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: Hyperspherical coordinates Surfaces embedded in E3: Monge chart Certain minimal surfaces: Asymptotic chart (see also asymptotic line)
Euclidean plane E2: Cartesian chart Sphere S2: Spherical coordinates Stereographic chart Central projection chart Axial projection chart Mercator chart 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): Central projection
Sn
Hopf chart
Hn
Upper half-space chart (Poincaré model) Hopf chart The following charts apply specifically to three-dimensional manifolds: Axially symmetric manifolds: Cylindrical chart Parabolic chart Hyperbolic chart Toroidal chart
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
See also
Coordinate chart
Coordinate system
Metric tensor
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