In this paper we introduce metric-based means for the space of positive-definite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a characterization of this mean.

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A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Integral formulas in Riemannian geometry

,by Darryl D. Holm, Tanya Schmah and Cristina Stoica, Oxford University Press,Oxford, 2009, xi + 515 pp., ISBN: 978-0-19-921290-3The purpose of the book is to provide the unifying viewpoint of Lagrangian andHamiltonian mechanics in the coordinate-free language of differential geometryin the spirit of the Marsden-Ratiu school. The book is similar in content - althoughless formal - to the book by J. Marsden and T. Ratiu [7]. One can also mentionthe companion two-volumes book by Holm [4,5] written at a more basic level,and that one can recommend as an introductory reading. The classical treatises onthe subject are the books by Abraham-Marsden [1], Arnold [2] and Libermann-Marle [6].Typical applications are N-particle systems, rigid bodies, continua such as u-ids and electromagnetic systems that illustrate the powerfulness of the adoptedpoint of view. The geometrical structure allows the covering of both the nite-dimensional conservative case (rst part of the book) and the innite dimensionalsituation in the second part. The notion of symmetry here is central, as it allowsa reduction of the number of dimensions of the mechanical systems, and furtherexploits the conserved quantities (momentum map) associated to symmetry. Liegroup symmetries, Poisson reduction and momentum maps are rst discussed.The concepts are introduced in a progressive and clear manner in the rst part ofthe book. The second part devoted to innite dimensional systems is motivatedby the identication of Euler’s ideal uid motion with the geodesic o w on thegroup of volume-preserving diffeomorphism. The Euler-PoincarO (EP) variationalprinciple for the Euler uid equations is exposed in the framework of geometricmechanics, in association with Lie-Poisson Hamiltonian structure of Noether’stheorem and momentum maps. Original applications of the Euler-PoincarO equa-tions to solitons, computational anatomy, image matching, or geophysical uiddynamics are given at the end of the second part of the book.Here the rst chapter recapitulates the Newtonian, Lagrangian and Hamiltonian117

Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions

Exact Solutions of Einstein's Field Equations: Notation

The object of the present paper is to obtain a necessary and sufficient condition for a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold to be locally $\phi $-symmetric in the sense of Takahashi and the condition is verified by an example. Next we characterize a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold satisfying certain curvature conditions on the concircular curvature tensor. Finally, we construct an example of a generalized $(\kappa,\mu)$-contact metric manifold to verify Theorem $1$ of our paper.

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On some classes of $3$-dimensional generalized $ (\kappa ,\mu )$-contact metric manifolds

In this article, we classify h-almost Ricci-Yamabe solitons in paracontact geometry. In particular, we characterize para-Kenmotsu manifolds satisfying h-almost Ricci-Yamabe solitons and 3-dimensional para-Kenmotsu manifolds obeying h-almost gradient Ricci-Yamabe solitons. Next, we classify para-Sasakian manifolds admitting h-almost Ricci-Yamabe solitons and h-almost gradient Ricci-Yamabe solitons. Besides these, we investigate h-almost Ricci-Yamabe solitons and h-almost gradient Ricci-Yamabe solitons in para-cosymplectic manifolds. Finally, we construct two examples to illustrate our results.

H-almost Ricci-Yamabe solitons in paracontact geometry

We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

On Interpolating Sesqui-Harmonic Legendre Curves in Sasakian Space Forms

The prime purpose of this paper is to characterize the f ( R ) -gravity, when Ricci solitons, gradient Ricci solitons and gradient Yamabe solitons, respectively are its metrics. At ﬁrst, the existence of gradient Yamabe solitons is established by some non-trivial examples. We prove that if ( g , ρ, λ) be a Ricci soliton in a perfect ﬂuid spacetime with constant Ricci scalar in f ( R ) -gravity, then the perfect ﬂuid spacetime represents a dark matter era under certain restrictions on the vector ﬁeld ρ . Besides, we investigate gradient Ricci solitons and gradient Yamabe solitons in f ( R ) -gravity, respectively. As a result, we acquire some signiﬁcant theorems. Finally, a ( t ) , the scale factor is obtained when spatially ﬂat Robertson-Walker spacetime admits Ricci solitons and gradient Yamabe solitons.

Investigations on solitons in f ( R ) -gravity

Let $M$ be a $3$-dimensional almost contact metric manifold satisfying $(*)$ condition. We denote such a manifold by $M^{*}$. At first we study symmetric and skew-symmetric parallel tensor of type $(0,2)$ in $M^{*}$. Next we prove that a non-cosymplectic manifold $M^{*}$ is Ricci semisymmetric if and only if it is Einstein. Also we study locally $\phi $-symmetry and $\eta $-parallel Ricci tensor of $M^{*}$. Finally, we prove that if a non-cosymplectic $M^{*}$ is Einstein, then the manifold is Sasakian.

Uday Chand De

Papers

On some classes of $3$-dimensional generalized $ (\kappa ,\mu )$-contact metric manifolds

H-almost Ricci-Yamabe solitons in paracontact geometry

On Interpolating Sesqui-Harmonic Legendre Curves in Sasakian Space Forms

Investigations on solitons in f ( R ) -gravity

Certain results on a type of contact metric manifold