# What is the name of this frame?

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?

• Those are Riemann normal coordinates. – Slereah Feb 22 '17 at 5:52
• 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. – Bence Racskó Feb 22 '17 at 5:52

## 1 Answer

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).'