Question about Christoffel symbol's value

[raopeng]

In my geometry textbook it is stated that intuitively we can choose a suitable basis of coordinates that the components Christoffel symbol vanishes locally at that point(= 0). However can one obtain a formal proof of it? For example if we use rectification theorem to rectify the geodesics passing through a point into straight lines, can we say under such diffeomorphism the Christoffel symbol vanishes since the geodesic is mapped into a line in the neighbourhood of that point?

 

[quasar987]

Formally, this is the existence of normal coordinates. See for instance Lee (Riemannian manifolds, p.76-78)

First you prove that the exponential map T_pM-->M v-->"geodesic through p with initial speed v evaluated at t=1" is a local diffeomorphism using the inverse function theorem. Then you pick a g_p-orthonormal basis of T_pM and use this together with the exponential map to define local coordinates of M around p (normal coordinates). In these coordinates, geodesics are straight lines. Then stare at the geodesic equation in these coordinates. Clearly, the Christofel symbols all vanish.

 

[raopeng]

Thank you, it puts everything in its place