# Normal coordinates

(Redirected from 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

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

${\displaystyle \exp _{p}:T_{p}M\supset V\rightarrow M}$

and an isomorphism

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

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 TpM, and expp acts as a diffeomorphism between U and V. On the normal neighborhood U of p in M, the chart is given by:

${\displaystyle \varphi :=E^{-1}\circ \exp _{p}^{-1}:U\rightarrow \mathbb {R} ^{n}}$

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.

### Properties

The properties of normal coordinates often simplify computations. In the following, assume that ${\displaystyle U}$ is a normal neighborhood centered at a point ${\displaystyle p}$ in ${\displaystyle M}$ and ${\displaystyle x^{i}}$ are normal coordinates on ${\displaystyle U}$.

• Let ${\displaystyle V}$ be some vector from ${\displaystyle T_{p}M}$ with components ${\displaystyle V^{i}}$ in local coordinates, and ${\displaystyle \gamma _{V}}$ be the geodesic at ${\displaystyle t=0}$ pass through the point ${\displaystyle p}$ with velocity vector ${\displaystyle V}$, then ${\displaystyle \gamma _{V}}$ is represented in normal coordinates by ${\displaystyle \gamma _{V}(t)=(tV^{1},...,tV^{n})}$ as long as it is in ${\displaystyle U}$.
• The coordinates of a point ${\displaystyle p}$ are ${\displaystyle (0,...,0)}$
• In Riemannian normal coordinates at a point ${\displaystyle p}$ the components of the Riemannian metric ${\displaystyle g_{ij}}$ simplify to ${\displaystyle \delta _{ij}}$, i.e., ${\displaystyle g_{ij}(p)=\delta _{ij}}$.
• The Christoffel symbols vanish at ${\displaystyle p}$, i.e., ${\displaystyle \Gamma _{ij}^{k}(p)=0}$. In the Riemannian case, so do the first partial derivatives of ${\displaystyle g_{ij}}$, i.e., ${\displaystyle {\frac {\partial g_{ij}}{\partial x^{k}}}(p)=0,\,\forall i,j,k}$.

## 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 TpM. That is, one introduces on TpM the standard spherical coordinate system (r,φ) where r ≥ 0 is the radial parameter and φ = (φ1,...,φn−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 ${\displaystyle \partial /\partial r}$. That is,

${\displaystyle \langle df,dr\rangle ={\frac {\partial f}{\partial r}}}$

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

${\displaystyle g={\begin{bmatrix}1&0&\cdots \ 0\\0&&\\\vdots &&g_{\phi \phi }(r,\phi )\\0&&\end{bmatrix}}.}$

## References

• 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