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sniffnoyEDIT Nov 19: Fixed erroneous description of co-Euclidean geometry.
Seriously, I am so confused about this.
OK, background. When I was back in high school, I went to some talk at a math competition -- was it HMMT? Well, it was a math competition where there was also a talk in the middle -- about the nine Cayley-Klein geometries on the plane. I didn't remember that name though; I only rediscovered the term "Cayley-Klein geometry" much later. But the basic idea stuck with me.
I'm going to describe the basic idea of Cayley-Klein geometries in two dimensions, but first I want to skip to the question: How the hell do these fit in with the rest of geometry?
Some of them are, straightforwardly, Riemannian manifolds. Others are pseudo-Riemannian manifolds. But others don't seem to fall under this as best I can tell. Moreover, even if we can situate the individual Cayley-Klein geometries within the usual framework of differential geometry, that doesn't answer the question of how we fit the Cayley-Klein framework as a whole within geometry as I know it.
The few expositions I can find online of Cayley-Klein geometries, well, are mostly not about Cayley-Klein geometries, and they seem to present them in some entirely different fashion; and while I haven't taken the time to sit down and work through these expositions, they don't seem like they would answer my questions. So I am really confused here.
So. Having stated roughly what I'm confused about, I'm now going to describe the idea of the nine Cayley-Klein geometries, as I understand them. I'm going to start with the general idea, then I'm going to go over them one by one. For the ones that fit into geometry as I know of it, I'll describe how they do. Afterwards, I'm going to discuss the problems.
Note that Cayley-Klein geometry is normally presented as a thing taking place in two dimensions, but at least some of this seems to generalize to higher dimensions. But I'll get back to this point.
OK, so, what is the basic idea of Cayley-Klein geometries? The idea is that we can vary how we measure distance, and we can vary how we measure angle. (Note: I'm discussing here angles between lines, not rays, so be careful of that.) Each can be elliptic, parabolic, or hyperbolic. Elliptic works a bit different from the others in that it's bounded. Note that there's only an angle between two lines if they intersect... and there's only a distance between two points if they have a line through them.
Yes, not all pairs of points need have a line through them! However, if one does exist, it's unique. Similarly (and more familiarly), not all pairs of lines need intersect, but if they do, the intersection is unique.
The choice of distance/angle measure affects what parallel postulate we get... as well was what dual of the parallel postulate we get.
The effects of the distance measure will be familiar. Elliptic distance: Any two lines intersect. Parabolic distance: Given a line and a point not on it, there's a unique line through that point not intersecting the original line. Hyperbolic distance: Given a line and a point not on it, there are many lines through that point not intersecting the original line.
But then we have angle. Elliptic angle: Through any two points there is a line. Parabolic angle: Given a point and a line not through it, there's a unique point on the line such that there is no line between that point and the original. Hyperbolic angle: Given a point and a line not through it, there are many points on the line such that there is no line between that point and the original.
So, let's review these one by one, in order from most familiar to least familiar.
Elliptic angle, parabolic distance: This is ordinary Euclidean geometry. This can be formalized in a number of different ways, but in particular it's a Riemannian manifold with no curvature, and its notions of distance and angle are compatible with that. (I mean, you can't found it that way because that's circular, but you know what I mean. :P ) Obviously this generalizes beyond two dimensions.
Elliptic angle, hyperbolic distance: Hyperbolic geometry. Formalized as a Riemannian manifold with constant negative curvature, again with distance and angle both coming from that. Again, generalizes beyond two dimensions.
Elliptic angle, elliptic distance: Elliptic/projective geometry; the projective plane. Again, a Riemannian manifold, this time with constant positive curvature. Still using the appropriate notions of distance and angle. Again, generalizes to higher dimensions. I guess there's not really a way to realize this one on the plane, per se? But whatever.
OK, so far so familiar. Also still pretty familiar will be...
Hyperbolic angle, parabolic distance: This is Minkowski space! One spatial dimension, one time dimension. It's pseudo-Euclidean space with signature (1,1), or, generalizing to higher dimensions (which this does), signature (n-1,1) (or (1,n-1) depending on your convention). And of course we can also think of it as a Lorentzian manifold (a pseudo-Riemannian manifold with that signature) with no curvature. Again, this one generalizes to higher dimensions.
