PeterDonis
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"Don't panic!" said:So the homeomorphism guarantees that we can assign n-tuples of real numbers to each point and then the metric on a given patch determines the type of coordinate maps we can construct from that patch to ##\mathbb{R}^{n}##
Not quite. An assignment of n-tuples of real numbers to each point is a coordinate map from the patch to ##\mathbb{R}^{n}##. The geometry of the manifold determines the form of the line element ##ds^2## when expressed in terms of that coordinate map.