scholarly journals Anisotropic tensor calculus

2019 ◽  
Vol 16 (supp02) ◽  
pp. 1941001 ◽  
Author(s):  
Miguel Angel Javaloyes
Keyword(s):  
Curvature Tensor ◽  
Finsler Metric ◽  
Lie Derivative ◽  
Classical Concept ◽  
Tensor Calculus ◽  
Chern Connection ◽  
Linear Connections ◽  
Anisotropic Curvature ◽  
Levi Civita Connection ◽  
Anisotropic Tensor

We introduce the anisotropic tensor calculus, which is a way of handling tensors that depends on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application we show how to define derivations using anisotropic linear connections in a manifold. In particular, we show that the Chern connection of a Finsler metric can be interpreted as the Levi-Civita connection and we introduce the anisotropic curvature tensor. We also relate the concept of anisotropic connection with the classical concept of linear connections in the vertical bundle. Furthermore, we also introduce the concept of anisotropic Lie derivative.

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.

Derivatives and curvature

2019 ◽  
pp. 37-51
Author(s):  
Steven Carlip
Keyword(s):  
General Relativity ◽  
Curvature Tensor ◽  
Differential Forms ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Tensor Calculus ◽  
Covariant Derivatives ◽  
Civita Connection ◽  
Physical Picture ◽  
Levi Civita Connection

This chapter develops tensor calculus: integration on manifolds, Cartan calculus for differential forms, connections and covariant derivatives, and the Levi-Civita connection used in general relativity. It then introduces the Riemann curvature tensor in several different ways, including the most directly physical picture of the curvature as a measure of the convergence of neighboring geodesics. The chapter concludes with a discussion of Cartan’s beautiful formulation of the connection and curvature in the language of differential forms.

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 THE GAUSS–BONNET FORMULA IN RIEMANN–FINSLER GEOMETRY

2002 ◽  
Vol 34 (3) ◽  
pp. 329-340 ◽  
Author(s):  
BRAD LACKEY
Keyword(s):  
Finsler Space ◽  
Finsler Geometry ◽  
Euler Class ◽  
Constant Volume ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Riemannian Case ◽  
Torsion Free

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

scholarly journals Covariant derivatives on Kähler C-spaces

1996 ◽  
Vol 143 ◽  
pp. 31-57
Author(s):  
Koji Tojo
Keyword(s):  
Curvature Tensor ◽  
Covariant Derivatives ◽  
Civita Connection ◽  
Levi Civita Connection

Let (M, g) be a Kähler C-space. R and ∇ denote the curvature tensor and the Levi-Civita connection of (M, g), respectively.In [6], Takagi have proved that there exists an integer n such that

scholarly journals NATURAL CONNECTION WITH TOTALLY SKEW-SYMMETRIC TORSION ON ALMOST CONTACT MANIFOLDS WITH B-METRIC

2012 ◽  
Vol 09 (05) ◽  
pp. 1250044 ◽  
Author(s):  
MANCHO MANEV
Keyword(s):  
Curvature Tensor ◽  
Torsion Tensor ◽  
Contact Manifolds ◽  
Natural Connection ◽  
Civita Connection ◽  
Levi Civita Connection

A natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric is constructed. The class of these manifolds, where the considered connection exists, is determined. Some curvature properties for this connection, when the corresponding curvature tensor has the properties of the curvature tensor for the Levi-Civita connection and the torsion tensor is parallel, are obtained.

scholarly journals On Sectional Curvature of Metric Connection with Vectorial Torsion

2020 ◽  
pp. 124-127
Author(s):  
E.D. Rodionov ◽  
V.V. Slavsky ◽  
O.P. Khromova
Keyword(s):  
Sectional Curvature ◽  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Civita Connection ◽  
Modern Physics ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
The Difference ◽  
Topology Of Manifolds ◽  
Conformal Deformations

Papers of many mathematicians are devoted to the study of semisymmetric connections or metric connections with vector torsion on Riemannian manifolds. This type of connectivity is one of the three main types discovered by E. Cartan and finds its application in modern physics, geometry, and topology of manifolds. Geodesic lines and the curvature tensor of a given connection were studied by I. Agricola, K. Yano, and other mathematicians. In particular, K. Yano proved an important theorem on the connection of conformal deformations and metric connections with vector torsion. Namely: a Riemannian manifold admits a metric connection with vector torsion and the curvature tensor being equal to zero if and only if it is conformally flat. Although the curvature tensor of a hemisymmetric connection has a smaller number of symmetries compared to the Levi-Civita connection, it is still possible to define the concept of sectional curvature in this case. The question naturally arises about the difference between the sectional curvature of a semisymmetric connection and the sectional curvature of a Levi-Civita connection.This paper is devoted to the study of this issue, and the authors find the necessary and sufficient conditions for the sectional curvature of the semisymmetric connection to coincide with the sectional curvature of the Levi-Civita connection. Non-trivial examples of hemisymmetric connections are constructed when possible.

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 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.

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