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.

On 3-dimensional almost Kenmotsu manifolds admitting certain nullity distribution

Abstract: The aim of this paper is to characterize 3-dimensional almost Kenmotsu manifolds with belonging to the (k;\mu )'-nullity distribution and h' eq 0 satisfying certain geometric conditions. Finally, we give an example to verify some results.

A note on almost Ricci soliton and gradient almost Ricci soliton on para-Sasakian manifolds

Abstract: The object of the offering exposition is to study almost Ricci soliton and gradient almost Ricci soliton in 3-dimensional para-Sasakian manifolds. At first, it is shown that if $(g, V,\lambda)$ be an almost Ricci soliton on a 3-dimensional para-Sasakian manifold $M$, then it reduces to a Ricci soliton and the soliton is expanding for $\lambda$=-2. Besides these, in this section, we prove that if $V$ is pointwise collinear with $\xi$, then $V$ is a constant multiple of $\xi$ and the manifold is of constant sectional curvature $-1$. Moreover, it is proved that if a 3-dimensional para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature $-1$ or it reduces to a gradient Ricci soliton. Finally, we consider an example to justify some results of our paper.

Classification of Ricci Semisymmetric Contact Metric Manifolds

Abstract: The object of the present paper is to study Ricci semisymmetric contact metric manifolds. As a consequence of the main result we deduce some important corollaries. Besides these we study contact metric manifolds satisfying the curvature condition Q.R = 0, where Q and R denote the Ricci operator and curvature tensor respectively. Also we study the symmetric properties of a second order parallel tensor in contact metric manifolds. Finally, we give an example to verify the main result.

Almost co-Kähler manifolds and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg" display="inline" id="d1e23"><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-quasi-Einstein solitons

Abstract: The present paper aims to investigate (m,ρ)-quasi-Einstein metrics on almost co-Kähler manifolds M. It is proven that if a (κ,μ)-almost co-Kähler manifold with κ<0 is (m,ρ)-quasi-Einstein manifold, then M represents a N(κ)-almost co-Kähler manifold and the manifold is locally isomorphic to a solvable non-nilpotent Lie group. Next, we study the three dimensional case and get the above mentioned result along with the manifold M3 becoming an η-Einstein manifold. We also show that there does not exist (m,ρ)-quasi-Einstein structure on a compact (κ,μ)-almost co-Kähler manifold of dimension greater than three with κ<0. Further, we prove that an almost co-Kähler manifold satisfying η-Einstein condition with constant coefficients reduces to a K-almost co-Kähler manifold, provided ma1≠(2n−1)b1 and m≠1. We also characterize perfect fluid spacetime whose Lorentzian metric is equipped with (m,ρ)-quasi Einstein solitons and acquired that the perfect fluid spacetime has vanishing vorticity, or it represents dark energy era under certain restriction on the potential function. Finally, we construct an example of an almost co-Kähler manifold with (m,ρ)-quasi-Einstein solitons.

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