This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article's tone or style may not reflect the encyclopedic tone used on Wikipedia. See Wikipedia's guide to writing better articles for suggestions. (June 2010) (Learn how and when to remove this template message) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: "List of coordinate charts" – news · newspapers · books · scholar · JSTOR (September 2023) (Learn how and when to remove this template message) (Learn how and when to remove this template message)

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 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
Anti-de Sitter space AdSn Half-space chart

See also

References

  1. ^ 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.
Category