Geodesic normal coordinates

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Contents

[edit] Introduction

Geodesic normal coordinates (also Riemannian normal coordinates or simply normal coordinates) are local coordinates on a Riemannian manifold implied by the exponential map

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

and an isomorphism

E: \mathbb{R}^n \rightarrow T_{p}M

where in the domain of E an orthonormal basis is assumed.

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.


[edit] Properties

The properties of normal coordinates my simplify some computations, so it is useful to keep them in mind. 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 γV be the geodesic with starting point p and velocity vector V, then γV is represented in normal coordinates by γV(t) = (tV1,...,tVn) as long as it is in U
  • The coodinates of p are (0, ... , 0)
  • At p the components of the Riemannian metric g simplify to δij
  • The Christoffel symbols vanish at p as well as the first partial derivatives of gij

[edit] See also

[edit] References

  • Lee, John M.; Introduction to Smooth Manifolds, Springer, 2003
  • Chern, S. S.; Chen, W. H.; Lam, K. S.; Lectures on Differential Geometry, World Scientific, 2000
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