Normal coordinates
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In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neigborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish.
A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t. This idea was implemented in a fundamental way by Albert Einstein in the general theory of relativity: the equivalence principle uses normal coordinates via inertial frames. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or Pseudo-Riemannian manifold. By contrast, there is no way to define normal coordinates for Finsler manifolds (Busemann 1955).
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[edit] Geodesic normal coordinates
Geodesic normal coordinates are local coordinates on a Riemannian manifold afforded by the exponential map
and an isomorphism
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 TpM and expp 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:
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 often simplify computations. 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 TpM with components Vi in local coordinates, and γV be the geodesic with starting point p and velocity vector V, then γV is represented in normal coordinates by γV(t) = (tV1,...,tVn) 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 δij
- The Christoffel symbols vanish at p, as do the first partial derivatives of gij
[edit] References
- Busemann, Herbert (1955), "On normal coordinates in Finsler spaces", Mathematische Annalen 129: 417–423, MR0071075, ISSN 0025-5831.
- Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1 (New ed.), Wiley-Interscience, ISBN 0471157333.
- Lee, John M. (2003), Introduction to Smooth Manifolds, Springer
- Chern, S. S.; Chen, W. H.; Lam, K. S.; Lectures on Differential Geometry, World Scientific, 2000
[edit] See also
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