Everywhere I see Cayley-Klein geometries discussed, they're presented in a manner that seems wholly disconnected from the rest of geometry. Just what sorts of mathematical object are Cayley-Klein geometries/spaces? What structure is considered to be part of the abstraction and what is considered accidental? And how do these fit in with the rest of geometry?
Like, the cases of elliptic angle -- elliptic/projective geometry, Euclidean geometry, hyperbolic geometry -- these are all familiar as Riemannian manifolds. And the cases of hyperbolic angle -- de Sitter / co-hyperbolic geometry, Minkowski geometry, anti-de-Sitter / doubly hyperbolic geometry -- these are all familiar as Lorentzian manifolds. But what about the cases of parabolic angle (co-Euclidean, Galilean, and co-Minkowski geometries)? Do we formalize these as manifolds where tangent spaces are equipped with a degenerate metric tensor? Does that even result in something usable, and how would one get the appropriate notion of angle from that?
Of course, ideally I'd like an answer that doesn't just work separately for individual Cayley-Klein geometries but provides a more unified answer. And again I'm asking above -- just what exactly is a Cayley-Klein geometry? They have points, lines, incidence, distance (sort of) and angle between lines (sort of), but is there more included in the abstraction, or is it really just incidence geometry but with distance and angle information?
Thank you all! I'm pretty confused by this...
Edit: At the suggestion of brainjam, I'm updating this question to explain why e.g. I don't find the Wikipedia article a sufficient answer.
So, this article talks a field of study called Cayley-Klein geometry, which involves looking at symmetries of Cayley-Klein metrics. But I'm asking about a mathematical object called a Cayley-Klein geometry; I don't want just the symmetries, I want the thing we're looking at the symmetries of! So the Cayley-Klein metric is what I want, then, right? Well, not so fast...
The problem is that this Cayley-Klein metric doesn't seem to explain any of what I think I know about Cayley-Klein geometries. What is a line, and how do we get the expected incidence properties (specifically, the appropriate variants of the parallel postulate and the dual parallel postulate)? Why does "angle" seem to mean something different from usual? (Especially in the case of parabolic angle.) And this article talks about a Cayley-Klein metric, but "distance" in Cayley-Klein spaces as I know them is not necessarily a metric -- it's not even defined between all pairs of points! -- and is only a metric in the cases where angle is elliptical. These are the sorts of questions I want answers to... hoping that clears things up!
Edit again: OK, it seems that I still haven't expressed the question clearly enough. Let me try again.
Very frequently in mathematics, we define an area of study by defining a set of features we want to study and what conditions those features are required to satisfy. E.g., groups -- the feature is a binary operation, the requirements are the group axioms. Or metric spaces -- the feature is a two-input function to nonnegative reals, the requirements are the metric space axioms. Etc. Obviously there may be multiple features; perhaps there are multiple operations, an operation and a topology, etc, in which case typically the requirements demand that those features be somehow compatible with one another. Some times some of the features have no requirements, as with e.g. graphs or incidence geometries. But you get what I'm talking about.
In geometry -- not of the algebraic sort -- there isn't one idea of what makes a geometry, but rather there are several of these abstractions, depending on what features we are interested in capturing. Incidence geometries. Metric spaces and length spaces. Riemannian manifolds and pseudo-Riemannian manifolds. Etc.
I want to know if it is possible to fit Cayley-Klein geometries into one of these (beyond incidence geometries, which are kind of, well, disappointing). As I said above -- the elliptic angle cases can be realized as Riemannian manifolds; the hyperbolic angle cases can be realized as Lorentzian manifolds. What I want to know is, is there some notion of a "geometry" which:
- Is broad enough that it can include all the Cayley-Klein geometries;
- Is also broader than that, and can serve as a useful general notion of a geometry, such that all sorts of other familiar spaces can be fit into it, rather than only these nine;
- Ideally, incorporates the Cayley-Klein geometries in a natural way, such that the relevant notions of angle and distance fall out naturally rather than needing definitions with explicit case distinctions.
So, e.g., just saying, well, here's the definition of a Cayley-Klein geometry, is not what I'm looking for, because it fails condition (2). I don't want just Cayley-Klein geometries, I want an abstraction that covers both those and other geometries. Plus, the definition of Cayley-Klein geometries is more of a construction than a set of requirements. I mean technically it can be phrased as a set of requirements, but morally it's more of a construction.
So e.g., if the only Cayley-Klein geometries were the ones with elliptic angle measure, then "Riemannian manifolds" would be a suitable answer. Since that's not the case, though, it isn't. If the only Cayley-Klein geometries were the ones with elliptic or hyperbolic angle measure, then "pseudo-Riemannian manifolds" would be a somewhat acceptable answer; it wouldn't be ideal though because it would fail condition (3), e.g. with having to define angle differently depending on the signature (and not having angle at all in most signatures).
If the above cannot be done, then that is what I would call being disconnected from the rest of geometry.
I am hoping that this will finally make it clear what I am asking.
Richter-Gebert's Perspectives on Projective Geometry - A Guided Tour Through Real and Complex Geometry devotes Chapters 20-23 to working out many details and examples of C-K (Cayley-Klein) geometries. The preceding two chapters build up Euclidean geometry from projective geometry, giving a foretaste of the general C-K methods and concepts. (There's an online draft here.) In the following, I'll assume the planar case.
To answer your question, a C-K geometry is the projective plane along with a fundamental conic (also called the absolute conic). There are dual recipes for deriving the distance between points and the angle between lines using the absolute conic. The allowable transformations are the projective transformations that preserve the reference conic.
section 20.5 discusses distance, which can be real, imaginary, or complex. As OP points out, if one is thinking of a real distance, then that may not be defined between all pairs of points.
parts of the space may be removed for various purposes. Often, for example, the fundamental conic is removed because it is at infinite distance from the rest of the points. If the fundamental conic has an interior (the hyperbolic geometry case), the non-interior can be removed, and the remaining line segments can be regarded as the lines of the hyperbolic plane (they are infinite according to the induced metric). In the Euclidean case removal of the fundamental conic (the circular $I$ and $J$ points) and the line they are on amounts to removing the line at infinity from the projective plane, thus allowing the existence of parallel lines that have no intersection in the geometry. I haven't thought about it much, but maybe these removals could also be implemented by removing all points that don't have a finite real distance from some chosen point.
section 2.6 is a census of the planar C-K geometries, and includes many (but not all) listed in the OP. There are frameworks other than C-K that include more geometries.
it's worth remembering that the C-K idea is a way of constructing models of various geometries. Perhaps this is why OP sees them as disconnected from the rest of geometry - because they are perhaps not the best way to reason within a given geometry. As for whether the various parallel postulates are part of a C-K geometry, I'd say it's more a matter of verifying which postulate applies to a given geometry.
the paper On Klein’s So-called Non-Euclidean geometry is worth a look. It discusses Klein's C-K paper in light of the ideas in the air at the time, and earlier mathematical ideas and mathematicians that led up it.
As a side note, it's interesting how projective geometry, which is famously devoid of measurements such as distance and angle, can generate them once a reference conic is given. It's reminiscent of the Poncelet–Steiner theorem, which states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given.