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Question: Is there any (absolute) geometry which is not "infinitesimally Euclidean"?

Context: All of the geometries listed on the Wikipedia page "Foundations of Geometry" (describing axiomatic formulations of geometry) seem to correspond to special cases of absolute geometry, and it seems like any absolute geometry is either hyperbolic, elliptic, or Euclidean (parabolic?) according to the version of the parallel postulate used, perhaps equivalently according to the type of curvature of the underlying geometric space. These all seem to have realizations or models as Riemannian manifolds of some sort.

Any geometry of (smooth) manifolds seems to be infinitesimally Euclidean, even for those without a Riemannian metric, since each neighborhood is (diffeomorphic) homeomorphic to Euclidean space.

Hyperbolic geometry seems to be the study of Riemannian manifolds with negative curvature, elliptic geometry the study of Riemannian manifolds with positive curvature, and Euclidean geometry the special case where there is no curvature. But obviously every neighborhood of a Riemannian manifold is diffeomorphic Euclidean space, thus even the Riemannian geometry of spaces like the torus, which is neither strictly elliptic nor hyperbolic, is infinitesimally Euclidean.

Thus it seems like to me that all of the elementary geometric axioms determine every aspect of the geometric space (e.g. that it must be a Riemannian manifold) except the curvature -- thus changes in the parallel postulate seem to correspond to different values of the curvature of the space.

Am I understanding this correctly? I had thought previously that the term geometry could be applied to spaces so abstract that they could not be embedded in any Euclidean space, and in particular were not infinitesimally Euclidean, but now I am not so sure. Any clarification would be appreciated. The question stems in part from my reading of Agricola and Friedrich's "Elementary Geometry" (also of the original German version), so perhaps if you have read some of that book as well you might understand better the source of my misunderstanding.

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Riemann in his famous essay of 1854 considered only metrics that are infinitesimal. For this reason Riemannian geometry properly speaking is only concerned with this type of manifold. A generalisation is known as Finsler spaces. Here infinitesimally the space looks like a Banach space, which is more general than Euclidean space.

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  • $\begingroup$ You are right that "infinitesimal" in the sense of geometry probably has to mean tangent spaces of a smooth manifold. Thus your mention of Finsler spaces seems to be exactly on target -- from the first paragraph of the Wikipedia article on Finsler manifolds (en.wikipedia.org/wiki/Finsler_manifold) -- "Finsler manifolds non-trivially generalize Riemannian manifolds in the sense that they are not necessarily infinitesimally Euclidean. This means that the (asymmetric) norm on each tangent space is not necessarily induced by an inner product (metric tensor)." $\endgroup$ – Chill2Macht Jan 27 '17 at 21:32
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The subject of topology, and its sub-subject metric spaces, contains many examples of "absolute geometry" spaces that are not infinitesmally Euclidean, neither in the metric sense, nor the smooth sense, nor the topological sense. These subjects abound with many natural examples.

For example, infinite dimensional Hilbert spaces, which have many examples amongst function spaces, are not locally homeomorphic to Euclidean space. And yet their geometry is of intense interest. Take a look at any functional analysis book.

For another example, there is an entire theory devoted to metric spaces of nonpositive curvature. See the book of Bridson and Haefliger for a better feel.

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    $\begingroup$ Building on this, I've always wondered: are there natural examples of classes of manifold type objects which instead of being locally homeomorphic to Euclidean space are locally homeomorphic to some other kind of well-understood topological space? $\endgroup$ – Vik78 Oct 16 '16 at 14:12
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    $\begingroup$ Banach manifolds are examples. Universal Menger compacta, whose existence was established in Bestvina's thesis, are other examples. $\endgroup$ – Lee Mosher Oct 16 '16 at 14:13
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    $\begingroup$ Thanks. I always just felt like the definition of manifold could easily be generalized using any topological space, and wondered why this approach wasn't more common. $\endgroup$ – Vik78 Oct 16 '16 at 14:15
  • $\begingroup$ There's also matchbox manifolds, which are locally cantor spaces (or the product with a Euclidean ball). Solenoids fall into this category as some of the simplest examples. $\endgroup$ – Dan Rust Oct 16 '16 at 14:19
  • $\begingroup$ @Vik78: Part of the trouble with such generalizations is that manifolds have a "homogeneity" property which is hard to mimic in situations that are not locally Euclidean. That, perhaps, might explain why it took so long for the theory of universal Menger compacta to be put on solid ground. But local homogeneity is evident for Hilbert spaces and other normed vector spaces, which perhaps explains why they have been around for much longer. $\endgroup$ – Lee Mosher Oct 16 '16 at 14:31
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Consider the geometry of the surface of a cone. The curvature is singular in one direction around the bottom edge, and in every direction at the tip. Thus the geometry is not even approximately Euclidean at those points. For a space that is not approximately Euclidean anywhere, you're getting into the domain of fractals, like would be formed by some higher dimension analogue of the Weierstrauss function. I'm unsure of whether the geometry of such a complicated space could be axiomatized into an absolute geometry. It is, I think, worth a look.

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I know this is an old question, but I just discovered it, and I'm a little confused by the responses from others, because there is an obvious problem with the following claim you make:

Any geometry of (smooth) manifolds seems to be infinitesimally Euclidean, even for those without a Riemannian metric, since each neighborhood is (diffeomorphic) homeomorphic to Euclidean space.

You are confusing the category of smooth manifolds with the category of Riemannian manifolds, or the subject of differential topology with the subject of differential geometry.

Yes, every smooth manifold is locally diffeomorphic to Euclidean space, and yes this has nothing to do with whether or not the manifold is endowed with a metric structure. But not every Riemannian manifold is locally isometric to Euclidean space. You're using the wrong notion of isomorphism in the category of Riemannian manifolds: you're forgetting the infinitesimal metric structure. There are no local invariants in differential topology, for the reason you state. But there are local invariants in Riemannian geometry. The word "geometry" typically refers to metric structure; if you say "the geometry of (smooth) manifolds" most people are going to hear you talking about differential geometry (either Riemannian or pseudo-Riemannian), not differential topology.

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    $\begingroup$ In particular, the geometry of the sphere, for example, is not "infinitesimally Euclidean." As you say, the sphere has positive curvature, but Euclidean space has zero curvature. The geometry around a point of zero curvature will be infinitesimally Euclidean, but not all points your torus, for example, have zero curvature: at those points the geometry won't be infinitesimally Euclidean. $\endgroup$ – symplectomorphic Mar 16 '17 at 0:56
  • $\begingroup$ Although, to be fair, I'm conflating "infinitesimal" with "local," partly because you did: by "infinitesimally Euclidean" you meant "around each point there is a diffeomorphism to $\mathbb{R}^n$," which is true; in geometry the analogous notion would be "around each point there is an isometry to $\mathbb{R}^n$ (with its standard metric)." But we should really call these facts "local" rather than "infinitesimal." Infinitesimally a Riemannian manifold is just a Euclidean vector space (a vector space with an inner product), and there the geometry definitely is Euclidean. $\endgroup$ – symplectomorphic Mar 16 '17 at 1:17
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Geometry is what mathematicians call geometry.

Absolute plane geometry, referring to the theory you've linked, is not an attempt at being a general theory of geometry — instead, it axiomatizes an extremely rigid structure that encompasses exactly two models: the euclidean plane and the hyperbolic plane.

To the best of my knowledge, the only real reasons for the theory is:

  • to provide the setting in which one can consider the question "can you prove the parallel postulate?"
  • a starting point for learning hyperbolic geometry
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