Christoffel Symbols, Covariant Derivative
In a smooth coordinate chart, the Christoffel symbols of the first kind are given by
and the Christoffel symbols of the second kind by
is the dimension of the manifold.
Christoffel symbols satisfy the symmetry relation
which is equivalent to the torsion-freeness of the Levi-Civita connection.
The contracting relations on the Christoffel symbols are given by
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:
and similarly the covariant derivative of a -tensor field with components is given by:
For a -tensor field with components this becomes
and likewise for tensors with more indices.
The covariant derivative of a function (scalar) is just its usual differential:
Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,
The geodesic starting at the origin with initial speed has Taylor expansion in the chart:
Read more about this topic: List Of Formulas In Riemannian Geometry
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