So what's up with angle and distance here? Well... pseudo-Euclidean spaces, or pseudo-Riemannian manifolds, don't really have a notion of "distance". Yeah, there's the interval, but taking the square root of that doesn't yield a metric, even if you only check triangles where all the square roots are defined. Nonetheless, that's what we're going to do.
But, I need to say something about lines here first. Because there's not going to be a line through every two points. The idea is that only the timelike lines count, because those are the only lines by which events can actually be connected. (No, lightlike doesn't count here; I guess the physical intuition only goes so far.) So if two points are timelike separated, there will be a distance between them equal to the proper time; while if they're spacelike or lightlike separated, there will be no line and no distance.
I guess you could say lightlike is distance zero, as opposed to spacelike having truly undefined distance? Similar to how in hyperbolic geometry, given a line and a point not on that line, most of the lines through that point not intersecting the line will truly not make any angle with it, but two of them, the "parallels", could be said to make angle zero. (And of course in Euclidean geometry two parallel lines could be said to have angle zero.) It's similar here but in dual: Given a [lightlike] line and a point not on it, there will be lots of points on that line that are spacelike separated from the original, but only two that are lightlike separated from it. You see?
I'm going to mostly ignore this point, but I may return to it occasionally. You can fill it in yourself when I don't.
A note here on convention -- I wasn't sure whether to say timelike-lines only or spacelike-lines only (in which case we'd be using proper distance, not proper time, as our distance). I went with the former because I remember the physical motivation as being pretty important (more on this later). Obviously, in two dimensions it doesn't matter; but in higher dimensions it does, and it seemed a little weird to use proper time rather than proper distance in higher dimensions. Still, gonna stick with the physics.
OK, but, what's up with angle? I still haven't talked about that. Well, this is actually a standard thing in pseudo-Euclidean spaces; our normal notion of angle doesn't work, but you can define the hyperbolic angle by taking the inverse hyperbolic cosine rather than the inverse cosine. (Note you have to take absolute value first, which means -- as I mentioned above! -- this is only good for measuring angle between lines, not angle between rays.) In relativity of course this corresponds to absolute difference in rapidity. So yeah, hyperbolic angle; it's not elliptic, it's not bounded! It can be as large as you want, just like hyperbolic or parabolic distance...
OK, that might have been a little unfamiliar, but it's not too strange; it's just Minkowski space after all. So next up is...
Hyperbolic angle, elliptic distance: This is de Sitter space, a Lorentzian manifold with constant positive curvature. Generalizes to higher dimensions, obviously. In the Cayley-Klein context it's often called "co-hyperbolic" because it's dual to hyperbolic geometry. Not going to detail the rest, it's as appropriate; again, we only allow timelike lines.
Hyperbolic angle, hyperbolic distance: Anti-de Sitter space, a Lorentzian manifold with constant negative curvature. Again, generalizes to higher dimensions. In the Cayley-Klein context it's often called "doubly hyperbolic". We once again only allow timelike lines, etc.
OK! So those are all ones that are fairly standard. But now it's time to talk about the case of parabolic angle. In this case the one we're starting with might be the least familiar, but it's also the most interesting. So let's talk about...
Parabolic angle, parabolic distance: This is known as Galilean geometry. Why? Because, just as Minkowski space is the geometry of spacetime in Einsteinian relativity, this is the geometry of spacetime in Galilean relativity!
So, again, we're going to interpret one dimension as space, one dimension as time. And once again we're only going to allow lines that are "timelike". However this looks pretty different from the Minkowski case. (Note: I'm going to follow the relativists' convention of putting time on the vertical axis, even though I find that horribly unintuitive and it was actually explained to me with the reverse convention, because, well, it's just easier to keep things consistent that way.)
In the Minkowski case, lightlike lines have a slope of ±1. But in the Galilean case, a lightlike line is horizontal; the "speed of light" is infinite. So, we only admit lines that are not horizontal! There is, obviously, no equivalent to two points being spacelike separated.
Distance, I guess, is just the time separation between events? That's kind of weird, but I guess not weirder than what we're doing in the Minkowski case I guess, so whatever. So yeah, two points can be "timelike separated" (not simultaneous), with the distance being the time between them; or they can be "lightlike separated" (simultaneous), with there being no line between them but the distance sort of being zero.
