The aim of the paper is to define a k-cosymplectic structure on the standard k-cosymplectic manifold associated to a regular Lagrangian and to reduce it via Marsden-Weinstein reduction.

Identities for the curvature tensor of the Levi-Cività connection on an almost para-cosymplectic manifold are proved. Elements of harmonic theory for almost product structures are given and a Bochner-type formula for the leaves of the canonical foliation is established.

The object of the present paper is to investigate the nature of Ricci solitons on D-homothetically deformed Kenmotsu manifold with generalized weakly symmetric and generalized weakly Ricci symmetric curvature restrictions.

In the present paper we give some properties of $f$-biharmonic hypersurfaces in real space forms. By using the $f$-biharmonic equation for a hypersurface of a Riemannian manifold, we characterize the $f$-biharmonicity of constant mean curvature and totally umbilical hypersurfaces in a Riemannian manifold and, in particular, in a real space form. As an example, we consider $f$-biharmonic vertical cylinders in ${S}^{2}\times \mathbb{R}$.

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