Riemann normal coordinates

Riemann normal coordinates may be thought of as a generalization of Cartesian coordinatesMathworldPlanetmath from Euclidean spaceMathworldPlanetmath to any manifoldMathworldPlanetmath (which should be at least twice differentiableMathworldPlanetmathPlanetmath) with affine connectionMathworldPlanetmath. (Including Riemannian manifoldsMathworldPlanetmath as a special case, of course!)

To define a system of Riemann normal coordinates, one needs to pick a point P on the manifold which will serve as origin and a basis for the tangent spaceMathworldPlanetmathPlanetmath at P. Suppose that the manifold is d dimensional. To any d-tuplet of real numbers (x1,xn), we shall assign a point Q of the manifold by the following procedure:

Let v be the vector whose components with respect to the basis chosen for the tangent space at P are x1,xn. There exists a unique affinely-parameterized geodesicMathworldPlanetmath C(t) such that C(0)=P and [dC(t)/dt]t=0=v. Set Q=C(1). Then Q is defined to be the point whose Riemann normal coordinates are (x1,xn).

Riemann normal coordinates enjoy several important properties:

  1. 1.

    The connection coefficients vanish at the origin of Riemann normal coordinates.

  2. 2.

    Covariant derivativesMathworldPlanetmath reduce to partial derivativesMathworldPlanetmath at the origin of Riemann normal coordinates.

  3. 3.

    The partial derivatives of the components of the connection evaluated at the origin of Riemann normal coordinates equals the components of the curvature tensor. In fact, some authors take this property as a definition of the curvature tensor.

To every point on the manifold one may associate an open neighborhood of that point in which Riemann normal coordinates based at the point provide a diffeomorphism between the neighborhood and a subset of d. In general, Riemann normal coordinates become singular when a conjugate point of P is encountered so they are typically more useful for studying local geometryMathworldPlanetmath than global geometry.

References: doCarmo 1992 (see bibliography for differential geometry)

Title Riemann normal coordinates
Canonical name RiemannNormalCoordinates
Date of creation 2013-03-22 14:35:35
Last modified on 2013-03-22 14:35:35
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type Definition
Classification msc 53B05