Hi:(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Riemannian Connections: Summary

**Physics Forums | Science Articles, Homework Help, Discussion**