List of coordinate charts

This article contains a non-exhaustive list of coordinate charts for Riemannian manifolds and pseudo-Riemannian manifolds. Coordinate charts are mathematical objects of topological manifolds, and they have multiple applications in theoretical and applied mathematics. When a differentiable structure and a metric are defined, greater structure exists, and this allows the definition of constructs such as integration and geodesics.

Charts for Riemannian and pseudo-Riemannian surfaces

The following charts (with appropriate metric tensors) can be used in the stated classes of Riemannian and pseudo-Riemannian surfaces:


Surface Class	Surface	Charts
$n$ -spheres	$n$ -sphere Sⁿ	Hopf chart Hyperspherical coordinates
	Sphere S²	Spherical coordinates Stereographic chart Central projection chart Axial projection chart Mercator chart
	3-sphere S³	Polar chart Stereographic chart Mercator chart
Euclidean spaces	$n$ -dimensional Euclidean space Eⁿ	Cartesian chart
	Euclidean plane E²	Bipolar coordinates Biangular coordinates Two-center bipolar coordinates
	Euclidean space E³	Polar spherical chart Cylindrical chart Elliptical cylindrical, hyperbolic cylindrical, parabolic cylindrical charts Parabolic chart Hyperbolic chart Prolate spheroidal chart (rational and trigonometric forms) Oblate spheroidal chart (rational and trigonometric forms) Toroidal chart
Hyperbolic spaces	$n$ -dimensional hyperbolic space Hⁿ	Upper half-space chart (Poincaré model) Hopf chart Central projection chart (Klein model)
Hyperbolic spaces	Hyperbolic plane H²	Polar chart Stereographic chart (Poincaré model)
Embedded surfaces	Embedded in E³	Monge chart^[1]
Minimal surfaces	Minimal surfaces	Asymptotic chart
Lorentzian manifolds	De Sitter space dSn	Static chart
Lorentzian manifolds	Anti-de Sitter space AdS_n	Half-space chart

^ Sotomayor, Jorge. "On Monge's and Principal Curvature Configurations, Revised Version, Essay dedicated to the memory of Carlos T. Gutiérrez Vidalón". researchgate.net. Retrieved 1 September 2023. He settled down to work in a small desk in my room, filling out pages and more pages, which quickly exceeded the space available at the table, spreading through the floor. He performed long calculations in Monge charts and drew beautiful pictures.