List of Coordinate Charts - Favorite Riemannian Three-manifolds

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:

(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 (PoincarĂ© model)
    • Hopf chart

Read more about this topic:  List Of Coordinate Charts

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