Harmonic coordinates

<p>In <a href="/facts/Riemannian_geometry/pPEzjyQC">Riemannian geometry</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, harmonic coordinates are a certain kind of <a href="/facts/Coordinate_chart/eLKXMzzR">coordinate chart</a> on a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a>, determined by a <a href="/facts/Riemannian_metric/5KYnmEgF">Riemannian metric</a> on the manifold. They are useful in many problems of <a href="/facts/Geometric_analysis/wLtqsMAe">geometric analysis</a> due to their regularity properties.
</p><p>In two dimensions, certain harmonic coordinates known as <a href="/facts/Isothermal_coordinates/SFBDbMcB">isothermal coordinates</a> have been studied since the early 1800s. Harmonic coordinates in higher dimensions were developed initially in the context of <a href="/facts/Lorentzian_manifold/VFXsVjqb">Lorentzian geometry</a> and <a href="/facts/General_relativity/jVsoIg8X">general relativity</a> by <a href="/facts/Albert_Einstein/CNTet8qu">Albert Einstein</a> and <a href="/facts/Cornelius_Lanczos/FRqrbJIb">Cornelius Lanczos</a> (see <a href="/facts/Harmonic_coordinate_condition/VLdJEdnG">harmonic coordinate condition</a>). Following the work of <a href="/facts/Dennis_DeTurck/GkTNBYWr">Dennis DeTurck</a> and <a href="/facts/Jerry_Kazdan/vKlyS1dt">Jerry Kazdan</a> in 1981, they began to play a significant role in the <a href="/facts/Geometric_analysis/wLtqsMAe">geometric analysis</a> literature, although Idzhad Sabitov and S.Z. Šefel had made the same discovery five years earlier.
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Harmonic coordinates open-in-new

Harmonic coordinates