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

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.

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.

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 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.

Riemannian manifolds

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Covariant Derivative ◽  
Einstein Equations ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Hilbert Action

This chapter is about Riemannian manifolds. It first discusses the metric manifold and the Levi-Civita connection, determining if the metric is Riemannian or Lorentzian. Next, the chapter turns to the properties of the curvature tensor. It states without proof the intrinsic versions of the properties of the Riemann–Christoffel tensor of a covariant derivative already given in Chapter 2. This chapter then performs the same derivation as in Chapter 4 by obtaining the Einstein equations of general relativity by varying the Hilbert action. However, this will be done in the intrinsic manner, using the tools developed in the present and the preceding chapters.

scholarly journals A Study on Conservative C-Bochner Curvature Tensor in K-Contact and Kenmotsu Manifolds Admitting Semisymmetric Metric Connection

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Gurupadavva Ingalahalli ◽  
C. S. Bagewadi
Keyword(s):  
Curvature Tensor ◽  
Civita Connection ◽  
Kenmotsu Manifolds ◽  
Metric Connection ◽  
Levi Civita Connection

The paper deals with the study on conservative C-Bochner curvature tensor in K-contact and Kenmotsu manifolds admitting semisymmetric metric connection, and it is shown that these manifolds are η-Einstein with respect to Levi-Civita connection, and the results are illustrated with examples.

scholarly journals On the geometry of the tangent bundle with gradient Sasaki metric

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Lakehal Belarbi ◽  
Hichem Elhendi
Keyword(s):  
Curvature Tensor ◽  
Tangent Bundle ◽  
Riemannian Curvature ◽  
Curvature Scalar ◽  
Sasaki Metric ◽  
Content Type ◽  
Civita Connection ◽  
Riemannian Curvature Tensor ◽  
Sectional Curvatures ◽  
Levi Civita Connection

PurposeLet (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature, scalar and sectional curvatures.Design/methodology/approachIn this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.FindingsThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.Originality/valueThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.

scholarly journals Levi-Civita connections and vector fields for noncommutative differential calculi

2020 ◽  
Vol 31 (08) ◽  
pp. 2050065
Author(s):  
Jyotishman Bhowmick ◽  
Debashish Goswami ◽  
Giovanni Landi
Keyword(s):  
Lie Algebra ◽  
Covariant Derivative ◽  
Vector Fields ◽  
Covariant Derivatives ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
One To One

We study covariant derivatives on a class of centered bimodules [Formula: see text] over an algebra [Formula: see text] We begin by identifying a [Formula: see text]-submodule [Formula: see text] which can be viewed as the analogue of vector fields in this context; [Formula: see text] is proven to be a Lie algebra. Connections on [Formula: see text] are in one-to-one correspondence with covariant derivatives on [Formula: see text]. We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived.

scholarly journals A quarter-symmetric metric connection on almost contact B-metric manifolds

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5181-5190
Author(s):  
Şenay Bulut
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection

The aim of this paper is to study the notion of a quarter-symmetric metric connection on an almost contact B-metric manifold (M,?,?,?,g). We obtain the relation between the Levi-Civita connection and the quarter-symmetric metric connection on (M,?,?,?,g).We investigate the curvature tensor, Ricci tensor and scalar curvature tensor with respect to the quarter-symmetric metric connection. In case the manifold (M,?,?,?,g) is a Sasaki-like almost contact B-metric manifold, we get some formulas. Finally, we give some examples of a quarter-symmetric metric connection.

Sign in / Sign up

Export Citation Format

Share Document