So, we get not only the usual parallel postulate, but also the dual parallel postulate, as discussed above.
Angle I think is just the absolute difference of the slopes? Or the inverse slopes, I guess, since we're putting time on the vertical axis. The absolute difference of the inverse slopes, then. Look, it's the absolute difference of the speeds, OK? And once again parallel lines may not meet but they sort of make a zero angle.
This seems to generalize to higher dimensions pretty straightforwardly, with multiple spatial dimensions and a single time dimension.
One last note: Just as the symmetries of Minkowski space are the coordinate transforms of special relativity, the symmetries of Galilean space are the coordinate transforms of Galilean relativity, of Newtonian physics. So like with Minkowski space, the symmetry group is generated by spatial symmetries (translation/rotation/reflection), time symmetries (translation/reflection... not much rotation in one dimension), and boosts that mix the two, but the form of those boosts is a little simpler to write down. :) In particular, these boosts don't change the time coordinate at all! Simultaneity isn't relative in Newtonian physics. :)
(Note that symmetries here should take lines to lines, and then also preserve distance and angle, I guess.)
OK, got that? Cool, now let's move on to the last two:
Parabolic angle, elliptic distance: This is called co-Euclidean geometry, because it's dual to Euclidean geometry. You can visualize it as Euclidean geometry, just interchanging the words "point" and "line" everywhere, as well as "distance" and "angle" everywhere, but let's give a more concrete description. Consider an infinite cylinder with a designated center point. Points will be pairs of antipodes on the cylinder. Lines will be "great circles" on the cylinder: The intersection of the cylinder with a plane through the center point; but, importantly, planes containing the cylinder's axis are disallowed. So it's sort of like Galilean geometry with circular time, but not quite. Distance I guess is measured purely in the "time" (circular) dimension? And angle I guess would be difference of slopes again?
Parabolic angle, hyperbolic distance: This is of course called co-Minkowski geometry. I don't know how to visualize this one, other than purely thinking of it as a dual. Maybe I would be able to if I understood hyperbolic geometry better. But then it's not like I know how to visualize de Sitter space or anti-de Sitter space either! I assume it too can be generalized to higher dimensions, but since I don't know how to think of this other than as a dual, I don't know how to visualize that.
OK, cool, got all that? Like, that's pretty neat, right? But now let's discuss the problems...
Problem #1: What the hell is Galilean geometry? Or the parabolic-angle geometries in general?
Let's just stick with Galilean geometry, because that one hopefully we can realize as some sort structure on a vector space, and not have to bring manifolds into the picture.
Like, Euclidean geometry, Minkowski geometry, these are both given by quadratic forms on R² (or more generally Rn). Euclidean is (+,+); Minkowski is (+,-). So what the hell is Galilean? (+,0)?
I mean, I guess that yields the right notion of distance as just being the time between two events... in two dimensions, anyway. In higher dimensions, using (+,...,+,0)... well, if you've got n-1 space dimensions and 1 time dimension... yeah, you see the problem. Maybe we should throw out the physical intuition, or maybe we should just disregard higher dimensions, IDK. I mean I suppose it could be (0,..,0,+), but that doesn't really seem right to me... but I guess maybe it has to be? More on this in a moment.
And where does the notion of angle come from? Is there any good way to derive that? Or is it just ad-hoc? I mean, I guess hyperbolic angle (for Minkowski space) seems kind of ad-hoc too. But the usual angle (for lines, not for rays!) and the hyperbolic angle at least both involve looking at |<v,w>|/(√<v,v>√<w,w>). This doesn't, since that ratio just works out to 1 here!
And that's just for Galilean geometry; I have no idea how workable it is to define something like pseudo-Riemannian manifolds but with a degenerate form... will "constant positive curvature" and "constant negative curvature" even make sense? Will we be able to get nonzero curvatures at all?
And then there's the symmetries. In two dimensions this works fine but again in higher dimensions... I mean, let's say we did go with (0,...,0,+) as I suggested above. That's just going to have way too many symmetries, right? I mean, sure, the lines-to-lines requirement means we will at least only be dealing with affine transformations. But with more than one spatial dimension, what will there be to constrain us to the usual isometries of space, if the spatial dimensions contribute a big fat 0 to the interval?
