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 prime object of this paper is to study the perfect fluid spacetimes obeying [Formula: see text]-gravity, when [Formula: see text]-Ricci solitons, gradient [Formula: see text]-Ricci solitons, gradient Einstein solitons and gradient [Formula: see text]-quasi Einstein solitons are its metrics. At first, the existence of the [Formula: see text]-Ricci solitons is proved by a non-trivial example. We establish conditions for which the [Formula: see text]-Ricci solitons are expanding, steady or shrinking. Besides, in the perfect fluid spacetimes obeying [Formula: see text]-gravity, when the potential vector field of [Formula: see text]-Ricci soliton is of gradient type, we acquire a Poisson equation. Moreover, we investigate gradient [Formula: see text]-Ricci solitons, gradient Einstein solitons and gradient [Formula: see text]-quasi Einstein solitons in [Formula: see text]-gravity, respectively. As a result, we establish some significant theorems about dark matter era. 

Characterizations of Perfect Fluid Spacetimes Obeying f(ℛ)-Gravity Equipped with Different Gradient Solitons

The object of the present paper is to prove that in a Kaehler manifold of dimension n ≥ 4, div R = 0 and div C = 0 are equivalent, where ’div’ denotes divergence and R and C denote the curvature tensor and Weyl conformal curvature tensor, respectively.

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A Note on Kaehler Manifolds

Almost Schouten solitons and almost cosymplectic manifolds

The aim of the present article is to analyze three-dimensional quasi-Sasakian
 manifolds admitting Riemann solitons and Ricci Bourguignon solitons.

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Riemann and Ricci bourguignon solitons on three-dimensional quasi-Sasakian manifolds

The object of the present paper is to study pseudosymmetric,Weyl-pseudosymmetric and Ricci-pseudosymmetric \(LP\)-Sasakian manifolds.

Uday Chand De

Papers

Characterizations of Perfect Fluid Spacetimes Obeying f(ℛ)-Gravity Equipped with Different Gradient Solitons

A Note on Kaehler Manifolds

Almost Schouten solitons and almost cosymplectic manifolds

Riemann and Ricci bourguignon solitons on three-dimensional quasi-Sasakian manifolds

On a class of pseudosymmetric LP-Sasakian manifolds