prove that the Ricci operator of M is transversely Killing if and only if M is locally isometric to a product space of an open interval and a surface of constant Gauss curvature, or a unimodular Lie group equipped with a left invariant almost cosymplectic structure. Some corollaries of this result and some examples illustrating main results are given.

AXIOM OF Φ-HOLOMORPHIC (2r+1)-PLANES FOR GENERALIZED KENMOTSU MANIFOLDS

In this paper we study generalized Kenmotsu manifolds (shortly, a GK-manifold) that satisfy the axiom of Φ-holomorphic (2r+1)-planes. After the preliminaries we give the definition of generalized Kenmotsu manifolds and the full structural equation group. Next, we define Φ- holomorphic generalized Kenmotsu manifolds and Φ-paracontact generalized Kenmotsu manifold give a local characteristic of this subclasses. The Φ-holomorphic generalized Kenmotsu manifold coincides with the class of almost contact metric manifolds obtained from closely cosymplectic manifolds by a canonical concircular transformation of nearly cosymplectic structure. A Φ- paracontact generalized Kenmotsu manifold is a special generalized Kenmotsu manifold of the second kind. An analytical expression is obtained for the tensor of Ф-holomorphic sectional curvature of generalized Kenmotsu manifolds of the pointwise constant Φ-holomorphic sectional curvature. Then we study the axiom of Φ-holomorphic (2r+1)-planes for generalized Kenmotsu manifolds and propose a complete classification of simply connected generalized Kenmotsu manifolds satisfying the axiom of Φ-holomorphic (2r+1)-planes. The main results are as follows. A simply connected GK-manifold of pointwise constant Φ-holomorphic sectional curvature satisfying the axiom of Φ-holomorphic (2r+1)-planes is a Kenmotsu manifold. A GK-manifold satisfies the axiom of Φ-holomorphic (2r+1)-planes if and only if it is canonically concircular to one of the following manifolds: (1) CPn×R; (2) Cn×R; and (3) CHn×R having the canonical cosymplectic structure.

MULTISYMPLECTIC GEOMETRY AND k-COSYMPLECTIC STRUCTURE FOR THE FIELD THEORIES AND THE RELATIVISTIC MECHANICS

The aim of this work is twofold: First, we extend the multisymplectic geometry already done for field theories to the relativistic mechanics by introducing an appropriate configuration bundle. In particular, we developed the model to obtain the Hamilton–De Donder–Weyl equations to the movement of a relativistic charged particle immerged in an electromagnetic field. Second, we have found a direct relationship between the multisymplectic geometry and the k-cosymplectic structure of a physical system.

Poisson Brackets with Prescribed Casimirs

We consider the problem of constructing Poisson brackets on smooth manifolds M with prescribed Casimir functions. If M is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on M, while, in the case where M is of odd dimension, our objective is achieved using a convenient almost cosymplectic structure. Several examples and applications are presented.

On Riemannian manifolds endowed with a locally conformal cosymplectic structure

We deal with a locally conformal cosymplectic manifoldM(φ,Ω,ξ,η,g)admitting a conformal contact quasi-torse-forming vector fieldT. The presymplectic2-formΩis a locally conformal cosymplectic2-form. It is shown thatTis a3-exterior concurrent vector field. Infinitesimal transformations of the Lie algebra of∧Mare investigated. The Gauss map of the hypersurfaceMξnormal toξis conformal andMξ×Mξis a Chen submanifold ofM×M.