List of Formulas in Riemannian Geometry - Christoffel Symbols, Covariant Derivative

Christoffel Symbols, Covariant Derivative

In a smooth coordinate chart, the Christoffel symbols of the first kind are given by

Gamma_{kij}=frac12 left( frac{partial}{partial x^j} g_{ki} +frac{partial}{partial x^i} g_{kj} -frac{partial}{partial x^k} g_{ij} right) =frac12 left( g_{ki,j} + g_{kj,i} - g_{ij,k} right) ,,

and the Christoffel symbols of the second kind by

begin{align} Gamma^m{}_{ij} &= g^{mk}Gamma_{kij}\ &=frac12, g^{mk} left( frac{partial}{partial x^j} g_{ki} +frac{partial}{partial x^i} g_{kj} -frac{partial}{partial x^k} g_{ij} right) =frac12, g^{mk} left( g_{ki,j} + g_{kj,i} - g_{ij,k} right) ,. end{align}

Here is the inverse matrix to the metric tensor . In other words,


delta^i{}_j = g^{ik}g_{kj}

and thus


n = delta^i{}_i = g^i{}_i = g^{ij}g_{ij}

is the dimension of the manifold.

Christoffel symbols satisfy the symmetry relation


Gamma^i{}_{jk}=Gamma^i{}_{kj} ,,

which is equivalent to the torsion-freeness of the Levi-Civita connection.

The contracting relations on the Christoffel symbols are given by

and

where |g| is the absolute value of the determinant of the metric tensor . These are useful when dealing with divergences and Laplacians (see below).

The covariant derivative of a vector field with components is given by:


v^i {}_{;j}=nabla_j v^i=frac{partial v^i}{partial x^j}+Gamma^i{}_{jk}v^k

and similarly the covariant derivative of a -tensor field with components is given by:


v_{i;j}=nabla_j_v_i=frac{partial v_i}{partial x^j}-Gamma^k{}_{ij} v_k

For a -tensor field with components this becomes


v^{ij}{}_{;k}=nabla_k v^{ij}=frac{partial v^{ij}}{partial x^k} +Gamma^i{}_{kell}v^{ell j}+Gamma^j{}_{kell}v^{iell}

and likewise for tensors with more indices.

The covariant derivative of a function (scalar) is just its usual differential:


nabla_i phi=phi_{;i}=phi_{,i}=frac{partial phi}{partial x^i}

Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,


nabla_k_g_{ij} = nabla_k g^{ij} = 0

The geodesic starting at the origin with initial speed has Taylor expansion in the chart:


X(t)^i=tv^i-frac{t^2}{2}Gamma^i{}_{jk}v^jv^k+O(t^2)

Read more about this topic:  List Of Formulas In Riemannian Geometry

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