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

In this paper, it is shown that for a 3-dimensional compact simply connected trans-Sasakian manifold of type ${(\alpha,\beta)}$, the smooth functions ${\alpha,\beta}$ satisfy the Poisson equations ${\Delta \alpha = \beta}$, ${\Delta \alpha = \alpha ^{2}\beta}$ and ${\Delta \beta = \alpha ^{2}\beta}$, respectively, if and only if it is homothetic to a Sasakian manifold. We also find a necessary and sufficient condition for a connected 3-dimensional trans-Sasakian manifold of type ${(\alpha,\beta)}$ in terms of a differential equation satisfied by the smooth function ${\alpha}$ to be homothetic to a Sasakian manifold.

Trans-Sasakian manifolds homothetic to Sasakian manifolds

The object of the present paper is to study N()-quasi Einstein manifolds. Existence of N()-quasi Einstein manifolds are proved. Physical example of N()-quasi Einstein manifold is also given. Finally, Weyl-semisymmetric N()-quasi Einstein manifolds have been considered.

ON A CLASS OF N(κ)-QUASI EINSTEIN MANIFOLDS

Generalized Robertson–Walker (GRW) spacetime is the generalization of the Robertson–Walker (RW) spacetime and a further generalization of GRW spacetime is the twisted spacetime. In this paper, we g...

Some characterizations of Lorentzian manifolds

Spacetimes with different forms of energy-momentum tensor

The object of the present paper is to study almost pseudo-Z-symmetric manifolds. Some geometric properties have been studied. Next we consider conformally flat almost pseudo-Z-symmetric manifolds. We obtain a sufficient condition for an almost pseudo-Z-symmetric manifold to be a quasi Einstein manifold. Also we prove that a totally umbilical hypersurface of a conformally flat $A(PZS)_{n}$ ($n>3$) is a manifold of quasi constant curvature. Finally, we give an example to verify the result already obtained in Section 5.

Uday Chand De

Papers

Trans-Sasakian manifolds homothetic to Sasakian manifolds

ON A CLASS OF N(κ)-QUASI EINSTEIN MANIFOLDS

Some characterizations of Lorentzian manifolds

Spacetimes with different forms of energy-momentum tensor

On Almost Pseudo-Z-symmetric Manifolds