![]() The Christoffel symbols are the means of correcting your flat-space, naive differentiation to account for the curvature of the space in which you're doing your calculations, between those two points. The affine connection is the conceptual link between two very nearby points where the vectors you would like to compare reside. ![]() Basically, you need to calculate some corrections when differentiating in a curved space or else you will get anomalous answers that depend on the details of your calculation. Misner, Thorne, and Wheeler's textbook Gravitation really works out the concepts in extreme detail to make them clear. They are very closely related- so much so that Christoffel symbols are commonly also called "Connection coefficients." In a curved space, comparing one vector (or other mathematical object- tensor, n-forms, etc.) to another is not so straightforward a task as it is in nice, flat, Euclidean space. ![]()
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