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

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 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:

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[edit]

The properties of normal coordinates often simplify computations. In the following, assume that is a normal neighborhood centered at a point in and are normal coordinates on .

  • Let be some vector from with components in local coordinates, and be the geodesic at pass through the point with velocity vector , then is represented in normal coordinates by as long as it is in .
  • The coordinates of a point are
  • In Riemannian normal coordinates at a point the components of the Riemannian metric simplify to , i.e., .
  • The Christoffel symbols vanish at , i.e., . In the Riemannian case, so do the first partial derivatives of , i.e., .

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 . That is,

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

References[edit]

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