Almost Cosymplectic $$(k,\mu )$$-metrics as $$\eta$$-Ricci Solitons

In this paper, we study $$\eta$$ η -Ricci solitons on almost cosymplectic $$(k,\mu )$$ ( k , μ ) -manifolds. As an application, it is proved that if an almost cosymplectic $$(k,\mu )$$ ( k , μ ) -metric with $$k<0$$ k < 0 represents a Ricci soliton, then the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and the corresponding almost cosymplectic manifold is locally isometric to a Lie group whose local structure is determined completely by $$k<0$$ k < 0 . In addition, a concrete example is constructed to illustrate the above result.

Locally Conformal Almost Cosymplectic Manifold of Φ-holomorphic Sectional Conharmonic Curvature Tensor

The aim of the present paper is to study the geometry of locally conformal almost cosymplectic manifold of Φ-holomorphic sectional conharmonic curvature tensor. In particular, the necessaryand sucient conditions in which that locally conformal almost cosymplectic manifold is a manifold of point constant Φ-holomorphic sectional conharmonic curvature tensor have been found. The relation between the mentioned manifold and the Einstein manifold is determined.

Curvature inequalities for C-totally real doubly warped products of locally conformal almost cosymplectic manifolds

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise ?-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some applications of obtained results are derived.

TRANSVERSAL HARMONIC THEORY FOR TRANSVERSALLY SYMPLECTIC FLOWS

We develop the transversal harmonic theory for a transversally symplectic flow on a manifold and establish the transversal hard Lefschetz theorem. Our main results extend the cases for a contact manifold (H. Kitahara and H. K. Pak, ‘A note on harmonic forms on a compact manifold’, Kyungpook Math. J.43 (2003), 1–10) and for an almost cosymplectic manifold (R. Ibanez, ‘Harmonic cohomology classes of almost cosymplectic manifolds’, Michigan Math. J.44 (1997), 183–199). For the point foliation these are the results obtained by Brylinski (‘A differential complex for Poisson manifold’, J. Differential Geom.28 (1988), 93–114), Haller (‘Harmonic cohomology of symplectic manifolds’, Adv. Math.180 (2003), 87–103), Mathieu (‘Harmonic cohomology classes of symplectic manifolds’, Comment. Math. Helv.70 (1995), 1–9) and Yan (‘Hodge structure on symplectic manifolds’, Adv. Math.120 (1996), 143–154).

Inequality for Ricci curvature of certain submanifolds in locally conformal almost cosymplectic manifolds

We establish inequalities between the Ricci curvature and the squared mean curvature, and also between thek-Ricci curvature and the scalar curvature for a slant, semi-slant, and bi-slant submanifold in a locally conformal almost cosymplectic manifold with arbitrary codimension.

On a class of exact locally conformal cosymlectic manifolds

An almost cosymplectic manifoldMis a(2m+1)-dimensional oriented Riemannian manifold endowed with a 2-formΩof rank2m, a 1-formηsuch thatΩm Λ η≠0and a vector fieldξsatisfyingiξΩ=0andη(ξ)=1. Particular cases were considered in [3] and [6].Let(M,g)be an odd dimensional oriented Riemannian manifold carrying a globally defined vector fieldTsuch that the Riemannian connection is parallel with respect toT. It is shown that in this caseMis a hyperbolic space form endowed with an exact locally conformal cosymplectic structure. MoreoverTdefines an infinitesimal homothety of the connection forms and a relative infinitesimal conformal transformation of the curvature forms.The existence of a structure conformal vector fieldConMis proved and their properties are investigated. In the last section, we study the geometry of the tangent bundle of an exact locally conformal cosymplectic manifold.