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I read somewhere the geodesic distance between an arbitrary point ##x## and the base point ##x_0## in normal coordinates is just the Euclidean distance. Why?! That's the part I don't understand. I know that one can write

[itex]

g_{\mu \nu} = \delta_{\mu \nu} - \frac{1}{6} (R_{\mu \rho \nu \sigma} + R_{\mu \sigma \nu \rho} ) (x^\rho-x_0^\rho) (x^\sigma-x_0^\sigma) + \dots

[/itex]

I've tried to figure out the square of the distance (which seems more simple than the distance) in normal coordinates. The calculations wasn't clear, what I get is

[itex]

d(x,x_0)^2=g_{\mu\nu} (x^\mu-x_0^\mu)(x^\nu-x_0^\nu) +1/3 R_{\mu\nu\rho\sigma}x^\rho x^\sigma(x^\mu-x_0^\mu)(x^\mu- x_0^\nu) + \dots

[/itex]

Where am I doing anything wrong?

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# Distance function in Riemannian normal coordinates

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