History of the notion of $(G,X)$-structure

Question

I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work.

So far, it appears that he was the first to set it. Many mathematicans cite Ehresmann's work, also I'm not sure to what precisely. It also seems that Kuiper (On compact conformally Euclidean spaces of dimension $> 2$, 1950) also had some ideas close to what we call a developing map, and related to the idea to put a structure on a manifold.

Maybe someone here already have made such researches, or could testify ?

Thanks

edit :

As requested, I specify a sense of $(G,X)$-structure. I mean by that a pair of $G$ a group (or pseudo-group if wanted) of analytic diffeomorphisms of $X$, a smooth analytic connected manifold.

The point is to have $(G,X)$-structure notion clean enough to get the pair of a developing and holonomy maps.

Answer 1

Here is a collection of remarks on the history; when and if I have more time, I will add more detail.

1. The ideas go back to 19th century (Poincare and others) who studied 2nd order holomorphic ODEs on Riemann surfaces and had notions of a linearly polymorphic function (our developing map) and monodromy (our holonomy). In the modern terminology, these were "complex-projective structures on Riemann surfaces". Prior to Thurston, most of the work was done in this class of geometric structures (especially Gunning, Kra, Maskit and Hejhal).

You can find many basic results and historic references in

D. Hejhal, Monodromy groups and linearly polymorphic functions. Acta Math. Volume 135 (1975), 1-55.

In the paper, Hejhal proves (among other things) what is known as "Thurston's Holonomy Theorem" for complex-projective structures on surfaces, i.e. that the holonomy map is a local diffeomorphism from the space of structures to the character variety. (This result is not quite true in general.)

2. The first proofs of the existence of a developing map and associated holonomy representation appear to be in Kuiper's papers

N. H. Kuiper, On Conformally-Flat Spaces in the Large, Annals of Mathematics, Vol. 50, No. 4 (Oct., 1949), pp. 916-924

(here he proves the existence of a developing map, which he calls "the development")

and

N. H. Kuiper, On Compact Conformally Euclidean Spaces of Dimension > 2. Annals of Mathematics, Vol. 52, No. 2 (Sep., 1950), pp. 478-490.

(here he proves the existence of the holonomy homomorphism and some other important results which were rediscovered and, sometimes incorrectly, reproved, by others in the next 40 or so years).

Kuiper's treatment was limited to flat conformal structures ($X=S^n$, $G\cong PO(n,1)$) but his proofs were general.

3. Another pre-Thurston reference is the book Kobayashi, Nomizu "Transformation Groups in Differential Geometry", 1972, where they discussed general pseudogrooup structures, their connections and curvature. The most relevant part of the paper is section I.8 where they define what amounts to a general $(𝑋,𝐺)$-structure in Thurston's sense, but prove very little in this generality. Their primary focus is the development of differential-geometric tools such as connections and curvature, and analysis of automorphism groups.

Note that their notion of a $G$-structure is even more general than Thurston's: Thurston's geometric structures are "flat structures" in their terminology.

4. In addition, there are papers (especially, Bensecri) on flat affine and flat projective structures from 1950s and 1960s. I will add more references when I have time. Two important open problems in the area (Auslander and Markus conjectures) go back to this time period.

Answer 2

Dennis Sullivan and W. P. Thurston. Manifolds with canonical coordinate charts:some examples. L’Enseignment Math ́ematique, 29(15-25), 1983

This is a note with Thurston from a collaboration during the fall semester of 1976 visiting Princeton from the IHES in France whose director was Nico Kuiper and where I was a professor till 1994. It was written first as a preprint of the Princeton Math Dept appearing january 1977 at the point I returned to the IHES.

In it we gave examples of G structures with interesting developing mappings. Some of these answered questions of Kuiper showing extensions of Gunning's Theorem (in his monograph) were not possible. Kuiper smoothed the text and eventually caused it to be published as indicated above.

In it we also review Ehresmann's viewpoint in terms of germs , sheaves and foliations which gives an elegant and transparent view of the geometry.

We also ask a general question about three manifolds during that fall that if true implied the poincare conjecture by a very well known argument.

