Favorite Riemannian Three-manifolds
Here are some charts which (with appropriate metric tensors) can be used in the stated classes of three-dimensional Riemannian manifolds:
- Diagonalizable manifolds:
- isothermal chart
(Note: not every three manifold admits an isothermal chart.)
- Axially symmetric manifolds:
- cylindrical chart
- parabolic chart
- hyperbolic chart
- prolate spheroidal chart (rational and trigonometric forms)
- oblate spheroidal chart (rational and trigonometric forms)
- toroidal chart
Here are some charts which can be used on some of the most useful Riemannian three-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
- polar chart
- stereographic chart
- Hopf chart
- Hyperbolic three-space H3
- polar chart
- upper half space chart (Poincaré model)
- Hopf chart
Read more about this topic: List Of Coordinate Charts
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