I learn that Geometry has several categories/subfields from Wikipedia. But I am still not clear about the standards according to which they are classified.

  1. It seems Euclidean Geometry, Affine Geometry and Projective Geometry are classified according some rule, while Hyperbolic Geometry, Elliptic Geometry and Riemann Geometry according to another, and Axiomatic, Analytic, Algebraic and Differential Geometry perhaps according to a different one? What rules are they?

  2. Are Affine Geometry, Projective Geometry, Hyperbolic Geometry, Elliptic Geometry and Riemann Geometry all Non-Euclidean Geometry? What are their common characteristics that make them Non-Euclidean Geometry?

Really appreciate if someone could clarify these questions for me and also hope you can provide more insights into the subfields of Geometry not necessarily the specific questions I asked.

Update: Although my major is not math, I have been involved in some projects requiring quite a few mathematics and have taken a lot of courses in mathematics on undergraduate/graduate level. Now looking back, I am confused about what I have learned and heard, and would like to get a big picture.

  • 2
    $\begingroup$ If you're a math student, wait some years to get some background, and then have a look to this: R.W.Sharpe, Differential Geometry: Cartans Generalization of Kleins Erlangen Program $\endgroup$
    – Qfwfq
    Apr 5 '10 at 13:54
  • 2
    $\begingroup$ Perhaps immediately you can have a look at: H.S.M. Coxeter, Introduction to Geometry $\endgroup$ Apr 5 '10 at 14:01

There are a couple of subtle differences. Some of the concepts are only relevant when talking about geometry from an axiomatic perspective.

When one is talking about geometry from an axiomatic perspective ( you want to talk about points, lines, planes, angles etc.) you are really looking at a model for your axioms. Here we might talk about Euclidean, Riemannian, Hyperbolic, Projective, Spherical and (perhaps) Elliptic geometries. Usually the main difference is whether or not we choose to take the parallel postulate as an axiom or one of its negations. If we take the parallel postulate then we are working in a model of Euclidean geometry, this is sort of flat geometry where things are what you expect. In Spherical geometry (sometimes called Riemannian, and maybe elliptical but i am not sure about that) is where we have no parallel lines, a model of this is the sphere where we take lines to be great circles. Note though that typically Riemannian geometry is about manifolds with a Riemannian structure, but that we can save until later. Hyperbolic geometry is when we have infinitely many parallel lines through a given point, a model of this is the Poincare disk. Projective geometry has every two parallel lines meet... at the point at infinity. There are much better references for this stuff but so far we have just been looking at models of axiom systems that one could work with and "do geometry" in.

Algebraic, differential, and Riemannian geometry are more complicated. Here are some "one line slogans" which i am sure others can improve upon. Differential geometry is about curves, surfaces and homogeneous objects that you want to study via calculus, there is a priori some smooth structure on that object. Algebraic geometry wants to study similar objects but only when you are concerned with what you can tell about an object via rational functions. For me, this starts with the Gelfand-Naimark result. it gets much much richer. Riemannian geometry is differential geometry except you have a well behaved notion of distance between points, distance ON the hypersurface itself! All of these could be fixed by someone with more knowledge then I on the respective subjects.

I didn't mention affine geometry, but I will take a stab and say it is like a geometry in any one of the above models except you only care about "flat" things, lines, planes, etc.

The above comment suggests looking at some good books. It mentions Klein's Erlangen program, which is where Klein proposed studying a geometry by understanding the group of symmetries that preserve it. So you can think of the first big paragraph as looking at groups and the geometries you get from them. The second big paragraph you can get by looking at different types of sheaves on various topological spaces (I think) where the sheaf keeps track of the type of structure you care about. The two classes are sort of a bit different, but they are similar in that you can look at objects that act as receptacles for the geometric content: groups and sheaves.

Any suggestions are very welcome!


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