Normal coordinates

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In Riemannian geometry, the normal coordinates at p consist of a chart such that locally the symmetric part of the Christoffel symbols vanish, i.e. $\Gamma^a_{(bc)} =0$ . Furthermore, at p, the following equations hold

$g_{ij}(p) = \delta_{ij}, \quad \frac{\partial g_{ij}}{\partial x^k}(p) = 0, \quad \Gamma^i_{jk}(p) = 0.$

Therefore, the covariant derivative reduces to a partial derivative, and the geodesics through p are locally linear functions of t. This idea was implemented by Einstein in his General Relativity using his Equivalence Principle and understanding the normal coordinates as an inertial frame.

Normal coordinates are specific to Riemannian and Pseudo-Riemannian geometry. In particular, they do not generalize to Finsler geometry (Rund, 1959).

1 Geodesic normal coordinates
- 1.1 Properties
2 References
3 See also

[edit] Geodesic normal coordinates

Geodesic normal coordinates are local coordinates on a Riemannian manifold implied by the exponential map

$\exp_p : T_{p}M \supset V \rightarrow M$

and an isomorphism

$E: \mathbb{R}^n \rightarrow T_{p}M$

where in the domain of E an orthonormal basis is assumed.

Normal coordinates exist on a normal neighborhood of a point p in M. A normal neighborhood U is a subset of M such that there is a proper neighborhood V of the origin in the tangent space T_pM and exp_p acts as a diffeomorphism between U and V. Now let U be a normal neighborhood of p in M then the chart is given by:

$\varphi := E^{-1} \circ \exp_p^{-1}: U \rightarrow \mathbb{R}^n$

The isomorphism E can be any isomorphism between both vectorspaces, so there are as many charts as different orthonormal bases exist in the domain of E.

[edit] Properties

The properties of normal coordinates may simplify some computations, so it is useful to keep them in mind. In the following, assume that U is a normal neighborhood centered at p in M and (xⁱ) are normal coordinates on U.

Let V be some vector from T_pM with components Vⁱ in local coordinates, and $γ V$ be the geodesic with starting point p and velocity vector V, then $γ V$ is represented in normal coordinates by $γ V (t) = (t V 1,..., t V n)$ as long as it is in U
The coordinates of p are (0, ... , 0)
At p the components of the Riemannian metric g simplify to $δ i j$
The Christoffel symbols vanish at p as well as the first partial derivatives of $g i j$

[edit] References

H. Rund. The Differential Geometry of Finsler Spaces, Springer-Verlag, 1959. ASIN B0006AWABG.
Lee, John M.; Introduction to Smooth Manifolds, Springer, 2003
Chern, S. S.; Chen, W. H.; Lam, K. S.; Lectures on Differential Geometry, World Scientific, 2000