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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 or an algebra?

Hans-Peter Stricker
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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 06 '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 06 '12 at 20:47
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    "Geometry" = "measurement of the earth"... by etymology – GEdgar Sep 06 '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 06 '12 at 21:42
  • algebra is about the structure of sets, where structure means a function defined on it. – Vicfred Sep 09 '12 at 22:44
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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. – Andrew Dudzik Dec 19 '14 at 21:05
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    Hilbert wrote that there's no difference b/w the methods of geometry and those of physics. Though physics requires time for motion, while geometry does not, dynamical systems can abstractly be defined as group or semigroup actions on a state space, so that's consistent w/ Hilbert's view. – alancalvitti Dec 20 '14 at 15:04

5 Answers5

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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).

MvG
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Peter Franek
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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 $X$.

user02138
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  • But what is a space? A metric space? A topological space? A manifold? Is there a general definition of "space"? – Hans-Peter Stricker Sep 06 '12 at 20:01
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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 06 '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-Peter Stricker Sep 06 '12 at 20:12
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    This answers the matter of what is a geometry. What is the branch of mathematics called geometry then? I'd say it is the study of group actions and their invariants, but I'm no geometer and I've always found that kind of confusing, so I'll leave it for others to answer. :) – tomasz Sep 06 '12 at 20:19
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    @user02138: If it's so simple and straight-forward: why is this not to be found prominently at the appropriate places, e.g. Wikipedia or most of the text books on geometry? – Hans-Peter Stricker Sep 06 '12 at 20:22
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    @tomasz: As a differential geometer, I would agree with you. Things like curvature and torsion are simply differential invariants of group actions. You need to consider the prolonged group action on the jet space of a space. Cartan made great strides in this direction. – Fly by Night Sep 06 '12 at 20:26
  • @HansStricker: Wikipedia has an article on Klein's Erlangen Program. See https://en.wikipedia.org/wiki/Erlangen_program – Fly by Night Sep 06 '12 at 20:31
  • @user02138: You are not totally consistent: shouldn't you have said "if said set is a metric space..."? – Hans-Peter Stricker Sep 06 '12 at 20:32
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    @tomasz: If user02138's definition is the right one (the most general and generally agreed upon one) geometry-as-a-branch-of-mathematics cannot be anything else than the general study of such geometries. There may be preferences - for examples on invariants of the group actions - but one cannot and should not restrict geometry-as-a-branch-of-mathematics to that. – Hans-Peter Stricker Sep 06 '12 at 20:39
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    I personally wouldn't call an arbitrary group action on a set a geometry or even a geometric object. On the other hand, even though I don't really know much about such things, I imagine that there are groupoid actions beyond group actions which could be considered to be geometries or at least geometric entities. – Michael Joyce Sep 06 '12 at 20:40
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    Things are getting interesting: not every group action gives rise to a geometry (so how to characterize those that actually do?) but what's more: it doesn't have to be a group action, but maybe another algebraic structure's action, e.g. a groupoid's. Which algebraic structures may be considered? – Hans-Peter Stricker Sep 06 '12 at 20:46
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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 06 '12 at 20:53
  • @Marek: I am not satisfied with this approach, it's too vague: Which kind of generalizations in which kinds of directions of which objects? – Hans-Peter Stricker Sep 06 '12 at 20:56
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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 06 '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-Peter Stricker Sep 06 '12 at 21:12
  • @HansStricker, Arkangelskii (sp?) said topology is geometry without a metric. – alancalvitti Dec 20 '14 at 15:01
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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).

Edit. Simons Center for Geometry and Physics has a page aptly named "What is Geometry?" which has several prominent geometers, topologists and physicists trying to answer the title question (Sullivan, Donaldson, Vafa...) and (not surprisingly) failing to come up with anything close to an answer. (Although, I'd say, Fukaya comes closest.)

Moishe Kohan
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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.

InfinitelyInquisitive
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    Actually, this is far from being the most general notion, e.g. It excludes symplectic geometry, algebraic geometry, etc. – Moishe Kohan Dec 19 '14 at 16:47
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The term geometric space is sometimes used as a synonym for locally ringed space; i.e., a topological space $X$ together with a sheaf $\mathcal F$ of rings on $X$ such that the stalks of $\mathcal F$ are local rings. This suggests that geometry should be the study of geometric spaces; i.e., locally ringed spaces. Manifolds, Riemann surfaces, varieties and schemes are all examples of locally ringed spaces.

Of course, this idea ignores the fact that the locally ringed space of, for example, a manifold has little bearing much of what we do in geometry (Riemannian metrics, angles, geodesics etc.) But the same could be said of much of topology.

Another possible flaw with this is that there are examples of interesting geometric objects, such as sheaves on sites and algebraic stacks which are not examples of locally ringed spaces.

John Gowers
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