In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In a broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are also called Christoffel symbols. The Christoffel symbols may be used for performing practical calculations in differential geometry. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives.
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Some articles on christoffel symbols:
... In a smooth coordinate chart, the Christoffel symbols of the first kind are given by and the Christoffel symbols of the second kind by Here is the inverse matrix to the metric tensor ... Christoffel symbols satisfy the symmetry relation which is equivalent to the torsion-freeness of the Levi-Civita connection ... The contracting relations on the Christoffel symbols are given by and where
... when equations of motion are expressed in a curvilinear coordinate system, Christoffel symbols appear in the acceleration of a particle expressed in this coordinate system, as described below in ... in which the coefficients of the unit vectors are the Christoffel symbols for the coordinate system ... The general notation and formulas for the Christoffel symbols are and the symbol is zero when all the indices are different ...
... The Christoffel symbols find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita connection ... spacetime in the presence of matter—contain the Ricci tensor, and so calculating the Christoffel symbols is essential ... are calculated by solving the geodesic equations in which the Christoffel symbols explicitly appear ...
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