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The Wikipedia's article for geometry is somehow overwhelming. To make things clear, allow me to ask some questions:

  1. I wonder if "geometry" can be defined as the study of a metric space (possibly with or without other structures)?

  2. Any thing more general than metric space (such as uniform spaces and topological spaces) is not in the scope of "geometry"?

  3. Does "geometry" assume the set under study to have some algebraic structure?

    Also there is algebraic geometries.

  4. Is the underlying set a topological vector space, normed space, inner product space, or even Euclidean space?

  5. Since projective and affine spaces are pure algebraic concepts without metrics, why are there "projective geometry" and "affine geometry"?

Thanks and regards!

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    Differential geometry doesn't generally assume a metric. Even if we insist that a differential structure alone is not sufficient to merit the label geometry, once we include a connection, we have geodesics, and I expect you will want to call it geometry. But there need not be a metric.2012-03-17
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    @yasmar: Thanks! (1) So are you saying it is connection that merit the label geometry? (2) Can this view be unified with geometries on inner product space, on (pure algebraic) projective and affine spaces?2012-03-17
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    Hi @Tim, I would certainly not go so far as to say that a connection is what is needed for geometry. In fact, I think the question in the title of your post is a good one, but I doubt there is any universally accepted answer. I would like to see more opinions expressed. I think 'geometry' can happen even in the absence of a metric, and a connection is an example, but there must be other examples of structures that don't imply a connection, or even a manifold. I can't competently answer your second question in your comment. I hope somebody else will.2012-03-17

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A partial (i.e. highly incomplete) answer to your questions on algebraic geometry:

Is the underlying set a topological vector space, normed space, inner product space, or even Euclidean space?

It depends - in a lot of algebraic geometry, you think about particular subsets of affine or projective spaces, which are not equipped with standard metrics, norms or inner products. In more abstract settings you might even be dealing with schemes, which are a big generalisation of these spaces. Very loosely, they can be thought of as being constructed by turning rings into geometric objects and then gluing them together. For example, $n$-dimensional affine space over a field $K$ is the geometric object associated to the polynomial ring $K[x_1,\ldots,x_n]$, and projective space can be obtained by gluing together some affine spaces in a particular way.

It is definitely too strong to insist that the underlying space is Euclidean, because that would be ignoring all of the interesting geometry on spheres and hyperbolic spaces, among others. (Here I mean geometry in the more traditional sense of theorems about distances, angles, intersections of lines and so on, but there is a lot more to spherical and hyperbolic geometry as well).

Since projective and affine spaces are pure algebraic concepts without metrics, why are there "projective geometry" and "affine geometry"?

I may have got the wrong idea from your question here, but I've taken it to be asking what kinds of geometric theorems you can have without metrics. A number of the big theorems in algebraic geometry are primarily algebraic in nature, but are motivated by geometric questions, while others are more recognizable as being geometric theorems.

One example is as follows. The main object of study in affine and projective algebraic geometry is an algebraic variety, which is a subset of affine or projective space defined by an ideal of polynomial equations. In particular, any curve in the plane ($2$-dimensional affine space) defined by a single polynomial is an algebraic variety. Up to worrying about degenerate cases, we can say that the degree of the curve is the degree of the polynomial defining it. Then Bézout's theorem states that two "general" curves of degrees $d_1$ and $d_2$ intersect in $d_1d_2$ points (there is a lot of interesting mathematics involved in saying exactly what "general" means). So this is an example of a fairly strong theorem, which says a lot about how curves behave, without any reference to a metric.

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    +1 Thanks! I wonder what "turning rings into geometric objects and then gluing them together" means? Especially what are "geometric objects"? Do they assume metric or further structures on their sets?2012-03-16
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    Here what I mean is to take the spectrum of a ring, which yields a set with a topology and a sheaf of functions, but no metric. See http://en.wikipedia.org/wiki/Spectrum_of_a_ring, although this article is fairly technical. Again the topology is the one which is supposed to make sets defined as the zero set of an ideal closed, so it is not induced by any metric.2012-03-16
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    Thanks! (1) So the kind of "geometry" in your reply is purely topological and algebraic. Why isn't it called "algebraic topology" instead of "algebraic geometry"? (2) Can it be unified with the other kind of "geometry" such as for inner product space?2012-03-16
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    One answer to (1) is that the notion of isomorphism for topological spaces is much weaker than that for the geometric objects I'm describing here. One way to think of this is to imagine the unit disk in the plane, and think of raising the origin up into a point, so you get a cone. Topologically, that space is homeomorphic to the disk, so the algebro-topological invariants are the same as for the disk, but the geometry somehow "knows" that the origin is now special, so the algebro-geometric invariants reflect this. As for (2), I don't know. It's certainly not something I've seen.2012-03-16
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    About (1), isomorphism for topological spaces means bijective continuous mapping with continuous inverse. But for the geometric objects you described earlier, what is the isomorphism?2012-03-16
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    It depends on the level of generality, but if we're talking about subsets of affine or projective space defined by polynomial equations, then it's a polynomial map with polynomial inverse. The cone I described is not such an object, but a similar example is given by the curve y^2=x^3 in the plane. This is homeomorphic to a line (if we give it the subspace topology induced by the usual topology on $\mathbb{R}^2$) but it's not isomorphic to the affine line as an algebraic variety because of the "spiky" point at the origin.2012-03-16
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I would say that "geometry" in a strict sense needs a notion of length AND some kind of inner product (I'd say a quadratic form). Thus the study of most manifolds is geometry since it's usually equipped with a standard Riemanian or Lorentz form (it's the case in particular for projective or affine spaces on $\mathbb{C}$ or $\mathbb{R})$.

