Omitted Examples
There are of course many important and interesting examples of Riemannian and semi-Riemannian manifolds which are not even mentioned here, including:
- Bianchi groups: there is a short list (up to local isometry) of three-dimensional real Lie groups, which when considered as Riemannian-three manifolds give homogeneous but (usually) non-isotropic geometries.
- other noteworthy real Lie groups,
- Lorentzian manifolds which (perhaps with some added structure such as a scalar field) serve as solutions to the field equations of various metric theories of gravitation, in particular general relativity. There is some overlap here; in particular:
- axisymmetric spacetimes such as Weyl vacuums possess various charts discussed here; the prolate spheroidal chart turns out to be particularly useful,
- de Sitter models in cosmology are, as manifolds, nothing other than H1,3 and as such possess numerous interesting and useful charts modeled after ones listed here.
In addition, one can certainly consider coordinate charts on complex manifolds, perhaps with metrics which arise from bundling Hermitian forms. Indeed, this natural generalization is just the tip of iceberg. However, these generalizations are best dealt with in more specialized lists.
Read more about this topic: List Of Coordinate Charts
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