The purpose of this article is to derive an integral inequality of Ricci curvature with respect to Reeb field in a Finsler space and give a new geometric characterization of Finsler metrics with constant flag curvature 1.

Integral formulas in riemannian geometry

A new characterization of Finsler metrics with constant flag curvature 1

The purpose of the paper is to study *-Ricci solitons and *-gradient Ricci solitons on three-dimensional normal almost contact metric manifolds. First, we prove that if a non-cosymplectic normal almost contact metric manifold with α, β = constant of dimension three admits a *-Ricci soliton, then the manifold is *-Ricci flat, provided β ≠ 0 and α ≠ ±β. Further, we prove that if a normal almost contact metric manifold with α, β = constant, of dimension three admits *-gradient Ricci soliton, then the manifold is *-Einstein, provided α2 − β2 ≠ 0.

*-Ricci Solitons on Three-dimensional Normal Almost Contact Metric Manifolds

We prove that if the metric of an η-Einstein contact metric manifold is a non-trivial Ricci soliton, then it is K-contact.

Notes on contact η-Einstein metrics as Ricci solitons

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,\rho)$-quasi-Einstein structure on a contact metric manifold. First, we prove that if a $K$-contact or Sasakian manifold $M^{2n+1}$ admits a closed $(m,\rho)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,\mu)$-contact structure -- it is locally isometric to the product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant curvature $4$. Next, we prove that if a compact contact or $H$-contact metric manifold admits an $(m,\rho)$-quasi-Einstein structure, whose potential vector field $V$ is collinear to the Reeb vector field, then it is a $K$-contact $\eta$-Einstein manifold.

On non-gradient $(m,\rho)$-quasi-Einstein contact metric manifolds.

An almost cosymplectic manifold M is a ( 2 m + 1 ) -dimensional oriented Riemannian
manifold endowed with a 2-form Ω of rank 2 m , a 1-form η such that Ω m  Λ  η ≠ 0 and a vector field ξ 
satisfying i ξ Ω = 0 and η ( 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
field T such that the Riemannian connection is parallel with respect to T . It is shown that in this case
 M is a hyperbolic space form endowed with an exact locally conformal cosymplectic structure. Moreover
 T defines an infinitesimal homothety of the connection forms and a relative infinitesimal conformal
transformation of the curvature forms.The existence of a structure conformal vector field C on M is 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.

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Integral formulas in riemannian geometry

Citations

A new characterization of Finsler metrics with constant flag curvature 1

*-Ricci Solitons on Three-dimensional Normal Almost Contact Metric Manifolds

Notes on contact η-Einstein metrics as Ricci solitons

On non-gradient $(m,\rho)$-quasi-Einstein contact metric manifolds.

On a class of exact locally conformal cosymlectic manifolds

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