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Community: Geometry vs. Metric Geometry

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    Could someone please distinguish for me the difference between the tags "geometry" and "metric geometry"? I fear I have not been classifying questions appropriately because I do not really understand what is non-metric geometry, distinguished, say, from topology. Apologies for my naiveté. Maybe an example or two in the set {geometry} \ {metric geometry} might help... Thanks!
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    Conformal geometry is non-metric geometry, right?

    • CommentAuthorRyan Budney
    • CommentTimeFeb 11th 2011 edited
     
    Building on Qiaochu's thread of ideas: manifolds tend to be given various structures corresponding to various structure groups -- symplectic structures, conformal structures, volume structures, contact structures, foliations, etc. Some of these have a more metrical nature, some don't. I suppose that's a vague distinguishing idea.
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    Joseph-

    Look at the arXiv headings: there's Algebraic Geometry, Differential Geometry and Symplectic Geometry, all different from Metric Geometry.

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    Thanks for the clarifications. I guess what confuses me is that most of differential geometry (certainly Riemannian geometry, even pseudo-Riemannian geometry and Finsler geometry), has a metric, and so could be considered metric geometry. But I guess this is not true of symplectic geometry. These distinctions aside, to come back to why I raised this issue: Is there any MO question that should be classified as geometry, or should one always classify in a more specific category? I ask because my geometry tags have often been changed to metric geometry. This is hardly a crucial issue!! I don't really care one way or the other, I just want
    to follow the appropriate MO convention. Maybe a better question is this: What type of question should be classified as _geometry_ (as opposed to metric-, Riemannian-, differential-, Euclidean-, symplectic-, etc.)?
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    I doubt your question is ever really going to be settled, even if there is a policy decision made in this thread. Geometry is a word that is used so loosely in mathematics by so many different schools of thought that attempting to enforce any policy might be light trying to stop the tides!

    Sorry for being kind of negative.
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    I can think of one question where geometry is a more appropriate tag than any of the above mentioned ones:
    http://mathoverflow.net/questions/20383/categories-of-geometry/

    I only know because at the time it was the only MO question I felt comfortable answering. If it was posted now it certainly would be closed as being off topic, and rightly so.

    (I the above is just to give an example, I think it would be unfortunate if this question were to become active again.)
    • CommentAuthorKevin Lin
    • CommentTimeFeb 11th 2011
     
    • CommentAuthordeane.yang
    • CommentTimeFeb 11th 2011
     
    The link Kevin gave does describe metric geometry reasonably well. But still a question like "what is metric geometry and where did it come from?" seems like a reasonable thing to ask on MO. Let me give my impressions. Although aspects of it are quite classical, I don't think anyone used the term "metric geometry" until the last 20 years. To me, it developed into a subject of its own after Gromov used ideas and tools from metric spaces (which up to then were studied mostly by functional analysts and not by differential geometers) and used them in Riemannian geometry. The seminal work was a hard-to-find out-of-print monograph called "Structures métriques pour les variétés riemanniennes" (I was able to buy one from Pierre Pansu only because he knew where all the remaining copies were) that has been translated and reprinted in English. The major theorems were the Cheeger-Gromov finiteness and convergence theorems, as well as the Cheeger-Gromov collapse theorems. After that the subject was developed further by many people, including for example Anton Petrunin who participates actively on MO. More recently, there has been work on so-called metric measure spaces on which a notion of Ricci curvature can be defined. Work on this in collaboration with Lott was one of the things cited for Villani's Fields Medal.
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    Discrete metric geometry, which is an outgrowth of the local (meaning finite dimensional) geometry of Banach spaces, has become quite fashionable, in part because of its use in the design of algorithms. One current program at INI is devoted to Discrete Analysis:

    http://www.newton.ac.uk/programmes/DAN/index.html

    Discrete and continuous metric geometry are the topics of a program at MSRI in the fall:

    http://www.msri.org/web/msri/scientific/programs

    As Deane mentioned, the continuous part got an impetus from Gromov, but in fact the flat continuous part (that is, the metric geometry of Banach spaces) was started by Lindenstrauss in 1964. (Joram; not Elon, who was not that precocious).
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    @Bill: given that Elon was born in 1970, that would be beyond precocious!

    Did you have a specific paper in mind, maybe "On nonlinear projections on Banach spaces"?

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    Thanks everyone for your comments! What for me started as a narrow question concerning how to tag my MO questions has turned into a highly informative discussion on the varieties of geometry. I appreciate the tutorials!
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    That one, Mark. It marked the beginning of the theory even if there were fragmentary results early, such as the Mazur map and Schoenberg's theorems.
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    I think that I might be in a position to discuss the most literal version of the question. I was a member of the math advisory committee for the arXiv when that committee put together the list of categories in 1997. The first official list back then is only slightly different from the list still in use now. math.DG, Differential Geometry, had already existed since 1994 as dg-ga. I proposed the category math.MG, Metric Geometry, as a category that would be the home of classical Euclidean geometry, convex geometry, packing and covering problems, etc. I wanted a short name that would convey the right idea, and I wanted it to sound modern enough to include new topics like CAT(0) spaces. On the other hand, differential geometry already had and clearly deserved its own home, so the rough idea for metric geometry was Euclidean geometry and certain generalizations, other than questions based on differential structures on manifolds.

    Note that arXiv categories are ideally a map of communities, or sometimes glued-together clusters of communities, of mathematicians. Is there an identifiable community of metric geometers, distinct from differential geometers? I think so; I think that it includes people like Robert Connelly, Karoly Boroczky, Tom Hales, etc. (And, not coincidentally, my dad.) I think it's fair to say that these people are in the same area as each other and not in the same area as Grigory Perelman and Chris Croke. Of course there is some overlap, occasionally a lot of overlap. arXiv categories aren't and can't be perfect; and to the extent that they could be entirely accurate, they are an open cover of research and not a tiling.

    On the other hand, I think it's cool that the list of categories turned out to be reasonable enough that it is useful for a distinctly different system, Math Overflow, 12 years after it was drafted.

    I also think that it's a bit limp to just label an MO question "geometry". Usually this label doesn't mean algebraic geometry or differential geometry; usually mg.metric-geometry is what is really meant.
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    @Greg: That is quite clarifying! I especially like the "map of communities" viewpoint. Thanks!

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