Normal coordinates

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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 (Busemann 1955).

Geodesic normal coordinates[edit]

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

 \exp_p : T_{p}M \supset V \rightarrow M

and an isomorphism

 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. Now let U be a normal neighborhood of p in M then the chart is given by:

 \varphi := E^{-1} \circ \exp_p^{-1}: U \rightarrow \mathbb{R}^n

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[edit]

The properties of normal coordinates often simplify computations. In the following, assume that U is a normal neighborhood centered at p in M and (xi) are normal coordinates on U.

  • Let V be some vector from TpM with components Vi in local coordinates, and  \gamma_V be the geodesic with starting point p and velocity vector V, then  \gamma_V is represented in normal coordinates by  \gamma_V(t) = (tV^1, ... , tV^n) 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  \delta_{ij} .
  • The Christoffel symbols vanish at p. In the Riemannian case, so do the first partial derivatives of  g_{ij} .

Polar coordinates[edit]

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 \partial/\partial r. That is,

\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

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

References[edit]

See also[edit]