I sometimes do a tensor calculations with setting $g_{\mu\nu}= \eta_{\mu\nu},\; \Gamma_\alpha{}^\lambda{}_\beta=0$. I usually call it the locally-flat frame but when I look for description about locally-flat am not sure. Am I right?

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    $\begingroup$ Those are Riemann normal coordinates. $\endgroup$
    – Slereah
    Feb 22, 2017 at 5:52
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    $\begingroup$ Riemann normal coordinates is what you are looking for. Locally inertial or locally flat coordinates are also used by physicists, but mathematicians call this Riemannian normal coordinates. $\endgroup$ Feb 22, 2017 at 5:52

1 Answer 1


You understand it correctly it is the Riemann normal coordinates or locally flat coordinate.

From nlab they described

`Around every point of a Riemannian manifold there is a coordinate system such that the Levi-Civita connection of the metric pulled back to these coordinates vanishes at that point. (Notice that the Riemann curvature will not in general vanish even at that point).'


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