Let ∇ be the connection of the Riemannian metric. Choose local coordinates and let be the Christoffel symbols with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry
The definition of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as
where as usual are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix .
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Other articles related to "christoffel symbols, symbol":
... 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 ... 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 ... of matter—contain the Ricci tensor, and so calculating the Christoffel symbols is essential ... of particles and light beams are calculated by solving the geodesic equations in which the Christoffel symbols explicitly appear ...