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List of coordinate charts

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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:

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!Surface Class

!Surface

!Charts

rowspan="3" |N-sphereN-sphere
spheres

|{{mvar|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

rowspan="3" |Euclidean spaces

|{{mvar|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

rowspan="2" |Hyperbolic spaces

|{{mvar|n}}-dimensional hyperbolic space Hn

|Upper half-space chart (Poincaré model)

Hopf chart

Central projection chart (Klein model)

Hyperbolic plane H2

|Polar chart

Stereographic chart (Poincaré model)

Embedded surfaces

|Embedded in E3

|Monge chart{{cite web |last1=Sotomayor |first1=Jorge |title=On Monge's and Principal Curvature Configurations, Revised Version, Essay dedicated to the memory of Carlos T. Gutiérrez Vidalón |url=https://www.researchgate.net/publication/328276602_On_Monge's_and_Principal_Curvature_Configurations_Revised_Version_Essay_dedicated_to_the_memory_of_Carlos_T_Gutierrez_Vidalon |access-date=1 September 2023 |website=researchgate.net |language=en |quote="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."}}

Minimal surfaces

|Minimal surfaces

|Asymptotic chart

rowspan="2" |Lorentzian manifolds

|De Sitter space dSn

|Static chart

Anti-de Sitter space AdSn

|Half-space chart

See also

References

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Coordinate charts