# 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, there is no way to define normal coordinates for Finsler manifolds Template:Harv.

## Geodesic normal coordinates

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

_{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:

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*.

### 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 *(x ^{i})* are normal coordinates on

*U*.

- Let
*V*be some vector from*T*with components_{p}M*V*in local coordinates, and be the geodesic with starting point^{i}*p*and velocity vector*V*, then is represented in normal coordinates by as long as it is in*U*. - The coordinates of
*p*are (0, ..., 0) - In Riemannian normal coordinates at
*p*the components of the Riemannian metric*g*simplify to . - The Christoffel symbols vanish at
*p*. In the Riemannian case, so do the first partial derivatives of .

## Polar coordinates

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*,φ) where *r* ≥ 0 is the radial parameter and φ = (φ_{1},...,φ_{n−1}) is a parameterization of the (*n*−1)-sphere. Composition of (*r*,φ) 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

## References

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- Chern, S. S.; Chen, W. H.; Lam, K. S.;
*Lectures on Differential Geometry*, World Scientific, 2000