Let $\sigma(u, v) = (u \cos v, u \sin v, 0)$ be the polar parametrization of a plane. Compute the Christofell symbols and show that it is not reparametrization invariant.
I have computed the Christofell symbols but how to show that it is not reparametrization invariant?
The Christofell symbols are $0$ and $\frac1u$.
Need some hints to proceed with the problem!
Isometric mappings leave Christoffel symbols unchanged.
Polar grid alone is not the only net to parametrize a plane.
Any arbitrary manner of changing distances on the plane by a new metric may leave zero Gauss curvature and integral curvature (trivial case) unchanged but not the parametrization itself !