I wouldn't say that the study of general, abstract metric spaces is "geometry" in that sense (and therefore much less so for general topological spaces), but I think it's not something set in stone.

Also there is a very broad acceptation of the term "geometry" as in "algebra, analysis, probabilities and geometry", and since general topology fits better in geometry than in analysis or algebra, you could say that it belongs there, but for me it's a slight abuse of language.

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    +1 Thanks! ["Metric spaces"](http://en.wikipedia.org/wiki/Metric_space) at Wikipedia says "the geometric properties of the space depend on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity." It leads me to think maybe "geometry" only or at least assumes metric spaces.2012-03-15
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    the metrics they speak of are derived from a quadratic form of signature (1,3) (a Lorentz form) which gives more structure than just a metric. also the term "non-euclidian geometries" refer to the notions of curvature and geodesics, for which you need more structure than just a distance. that's why I said that in my answer2012-03-15
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    So "projective geometry" and "affine geometry" don't just assume pure algebraic projective and affine spaces, do they? How about algebraic geometry?2012-03-15
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    I would say that algebraic geometry does have a "pure" notion of affine and projective spaces, without metrics or preferred co-ordinates; in fact the first definition of affine space I was given as an undergraduate was that it is a Euclidean space with no metric and no distinguished origin. I've tried to give a flavour of this in my answer, but there's an enormous amount that could be said on this subject so it's certainly not a complete answer.2012-03-15
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    I don't know too much about algebraic geometry, and Matt Presland gave an extensive answer below for that. as for projective geometry and affine geometry, as I said, it's not so much that you "assume", but when the fields are $\mathbb{C}$ or $\mathbb{R}$ (ie with a nice topology), you have some natural metric.2012-03-15
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    Yes, if you use $\mathbb{C}$ or $\mathbb{R}$ you will get a nice metric - but it predominantly isn't used in algebraic geometry! Partly this could be motivated by wanting statements about geometry over general fields, but it probably also has a lot to do with the fact that the standard algebro-geometric topology on affine and projective space isn't the usual one, and it's not even metrizable. It's the topology in which a set is closed if and only if it is a variety (or a union of varieties if your definition of variety includes irreducibility).2012-03-15
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    @Matt: So for a vector field with $\mathbb{R}$ or $\mathbb{C}$ as base field, how is the natural metric on the vector field induced?2012-03-16
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    I assume you mean what I would call a vector space rather than a vector field. This isn't really my area, but I would say a natural metric on $\mathbb{R}^n$ or $\mathbb{C}^n$ is one induced by any norm, by the formula $d(x,y)=|x-y|$. In finite dimension, all of these metrics induce the same topology, called the Euclidean topology. But as I say, in the algebraic geometry setting, you don't use this topology, so the metric doesn't help you at all.2012-03-16
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    @MattPressland: My bad, I meant a general vector space with $\mathbb{R}$ or $\mathbb{C}$ as its base field, not just a Euclidean vector space. How is a metric induced on the vector space from the base field?2012-03-16
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    No problem. I'm not really sure what it has to do with the base field, as it's the same formula for both $\mathbb{R}$ and $\mathbb{C}$, and this will definitely work for $\mathbb{Q}$ as well, or indeed any field extension of $\mathbb{Q}$. Over finite fields you might have some problems defining a norm that doesn't give non-zero vectors zero length, but if you can then the same formula works again - it's a standard way of getting a metric out of a norm. Where the norms come from is really not my area!2012-03-16
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    @Glougloubarbaki: I must disagree. I think of geometry as the study of locality; one does not need a notion of length or angle for this.2012-03-16
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    @ZhenLin: Thanks! Could you elaborate on what "locality" geometry studies is? It will be great if you could post a new reply, given the number of comments?2012-03-16
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    @ZhenLin : some very geometrical questions are global, not local (e.g., behaviour of geodesics)2012-03-16