On the reconstruction of the variation of the metric tensor of a surface on the basis of a given variation of christoffel symbols of the second kind under infinitesimal deformations of surfaces in the euclidean space E 3

2011 ◽  
Vol 63 (4) ◽  
pp. 609-616 ◽  
Author(s):  
I. V. Potapenko
Keyword(s):  
Euclidean Space ◽  
Metric Tensor ◽  
Christoffel Symbols ◽  
Infinitesimal Deformations

Metric Spaces and Geodesic Motion

2019 ◽  
pp. 355-384
Author(s):  
David D. Nolte
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Parallel Transport ◽  
Variational Calculus ◽  
Geodesic Equation ◽  
Geodesic Curve ◽  
Geodesic Motion ◽  
Metric Tensor ◽  
Christoffel Symbols ◽  
Derivatives Of

The metric tensor uniquely defines the geometric properties of a metric space, while differential geometry is concerned with the derivatives of vectors and tensors within the metric space. Derivatives of vector quantities include the derivatives of basis vectors, which lead to Christoffel symbols that are directly connected to the metric tensor. This connection between tensor derivatives and the metric tensor provides the tools necessary to define geodesic curves. The geodesic equation is derived from variational calculus and from parallel transport. Geodesic motion is the trajectory of a particle through a metric space defined by an action metric that includes a potential function. In this way, dynamics converts to geometry as a trajectory in a potential converts to a geodesic curve through an appropriately defined metric space.

scholarly journals 2 + 1 EINSTEIN GRAVITY AS A DEFORMED CHERN-SIMONS THEORY

1998 ◽  
Vol 13 (23) ◽  
pp. 4023-4047 ◽  
Author(s):  
G. BIMONTE ◽  
R. MUSTO ◽  
P. VITALE ◽  
A. STERN
Keyword(s):  
Gauge Group ◽  
Einstein Gravity ◽  
Parameter Family ◽  
Group Symmetry ◽  
Simons Theory ◽  
Spin Connection ◽  
Field Equations ◽  
Metric Tensor ◽  
Christoffel Symbols ◽  
Chern Simons

The usual description of (2+1)-dimensional Einstein gravity as a Chern–Simons (CS) theory is extended to a one parameter family of descriptions of 2+1 Einstein gravity. This is done by replacing the Poincaré gauge group symmetry by a q-deformed Poincaré gauge group symmetry, with the former symmetry recovered when q → 1. As a result, we obtain a one parameter family of Hamiltonian formulations for 2+1 gravity. Although formulated in terms of noncommuting dreibeins and spin-connection fields, our expression for the action and our field equations, appropriately ordered, are identical in form to the ordinary ones. Moreover, starting with a properly defined metric tensor, the usual metric theory can be built; the Christoffel symbols and space–time curvature having the usual expressions in terms of the metric tensor, and being represented by c-numbers. In this article, we also couple the theory to particle sources, and find that these sources carry exotic angular momentum. Finally, problems related to the introduction of a cosmological constant are discussed.

Curved Space

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Arbitrary Dimension ◽  
Spherical Geometry ◽  
Flat Space ◽  
Geodesic Equation ◽  
Metric Tensor ◽  
Curved Space ◽  
Covariant Derivatives ◽  
Christoffel Symbols ◽  
Geometry Configuration ◽  
Fisheye Lenses

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that describes general curved spaces of arbitrary dimension. The chapter begins with a comparison of Euclidean geometry and spherical geometry. The concept of the geodesic is introduced. The discovery of hyperbolic geometry is discussed. Gaussian curvature is defined. Tensors are introduced. The metric tensor is defined and simple examples are given. This leads to the use of covariant derivatives, expressed in terms of Christoffel symbols, the Riemann curvature tensor and all machinery of Riemannian geometry, with each step illustrated by simple examples. The geodesic equation and the equation of geodesic deviation are derived. The final section considers some applications of curved geometry: configuration space, mirages and fisheye lenses.

Estimates for the constant in two nonlinear Korn inequalities

2020 ◽  
pp. 108128652095770
Author(s):  
Maria Malin ◽  
Cristinel Mardare
Keyword(s):  
Euclidean Space ◽  
Open Subset ◽  
Rigid Motion ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Square Roots ◽  
Korn Inequality ◽  
New Inequalities

A nonlinear Korn inequality estimates the distance between two immersions from an open subset of [Formula: see text] into the Euclidean space [Formula: see text], [Formula: see text], in terms of the distance between specific tensor fields that determine the two immersions up to a rigid motion in [Formula: see text]. We establish new inequalities of this type in two cases: when k = n, in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when k = 3 and n = 2, in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.

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