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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.

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# Riemannian Connections: Summary

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