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Assume $U\subset {\mathbb ℝ}^2$, $g$ and $h$ two metrics on $U$. Assume that the Christoffel symbols $Γ^i_{jk}(g)\equiv Γ^i_{jk}(h)$, as a pointwise identity, for all sets of indices. Does it follow that $g=h$? If not, is there a counter-example? Will then the result hold true if $g$ and $h$ coincide in a point or on the boundary or a region? In general, what can be said about the relationship between g and h under the above hypothesis?

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As a quick response:

Remember how we can calculate the Christoffel symbols from the metric, we use there first order derivatives, thus roughly speaking the opposite relation will be kind of first order PDE, which in turn has not one solution.

So knowing Christoffel don't identify the metric uniquely, because they contain "less" information about the manifold.

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Replacing metric with its scalar multiple results in the same Levi-Civita connection, hence, the same Christoffel symbols.

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