Uday Chand De

Bio: Uday Chand De is an academic researcher from University of Calcutta. The author has contributed to research in topics: Ricci curvature & Riemann curvature tensor. The author has an hindex of 24, co-authored 225 publications receiving 2257 citations. Previous affiliations of Uday Chand De include Yahoo! & Uludağ University.

Riemann solitons on almost co-Kähler manifolds

Abstract: The aim of the present paper is to characterize almost co-K?hler manifolds whose metrics are the Riemann solitons. At first we provide a necessary and sufficient condition for the metric of a 3-dimensional manifold to be Riemann soliton. Next it is proved that if the metric of an almost co-K?hler manifold is a Riemann soliton with the soliton vector field ?, then the manifold is flat. It is also shown that if the metric of a (?, ?)-almost co-K?hler manifold with ? < 0 is a Riemann soliton, then the soliton is expanding and ?, ?, ? satisfies a relation. We also prove that there does not exist gradient almost Riemann solitons on (?, ?)-almost co-K?hler manifolds with ? < 0. Finally, the existence of a Riemann soliton on a three dimensional almost co-K?hler manifold is ensured by a proper example.

Mixed Generalized Quasi-Einstein Warped Product Manifolds

Abstract: In this paper, warped product mixed generalized quasi-Einstein manifolds MG(QE) $$_{n}$$ are studied. It is shown that a Ricci recurrent MG(QE) $$_{n}$$ manifold is a product manifold whose factors are Ricci recurrent. Sufficient conditions for a MG(QE) $$_{n}$$ to be a G(QE) $$_{n}$$ are derived. Then, we classify three types of warped product MG(QE) $$_{n}$$ . For example, it is proved that the fiber manifold is Einstein in the first type, (QE) $$_{n}$$ in the second type and MG(QE) $$_{n}$$ in the third type.

A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices

Abstract: 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.

Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions

Abstract: ,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