scholarly journals On The Formulation of Bianchi Identity from Action Principle

2021 ◽  
Author(s):  
Shiladittya Debnath
Keyword(s):  
Curvature Tensor ◽  
Basic Property ◽  
Covariant Derivative ◽  
Bianchi Identity ◽  
Action Principle ◽  
Lie Derivative ◽  
General Covariance ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metric Tensor

Abstract In this letter, we investigate the basic property of the Hilbert-Einstein action principle and its infinitesimal variation under suitable transformation of the metric tensor. We find that for the variation in action to be invariant, it must be a scalar so as to obey the principle of general covariance. From this invariant action principle, we eventually derive the Bianchi identity (where, both the 1st and 2nd forms are been dissolved) by using the Lie derivative and Palatini identity. Finally, from our derived Bianchi identity, splitting it into its components and performing cyclic summation over all the indices, we eventually can derive the covariant derivative of the Riemann curvature tensor. This very formulation was first introduced by S Weinberg in case of a collision less plasma and gravitating system. We derive the Bianchi identity from the action principle via this approach; and hence the name ‘Weinberg formulation of Bianchi identity’.

scholarly journals Dislocation Density Tensor of Thin Elastic Shells at Finite Deformation

2018 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Dislocation Density ◽  
Curvature Tensor ◽  
Finite Deformation ◽  
Middle Surface ◽  
Gauss Curvature ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metric Tensor ◽  
Elastic Shells ◽  
Dislocation Density Tensor

The dislocation density tensors of thin elastic shells have been formulated explicitly in terms of the Riemann curvature tensor. The formulation reveals that the dislocation density of the shells is  proportional to KA3=2, where K is the Gauss curvature and A is the determinant of metric tensor ofthe middle surface.

scholarly journals Geometry of Nonholonomic Kenmotsu Manifolds

2021 ◽  
pp. 84-87
Author(s):  
A.V. Bukusheva
Keyword(s):  
Curvature Tensor ◽  
Einstein Manifold ◽  
Lie Derivative ◽  
Riemann Curvature Tensor ◽  
Intrinsic Geometry ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Schouten Tensor ◽  
Kenmotsu Manifolds ◽  
Formal Properties

The concept of the intrinsic geometry of a nonholonomic Kenmotsu manifold M is introduced. It is understood as the set of those properties of the manifold that depend only on the framing  of the D^ distribution D of the manifold M, on the parallel transformation of vectors belonging to the distribution D along curves tangent to this distribution. The invariants of the intrinsic geometry of the nonholonomic Kenmotsu manifold are: the Schouten curvature tensor; 1-form η generating the distribution D; the Lie derivative  of the metric tensor g along the vector field ; Schouten — Wagner tensor field P, whose components in adapted coordinates are expressed using the equalities . It is proved that, as in the case of the Kenmotsu manifold, the Schouten — Wagner tensor of the manifold M vanishes. It follows that the Schouten tensor of a nonholonomic Kenmotsu manifold has the same formal properties as the Riemann curvature tensor. It is proved that the alternation of the Ricci — Schouten tensor coincides with the differential of the structural form. This property of the Ricci — Schouten tensor is used in the proof of the main result of the article: a nonholonomic Kenmotsu manifold cannot carry the structure of an η-Einstein manifold.

scholarly journals Projective-symmetric spaces

1967 ◽  
Vol 7 (1) ◽  
pp. 48-54 ◽  
Author(s):  
R. F. Reynolds ◽  
A. H. Thompson
Keyword(s):  
Curvature Tensor ◽  
Covariant Derivative ◽  
Symmetric Spaces ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor ◽  
Riemannian Spaces

Gy. Soos [1] and B. Gupta [2] have discussed the properties of Riemannian spaces Vn (n > 2) in which the first covariant derivative of Weyl's projective curvature tensor is everywhere zero; such spaces they call Protective-Symmetric spaces. In this paper we wish to point out that all Riemannian spaces with this property are symmetric in the sense of Cartan [3]; that is the first covariant derivative of the Riemann curvature tensor of the space vanishes. Further sections are devoted to a discussion of projective-symmetric af fine spaces An with symmetric af fine connexion. Throughout, the geometrical quantities discussed will be as defined by Eisenhart [4] and [5].

scholarly journals Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

On geometry of Kenmotsu manifolds with N-connection

2019 ◽  
pp. 48-60
Author(s):  
A. Bukusheva
Keyword(s):  
Vector Field ◽  
Coordinate System ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Lie Derivative ◽  
Kenmotsu Manifold ◽  
Metric Tensor ◽  
Civita Connection ◽  
Kenmotsu Manifolds ◽  
Levi Civita Connection

A Kenmotsu manifold with a given N-connection is considered. From the integrability of the distribution of a Kenmotsu manifold it follows that the N-connection belongs to the class of the quarter-symmetric connections. Among the N-connections, the class of connections adapted to the structure of the Kenmotsu manifold is specified. In particular, it is proved that an N-connection preserves the structure endomorphism φ of the Kenmotsu manifold if and only if the endomorphisms N and φ commute. A formula expressing the N-connection in terms of the Levi-Civita connection is obtained. The Chrystoffel symbols of the Levi-Civita connection and of the N-connection of the Kenmotsu manifold with respect to the adapted coordinates are computed. The properties of the invariants of the interior geometry of the Kenmotsu manifolds are investigated. The invariants of the interior geometry are the following: the Schouten curvature tensor; the 1-form  defining the distribution D; the Lie derivative 0   L g of the metric tensor g along the vector field ;  the tensor field P with the components given with respect to the adapted coordinate system by the formula Pacd  ncad . The field P is called in the work the Schouten — Wagner tensor. It is proved that the Schouten — Wagner tensor of the interior connection of the Kenmotsu manifold is zero. The conditions that satisfies the endomorphism N defining the metric N-connection are found. At the end of the work, an example of a Kenmotsu manifold with a metric N-connection preserving the structure endomorphism φ is given.

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

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