I Christoffel symbols

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What is the general difference or importance between using christoffel symbols of the first kind and those of the second kind in terms of geometry and their application. The christoffel symbols of the second are identical to those of the first except with the inverse metric tensor in front, correct?Why have the both the first and second kind? what are the first used for and the second used for in general? I know the importance of the second kind in terms of the covariant derivative but what use does the first kind have in physics and math.
 

Orodruin

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This depends on what you mean by ”use”. Some things, such as arriving at the expression for the Christoffel symbols in terms of the metric, are easier to do using the first kind. Geometrically and computationally, the second kind is typically more intuitive and useful.
 
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The reason for the distinction between the first and second kind may be that neither are tensors. If they were tensors then the difference would be the trivial lowering of an index, and we would regard them as manifestations of the same tensor. But they are not tensors, so we have to be careful and treat them separately. The first kind is useful for defining the Riemann curvature, second is useful for torsion and contorsion.
 

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