Harmonic coordinates

In Riemannian geometry, a branch of mathematics, harmonic coordinates are a coordinate system (x¹,...,xⁿ) on a Riemannian manifold each of whose coordinate functions xⁱ is harmonic, meaning that it satisfies Laplace's equation

$\Delta x^{i}=0.\,$

Here Δ is the Laplace–Beltrami operator. Equivalently, regarding a coordinate system as a local diffeomorphism φ : M → Rⁿ, the coordinate system is harmonic if and only if φ is a harmonic map of Riemannian manifolds, roughly meaning that it minimizes the elastic energy of "stretching" M into Rⁿ. The elastic energy is expressed via the Dirichlet energy functional

$E[\varphi ]=\int _{M}|d\varphi |^{2}\,dV.$

Overview

In two dimensions, harmonic coordinates have been well understood for more than a century, and are closely related to isothermal coordinates, the latter being a special case of the former. Harmonic coordinates in higher dimensions were developed initially in the context of general relativity by Einstein (1916) (see harmonic coordinate condition). They were then introduced into Riemannian geometry by Sabitov & Šefel (1976) and later were studied by DeTurck & Kazdan (1981). The essential motivation for introducing harmonic coordinate systems is that the metric tensor is especially smooth when written in these coordinate systems.

Harmonic coordinates are characterized in terms of the Christoffel symbols by means of the relation

$g^{ij}\Gamma _{ij}^{k}=0\,$

and indeed, for any coordinate system at all,

$\Delta x^{k}=-g^{ij}\Gamma _{ij}^{k}.$

Harmonic coordinates always exist (locally), a result which follows easily from standard results on the existence and regularity of solutions of elliptic partial differential equations. In particular, the equation

$\Delta u^{j}=0\,$

has a solution in a ball around any given point p, such that u^j(p) and $\partial u^{j}/\partial x^{i}(p)$ are all prescribed.

The basic regularity theorem concerning the metric in harmonic coordinates is that if the components of the metric are in the Hölder space C^k,α when expressed in some coordinate system, then they are in that same Hölder space when expressed in harmonic coordinates. Harmonic mapping to generate harmonic coordinates in regions with boundary is one of the original well known methods for grid generation in the field of computational fluid dynamics. Here the goal is to find a harmonic map of a given region (in Euclidean space or in a Riemannian manifold) to a convex region (very often a rectangle or a box in the case of grid generation in Euclidean space) with the additional requirement that the boundary map should be a homeomorphism (see the works of S. S. Sritharan in the reference list below).

In general relativity, harmonic coordinates are solutions of the wave equation instead of the Laplace . This is known as the harmonic coordinate condition in physics.

References

• DeTurck, Dennis M.; Kazdan, Jerry L. (1981), "Some regularity theorems in Riemannian geometry", Annales Scientifiques de l'École Normale Supérieure, Série 4, 14 (3): 249–260, ISSN 0012-9593, MR 0644518.
• Einstein, Albert (1916), "Näherungsweise Integration der Feldgleichungen der Gravitation", S.-B. Preuss. Akad. Wiss.: 688–696 [Approximative Integration of the Field Equations of Gravitation].
• Lee, John; Parker, Thomas (1987), "The Yamabe problem", Bull. Amer. Math. Soc., 17: 37–81, doi:10.1090/s0273-0979-1987-15514-5.
• Smith, P.; Sritharan, S. S. (1988), "Theory of Harmonic Grid Generation" (PDF), Complex Variables, 10: 359–369., doi:10.1080/17476938808814314
• Sritharan, S. S. (1992), "Theory of Harmonic Grid Generation-II", Applicable Analysis, 44: 127–149., doi:10.1080/00036819208840072

• Sabitov, I. H.; Šefel, S. Z. (1976), "Connections between the order of smoothness of a surface and that of its metric", Akademija Nauk SSSR. Sibirskoe Otdelenie. Sibirskii Matematičeskii Žurnal, 17 (4): 916–925, ISSN 0037-4474, MR 0425855.