Geodesic Normal Coordinates

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## Geodesic normal coordinates

### Properties

## Polar coordinates

## References

## See also

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Geodesic Normal Coordinates

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 neighborhood 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* (the affine parameter). 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, in general there is no way to define normal coordinates for Finsler manifolds in a way that the exponential map are twice-differentiable (Busemann 1955).

Geodesic normal coordinates are local coordinates on a manifold with an affine connection afforded by the exponential map

and an isomorphism

given by any basis of the tangent space at the fixed basepoint *p* ? *M*. If the additional structure of a Riemannian metric is imposed, then the basis defined by *E* may be required in addition to be orthonormal, and the resulting coordinate system is then known as a **Riemannian normal coordinate system**.

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 _{p}M*, and exp

The isomorphism *E* can be any isomorphism between the two vector spaces, so there are as many charts as there are different orthonormal bases in the domain of *E*.

The properties of normal coordinates often simplify computations. In the following, assume that is a normal neighborhood centered at a point in and are normal coordinates on .

- Let be some vector from with components in local coordinates, and be the geodesic at pass through the point with velocity vector , then is represented in normal coordinates by as long as it is in .
- The coordinates of a point are
- In Riemannian normal coordinates at a point the components of the Riemannian metric simplify to , i.e., .
- The Christoffel symbols vanish at , i.e., . In the Riemannian case, so do the first partial derivatives of , i.e., .

On a Riemannian manifold, a normal coordinate system at *p* facilitates the introduction of a system of spherical coordinates, known as **polar coordinates**. These are the coordinates on *M* obtained by introducing the standard spherical coordinate system on the Euclidean space *T*_{p}*M*. That is, one introduces on *T*_{p}*M* the standard spherical coordinate system (*r*,?) where *r* >= 0 is the radial parameter and ? = (?_{1},...,?_{n−1}) is a parameterization of the (*n*−1)-sphere. Composition of (*r*,?) with the inverse of the exponential map at *p* is a polar coordinate system.

Polar coordinates provide a number of fundamental tools in Riemannian geometry. The radial coordinate is the most significant: geometrically it represents the geodesic distance to *p* of nearby points. Gauss's lemma asserts that the gradient of *r* is simply the partial derivative . That is,

for any smooth function *ƒ*. As a result, the metric in polar coordinates assumes a block diagonal form

- Busemann, Herbert (1955), "On normal coordinates in Finsler spaces",
*Mathematische Annalen*,**129**: 417-423, doi:10.1007/BF01362381, ISSN 0025-5831, MR 0071075. - Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
*Foundations of Differential Geometry*, Vol. 1 (New ed.), Wiley Interscience, ISBN 0-471-15733-3. - Chern, S. S.; Chen, W. H.; Lam, K. S.;
*Lectures on Differential Geometry*, World Scientific, 2000

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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