Relationship To Index-free Notation
Let X and Y be vector fields with components and . Then the kth component of the covariant derivative of Y with respect to X is given by
Here, the Einstein notation is used, so repeated indices indicate summation over indices and contraction with the metric tensor serves to raise and lower indices:
Keep in mind that and that, the Kronecker delta. The convention is that the metric tensor is the one with the lower indices; the correct way to obtain from is to solve the linear equations .
The statement that the connection is torsion-free, namely that
is equivalent to the statement that —in a coordinate basis— the Christoffel symbol is symmetric in the lower two indices:
The index-less transformation properties of a tensor are given by pullbacks for covariant indices, and pushforwards for contravariant indices. The article on covariant derivatives provides additional discussion of the correspondence between index-free notation and indexed notation.
Read more about this topic: Christoffel Symbols
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