There are theorems of Euclidean, hyperbolic, elliptic and Minkowski geometry. I'm wondering about planar de Sitter geometry.
Regarding planar anti-de Sitter geometry, based on my understanding of Cayley-Klein models, the theorems of planar anti-de Sitter geometry are the same as those of planar hyperbolic geometry but with points and lines exchanged. The Cayley-Klein construction of de Sitter geometry chooses the absolute to be the unit circle, the points to be those outside the unit disk, and the lines to be those that meet the interior of the disk. From this, some point-pairs have no lines joining them, and some line-pairs have no point meeting them both. The former fact is related to the finite speed of light, and the latter fact is related to the accelerating expansion of the universe. [EDIT: My original description of the Cayley-Klein model had an error.]
[EDIT: LOL I got de Sitter and anti-de Sitter mixed up. I've changed every occurrence of the word "anti-de Sitter" to "de Sitter", and vice versa.]