Christoffel Symbol of the Second Kind

Christoffel symbols of the second kind are the coefficients of the Levi-Civita connection of a Riemannian metric in a chosen basis. They are variously denoted as (Walton 1967) or (Misner et al. 1973, Arfken 1985). They are also called connection coefficients (Misner et al. 1973, p. 210) or affine connection coefficients (Weinberg 1972, p. 71).

The coordinate-basis and noncoordinate-basis formulas below give coefficients of the same Levi-Civita connection in different bases. In a coordinate basis, the coefficients can be expressed in terms of the metric tensor alone.

For a coordinate basis, Arfken (1985, p. 161) gives

where is a partial derivative, is the metric tensor, and is the corresponding inverse metric,

where is the radius vector, and

Therefore, for an orthogonal curvilinear coordinate system, by this definition,

Because the coordinate basis vectors commute and the Levi-Civita connection is torsion-free,

(Walton 1967).

This Christoffel symbol of the second kind is related to the Christoffel symbol of the first kind by

Walton (1967) lists Christoffel symbols of the second kind for the 12 basic orthogonal coordinate systems.

For an orthonormal basis that need not be a coordinate basis, Misner et al. (1973, p. 209) give the coefficients as

where denotes the covariant derivative in the direction of . These coefficients need not be symmetric in and because the basis vectors need not commute.

Christoffel symbols of the second kind are not tensors and do not transform as tensor components. For two coordinate bases related by a change from to , the transformation law is

Here the lowered-index convention is . For the Levi-Civita connection in a general basis, these coefficients are

(Misner et al. 1973, p. 210), where the s are components of the metric tensor, for the commutation coefficients , and the commas indicate the comma derivative. In a coordinate basis, the commutation coefficients vanish and this reduces to a Christoffel symbol of the first kind. In an orthonormal basis, the metric components are constant, so and . Hence

For an orthogonal coordinate basis, in contrast, the metric is diagonal but its diagonal components need not be constant. In this setting,

In three dimensions, the Christoffel symbols of the second kind may be arranged in matrix form:

The Christoffel symbols are given in terms of the coefficients of the first fundamental form , , and by

and and . If , the Christoffel symbols of the second kind simplify to

(Gray 1997).

The following relationships hold between the Christoffel symbols of the second kind and coefficients of the first fundamental form,

(Gray 1997).

For a surface given in Monge's form ,

Christoffel symbols of the second kind arise in the computation of geodesics. The geodesic equation of free motion is

or

Expanding,

But

so

where