Geodesic coordinates

at a point $ P $ of a space with an affine connection with connection coefficients $ \Gamma _ {ij} ^ {k} $

Any system of coordinates in which all $ \Gamma _ {ij} ^ {k} = 0 $ at $ P $. If the equality $ \Gamma _ {ij} ^ {k} = 0 $ is satisfied at all points of a given curve, one speaks of geodesic coordinates along the curve (Fermi coordinates). In a Riemannian space with metric tensor $ g _ {ij} $, geodesic coordinates $ x ^ {k} $ are often defined by the conditions $ \partial g _ {ij} / \partial x ^ {k} = 0 $, which in this case are equivalent to the condition $ \Gamma _ {ij} ^ {k} = 0 $. For a symmetric connection, in particular, a Riemannian connection, geodesic coordinates exist at any point and along any regular arc without self-intersections. For a surface $ F $ in Euclidean space, geodesic coordinates are given by rectangular Cartesian coordinates of the tangent plane of $ F $ at $ P $; if the projection is effected onto the developable surface $ Q $, enveloped by the planes tangent to $ F $ along a curve, then the intrinsic Cartesian coordinates on $ Q $ will be Fermi coordinates on $ F $.

In geodesic coordinates the components at a point $ P $ of the covariant derivatives of a tensor field at $ P $ are equal to the ordinary derivatives of the tensor components. This may be taken as the definition of a covariant derivative, following E. Cartan's idea on the transfer of geometric objects or the operations of Euclidean geometry into more general spaces using special coordinate systems in which the effects of their non-Euclidean nature are eliminated to the greatest extent. This idea also forms the base of the use of geodesic coordinates in general relativity theory, where they are connected with locally inertial reference systems in space-time; the study of such systems plays an important role in the physical interpretation of the theory.

Geometric conditions $ \Gamma _ {ij} ^ {k} = 0 $ mean that there is a correspondence between the straight lines $ x ^ {i} = \xi ^ {i} t $( $ \xi ^ {i} = \textrm{ const } $, $ t $ is a parameter) in the domain where the coordinates are defined and curves $ \gamma ( t) $ in the space under consideration with initial at a point $ P $,

$$ \frac{D}{dt } \left ( \frac{d \gamma }{dt } \right ) = \left \{ \frac{d ^ {2} x ^ {i} }{dt ^ {2} } + \Gamma _ {jk} ^ {i} \frac{dx ^ {j} }{dt } \frac{dx ^ {k} }{dt } \right \} . $$

If the geodesic coordinates are such that to the straight lines issuing from a point $ P $ in all directions there correspond geodesics on which $ ( D / d t ) ( d \gamma / d t ) = 0 $, then they are called Riemannian coordinates.

Comments

Riemannian coordinates are better known as normal coordinates or geodesic polar coordinates in the West.

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
[a3]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)