When self-studying differential geometry, I find my book involves some clumsy, troublesome calculation about Christoffel symbol when proving theorem, which in fact doesn't have the symbols. I wonder if the symbol is actually useful when for doing calculation stuff of understand concept. Can someone explain to me the role of Christoffel symbol? Is it really important? Do I really need to go through all of them?
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It is possible and highly recommended to learn how to do the calculations without using Christoffel symbols, in a coordinate free manner. Personally, I like the abstract index notation, for instance.
On the other hand, a good understanding of the coordinate calculations is very helpful when one attempts to read the legacy papers.
The role of the Christoffel symbols is easy to explain: they serve as the components of the connection in a local coordinate patch.
More precisely, the Christoffel symbols are the components of the difference tensor between the given connection and the standard connection, which is available in this coordinate patch (and has all the Christoffels vanishing!). Here is my answer to a related question.
If you need more references or further explanations, I am happy to add them to my answer.
From a physics perspective, Christoffel Symbols are fundamental in General Relativity where they appear in the Geodesic equation:
Which describes the shortest path between two points in curved space. They also appear in the curvature Tensor, the covariant derivative and many other important geometric objects. They arise naturally in describing the effects of parallel transport in manifolds. Often due to the symmetry in General Relativity the number of independent Christoffel symbols is reduced to a much more manageable amount.