Geodesic Deviation and Curvature Tensor

2016 ◽  
pp. 95-114
Author(s):  
Maurizio Gasperini
Keyword(s):  
Curvature Tensor ◽  
Geodesic Deviation

Curvature

2021 ◽  
pp. 189-212
Author(s):  
Andrew M. Steane
Keyword(s):  
Curvature Tensor ◽  
Torsion Tensor ◽  
Parallel Transport ◽  
Lie Derivative ◽  
Riemannian Curvature ◽  
Riemann Curvature Tensor ◽  
Geodesic Deviation ◽  
Minkowski Metric ◽  
Covariant Derivatives ◽  
Global Properties

The mathematics of Riemannian curvature is presented. The Riemann curvature tensor and its role in parallel transport, in the metric, and in geodesic deviation are expounded at length. We begin by defining the curvature tensor and the torsion tensor by relating them to covariant derivatives. Then the local metric is obtained up to second order in terms of Minkowski metric and curvature tensor. Geometric issues such as the closure or non-closure of parallelograms are discussed. Next, the relation between curvature and parallel transport around a loop is derived. Then we proceed to geodesic deviation. The influence of global properties of the manifold on parallel transport is briefly expounded. The Lie derivative is then defined, and isometries of spacetime are discussed. Killing’s equation and properties of Killing vectors are obtained. Finally, the Weyl tensor (conformal tensor) is introduced.

Generalized Riemann spaces

1956 ◽  
Vol 52 (4) ◽  
pp. 623-625 ◽  
Author(s):  
John Moffat
Keyword(s):  
Curvature Tensor ◽  
Physical Theory ◽  
Dimensional Space ◽  
Riemann Space ◽  
Recent Attempt ◽  
Fundamental Tensor ◽  
Fundamental Properties ◽  
Complex Symmetric ◽  
Definition Of ◽  
Generalized Curvature Tensor

ABSTRACTThe recent attempt at a physical interpretation of non-Riemannian spaces by Einstein (1, 2) has stimulated a study of these spaces (3–8). The usual definition of a non-Riemannian space is one of n dimensions with which is associated an asymmetric fundamental tensor, an asymmetric linear affine connexion and a generalized curvature tensor. We can also consider an n-dimensional space with which is associated a complex symmetric fundamental tensor, a complex symmetric affine connexion and a generalized curvature tensor based on these. Some aspects of this space can be compared with those of a Riemann space endowed with two metrics (9). In the following the fundamental properties of this non-Riemannian manifold will be developed, so that the relation between the geometry and physical theory may be studied.

On Nye’s Lattice Curvature Tensor

2021 ◽  
pp. 103696
Author(s):  
Fabio Sozio ◽  
Arash Yavari
Keyword(s):  
Curvature Tensor ◽  
Lattice Curvature

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.

Analysis of the Curvature Tensor from the Viewpoint of Signal Processing

2008 ◽  
Author(s):  
Lennart Wietzke ◽  
Gerald Sommer ◽  
Christian Schmaltz ◽  
Joachim Weickert ◽  
Theodore E. Simos ◽  
...  
Keyword(s):  
Signal Processing ◽  
Curvature Tensor

scholarly journals Traceless component of the conformal curvature tensor in Kähler manifold

2006 ◽  
Vol 56 (3) ◽  
pp. 857-874 ◽  
Author(s):  
S. Funabashi ◽  
H. S. Kim ◽  
Y.-M. Kim ◽  
J. S. Pak
Keyword(s):  
Curvature Tensor ◽  
Kähler Manifold ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Kahler Manifold

scholarly journals Geodesics and geodesic deviation in a two-dimensional black hole

2003 ◽  
Vol 71 (10) ◽  
pp. 1037-1042 ◽  
Author(s):  
Ratna Koley ◽  
Supratik Pal ◽  
Sayan Kar
Keyword(s):  
Black Hole ◽  
Two Dimensional ◽  
Geodesic Deviation ◽  
Dimensional Black Hole
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