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.
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 (Busemann 1955).
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.
The existence of these sort of open neighborhoods (they form a topological basis) has been established by J.H.C.
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 On a Riemannian manifold, a normal coordinate system at p facilitates the introduction of a system of spherical coordinates, known as polar coordinates.
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
As a result, the metric in polar coordinates assumes a block diagonal form