How do we mathematically define an inertial frame in GR?

[Kontilera]

How do we mathematically define a inertial frame in GR?
Is it only a basis in some tangentspace or does it have to be induced by a coordinatechart? :/

 

[Ben Niehoff]

It's only a basis in the tangent space. But you can always find local coordinates that induce this basis at a single point. They are called normal coordinates.