Application of Christoffel symbol in differential geometry 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?

From a physics perspective, Christoffel Symbols are fundamental in General Relativity where they appear in the Geodesic equation: 
$$\frac{d^2x^\lambda}{d\sigma^2}+\Gamma^\lambda_{\alpha\beta}\frac{dx^\alpha}{d\sigma}\frac{dx^\beta}{d\sigma}=0$$
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.
