Can we use vielbeins on curved space?

Question

My question is: Can we use vielbeins on (let's say ) an anti de-sitter space?

This is why I am confused:

1. To couple fermions with gravity in curved space, using vielbeins is a well-known approach. So the answer to my question should be yes. AdS is just another curved space after all.

2. But a vielbein allows us to express a curved metric in terms of a flat metric $g_{\mu\nu}(x) = \eta_{ab} e ^{a} _\mu(x) e ^{b} _\nu(x) $. This kinda feels that we can express an curved space as a flat space close to a point. But how can then AdS space have constant negative curvature if I can make it flat around a point?

I think my second point is wrong. I am not sure if I can interpret vielbeins as actually changing the curvature to zero. Can someone shed some light as to why my second point ( or something else ) is incorrect?

Answer 1

It is perhaps helpful to mention that given a vielbein $e^a{}_{\mu}$ one can generically not find a local coordinate transformation $$x^{\mu}\quad\longrightarrow\quad y^a~=~f^a(x)\tag{1}$$ such that the vielbein is the Jacobian matrix $$e^a{}_{\mu}=\frac{\partial y^a}{\partial x^{\mu}}.\tag{2}$$ In fact eq. (2) is only possible if the integrability condition $$\frac{\partial e^a{}_{\mu}}{\partial x^{\nu}}~=~(\mu\leftrightarrow \nu)\tag{3}$$ is fulfilled, which is special (=not generic).

In particular, the existence of a vielbein on a pseudo-Riemannian manifold does not imply that the Levi-Civita curvature vanishes, cf. OP's question.