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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.
where the coordinates x a (s) are regarded as the coordinates of a curve γ(s) in and are the Christoffel symbols. The Christoffel symbols are functions of the metric and are given by: Γ b c a = 1 2 g a d ( g c d , b + g b d , c − g b c , d ) {\displaystyle \Gamma _{bc}^{a}={\frac {1}{2}}g^{ad}\left(g_{cd,b}+g_{bd,c}-g_{bc,d}\right)}
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 ...
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection.
The key is that when one regards X 1 ∂f / ∂u + X 2 ∂f / ∂v as a ℝ 3-valued function, its differentiation along a curve results in second partial derivatives ∂ 2 f; the Christoffel symbols enter with orthogonal projection to the tangent space, due to the formulation of the Christoffel symbols as the tangential ...
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.