The Cartan structure equations

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Riemannian Manifold ◽  
Riemann Tensor ◽  
Structure Equation ◽  
Exterior Derivative ◽  
Schwarzschild Metric ◽  
Civita Connection ◽  
Structure Equations ◽  
Levi Civita Connection

This chapter focuses on Cartan structure equations. It first introduces a 1-form and its exterior derivative, before turning to a study of the connection and torsion forms, thereby expressing the torsion as a function of the connection forms and establishing the torsion differential 2-forms. It then turns to the curvature forms drawn from Chapter 23 and Cartan’s second structure equation, along with the curvature 2-forms. It also studies the Levi-Civita connection. The components of the Riemann tensor are then studied, with a Riemannian manifold, or a metric manifold with a torsion-less connection. The Riemann tensor of the Schwarzschild metric are finally discussed.

2D RICCI FLAT GRADIENT SOLITONS ARISING FROM REMARKABLE MODELS IN PHYSICS

2013 ◽  
Vol 10 (10) ◽  
pp. 1350059
Author(s):  
GABRIEL BERCU
Keyword(s):  
Riemannian Manifold ◽  
Local Computation ◽  
Ricci Flat ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Gradient Solitons

On a pseudo-Riemannian manifold (M, g) we consider ∇ the Levi-Cività connection associated to metric g and a function f : M → ℝ whose pseudo-Riemannian Hessian [Formula: see text] is non-degenerate and with constant signature. We study properties of the pseudo-Riemannian manifold (M, h), [Formula: see text] in terms of local computation. Investigating the conditions for existence of the equal connections [Formula: see text], produced by g and h, we determine classes of explicit Ricci flat gradient solitons for some particular forms of g, arising from remarkable physics models.

EXPONENTIAL INTEGRABILITY FOR IMAGES OF ORNSTEIN–UHLENBECK OPERATORS ACTING ON CYLINDER FUNCTIONS ON LOOP SPACES

2002 ◽  
Vol 05 (04) ◽  
pp. 541-553
Author(s):  
FUZHOU GONG
Keyword(s):  
Riemannian Manifold ◽  
Loop Space ◽  
Loop Spaces ◽  
Exponential Integrability ◽  
Cylinder Function ◽  
Civita Connection ◽  
Levi Civita Connection

Let E be the loop space over a compact connected Riemannian manifold with a torsion skew symmetric (TSS) connection. Let L be the Ornstein–Uhlenbeck (O-U) operator on the loop space E, and f be a cylinder function on E. We first extend the expression of Lf, proved by Enchev and Stroock for the Levi–Cività connection, to a general TSS connection, and then prove that if [Formula: see text], ε |Lf|2 is exponential integrable for some constant ε := ε (f)>0.

On geometry of sub-Riemannian η-Einstein manifolds

2018 ◽  
pp. 68-81
Author(s):  
S. Galaev
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Einstein Manifold ◽  
Riemannian Structure ◽  
Einstein Manifolds ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
Curvature Tensors ◽  
Sasaki Manifold

On a sub-Riemannian manifold of contact type a connection  with torsion is considered, called in the work a Ψ-connection. A Ψ- connection is a particular case of an N-connection. On a sub-Riemannian manifold, a Ψ-connection is defined up to an endomorphism  :DD of a distribution D, this endomorphism is called in the work the structure endomorphism. The endomorphism ψ is uniquely defined by the following relations:  0,   (x, y)  g( x, y), x, yD. If the distribution of a sub-Riemannian manifold is integrable, then the Ψ-connection is of the class of the quarter-symmetric connections. It is proved that the Ψ- connection is a metric connection if and only if the structure vector field of the sub-Riemannian structure is integrable. A formula expressing the Ψ-connections in terms of the Levi-Civita connection of the sub- Riemannian manifold is obtained. The components of the curvature tensors and the Ricci-tensors of the Ψ-connection and of the Levi-Civita connection are computed. It is proved that if a sub-Riemannian manifold is an η-Einstein manifold, then it is also an η-Einstein manifold with respect to the Ψ-connection. The converse holds true only under the condition that the trace of the structure endomorphism Ψ is a constant not depending on a point of the manifold. The paper is completed by the theorem stating that a Sasaki manifold is an η-Einstein manifold if and only if M is an η-Einstein manifold with respect to the Ψ-connection.

Riemannian manifolds

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Riemannian Manifolds ◽  
Parallel Transport ◽  
Riemann Tensor ◽  
Metric Tensor ◽  
Geodesic Deviation ◽  
Common Space ◽  
Civita Connection ◽  
Inertial Frames ◽  
Levi Civita Connection

This chapter introduces the Riemann tensor characterizing curved spacetimes, and then the metric tensor, which allows lengths and durations to be defined. As shown in the preceding chapter, ‘absolute, true, and mathematical’ spacetimes representing ‘relative, apparent, and common’ space and time in Einstein’s theory are Riemannian manifolds supplied with a metric and its associated Levi-Civita connection. Moreover, this metric simultaneously describes the coordinate system chosen to reference the events. The chapter begins with a study of connections, parallel transport, and curvature; the commutation of derivatives, torsion, and curvature; geodesic deviation and curvature; the metric tensor and the Levi-Civita connection; and locally inertial frames. Finally, it discusses Riemannian manifolds.

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.

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.

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.

The Levi-Civita Connection

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