My take on the history is that all of this was quite well known and pursued all through the 70's ( the foliation decade) partly because of the foliation interpretation made clear already in Ehresmann.

Thurston should be singled out though because he understood it better and made it alive through examples , as in the paper, especially the real projective examples there....pure Thurston....

With a former phd student we are finishing a paper now where we answer the
question from the '77 preprint. While searching exact references to Ehresmann your discussion came up.

Dennis Sullivan.

Answer 3

The notion of (G,X)-structure is treated in Ehresmann's "Sur les espaces localement homogènes" (On locally homogeneous spaces), L'Enseignement Mathématique 35 ( 1936), 317-333. It is the text of a conference given by him in October 1935. You will also find this paper online in his complete works: https://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/C.E_Works.htm Part I (1 volume, 630 pages): Topologie Algébrique et Géométrie Différentielle, from page 87 on. His notion of locally homogeneous space is actually more general than the modern one. But in paragraphs 5 to 9, he treats of the modern one; in these paragraphs, he defines the developing map in the universal cover; the use of the holonomy morphism is clear; in the compact case, he notices that the developing map is not always a diffeomorphism onto the simply-connected model X; then he considers the cases of locally affine and locally projective spaces, both real and complex; and under a hypothesis of convexity, he makes a complete classification in dimension 2. All this in 15 pages and in 1935...

One should be aware that the text is not easy to read, even if you know well about (G,X)-structures and if French is your native language! Everything is said in a litterary style, the paper hardly contains any formula at all, and his mathematical vocabulary is archaic, still influenced by Elie Cartan... If you are interested, I would be glad to be of some help to decypher.

Added - Ehresmann's bibliography is much interesting too. Actually, for example, Whitehead's 1932 paper "Locally Homogeneous Spaces in Differential Geometry" deserves to be visited.

«Nos enfants croiront avoir de l'imagination, ils n'auront que des réminiscences.» (Henri de St. Simon).

Answer 4

" if Ehresmann had the idea to compare on one hand (G,X)-structures and on the other hand manifolds or geometrical objects to be modeled on them"?

After a new look at the text, I believe that the answer to your question is "yes" to a large extent; this requires some precision. Ehresmann is interested in the "relations between Group Theory and Topology: between the global properties and the infinitesimal properties of a space" (p. 87). Of course, this paper was published in the same year as Whitney's definition of smooth manifolds by atlases (Whitney, Hassler (1936). "Differentiable Manifolds". Annals of Mathematics. Annals of Mathematics. 37 (3): 645–680. doi:10.2307/1968482. JSTOR 196848) Ehresmann has no such definition, but the objects he treats of are on the one hand topological manifolds, called by him "manifolds" (first lines of p. 88); and on the other hand (local) Lie groups defined in the (real) analytic category (p. 88). At the end of p. 89 and beginning of p. 90 he defines a homogeneous space (resp. a locally homogeneous space) as a manifold together with an action (resp. a local action) of a Lie group with some transitivity condition; and then he remarks that the manifold then becomes (real) analytic in the sense that it admits local coordinates such that the coordinate changes are analytic (p. 90). At the beginning of p. 91 he defines his program as "find all the locally homogeneous spaces which are locally isomorphic to a given (locally) homogeneous space". I'm not a specialist in the history of these notions in that period (being just a mathematical grandson of Ehresmann); probably a study of the papers that he cites in the bibliography would allow one to understand which part of these ideas was actually original, and which part was already known or at least "in the air".

Answer 5

how do i guess n. kuiper learned about the developing mapping of a G structure.? kuiper and everyone before the war learned and used projective geometry. also in holland brouwers tradition was felt and he and other differential geometers thought geometrically in terms of developable surfaces, and the developing idea of curvature. I think ehreshmann was formalizing this intuition. kuipers papers 49-50 were also formalizing this in a more geometric way....ereshmanns lectures were very formal and abstract ( reported by tony phillips who attended them in early 60's as a fulbright scholar in france.) covering spaces and these ideas are taught everywhere but they are non trivial. one sees that in the work of thurstion who found important new phenomena of holonomy of developments in very classical examples like gluing two ideal noneuclidean triangle together. dennis sullivan