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In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. [1] The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric , allowing distances to be measured on that surface.
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.
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
[citation needed] Correction terms were introduced by Elwin Bruno Christoffel (following ideas of Bernhard Riemann) in the 1870s so that the (corrected) derivative of one vector field along another transformed covariantly under coordinate transformations — these correction terms subsequently came to be known as Christoffel symbols.
the Christoffel symbols that describe components of a metric connection; the stack alphabet in the formal definition of a pushdown automaton, or the tape-alphabet in the formal definition of a Turing machine; the Feferman–Schütte ordinal Γ 0; represents: the specific weight of substances; the lower incomplete gamma function
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 ...
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.