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

In this paper, we investigate almost Schouten solitons and almost gradient Schouten solitons in spacetimes of general relativity. At first, it is proven that if a generalized Robertson–Walker spacetime permits an almost Schouten soliton, then it becomes a perfect fluid spacetime as well as the spacetime represents a dark matter era. Besides this, we investigate almost gradient Schouten solitons in generalized Robertson–Walker spacetimes. Moreover, a spacetime obeying almost Schouten solitons whose potential vector field is a non-homothetic conformal vector field is of Petrov type [Formula: see text] or [Formula: see text]. 

Relativistic spacetimes admitting almost Schouten solitons

A conformally flat GRW space-time is a perfect fluid RW space-time. In this note, we investigated the influence of many differential curvature conditions, such as the existence of recurrent and semi-symmetric curvature tensors. In each case, the form of the Ricci curvature tensor, the energy–momentum tensor, the energy density, the pressure of the fluid, and the equation of state are determined and interpreted. For example, it is demonstrated that a Ricci semi-symmetric RW space-time reduces to Einstein space-time or a Ricci recurrent RW space-time, and the perfect fluid space-time is referred to as Yang pure space-time or dark matter era.

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A Note on the Geometry of RW Space-Times

In this short note we prove that there does not exist Ricci soliton $(g,\xi )$ in an almost Kenmotsu manifold.

Non-existence of Ricci solitons in almost Kenmotsu manifolds

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Uday Chand De

Papers

Relativistic spacetimes admitting almost Schouten solitons

A Note on the Geometry of RW Space-Times

Non-existence of Ricci solitons in almost Kenmotsu manifolds

Almost quasi-Yamabe and gradient almost quasi-Yamabe solitons on quasi-Sasakian manifolds

Static perfect fluid spacetimes on GRW spacetimes