Are the values of the christoffel symbols the same for all coordinate systems on a surface/manifold? I would love to see an example for the cone in two different parametrizations.
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Around every point you can choose normal coordinates in which the christoffel symbols vanish. – gofvonx Sep 15 '13 at 10:31
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The answer is no. The reason is that the Christoffel symbols are not scalar fields, nor tensor fields, they might vanish completely in a coordinate system and yet be non-vanishing in another.
As a simple example consider the plane in cartesian coordinates: All Christoffel symbols vanish. Now consider polar coordinates, there will be some which are non-vanishing.
Rogelio Molina
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