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 .

Geodesic normal coordinates

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

: $\exp_p : T_M \supset V \rightarrow M$

and an isomorphism

: $E: \mathbb^n \rightarrow T_M$

given by any basis of the tangent space at the fixed basepoint $p\in M$. 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 ''p'' in ''M''. 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 ''U'' of ''p'' in ''M'', the chart is given by:

: $\varphi := E^ \circ \exp_p^: U \rightarrow \mathbb^n$

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

Properties

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

* Let $V$ be some vector from $T_p M$ with components $V^i$ in local coordinates, and $\gamma_V$ be the geodesic with $\gamma_V(0) = p$ and $\gamma_V'(0) = V$. Then in normal coordinates, $\gamma_V(t) = (tV^1, ... , tV^n)$ as long as it is in $U$. Thus radial paths in normal coordinates are exactly the geodesics through $p$.
* The coordinates of the point $p$ are $(0, ..., 0)$
* In Riemannian normal coordinates at a point $p$ the components of the Riemannian metric $g_$ simplify to $\delta_$, i.e., $g_(p)=\delta_$.
* The Christoffel symbols vanish at $p$, i.e., $\Gamma_^k(p)=0$. In the Riemannian case, so do the first partial derivatives of $g_$, i.e., $\frac(p) = 0,\,\forall i,j,k$.

Explicit formulae

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

: $g_(x)= \delta_ - \frac R_(0) x^\sigma x^\tau + O(, x, ^3).$

The corresponding Levi-Civita connection Christoffel symbols are

: $_(x) = -\frac (R_(0)+R_(0))x^\tau+ O(, x, ^2).$

Similarly we can construct local coframes in which

: $e^_\mu(x)= \delta_ - \frac R_(0) x^\sigma x^\tau +O(x^2),$

and the spin-connection coefficients take the values

: $_(x)= - \frac _(0)x^\tau+O(, x, ^2).$

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 = (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 gradient of ''r'' is simply the partial derivative $\partial/\partial r$. That is,
:$\langle df, dr\rangle = \frac$
for any smooth function ''ƒ''. As a result, the metric in polar coordinates assumes a block diagonal form
:$g = \begin 1&0&\cdots\ 0\\ 0&&\\ \vdots &&g_(r,\phi)\\ 0&& \end.$

References

* .
* {{citation , last1=Kobayashi, first1=Shoshichi, last2=Nomizu, first2=Katsumi , title = Foundations of Differential Geometry, volume=1, publisher= Wiley Interscience , year=1996, edition=New, isbn=0-471-15733-3.
* Chern, S. S.; Chen, W. H.; Lam, K. S.; ''Lectures on Differential Geometry'', World Scientific, 2000

See also

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

Riemannian geometry
Coordinate systems in differential geometry