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 with V an open neighborhood of 0 in T_{p}M, and an isomorphism 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 TpM, and expp acts as a diffeomorphism between U and V. On a normal neighborhood U of p in M, the chart is given by: 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 basis) 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.

Explicit formulae

In the neighbourhood of any point equipped with a locally orthonormal coordinate system in which and the Riemann tensor at p takes the value we can adjust the coordinates x^\mu so that the components of the metric tensor away from p become The corresponding Levi-Civita connection Christoffel symbols are Similarly we can construct local coframes in which and the spin-connection coefficients take the values

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