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.

Certain results on a type of contact metric manifold

Abstract: Let \(M\) be a \(3\)-dimensional almost contact metric manifold satisfying \((*)\) condition. We denote such a manifold by \(M^{*}\). At first we study symmetric and skew-symmetric parallel tensor of type \((0,2)\) in \(M^{*}\). Next we prove that a non-cosymplectic manifold \(M^{*}\) is Ricci semisymmetric if and only if it is Einstein. Also we study locally \(\phi \)-symmetry and \(\eta \)-parallel Ricci tensor of \(M^{*}\). Finally, we prove that if a non-cosymplectic \(M^{*}\) is Einstein, then the manifold is Sasakian.

On almost Kenmotsu manifolds with nullity distributions

Abstract: The object of this paper is to characterize the curvature conditions $R\cdot P=0$ and $P\cdot S=0$ with its characteristic vector field $\xi$ belonging to the $(k,\mu)'$-nullity distribution and $(k,\mu)$-nullity distribution respectively, where $P$ is the Weyl projective curvature tensor. As a consequence of the main results we obtain several corollaries.

Pseudo B-symmetric manifolds

Abstract: In this paper, we introduce a new tensor named B-tensor which generalizes the Z-tensor introduced by Mantica and Suh [Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-B-symmetric manifolds (PBS)n which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a (PBS)n. Next, we prove that a pseudo-Riemannian manifold is B-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a (PBS)n to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a (PBS)n if the B-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-B-symmetric manifolds and prove that a (PBS)n(n > 3) spacetime is a pp-wave under certain conditions.

Spheres and Euclidean Spaces Via Concircular Vector Fields

Abstract: In this paper, we exhibit that non-trivial concircular vector fields play an important role in characterizing spheres, as well as Euclidean spaces. Given a non-trivial concircular vector field $$\xi $$ on a connected Riemannian manifold (M, g), two smooth functions $$\sigma $$ and $$\rho $$ called potential function and connecting function are naturally associated to $$\xi $$ . We use non-trivial concircular vector fields on n-dimensional compact Riemannian manifolds to find four different characterizations of spheres $$ {\mathbf {S}}^{n}(c)$$ . In particular, we prove an interesting result namely an n-dimensional compact Riemannian manifold (M, g) that admits a non-trivial concircular vector field $$\xi $$ such that the Ricci operator is invariant under the flow of $$\xi $$ , if and only if, (M, g) is isometric to a sphere $$ {\mathbf {S}}^{n}(c)$$ . Similarly, we find two characterizations of Euclidean spaces $${\mathbf {E}}^{n}$$ . In particular, we show that an n-dimensional complete and connected Riemannian manifold (M, g) admits a non-trivial concircular vector field $$\xi $$ that annihilates the Ricci operator, if and only if, (M, g) is isometric to the Euclidean space $${\mathbf {E}}^{n}$$ .

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