So I guess two dimensions isn't too bad. But beyond that... yikes. Still, maybe we can restrict ourselves to two dimensions? That's how this was originally presented, after all.
Problem #2: The overall framework and dualization
But even if we understand how to fit Galilean geometry into the rest of mathematics, there's still the problem of, what's up with this overall framework? In particular, the idea of dualization?
Like a key thing in Cayley-Klein geometry is the symmetric treatment of points and of lines, and of distance and of angle. And because of this symmetric treatment, there seems to be this dualization operation, where you can take one Cayley-Klein geometry, swap "point" with "line" and "distance" with "angle", and get another.
But most math does not treat points and lines symmetrically. I guess, uh, incidence geometry does. But, well, incidence geometry is kind of crap (OK, finite projective planes are cool, but those are more combinatorics than geometry). It strips out most of the, y'know, geometry. And it gets pretty difficult to take this approach in more than two dimensions.
So what sort of structures are these, fundamentally? Are they a type of manifolds -- do they have notions of, say, area, sticking to two dimensions? Or are they just, like, incidence structures but with some extra distance and angle information attached? (Not clear how that would be formalized, especially with the varying ways distance and angle work, and especially how different elliptic is from the other two.) That'd be kind of crappy, wouldn't it? I mean Riemannian manifolds and pseudo-Riemannian manifolds have way more information than just incidence, distance, and angle. Is that all that's going on here, or is there more?
(There's another question for you -- what does the distinction between these different sorts of distance and angle measures actually mean? OK, I gather it has some meaning in the weird way that Cayley-Klein geometries seem to be so often presented, but what about in more usual terms?)
Also, notice how all 9 of these geometries are, essentially, uniform. If we make the general definition of what sort of mathematical objects these are, will we get non-uniform examples as well? There's plenty of non-uniform Riemannian and Lorentzian manifolds, so those will make for non-uniform distance, but is there a way of getting non-uniform angle as well?
Anyway yeah. I'm confused about Cayley-Klein geometries. I'm hoping somebody knows.
-Harry
Seriously, I am so confused about this.
OK, background. When I was back in high school, I went to some talk at a math competition -- was it HMMT? Well, it was a math competition where there was also a talk in the middle -- about the nine Cayley-Klein geometries on the plane. I didn't remember that name though; I only rediscovered the term "Cayley-Klein geometry" much later. But the basic idea stuck with me.
I'm going to describe the basic idea of Cayley-Klein geometries in two dimensions, but first I want to skip to the question: How the hell do these fit in with the rest of geometry?
Some of them are, straightforwardly, Riemannian manifolds. Others are pseudo-Riemannian manifolds. But others don't seem to fall under this as best I can tell. Moreover, even if we can situate the individual Cayley-Klein geometries within the usual framework of differential geometry, that doesn't answer the question of how we fit the Cayley-Klein framework as a whole within geometry as I know it.
The few expositions I can find online of Cayley-Klein geometries, well, are mostly not about Cayley-Klein geometries, and they seem to present them in some entirely different fashion; and while I haven't taken the time to sit down and work through these expositions, they don't seem like they would answer my questions. So I am really confused here.
So. Having stated roughly what I'm confused about, I'm now going to describe the idea of the nine Cayley-Klein geometries, as I understand them. I'm going to start with the general idea, then I'm going to go over them one by one. For the ones that fit into geometry as I know of it, I'll describe how they do. Afterwards, I'm going to discuss the problems.
Note that Cayley-Klein geometry is normally presented as a thing taking place in two dimensions, but at least some of this seems to generalize to higher dimensions. But I'll get back to this point.
OK, so, what is the basic idea of Cayley-Klein geometries? The idea is that we can vary how we measure distance, and we can vary how we measure angle. (Note: I'm discussing here angles between lines, not rays, so be careful of that.) Each can be elliptic, parabolic, or hyperbolic. Elliptic works a bit different from the others in that it's bounded. Note that there's only an angle between two lines if they intersect... and there's only a distance between two points if they have a line through them.
