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From my reading Wikipedia, I understand there are several branches of classical geometry (if the ordering is off, or I'm missing a few things, let me know):

  • Absolute
    • Euclidean
    • Non-Euclidean
      • Spherical
      • Hyperbolic
  • Projective
    • Affine
  • Transformational
  • Inversive

These can be studied from:

  • Coordinate geometry (aka algebraic) view
  • Axiomatic (aka synthetic) view
  • Analytic view

The combination of these branches and points of view generate many possible avenues of study.

  • How many of these are useful for a modern geometer to know? By useful, I mean fairly likely to lend intuitive insights into their work (This may seem vague without specifying what kind of work they do. But since most geometers aren't working in classical geometry, I guess the work of any such geometer could be considered relevant to the question).

  • How deeply must they learn?

I've got a good high school level knowledge of math, and some college level math, and I have an interest in pure mathematics for a career.

EDIT: Added tags relating to modern branches of geometry, so that these kinds of geometers see this question.

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  • $\begingroup$ Taking a stab at this, I would say that a modern geometer is probably an expert in all of those topics you listed. As you advance in your studies, you will become quite familiar with them. They are all part of a basic undergraduate education in math or beginning graduate studies $\endgroup$ – ClassicStyle Apr 17 '16 at 6:53
  • $\begingroup$ @TylerHG Now I'm wondering if these topics are elementary enough that one can master them at the undergraduate level? i.e. they're fields which have more taken a definitive form, and at their most advanced are still accessible to undergrads. This relates to the second question about "depth". $\endgroup$ – stranger Apr 17 '16 at 7:22
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    $\begingroup$ You pick up pieces of these things, and many theorems you learn without without necessarily being able to prove them. For example, I'm in graduate school, and I have seen the fact that three points determine a square before (it was inspired by the fact that three points determine a circle, something I saw in complex analysis). I think the square fact is a hard theorem in Euclidean geometry.. the proof is not at all obvious to me at least. But still I picked it up. I imagine with many more years, I'll learn all kinds of stuff about all of those branches of geometry. $\endgroup$ – Alfred Yerger Apr 17 '16 at 21:30
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    $\begingroup$ @tylerHG I don't agree with that at all. I have never seen a course specifically in any of these areas offered at the undergraduate or graduate level in the US, unless you really stretch and count an introduction to complex projective varieties-but in the 19th century you'd certainly have done the linear projective geometry thoroughly first, so I wouldn't count that. I'd say it's not clear that any of these areas need to be studied for their own sake to do research in modern geometry. One hopes that much of the intuition of past generations has been baked into the formalisms we now use. $\endgroup$ – Kevin Carlson Apr 17 '16 at 21:34
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    $\begingroup$ On this list, hyperbolic geometry is special, as it has turned out to be a crown jewel and wonderful tool for understanding a variety of areas in topology and geometry. The rest of the many flavors of geometry you have listed, while fascinating, are rarely essential to any flavor of modern geometry, and as Kevin says much of this is "baked in" how we do geometry now, eg the algebraic/complex/differential geometry of projective space. $\endgroup$ – user98602 Apr 17 '16 at 21:47
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If you are planning to do research in modern geometry, pretty much nothing on your list is relevant. Instead:

  1. If you are planning to work in algebraic geometry, you should learn commutative algebra or/and complex differential geometry.

  2. If you are planning to work in Riemannian geometry, most likely you should spend time learning differential equations, functional analysis, Sobolev spaces, etc.

I know, this sounds a bit sad, but the reality is that you have only that much time. I do not mean that learning "classical" geometry is waste of time, not at all, but you have to prioritize...

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  • $\begingroup$ Spot on, studiosus: +1. And indeed this is a bit sad... $\endgroup$ – Georges Elencwajg Apr 18 '16 at 19:09
  • $\begingroup$ Dear downvoter, I am a bit curious what did you find disagreeable in my answer. $\endgroup$ – Moishe Kohan Jul 5 '17 at 18:50

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