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.

/pdf/a-differential-geometric-approach-to-the-geometric-mean-of-50vfibj9i9.pdf

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

This study aims mainly at investigating the effects of concircular flatness and concircular symmetry of a warped product manifold on its fibre and base manifolds. Concircularly flat and concircularly symmetric warped product manifolds are investigated. The divergence free concircular curvature tensor on warped product manifolds is considered. Finally, we apply some of these results to generalized Robertson-Walker and standard static space-times.

/pdf/concircular-curvature-on-warped-product-manifolds-and-4xjldogojc.pdf

Concircular curvature on warped product manifolds and applications.

The object of the present paper is to characterize 𝑁(𝑘)-contact metric manifolds satisfying certain curvature conditions on the conharmonic curvature tensor. In this paper we study conharmonically symmetric, 𝜉-conharmonically flat, and 𝜙-conharmonically flat 𝑁(𝑘)-contact metric manifolds.

/pdf/conharmonic-curvature-tensor-on-contact-metric-manifolds-3wu60ngbii.pdf

Conharmonic Curvature Tensor on ()-Contact Metric Manifolds

The object of the present paper is to study mixed quasi-Einstein manifolds. Some geometric properties of mixed quasi-Einstein manifolds have been studied. We also discuss M(QE)4 spacetime with spac...

Characterizations of mixed quasi-Einstein manifolds

The purpose of the offering exposition is to characterize gradient Yamabe, gradient Einstein and gradient m-quasi Einstein solitons within the framework of 3-dimensional para-Kenmotsu manifolds. Fi...

A note on gradient solitons on para-Kenmotsu manifolds

We study locally and globally \phi -concircularly symmetric Kenmotsu manifolds. At first we show that in a Kenmotsu manifold globally \phi -symmetry and globally \phi -concircularly symmetry are equivalent. Next we study 3-dimensional locally \phi -concircularly symmetric Kenmotsu manifolds. Finally, we give some examples of \phi -concircularly symmetric Kenmotsu manifolds.

Uday Chand De

Papers

Concircular curvature on warped product manifolds and applications.

Conharmonic Curvature Tensor on ()-Contact Metric Manifolds

Characterizations of mixed quasi-Einstein manifolds

A note on gradient solitons on para-Kenmotsu manifolds

On \phi-Concircularly Symmetric Kenmotsu Manifolds