Approximate local flatness = Approximate local symmetries?

[Suekdccia]

TL;DR Summary

Approximate local flatness = Approximate local symmetries?

Pseudo-Riemannian manifolds (such as spacetime) are locally Minkowskian and this is very important for relativity since even in a highly curved spacetime, one could locally approximate the spacetime into a flat minkowski one.

However, this would be an approximation. Perhaps this is a naive question but, would this mean that the local symmetries (such as Poincaré, Lorentz...) hold also only approximately?

 

[PeterDonis]

would this mean that the local symmetries (such as Poincaré, Lorentz...) hold also only approximately?

No. The "local symmetries" you refer to are symmetries of the tangent space at each event. They are not symmetries of the spacetime. In the tangent space those symmetries are exact.