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Christoffel Symbol of the First Kind

Christoffel symbols of the first kind are the metric-lowered coefficients of the Levi-Civita connection of a Riemannian metric g. For coordinate vector fields e_i=partial/partialq^i, they satisfy [ij,k]=g(del _(e_j)e_i,e_k) (Meinrenken 2002, pp. 40-41). Christoffel symbols of the first kind are variously denoted [ij,k], [i j; k], Gamma_(abc), or {ab,c}. They are also known as connections coefficients (Misner et al. 1973, p. 210).

For a smooth submanifold in Euclidean space with its induced Riemannian metric, the position-vector calculation gives

[ij,k]=g_(mk)Gamma_(ij)^m
(1)
=g_(mk)e^->^m·(partiale^->_i)/(partialq^j)
(2)
=e^->_k·(partiale^->_i)/(partialq^j),
(3)

where g_(mk) is the metric tensor, Gamma_(ij)^m is a Christoffel symbol of the second kind, and

e^->_i=(partialr^->)/(partialq^i)=h_ie^^_i.
(4)

Here e^->^m=g^(ml)e^->_l is the reciprocal tangent basis. Differentiating the induced metric tensor gives

(partialg_(ij))/(partialq^k)=partial/(partialq^k)(e^->_i·e^->_j)
(5)
=(partiale^->_i)/(partialq^k)·e^->_j+e^->_i·(partiale^->_j)/(partialq^k)
(6)
=[ik,j]+[jk,i],
(7)

so

[ab,c]=1/2(g_(ac,b)+g_(bc,a)-g_(ab,c)).
(8)

See also

Christoffel Symbol, Christoffel Symbol of the Second Kind

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 160-167, 1985.Meinrenken, E. Riemannian Geometry. Lecture notes, Spring 2002. https://www.math.toronto.edu/mein/teaching/LectureNotes/rieall.pdf.Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco, CA: W. H. Freeman, 1973.

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Christoffel Symbol of the First Kind

Cite this as:

Weisstein, Eric W. "Christoffel Symbol of the First Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChristoffelSymboloftheFirstKind.html

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