# Christoffel symbol

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of a differential quadratic form An abbreviated notation for the expression The symbols are called the Christoffel symbols of the first kind, in contrast to the Christoffel symbols of the second kind, , defined by where is defined as follows: These symbols were introduced by E.B. Christoffel in 1869.

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Let , , be a linear connection on a manifold , where denotes the space of vector fields on . Let be a chart of . Then on , is completely determined by , where are coordinates on . The Christoffel symbols of the connection are now given by It is important to note that the are not the components of a tensor field. In fact if the denote the Christoffel symbols of with respect to a second set of coordinates on , then Let now be the Riemannian connection (cf. Riemannian geometry) defined by a (local) Riemannian metric . Then the Christoffel symbols of this quadratic differential form are those of the connection . I.e., so that indeed where the are the Christoffel symbols of the second kind of the quadratic differential form as defined above.

How to Cite This Entry:
Christoffel symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Christoffel_symbol&oldid=18161