Confusion about curved manifolds

[Kontilera]

Hello fellow mathematicians!
I am currently confused about the mathematics of curved manifolds.
When introducing the affine connection on our bundle it seems to be an object totally determined by our coordinate chart.
But since we can compute objects describing the curvature from the Christoffel symbol, e.g. the Riemann tensor, the curvature of our manifold should be encoded in our coordinate chart.
This is wrong.

Let me try to describe my way of thinking about the Christoffel symbol and maybe someone here can tell me where my logic fails.
The Christoffer symbols is introduced to "connect" nearby (co)tangentspace to each other. It determines the twisting and turning of the basis vectors when we move along the manifold.
So if you give me a coordinate chart. Wouldnt I just be able to take two nearby tangentspaces, look at the induced bases and then use:
$$\nabla_{E_j}E_i = \Gamma_{ij}^k E_k,$$
in order to get my coefficients?

 

[pat_connell]

Answer: No, the Christoffel symbols are not determined by your coordinate chart. Instead, the Christoffel symbols represent the affine connection on your bundle (manifold) which is independent of your choice of coordinates. The affine connection is an object that describes the parallel transport of vector fields along the manifold. It can be used to compute the curvature of the manifold, but is not determined by the coordinates itself. To calculate the Christoffel symbols you need to use the metric tensor and its derivatives. In particular, if you have a Riemannian metric, then the Christoffel symbols can be obtained from the metric tensor and its derivatives via the Christoffel equation. Hope this helps!