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 is a survey article on weakly symmetric structures on a Riemannian manifold. The study of weakly symmetric and weakly projective symmetric Riemannian manifold were initiated by Tamassy and Binh. Later on several authors studied weakly symmetric Riemannian manifold and analogous structures, viz. weakly Ricci symmetric, weakly projective symmetric and weakly conformally symmetric Riemannian manifolds. Here we present a brief survey of results on weakly symmetric structures on a Riemannian manifold and some applications in the theory of Relativity.

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On weakly symmetric structures on a Riemannian manifold

The aim of this paper is to characterize $3$-dimensional $N(k)$-paracontact metric manifolds satisfying certain curvature conditions. We prove that a $3$-dimensional $N(k)$-paracontact metric manifold $M$ admits a Ricci soliton whose potential vector field is the Reeb vector field $xi$ if and only if the manifold is a paraSasaki-Einstein manifold. Several consequences of this result are discussed. Finally, an illustrative example is constructed.

http://bims.iranjournals.ir/article_1040_a446023e2cc7956949c8c04a16793c63.pdf

On three-dimensional $N(k)$-paracontact metric manifolds and Ricci solitons

The object of the present paper is to study N(k)-quasi Einstein manifolds satisfy- ing certain curvature conditions. Two examples have been constructed to prove the existence of such a manifold. Finally, a physical example of an N(k)-quasi Einstein manifold is given.

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N(k)-quasi einstein manifolds satisfying certain curvature conditions

The purpose of this paper is to investigate the Fischer-Marsden conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional non-Kenmotsu (k, µ)′-almost Kenmotsu manifold sa...

The Fischer-Marsden conjecture on almost Kenmotsu manifolds

The object of the present paper is to study a transformation called the D-homothetic deformation of normal almost contact metric manifolds. In particular, it is shown that, in a (2n+1)-dimensional normal almost contact metric manifold, the Ricci operator Q commutes with the structure tensor ϕ under certain conditions, and the operator Qϕ – ϕQ is invariant under a D-homothetic deformation. We also discuss the invariance of η-Einstein manifolds, ϕ-sectional curvature, and the local ϕ-Ricci symmetry under the D-homothetic deformation. Finally, we prove the existence of these manifolds by a concrete example.

/pdf/d-homothetic-deformation-of-normal-almost-contact-metric-1x8o8hp0d2.pdf

Uday Chand De

Papers

On weakly symmetric structures on a Riemannian manifold

On three-dimensional $N(k)$-paracontact metric manifolds and Ricci solitons

N(k)-quasi einstein manifolds satisfying certain curvature conditions

The Fischer-Marsden conjecture on almost Kenmotsu manifolds

D -Homothetic Deformation of Normal Almost Contact Metric Manifolds