The affine connection

2021 ◽  
pp. 121-132
Author(s):  
Andrew M. Steane
Keyword(s):  
Covariant Derivative ◽  
Affine Connection ◽  
Connection Coefficients ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Coordinate Basis ◽  
Basis Vectors

The connection and the covariant derivative are treated. Connection coefficients are introduced in their role of expressing the change in the coordinate basis vectors between neighbouring points. The covariant derivative of a vector is then defined. Next we relate the connection to the metric, and obtain the Levi-Civita connection. The logic concerning what is defined and what is derived is explained carefuly. The notion of a derivative along a curve is defined. The emphasis through is on clarity and avoiding confusions arising from the plethora of concepts and symbols.

CAN TORSION BE TREATED AS JUST ANOTHER TENSOR FIELD?

2012 ◽  
Vol 07 ◽  
pp. 158-164 ◽  
Author(s):  
JAMES M. NESTER ◽  
CHIH-HUNG WANG
Keyword(s):  
Gauge Theory ◽  
Tensor Field ◽  
Affine Connection ◽  
Cartan Geometry ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Alternative Gravity Theories ◽  
Theory Of Gravity ◽  
Gauge Theory Of Gravity ◽  
Alternative Gravity

Many alternative gravity theories use an independent connection which leads to torsion in addition to curvature. Some have argued that there is no physical need to use such connections, that one can always use the Levi-Civita connection and just treat torsion as another tensor field. We explore this issue here in the context of the Poincaré Gauge theory of gravity, which is usually formulated in terms of an affine connection for a Riemann-Cartan geometry (torsion and curvature). We compare the equations obtained by taking as the independent dynamical variables: (i) the orthonormal coframe and the connection and (ii) the orthonormal coframe and the torsion (contortion), and we also consider the coupling to a source. From this analysis we conclude that, at least for this class of theories, torsion should not be considered as just another tensor field.

scholarly journals ON METRIZABILITY OF INVARIANT AFFINE CONNECTIONS

2012 ◽  
Vol 09 (01) ◽  
pp. 1250014 ◽  
Author(s):  
ERICO TANAKA ◽  
DEMETER KRUPKA
Keyword(s):  
Affine Connection ◽  
Rotation Group ◽  
Explicit Description ◽  
Invariant Metric ◽  
Euclidean Spaces ◽  
Affine Connections ◽  
Group Of Diffeomorphisms ◽  
Civita Connection ◽  
All Solutions ◽  
Levi Civita Connection

The metrizability problem for a symmetric affine connection on a manifold, invariant with respect to a group of diffeomorphisms G, is considered. We say that the connection is G-metrizable, if it is expressible as the Levi-Civita connection of a G-invariant metric field. In this paper we analyze the G-metrizability equations for the rotation group G = SO (3), acting canonically on three- and four-dimensional Euclidean spaces. We show that the property of the connection to be SO(3)-invariant allows us to find complete explicit description of all solutions of the SO(3)-metrizability equations.

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.

The metrizability problem for Lorentz-invariant affine connections

2016 ◽  
Vol 13 (08) ◽  
pp. 1650110
Author(s):  
Zbyněk Urban ◽  
Jana Volná
Keyword(s):  
Euclidean Space ◽  
Lie Groups ◽  
Affine Connection ◽  
Explicit Solutions ◽  
Invariant Metric ◽  
Affine Connections ◽  
Civita Connection ◽  
Levi Civita Connection

The invariant metrizability problem for affine connections on a manifold, formulated by Tanaka and Krupka for connected Lie groups actions, is considered in the particular cases of Lorentz and Poincaré (inhomogeneous Lorentz) groups. Conditions under which an affine connection on the open submanifold [Formula: see text] of the Euclidean space [Formula: see text] coincides with the Levi-Civita connection of some [Formula: see text], respectively [Formula: see text]-invariant metric field are studied. We give complete description of metrizable Lorentz-invariant connections. Explicit solutions (metric fields) of the invariant metrizability equations are found and their properties are discussed.

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 On singular semi-Riemannian manifolds

2014 ◽  
Vol 11 (05) ◽  
pp. 1450041 ◽  
Author(s):  
O. C. Stoica
Keyword(s):  
Riemannian Manifold ◽  
Einstein Equation ◽  
Riemannian Manifolds ◽  
Covariant Derivative ◽  
Riemannian Geometry ◽  
Riemann Curvature ◽  
Tensor Fields ◽  
Civita Connection ◽  
Einstein's Equation ◽  
Levi Civita Connection

On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this paper, we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.

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.

The Levi-Civita Connection

2022 ◽  
pp. 117-167
Author(s):  
Joel W. Robbin ◽  
Dietmar A. Salamon
Keyword(s):  
Civita Connection ◽  
Levi Civita Connection
Sign in / Sign up

Export Citation Format

Share Document