Christoffel Symbol of the Second Kind

Christoffel symbols of the second kind are the second type of tensor-like object derived from a Riemannian metric which is used to study the geometry of the metric. Christoffel symbols of the second kind are variously denoted as (Walton 1967) or (Misner et al. 1973, Arfken 1985). They are also known as affine connections (Weinberg 1972, p. 71) or connection coefficients (Misner et al. 1973, p. 210).

Unfortunately, there are two different definitions of the Christoffel symbol of the second kind.

Arfken (1985, p. 161) defines

where is a partial derivative, is the metric tensor,

where is the radius vector, and

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

The symmetry of definition (6) means that

(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.

A different definition of Christoffel symbols of the second kind is given by

(Misner et al. 1973, p. 209), where denotes a gradient. Note that this kind of Christoffel symbol is not symmetric in and .

Christoffel symbols of the second kind are not tensors, but have tensor-like contravariant and covariant indices. Christoffel symbols of the second kind also do not transform as tensors. In fact, changing coordinates from to gives

However, a fully covariant Christoffel symbol of the second kind is given by

(Misner et al. 1973, p. 210), where the s are the metric tensors, the s are commutation coefficients, and the commas indicate the comma derivative. In an orthonormal basis, and , so

and

For tensors of tensor rank 3, the Christoffel symbols of the second kind may be concisely summarized 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