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So, given two points, [itex]x[/itex] and [itex]x'[/itex], in a Lorentzian manifold (although I think it's the same for a Riemannian one). If in [itex]x[/itex] the determinant of the metric is [itex]g[/itex] and in the point [itex]x'[/itex] is [itex]g'[/itex]. How are [itex]g[/itex] and [itex]g'[/itex] related?By any means can [itex]g=g'[/itex]? In what conditions?

I'm sorry if this is a dumb question but when prooving an equation I found out that it holds only if and only if [itex]g=g'[/itex] and I don't think that this is always true.

Thank you.