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Every elementary book on abstract algebra usually begins with giving a definition of algebraic structures; generally speaking one or several functions on cartesian product of a point-set to the set. My question is this: Is there a property that unifies different geometric structures like topology(I consider it a geometric structure), differential structure, incidence structure and so on? Can one say a geometric structure on a set one way or another involves a subset of its powerset ?

Buddha
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  • There are several concepts of a general geometric structure, but they are more restrictive than the one you are asking about. All involve geometric structures on manifolds; none is a subset of the powerset; all are useful and important. If you are interested in these, I can explain more. – Moishe Kohan Aug 29 '14 at 18:14
  • @ you are referring to structures like defining some special kind of tensor on the tangent space, symplectic , riemannian ....? – Buddha Aug 29 '14 at 18:23
  • The definition of a topology is based on subsets of the powerset because topology says something about the "internal organization" of the set. The Kleinian perspective on geometry is that you're observing a fixed set of transformations on a set, and seeing what is invariant. Since a good notion of geometric structure would have to invite the Kleinian perspective to the party, I'm not sure how one would unite the two under one definition. – rschwieb Aug 29 '14 at 18:37
  • You might be interested in this question: http://math.stackexchange.com/questions/896846/is-category-theory-geometric – rschwieb Aug 29 '14 at 18:39
  • @rschwieb .topology is about continuity which is in my opinion very geometric as is smoothness. Everyone has a visual understanding of smoothness and continuity. – – Buddha Aug 29 '14 at 18:58
  • @Buddha So giving $\Bbb R$ the discrete topology, every function out of this space is continuous. Would you consider the characteristic function of the irrationals "smooth" geometrically? (You might consider it so, but this example kind of highlights the difference between "shape" and "continuity") – rschwieb Aug 29 '14 at 19:00
  • @rschwieb Topology is a "generalization" of our intuition of continuity and gradual(no gluing no tearing) deformation. – Buddha Aug 29 '14 at 19:04
  • @Buddha This is why I included the link to the "is category theory geometric?" question. Every category can be viewed this way as objects with morphisms between them that preserve the salient features of each object. At that level, yes, it can be considered geometric. I don't think one would go about defining geometric objects at that level of generality, though – rschwieb Aug 29 '14 at 19:30
  • also http://math.stackexchange.com/questions/192055/what-is-a-geometry – betoche Aug 30 '14 at 04:49
  • @rschwieb From one point of view, Cartan's approach to geometry (which today is formalized as the theory of Cartan geometries) treated certain classes of geometric structures as curved deformations of Klein geometries (homogeneous spaces) $G/H$ in a way that generalizes the way that Riemannian manifolds $(M, g)$ generalize the homogeneous space $\mathbb{R}^n \cong AO(n) / O(n)$ endowed with the structure that (conjugates) if $O(n)$ preserve at each point, i.e., the flat metric $\bar{g}$. Different choices of $G$ and $H$ lead to different kinds of geometric structure. – Travis Willse Aug 30 '14 at 05:56
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    @Buddha As a working geometer my point of view is that the topology (and, for me anyway, the smooth structure) are already fixed and is a prerequisite for a geometric structure but not a geometric structure itself, and geometric structures are data defined on a topological (differentiable) manifold that break topological (smooth) invariance, i.e., invariance under homeomorphism (diffeomorphism). – Travis Willse Aug 30 '14 at 06:05

2 Answers2

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I believe, you are looking in a wrong place. Geometry and Topology are related, but different fields of mathematics. Same with Analysis, which you are trying to put under the same umbrella. You might eventually find a definition which is broad enough to cover all three areas, but then it will cover so much in mathematics that it becomes useless.

Here are some notions of geometric structures (on smooth manifolds) that people working in geometry and topology actually use and quite successfully. This list will not answer your question, but, hopefully, will be useful (to somebody).

  1. Geometric structure (in the sense of Cartan, I think). If I remember correctly, these are discussed in detail in the book of Kobayashi and Nomizu "Foundations of Differential Geometry". Let $M$ be a smooth manifold. Then the geometric structure on $M$ is a reduction of the structure group of the frame bundle of $M$ from $G=GL(n, {\mathbb R})$ to a certain subgroup $H<G$. For instance, a Riemannian metric is a reduction to the orthogonal subgroup. An almost complex structure is a reduction to the subgroup $Gl(n,C)$.

  2. Geometric structure in the sense of Ehresmann (see here), or an $(X,G)$-structure. Let $X$ be an $n$-dimensional manifold and $G$ a group (or pseudogroup) of transformations of $X$. One usually assumes that $G$ acts transitively and real-analytically, but let's ignore this. Then an $(X,G)$-structure on an $n$-dimensional manifold $M$ is an atlas on $M$ with values in $X$ and transition maps equal to restrictions of elements of $G$. For instance, complex structure, symplectic structure, flat affine structure, hyperbolic structure etc, appear this way. This notion was successfully extended to cover spaces which are not manifolds, where one relaxes the assumption that charts are defined on open subsets: These extensions appear in algebraic geometry and theory of buildings.

  3. There is an important variation on these concepts due to Gromov, called rigid geometric structures, see:

Gromov, Michael, Rigid transformations groups, Géométrie différentielle, Colloq. Géom. Phys., Paris/Fr. 1986, Trav. Cours 33, 65-139 (1988). ZBL0652.53023.

Quiroga-Barranco, R.; Candel, A., Rigid and finite type geometric structures, Geom. Dedicata 106, 123-143 (2004). ZBL1081.53027.

An, Jinpeng, Rigid geometric structures, isometric actions, and algebraic quotients, Geom. Dedicata 157, 153-185 (2012). ZBL1286.57032.

and

Feres, Renato, Rigid geometric structures and actions of semisimple Lie groups, Foulon, Patrick (ed.), Rigidity, fundamental group and dynamics. Paris: Société Mathématique de France (ISBN 2-85629-134-1/pbk). Panor. Synth. 13, 121-167 (2002). ZBL1058.53037.

Moishe Kohan
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  • your answer is indeed very useful. – Buddha Aug 31 '14 at 07:04
  • @MoisheKohan It seems the third link is dead; it's likely https://arxiv.org/abs/1005.1423 is a replacement. Also for further reference Feres' "Rigid Geometric Structures and Actions of Semisimple Lie Groups" (https://www.math.wustl.edu/~feres/strassbourg.pdf) is worth having a look. – Alp Uzman Feb 15 '22 at 09:51
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    @AlpUzman: Thank you, I updated. – Moishe Kohan Feb 15 '22 at 16:29
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It is known that, for a (compact) topological space, the continuous functions into $\mathbb{C}$ characterize the topology on the space. As far as I know similar statements hold for smooth manifolds (using smooth functions) and algebraic varieties (using polynomials). So one possible answer is that a geometric structure is an algebra of functions on your space.

Owen Sizemore
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