Normal coordinates

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

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

with ${\displaystyle V}$  an open neighborhood of 0in${\displaystyle T_{p}M}$ , and an isomorphism

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

given by any basis of the tangent space at the fixed basepoint ${\displaystyle p\in M}$ . 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 pinM. A normal neighborhood U is an open 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. On a normal neighborhood UofpinM, the chart is given by:

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

The isomorphism E, and therefore the chart, is in no way unique. A convex normal neighborhood U is a normal neighborhood of every pinU. The existence of these sort of open neighborhoods (they form a topological basis) has been established by J.H.C. Whitehead for symmetric affine connections.

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 with ${\displaystyle \gamma _{V}(0)=p}$  and ${\displaystyle \gamma _{V}'(0)=V}$ . Then in normal coordinates, ${\displaystyle \gamma _{V}($t)=(tV^{1},...,tV^{n})}   as long as it is in ${\displaystyle U}$ . Thus radial paths in normal coordinates are exactly the geodesics through ${\displaystyle p}$ .
• The coordinates of the 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}  .

In the neighbourhood of any point ${\displaystyle p=(0,\ldots 0)}$  equipped with a locally orthonormal coordinate system in which ${\displaystyle g_{\mu \nu }(0)=\delta _{\mu \nu }}$  and the Riemann tensor at ${\displaystyle p}$  takes the value ${\displaystyle R_{\mu \sigma \nu \tau }(0)}$  we can adjust the coordinates ${\displaystyle x^{\mu }}$  so that the components of the metric tensor away from ${\displaystyle p}$  become

${\displaystyle g_{\mu \nu }($x)=\delta _{\mu \nu }-{\tfrac {1}{3}}R_{\mu \sigma \nu \tau }(0)x^{\sigma }x^{\tau }+O(|x|^{3}).}  

The corresponding Levi-Civita connection Christoffel symbols are

${\displaystyle {\Gamma ^{\lambda }}_{\mu \nu }($x)=-{\tfrac {1}{3}}{\bigl [}R_{\lambda \nu \mu \tau }(0)+R_{\lambda \mu \nu \tau }(0){\bigr ]}x^{\tau }+O(|x|^{2}).}  

Similarly we can construct local coframes in which

${\displaystyle e_{\mu }^{*a}($x)=\delta _{a\mu }-{\tfrac {1}{6}}R_{a\sigma \mu \tau }(0)x^{\sigma }x^{\tau }+O(x^{2}),}  

and the spin-connection coefficients take the values

${\displaystyle {\omega ^{a}}_{b\mu }($x)=-{\tfrac {1}{2}}{R^{a}}_{b\mu \tau }(0)x^{\tau }+O(|x|^{2}).}  

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_pM. That is, one introduces on T_pM the standard spherical coordinate system (r,φ) where r ≥ 0 is the radial parameter and φ = (φ₁,...,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 gradientofr 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

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

• 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

• Gauss Lemma
• Fermi coordinates
• Local reference frame
• Synge's world function