Definition of Inertial Frame in GR: Math Explained

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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? :/
 
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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.
 
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