What is a geometric structure?

Question

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 ?

Answer 1

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.

Answer 2

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.