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

In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that spacetimes are Ricci flat in trivial, , and subspaces, whereas perfect fluid in subspaces , , and , and have zero scalar curvature in subspace . Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.

/pdf/investigation-of-pseudo-ricci-symmetric-spacetimes-in-gray-s-1nv8cn6jjd.pdf

Investigation of Pseudo-Ricci Symmetric Spacetimes in Gray’s Subspaces

CONTACT RIEMANNIAN MANIFOLDS SATISFYING R (ksi, X). C= 0

Investigations on solitons in $$f({\mathcal {R}})$$-gravity

In this paper we study the properties of conformally recur- rent pseudo Riemannian manifolds admitting a proper conformal vector field with respect to the scalar field �, focusing particularly on the 4- dimensional Lorentzian case. Some general properties already proven by one of the present authors for pseudo conformally symmetric manifolds endowed with a conformal vector field are proven also in the case, and some new others are stated. Moreover interesting results are pointed out; for example, it is proven that the Ricci tensor under certain conditions is Weyl compatible: this notion was recently introduced and investigated by one of the present authors. Further we study conformally recurrent 4-dimensional Lorentzian manifolds (space-times) admitting a conformal vector field: it is proven that the covectorj is null and unique up to scaling; moreover it is shown that the same vector is an eigenvector of the Ricci tensor. Finally, it is stated that such space-time is of Petrov type N with respect toj.

http://koreascience.or.kr:80/article/JAKO201414938219890.pdf

Conformally recurrent space-times admitting a proper conformal vector field

We study the critical point equation $(CPE)$ conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional $(k,\mu)'$-almost Kenmotsu manifold satisfies the $CPE,$ then the manifold is either locally isometric to the product space $\mathbb H^2(-4)\times\mathbb R$ or the manifold is Kenmotsu manifold. Further, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation satisfies the $CPE$ conjecture, then the manifold is Einstein.

Uday Chand De

Papers

Investigation of Pseudo-Ricci Symmetric Spacetimes in Gray’s Subspaces

CONTACT RIEMANNIAN MANIFOLDS SATISFYING R (ksi, X). C= 0

Investigations on solitons in $$f({\mathcal {R}})$$-gravity

Conformally recurrent space-times admitting a proper conformal vector field

Critical Point Equation on Almost Kenmotsu Manifolds