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

SOME PROPERTlES OF A SEMI-SYMMETRIC METRIC CONNECTION ON A RIEMANNIAN MANIFOLD

We study pseudo symmetric (brie∞y (PS)n) and pseudo Ricci symmetric (brie∞y (PRS)n) warped product manifolds M £F N. If M is (PS)n, then we give a condition on the warping function that M is a pseudosymmetric space and N is a space of constant curvature. If M is (PRS)n, then we show that (i) N is Ricci symmetric and (ii) M is (PRS)n if and only if the tensor T deflned by (2.6) satisfles a certain condition.

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Pseudo symmetric and pseudo ricci symmetric warped product manifolds

A perfect-fluid space-time of dimension n>3 with 1) irrotational velocity vector field, 2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with Einstein fiber. Condition 1) is verified whenever pressure and energy density are related by an equation of state. The contraction of the Weyl tensor with the velocity vector field is zero. Conversely, a generalized Robertson-Walker space-time with null divergence of the Weyl tensor is a perfect-fluid space-time.

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A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time

In this paper we prove that a Φ-recurrent N(k)-contact metric manifold is an η-Einstein manifold with constant coefficients. Next, we prove that a 3-dimensional Φ-recurrent N(k)-contact metric manifold
is of constant curvature. The existence of a Φ-recurrent N(k)-contact metric manifold is also proved.

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On Φ-recurrent N(k)-contact Metric Manifolds

The object of the present paper is to study a type of non-conformally flat Riemannian manifold called almost pseudo conformally symmetric manifold. The existence of an almost pseudo conformally symmetric manifold is also shown by a non-trivial concrete example.

Uday Chand De

Papers

SOME PROPERTlES OF A SEMI-SYMMETRIC METRIC CONNECTION ON A RIEMANNIAN MANIFOLD

Pseudo symmetric and pseudo ricci symmetric warped product manifolds

A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time

On Φ-recurrent N(k)-contact Metric Manifolds

On almost pseudo conformally symmetric manifolds