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Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct the quotient space ## M^1\times S^1## by ##q:M^2 \to M^1\times S^1##. Now assuming the the metric tensor respect the equivalence relation there will be uniqe components, in say coordinate basis, ##\tilde g_{ij}## on ##M^1\times S^1## such that basis, ##\tilde g_{ij} \circ q =g_{ij} ##.
Now to my question: Is there any reason why we might not be able to transform away the off-diagonal elements of ##\tilde g_{ij}##?
Now to my question: Is there any reason why we might not be able to transform away the off-diagonal elements of ##\tilde g_{ij}##?