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I was trying to construct locally Euclidean metrics. Consider the sphere with the usual coordinate system induced from spherical coordinates in ##\mathbb R^3##. Consider a point ##p## in the Equator having coordinates ##(\theta_0, \phi_0) = (\pi/2, 0)##. If you make the coordinate change ##\xi^1 = \theta - \pi/2## and ##\xi^2 = \phi##, you find that the metric will be ##(d \xi^1)^2 + \sin^2 (\xi^1 + \pi/2) (d\xi^2)^2## in these coordinates. Close enough to ##p##, ##\xi^1 \approx 0## and the metric is flat.
But if I try to employ the same procedure for the north pole ##(\theta_0, \phi_0) = (0,0)## this doesnt work. Why?
But if I try to employ the same procedure for the north pole ##(\theta_0, \phi_0) = (0,0)## this doesnt work. Why?