Yes, not all pairs of points need have a line through them! However, if one does exist, it's unique. Similarly (and more familiarly), not all pairs of lines need intersect, but if they do, the intersection is unique.
The choice of distance/angle measure affects what parallel postulate we get... as well was what dual of the parallel postulate we get.
The effects of the distance measure will be familiar. Elliptic distance: Any two lines intersect. Parabolic distance: Given a line and a point not on it, there's a unique line through that point not intersecting the original line. Hyperbolic distance: Given a line and a point not on it, there are many lines through that point not intersecting the original line.
But then we have angle. Elliptic angle: Through any two points there is a line. Parabolic angle: Given a point and a line not through it, there's a unique point on the line such that there is no line between that point and the original. Hyperbolic angle: Given a point and a line not through it, there are many points on the line such that there is no line between that point and the original.
So, let's review these one by one, in order from most familiar to least familiar.
Elliptic angle, parabolic distance: This is ordinary Euclidean geometry. This can be formalized in a number of different ways, but in particular it's a Riemannian manifold with no curvature, and its notions of distance and angle are compatible with that. (I mean, you can't found it that way because that's circular, but you know what I mean. :P ) Obviously this generalizes beyond two dimensions.
Elliptic angle, hyperbolic distance: Hyperbolic geometry. Formalized as a Riemannian manifold with constant negative curvature, again with distance and angle both coming from that. Again, generalizes beyond two dimensions.
Elliptic angle, elliptic distance: Elliptic/projective geometry; the projective plane. Again, a Riemannian manifold, this time with constant positive curvature. Still using the appropriate notions of distance and angle. Again, generalizes to higher dimensions. I guess there's not really a way to realize this one on the plane, per se? But whatever.
OK, so far so familiar. Also still pretty familiar will be...
Hyperbolic angle, parabolic distance: This is Minkowski space! One spatial dimension, one time dimension. It's pseudo-Euclidean space with signature (1,1), or, generalizing to higher dimensions (which this does), signature (n-1,1) (or (1,n-1) depending on your convention). And of course we can also think of it as a Lorentzian manifold (a pseudo-Riemannian manifold with that signature) with no curvature. Again, this one generalizes to higher dimensions.
So what's up with angle and distance here? Well... pseudo-Euclidean spaces, or pseudo-Riemannian manifolds, don't really have a notion of "distance". Yeah, there's the interval, but taking the square root of that doesn't yield a metric, even if you only check triangles where all the square roots are defined. Nonetheless, that's what we're going to do.
But, I need to say something about lines here first. Because there's not going to be a line through every two points. The idea is that only the timelike lines count, because those are the only lines by which events can actually be connected. (No, lightlike doesn't count here; I guess the physical intuition only goes so far.) So if two points are timelike separated, there will be a distance between them equal to the proper time; while if they're spacelike or lightlike separated, there will be no line and no distance.
I guess you could say lightlike is distance zero, as opposed to spacelike having truly undefined distance? Similar to how in hyperbolic geometry, given a line and a point not on that line, most of the lines through that point not intersecting the line will truly not make any angle with it, but two of them, the "parallels", could be said to make angle zero. (And of course in Euclidean geometry two parallel lines could be said to have angle zero.) It's similar here but in dual: Given a [lightlike] line and a point not on it, there will be lots of points on that line that are spacelike separated from the original, but only two that are lightlike separated from it. You see?
I'm going to mostly ignore this point, but I may return to it occasionally. You can fill it in yourself when I don't.
A note here on convention -- I wasn't sure whether to say timelike-lines only or spacelike-lines only (in which case we'd be using proper distance, not proper time, as our distance). I went with the former because I remember the physical motivation as being pretty important (more on this later). Obviously, in two dimensions it doesn't matter; but in higher dimensions it does, and it seemed a little weird to use proper time rather than proper distance in higher dimensions. Still, gonna stick with the physics.
OK, but, what's up with angle? I still haven't talked about that. Well, this is actually a standard thing in pseudo-Euclidean spaces; our normal notion of angle doesn't work, but you can define the hyperbolic angle by taking the inverse hyperbolic cosine rather than the inverse cosine. (Note you have to take absolute value first, which means -- as I mentioned above! -- this is only good for measuring angle between lines, not angle between rays.) In relativity of course this corresponds to absolute difference in rapidity. So yeah, hyperbolic angle; it's not elliptic, it's not bounded! It can be as large as you want, just like hyperbolic or parabolic distance...
