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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:
| Surface Class | Surface | Charts |
|---|---|---|
| n-spheres | n-sphere S | Hopf chart |
| Sphere S | Spherical coordinates
Stereographic chart Central projection chart Axial projection chart Mercator chart | |
| 3-sphere S | Polar chart | |
| Euclidean spaces | n-dimensional Euclidean space E | Cartesian chart |
| Euclidean plane E | Bipolar coordinates | |
| Euclidean space E | Polar spherical chart
Prolate spheroidal chart (rational and trigonometric forms) Oblate spheroidal chart (rational and trigonometric forms) | |
| Hyperbolic spaces | n-dimensional hyperbolic space H | Upper half-space chart (Poincaré model)
Hopf chart Central projection chart (Klein model) |
| Hyperbolic plane H | Polar chart
Stereographic chart (Poincaré model) | |
| Embedded surfaces | Embedded in E | Monge chart |
| Minimal surfaces | Minimal surfaces | Asymptotic chart |
| Lorentzian manifolds | De Sitter space dSn | Static chart |
| Anti-de Sitter space AdSn | Half-space chart |
See also
- Coordinate chart
- Coordinate system
- Metric tensor
- List of mathematics lists, particularly:
References
- 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.