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Diffgeom β

Fermi coordinates

This article defines a property that is evaluated for a choice of local coordinates at a point on a Riemannian manifold

Definition

Let (M,g) be a Riemannian manifold. Let p \in M and \gamma a geodesic curve through p. Then a choice of local coordinates at p is said to give Fermi coordinates if:

  • For small changes of the first variable, we get the geodesic in the neighbourhood of p. In other words, the geodesic locally is the axis for the first coordinate
  • The metric tensor, restricted to that geodesic, is the Euclidean metric
  • All Christoffel symbols vanish on the geodesic

Relation with other properties

Weaker properties