OK, that might have been a little unfamiliar, but it's not too strange; it's just Minkowski space after all. So next up is...
Hyperbolic angle, elliptic distance: This is de Sitter space, a Lorentzian manifold with constant positive curvature. Generalizes to higher dimensions, obviously. In the Cayley-Klein context it's often called "co-hyperbolic" because it's dual to hyperbolic geometry. Not going to detail the rest, it's as appropriate; again, we only allow timelike lines.
Hyperbolic angle, hyperbolic distance: Anti-de Sitter space, a Lorentzian manifold with constant negative curvature. Again, generalizes to higher dimensions. In the Cayley-Klein context it's often called "doubly hyperbolic". We once again only allow timelike lines, etc.
OK! So those are all ones that are fairly standard. But now it's time to talk about the case of parabolic angle. In this case the one we're starting with might be the least familiar, but it's also the most interesting. So let's talk about...
Parabolic angle, parabolic distance: This is known as Galilean geometry. Why? Because, just as Minkowski space is the geometry of spacetime in Einsteinian relativity, this is the geometry of spacetime in Galilean relativity!
So, again, we're going to interpret one dimension as space, one dimension as time. And once again we're only going to allow lines that are "timelike". However this looks pretty different from the Minkowski case. (Note: I'm going to follow the relativists' convention of putting time on the vertical axis, even though I find that horribly unintuitive and it was actually explained to me with the reverse convention, because, well, it's just easier to keep things consistent that way.)
In the Minkowski case, lightlike lines have a slope of ±1. But in the Galilean case, a lightlike line is horizontal; the "speed of light" is infinite. So, we only admit lines that are not horizontal! There is, obviously, no equivalent to two points being spacelike separated.
Distance, I guess, is just the time separation between events? That's kind of weird, but I guess not weirder than what we're doing in the Minkowski case I guess, so whatever. So yeah, two points can be "timelike separated" (not simultaneous), with the distance being the time between them; or they can be "lightlike separated" (simultaneous), with there being no line between them but the distance sort of being zero.
So, we get not only the usual parallel postulate, but also the dual parallel postulate, as discussed above.
Angle I think is just the absolute difference of the slopes? Or the inverse slopes, I guess, since we're putting time on the vertical axis. The absolute difference of the inverse slopes, then. Look, it's the absolute difference of the speeds, OK? And once again parallel lines may not meet but they sort of make a zero angle.
This seems to generalize to higher dimensions pretty straightforwardly, with multiple spatial dimensions and a single time dimension.
One last note: Just as the symmetries of Minkowski space are the coordinate transforms of special relativity, the symmetries of Galilean space are the coordinate transforms of Galilean relativity, of Newtonian physics. So like with Minkowski space, the symmetry group is generated by spatial symmetries (translation/rotation/reflection), time symmetries (translation/reflection... not much rotation in one dimension), and boosts that mix the two, but the form of those boosts is a little simpler to write down. :) In particular, these boosts don't change the time coordinate at all! Simultaneity isn't relative in Newtonian physics. :)
(Note that symmetries here should take lines to lines, and then also preserve distance and angle, I guess.)
OK, got that? Cool, now let's move on to the last two:
Parabolic angle, elliptic distance: This is called co-Euclidean geometry, because it's dual to Euclidean geometry. You can visualize it as Euclidean geometry, just interchanging the words "point" and "line" everywhere, as well as "distance" and "angle" everywhere, but let's give a more concrete description. Consider an infinite cylinder with a designated center point. Points will be pairs of antipodes on the cylinder. Lines will be "great circles" on the cylinder: The intersection of the cylinder with a plane through the center point; but, importantly, planes containing the cylinder's axis are disallowed. So it's sort of like Galilean geometry with circular time, but not quite. Distance I guess is measured purely in the "time" (circular) dimension? And angle I guess would be difference of slopes again?
Parabolic angle, hyperbolic distance: This is of course called co-Minkowski geometry. I don't know how to visualize this one, other than purely thinking of it as a dual. Maybe I would be able to if I understood hyperbolic geometry better. But then it's not like I know how to visualize de Sitter space or anti-de Sitter space either! I assume it too can be generalized to higher dimensions, but since I don't know how to think of this other than as a dual, I don't know how to visualize that.
