Hi: I have been aching over figuring out Riemannian Connections Conceptually ( It's a Heartache! :) ) Please give me some input (comments, corrections) here: We use connections to make sense of the difference quotient in differentiation: Lim _h->0: ( F(x+h)-F(x)/h ) In IR^n, there is no problem, because the tangent spaces at any two points are naturally isomorphic, i.e., the fact that F(x+h) and F(x) live in different tangent spaces ( T_x and T_(x+h) , resp. ) is not a serious problem: we translate F(x+h) to the tangent space based at x ; translation gives us the isomorphism. Now: If we are working on a non-flat manifold , then vector spaces are not naturally isomorphic anymore. If the manifold M is emnbedded in IR^n , then we make sense of the difference quotient by seeing it as the directional derivative of one vector field in the direction of another V.Field. ( seeing T_x, the tangent space of x in M , as a point in IR^n , so that T_xM ~ T_x IR^n ). If the directional derivative does not live in the tangent space of M , we decompose IR^n as T_p (+) (T_p )^Perp. , i.e., the orthogonal decomposition of the space. Then the covariant derivative of X in the direction of Y is the projection of the derivative into T_p. Main Case: If M is stand-alone , i.e., not embedded in any ambient space , and M not flat. Then we need to specify how we will be tranlation the vector F(x+h) based at T_(x+h)M , ( tangent space of the point x+h in M , into the tangent space T_x M, of the point x in M ) . We do this , by specifying an isomorphism between the two tangent spaces . The choice of the isomorphism between any two points is the Christoffel symbol, and it determines the connection. Thanks For Any Comment/Suggestion/ Correction. P.S: I am kind of used to using ASCII. If others are not, let me know, and I will use Latex.