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There is no question what topology is and what it's about: it's about topologies (= topological spaces), and that's it.

There is also no question what (universal) algebra is and what it's about. (Among other things, it's about algebras.)

But what is geometry and what is it about? Is there a thorough and generally agreed upon definition of a geometry (= geometric structure) comparable to the unequivocal definition of a topology?

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I've heard it argued that the "correct setting" for geometry is locally ringed spaces. I'd elaborate further, but I don't yet know enough to do that perspective justice. –  Jesse Madnick Sep 6 '12 at 20:45
    
I don't think the question really makes sense for the following reason: both of the words topology and geometry have many meanings and you are comparing the meanings that are quite incomparable. Note, that topology is not just a certain collection of sets, but it is also a subject as such and it is also a phenomenon of ignoring the local details. Similarly with algebra. Now, geometry is again a subject, but it is certainly not a mathematical object in the same way topology (as a collection of sets) is! –  Marek Sep 6 '12 at 20:47
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"Geometry" = "measurement of the earth"... by etymology –  GEdgar Sep 6 '12 at 21:36
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Cf. Lurie's Derived Algebraic Geometry V, where he defines "geometries" in full generality. –  Aaron Mazel-Gee Sep 6 '12 at 21:42
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The definition of a "topology" is not necessarily as set in stone as it appears: the field of pointless topology considers objects that cannot be described with traditional spaces. More generally, there are Grothendieck topologies for dealing with, for example, étale coverings. –  Slade yesterday

4 Answers 4

up vote 18 down vote accepted

According to Klein, geometry can be viewed as the action of a group on a space, be it smooth or finite. See this. That is, a geometry on a set $X$ is a triple $(X,G,A)$, where $G$ is a group with action $A$ on $X$.

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In this context, a space may be taken to be a set. If extra structure is given, then we append more to the word "geometry". For example, if said space is metric, then we study Riemannian geometry, etc. –  user02138 Sep 6 '12 at 20:06
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So would you subscribe to the following definition: "A geometry is a triple $\langle X, G, a \rangle$ with $X$ a set, $G$ a group and $a$ a group action $a: G \times X \rightarrow X$." That's it? –  Hans Stricker Sep 6 '12 at 20:12
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This answer gives only one (pretty, but rather constrained) outlook on what a geometry is. A differential geometer or algebraic geometer (not to mention Banach geometer...) certainly wouldn't agree with you. I think we all know what geometry is from our high school study of triangles and squares and the best definition of geometry I can think of is: the subject that studies generalizations (in all kind of directions) of these objects. –  Marek Sep 6 '12 at 20:53
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@Hans: it is necessarily vague, since I claim there is no real answer to your question. It's the same as asking "what is math?". There is just too many geometrical subareas whose only common point seems to be what I have already said: they study some kind of generalization of those basic objects like circles and triangles in the plane. Whether those generalizations are varieties, manifolds, schemes, Banach spaces or whatever doesn't really matter. –  Marek Sep 6 '12 at 21:06
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@Marek: you may be right. Nevertheless geometry and topology can be seen on a par, and there is algebraic topology opposed to algebraic geometry and the question may hope for a definite answer: what geometry is opposed to topology? –  Hans Stricker Sep 6 '12 at 21:12

The answer to your question, "Is there a thorough and generally agreed upon definition of a geometry", is negative: There is no such definition. For instance, Klein's viewpoint (from 1872), was outdated by the time it was proposed, as it did not cover the emerging Riemannian geometry which was (at the time) in its infancy, as well as algebraic geometry which, at the time, was vigorously developed by the Italian school (and Cayley and many others). What's worse, Klein did not even cover Gauss' intrinsic geometry of surfaces which was, by that time, reasonable well-established.

At best, one can give an (admittedly incomplete) list several branches of mathematics, which name themselves geometry:

  1. Metric geometry.

  2. Riemannian geometry.

  3. Pseudo-Riemannian geometry.

  4. Symplectic geometry.

  5. Contact geometry.

  6. Geometry of foliations.

  7. Study of locally-homogeneous geometric structures in the sense of Ehresmann (e.g., flat projective structures, flat affine structures, etc).

  8. Incidence geometry and geometry of buildings a la J.Tits.

  9. Algebraic geometry.

  10. Noncommutative geometry of A.Connes.

Many (items 2, 3, 4, 5, 6), but definitely not not all, of these geometries, can be put under the umbrella of Cartan's definition of a geometric structure as a smooth $n$-manifold $M$ equipped with a reduction of the frame bundle to its $G$-subbundle, where $G$ is a closed subgroup of $GL(n,R)$.

(Klein's proposed definition of geometry fits as a small subfield of all of these items; it deals exclusively with, what we now call, homogeneous spaces.)

All these fields have some common features and, yet, resist a common definition. The suggested definition by Lurie, is primarily driven by algebro-geometric considerations and applications and is too broad to separate "geometry" from "topology" (the category of topological spaces will fit comfortably into Lurie's framework).

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Usually, geometry consists of an underlying topological space (a manifold, for example) and some structure on this space. The structure is an analogy of some tool -- such as a ruler or compass -- that enables you to see more than what the topology sees. It might be something that enables you, for example, to "measure angles and distances" (Riemannian geometry), or "just to measure angles" (Conformal geometry), or "to see what are lines and what are not lines" (Projective geometry), or some other, more abstract analog of a "ruler and compass."

From what I have learned, the Cartan geometry -- which defines geometry as a principal bundle over a manifold with some Cartan connection -- generalizes both Kleinian and Riemannian geometry in some sense; one book where this is explained is Sharpe: Cartan's generalization of Klein's Erlangen program.

The reason why there is not a single universal definition, unlike in topology, is the immense history of geometry (2500 years, compared to 100 years of topology).

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J.W. Cannon also gave a definition:

"A geometry is a topological space endowed with a proper path metric."

He also gave a definition of a geometric group action:

"A [group] action is geometric [on a set S] if S is a geometry and the action is isometric, cocompact and properly discontinuous."

These definitions can be appropriate to work in geometric group theory.

See: J. W. Cannon. Geometric group theory. In Handbook of Geometric Topology. Elsevier, 2002. (In particular p. 271-272.)

(Added more than a year later:)

Actually I would even say that "a geometry" is the same thing as a metric space. This seems to me the most generic notion of "a geometry", of which all other particular geometries are specializations.

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Actually, this is far from being the most general notion, e.g. It excludes symplectic geometry, algebraic geometry, etc. –  studiosus yesterday

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