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Most of the algebraic properties of the Christoffel symbols follow from their relationship to the affine connection; only a few follow from the fact that the structure group is the orthogonal group O(m, n) (or the Lorentz group O(3, 1) for general relativity). Christoffel symbols are used for performing practical calculations.
Christoffel symbols satisfy the symmetry relations = or, respectively, =, the second of which is equivalent to the torsion-freeness of the Levi-Civita connection. The contracting relations on the Christoffel symbols are given by
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.Levi-Civita, [1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols [2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold , one can additionally arrange that the metric tensor is the Kronecker delta at the point p , and that the first ...
On an n-dimensional Riemannian manifold, the geodesic equation written in a coordinate chart with coordinates is: + = where the coordinates x a (s) are regarded as the coordinates of a curve γ(s) in and are the Christoffel symbols.
Objects appearing in the formulas of Gauss, such as the Christoffel symbols, can be given a natural geometric interpretation in this framework. Unlike the more intuitive normal bundle , easily visualised as a tubular neighbourhood of an embedded surface in E 3 , the frame bundle is an intrinsic invariant that can be defined independently of an ...
The metric tensor relative to x is obtained from the metric tensor relative to y by a local calculation having to do with the first derivatives of x ∘ y −1, and hence the Christoffel symbols relative to x are calculated from second derivatives of x ∘ y −1.
where = + is the Riemann curvature tensor and is the Christoffel symbol. Because it is a sum of squares of tensor components, this is a quadratic invariant. Einstein summation convention with raised and lowered indices is used above and throughout the article.