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:
- Radially symmetric surfaces:
- Surfaces embedded in E3:
- Certain minimal surfaces:
- Asymptotic chart (see also asymptotic line)
- Euclidean plane E2:
- Sphere S2:
- Hyperbolic plane H2:
- Polar chart
- Stereographic chart (Poincaré model)
- Upper half-space chart (Poincaré model)
- Central projection chart (Klein model)
- Mercator chart
- AdS2 (or S1,1) and dS2 (or H1,1):
- Sn
- Hn
- Upper half-space chart (Poincaré model)
- Hopf chart
The following charts apply specifically to three-dimensional manifolds:
- Axially symmetric manifolds:
- Three-dimensional Euclidean space E3:
- Cartesian
- 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
- Cassini toroidal chart and Cassini bipolar chart
- Three-sphere S3
- Hyperbolic three-space H3
- Polar chart
- Upper half space chart (Poincaré model)
- Hopf chart
See also
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.
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