Following up another suggestion from Dan, it looks as if the notion of a "linear homogeneous space" --- aka "linear G-space" --- might make a pretty good fit with what I had in mind: more or less, this is a topological space which can be kitted out with a Lie group of symmetries, a matrix representation, all the familiar machinery. With one caveat --- it has been impressed on me more than once in the past that a good deal of constraint is imposed on geometers' imaginations by unconscious identification of topological points with geometric points. [Probably this solecism is committed less often by topologists than geometers.] A good example which had me puzzled for a while is A.~F.~Beardon and D.~Minda "Sphere-preserving Maps in Inversive Geometry" Proc. Amer. Math. Soc. 130 (2001) 987--998 In many ways this is an excellent paper --- careful, informative, and interesting. But its central message --- roughly, that the Moebius group is the largest which bijectively preserves spheres (etc) in inversive space --- is quite simply untrue: the Lie-sphere group, with dimension 15, and including the Moebius group with 10, provides an ancient and obvious counter-example. I'm fairly sure that the problem here involves an unstated assumption that geometric points are preserved. However, the topological points in inversive space are actually geometrical spheres, including both points and (hyper)planes as limiting cases! Fred Lunnon On 2/21/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
While recently attempting to compile a glossary of terms related to geometric algebra, I found myself in several places wanting to use the term "geometric space". Finally, of course, it too needed to be defined: at which point it became embarrassingly clear that I am unable to do so.
The spaces/ symmetry groups with which GA is principally concerned are pretty standard: e.g. Euclidean, inversive/ conformal/ Moebius, spherical/ orthogonal, equilong/ Laguerre, contact/ Lie-sphere, 3-space line geometry. The constraints of the formalism actually restrict these to "quadratic space" --- but what I have in mind is more general than that --- for instance, geometry of lines in n-space (involving multiple quadratics).
Conventional classifications based on topology, such as "metric space" --- as Dan Asimov recently pointed out --- rely on point-based notions which simply fail to apply in most of the cases above, even the familiar inversive geometry, because the "metric" (a quadratic form) is indefinite.
Geometry of points on a real manifold bears some relation to what's required: but that relies on too much prior machinery for comfort. And the mind-numbingly abstract approach of "schemes", "stacks" and their like opens a whole counter-intuitive can of worms, which I have no intention of attempting to broach.
So, I need a helping hand here, folks --- can somebody please come up with simple, elementary, intuitively plausible definition for what I am trying to discuss here?
Fred Lunnon