OK, cool, got all that? Like, that's pretty neat, right? But now let's discuss the problems...
Problem #1: What the hell is Galilean geometry? Or the parabolic-angle geometries in general?
Let's just stick with Galilean geometry, because that one hopefully we can realize as some sort structure on a vector space, and not have to bring manifolds into the picture.
Like, Euclidean geometry, Minkowski geometry, these are both given by quadratic forms on R² (or more generally Rn). Euclidean is (+,+); Minkowski is (+,-). So what the hell is Galilean? (+,0)?
I mean, I guess that yields the right notion of distance as just being the time between two events... in two dimensions, anyway. In higher dimensions, using (+,...,+,0)... well, if you've got n-1 space dimensions and 1 time dimension... yeah, you see the problem. Maybe we should throw out the physical intuition, or maybe we should just disregard higher dimensions, IDK. I mean I suppose it could be (0,..,0,+), but that doesn't really seem right to me... but I guess maybe it has to be? More on this in a moment.
And where does the notion of angle come from? Is there any good way to derive that? Or is it just ad-hoc? I mean, I guess hyperbolic angle (for Minkowski space) seems kind of ad-hoc too. But the usual angle (for lines, not for rays!) and the hyperbolic angle at least both involve looking at |<v,w>|/(√<v,v>√<w,w>). This doesn't, since that ratio just works out to 1 here!
And that's just for Galilean geometry; I have no idea how workable it is to define something like pseudo-Riemannian manifolds but with a degenerate form... will "constant positive curvature" and "constant negative curvature" even make sense? Will we be able to get nonzero curvatures at all?
And then there's the symmetries. In two dimensions this works fine but again in higher dimensions... I mean, let's say we did go with (0,...,0,+) as I suggested above. That's just going to have way too many symmetries, right? I mean, sure, the lines-to-lines requirement means we will at least only be dealing with affine transformations. But with more than one spatial dimension, what will there be to constrain us to the usual isometries of space, if the spatial dimensions contribute a big fat 0 to the interval?
So I guess two dimensions isn't too bad. But beyond that... yikes. Still, maybe we can restrict ourselves to two dimensions? That's how this was originally presented, after all.
Problem #2: The overall framework and dualization
But even if we understand how to fit Galilean geometry into the rest of mathematics, there's still the problem of, what's up with this overall framework? In particular, the idea of dualization?
Like a key thing in Cayley-Klein geometry is the symmetric treatment of points and of lines, and of distance and of angle. And because of this symmetric treatment, there seems to be this dualization operation, where you can take one Cayley-Klein geometry, swap "point" with "line" and "distance" with "angle", and get another.
But most math does not treat points and lines symmetrically. I guess, uh, incidence geometry does. But, well, incidence geometry is kind of crap (OK, finite projective planes are cool, but those are more combinatorics than geometry). It strips out most of the, y'know, geometry. And it gets pretty difficult to take this approach in more than two dimensions.
So what sort of structures are these, fundamentally? Are they a type of manifolds -- do they have notions of, say, area, sticking to two dimensions? Or are they just, like, incidence structures but with some extra distance and angle information attached? (Not clear how that would be formalized, especially with the varying ways distance and angle work, and especially how different elliptic is from the other two.) That'd be kind of crappy, wouldn't it? I mean Riemannian manifolds and pseudo-Riemannian manifolds have way more information than just incidence, distance, and angle. Is that all that's going on here, or is there more?
(There's another question for you -- what does the distinction between these different sorts of distance and angle measures actually mean? OK, I gather it has some meaning in the weird way that Cayley-Klein geometries seem to be so often presented, but what about in more usual terms?)
Also, notice how all 9 of these geometries are, essentially, uniform. If we make the general definition of what sort of mathematical objects these are, will we get non-uniform examples as well? There's plenty of non-uniform Riemannian and Lorentzian manifolds, so those will make for non-uniform distance, but is there a way of getting non-uniform angle as well?
Anyway yeah. I'm confused about Cayley-Klein geometries. I'm hoping somebody knows.
-Harry
