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 study a type of Riemannian manifold called generalized quasi Einstein manifold

On Generalized Quasi Einstein Manifolds

On generalized ricci-recurrent manifolds

. The objective of the present paper is to study φ –recurrentSasakian manifolds. AMS Mathematics Subject Classiﬁcation (2000): 53C05, 53C20, 53C25 Key words and phrases: φ –recurrent Sasakian manifold, Einstein mani-fold, sectional curvature 1. Introduction In 1977, T. Takahashi [2] introduced the notion of locally φ –symmetric Sasa-kian manifolds and studied their several interesting properties. In this paperwe introduce the notion of φ –recurrent Sasakian manifolds which generalizesthe notion of locally φ –symmetric Sasakian manifolds. After preliminaries, inSection 3, we study φ –recurrent Sasakian manifolds and show that such a man-ifold is always an Einstein manifold. Again, it is proved that in a φ –recurrentSasakian manifold, the characteristic vector ﬁeld ξ and the vector ﬁeld ρ asso-ciated to the 1–form A are codirectional. Also we obtain some other interestingresults of this manifold. 2. Preliminaries Let M 2 n +1 ( φ,ξ,η,g ) be a Sasakian manifold with the structure (

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On -recurrent Sasakian manifolds.

A perfect-fluid space-time of dimension n ≥ 4, with (1) irrotational velocity vector field and (2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with an 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

where 21 is a non-zero vector and comma denotes covariant differentiation with respect to the metric tensor of the space. Such a space was called by him a pseudo symmetric space and 2t was called its associated vector. An n-space of this kind was denoted by (PS),. The name pseudo symmetric was chosen, because if in (1) 2t is taken as zero, then the equation (1) takes the form Rhijk.t=0 and the space reduces to a symmetric space in the sense of Caftan. In the present paper some results on a (PS), are established. In Section 3 it is shown that if a (PS), is a decomposable space V~• (r_->2, n-r>=2), then one of the decomposition spaces is flat and the other is a pseudo symmetric space. In Section 4 the Ricci-associate of the vector field 21 is defined and some theorems relating to it are proved. Section 5 deals with (PS), (n>3) having cyclic Ricci tensor. The last section is concerned with (PS), admitting a concurrent or a recurrent vector field [4].

Uday Chand De

Papers

On Generalized Quasi Einstein Manifolds

On generalized ricci-recurrent manifolds

On -recurrent Sasakian manifolds.

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

On pseudo symmetric spaces