Is there another kind of two-dimensional geometry?

Question

I learned that in two-dimensional geometry, there are Euclidean geometry, hyperbolic geometry and spherical geometry. These geometries are homogeneous and isotropic. Is there another kind of two-dimensional geometry that is homogeneous and isotropic?

Answer 1

I. M. Yaglom in his book A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity (Springer, 1979) lists and explains the nine plane geometries described by Klein.

Answer 2

12 May 2021

There are nine possible two dimensional geometries.

These result from elaborating the different possible permutations of lineal and angular metric curvature.

For example, ordinary Euclidean geometry has a lineal metric with zero curvature and an angular metric with positive curvature.

In the Riemannian geometry of a sphere or ellipsoid, both metrics have positive curvature. In Hyperbolic geometry the lineal metric has negative curvature and the angular metric has positive curvature.

The Minkowski space of special relativity has an angular metric with negtive curvature and a lineal metric with zero curvature.

The other possibilities are more obscure.

Useful references:

https://arxiv.org/pdf/1406.7309.pdf

https://www-m10.ma.tum.de/foswiki/pub/Lehre/WS0910/ProjektiveGeometrieWS0910/GeomBook.pdf

https://link.springer.com/content/pdf/bbm:978-1-4612-6135-3/1.pdf

The last in particular, in its Table I on page 5, catalogs and names all nine possibilities.

The book cited in the previous answer may be obtained online here:

https://fdocuments.in/download/a-simple-non-euclidean-geometry-and-its-physical-basis-yaglom.html

Answer 3

In addition to the nine following a specific set of rules,

There are also more exotic versions such as projective geometry (think twisting the plane into a Mobius strip) and finite geometries.

If you relax the isotropic requirement you open up more possibilites such as the Manhattan metric aka Taxicab Geometry.