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I have often heard that Lie groups classify geometry. For example that $O(n)$ is about real manifolds, $U$ is about almost complex manifolds, $SO(n)$ about orientable real manifolds and so on.

I have also heard that manifolds can be defined very generally by patching local pieces via a pseduogroup of morphisms.

My questions are

1) Does this pseduogroup relate to the Lie group?

2) How do Lie groups classify geometry?

3) Is there a geometry for every Lie group or only some of them?

and maybe even

4) Does this classification seem work for algebraic (i.e. non-differential) geometry?

I am wondering whether this Erlangen program classification is actually formalized, or only serves as a slogan or principle.

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    $\begingroup$ Defining structures on mnflds using pseudogroups of morphisms is a very general technique, and the pseudogroups of morphisms don't all need to come from Lie groups (eg a smooth manifold uses the pseudogroup of smooth diffeomorphisms between open subsets of Euclidean spaces, a foliated manifold uses the pseudogroup of diffeomorphisms between open subsets of Euclidean spaces that can be expressed in the form $\phi(x,y)=(\phi_1(x,y),\phi_2(y))$, etc.). Probably the best way to get introduced to this point of view is to read Chapter 3 of Thurston's book "Three-Dimensional Geometry and Topology". $\endgroup$
    – Andy Putman
    Commented Oct 15, 2010 at 5:14
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    $\begingroup$ Taking up Andy's point, I think your question mixes three distinct ideas. (1) Pseudogroups. (2) Homogeneous spaces and their symmetry groups, Erlangen program, geometric structures on manifolds in Thurston's sense. (3) The structure group of the tangent bundle. The bundle of tangent frames on a smooth $n$-manifold is a principal $GL(n,\mathbb{R})$-bundle, but e.g. a choice of orientation and Riemannian metric gives a reduction of the structure group to $SO(n)$ (namely, the bundle of orthonormal oriented frames). $\endgroup$
    – Tim Perutz
    Commented Oct 15, 2010 at 13:42

3 Answers 3

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R.W.Sharpe, Differential Geometry - Cartan's Generalization of Klein's Erlangen Program

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The examples you give of Lie groups associated to geometric objects are the structure groups of the tangent bundles of manifolds. The tangent bundle is locally trivial, so you have a covering of your manifold and isomorphisms over the $U_i$ (i.e. commuting with the projections to the $U_i$) $\varphi_i:TM|_U \rightarrow \mathbb{R}^n \times U_i$. On the intersections this gives you isomorphisms over $U_i \cap U_j$ $\varphi_i \circ \varphi_j^{-1}: \mathbb{R}^n \times U_i \cap U_j \leftarrow TM|_{U_i \cap U_j} \rightarrow\mathbb{R}^n \times U_i \cap U_j$, i.e. isomorphisms $g_{ij}: \mathbb{R}^n \rightarrow \mathbb{R}^n$. These can be seen as gluing data of your bundle. The "structure group" of your bundle tells you what gluing data you allow: You could require the $g_{ij}$ to be in $GL(n), O(n), SO(n), U(n), Sp(n)$ and so on. The smaller this group of gluing data is, the more special is your bundle, e.g. if you can glue your tangent bundle only using maps from $SO(n)$, then your manifold will be orientable. The buzz word to google for is "reduction of structure group".

I am not sure what you have heard about pseudogroups but there is one consisting of the $\varphi_i$ - open patches homeomorphic to $\mathbb{R}^n$ are such that the tangent bundle is trivial over them, so if you know how to glue your manifold from such patches, you get the relation to the Lie group as above.

Of course you can embed every Lie group into some $GL_n$ and require the $g_{ij}$ to lie in that subgroup, but that seems somewhat arbitrary.

Finally some references on the Erlangen program are Sharpe's book and this, where you can look for more ingredients of the big picture.

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I will just quote from the Wikipedia article on holonomy:

In 1952 Georges de Rham proved the de Rham decomposition theorem, a principle for splitting a Riemannian manifold into a Cartesian product of Riemannian manifolds by splitting the tangent bundle into irreducible spaces under the action of the local holonomy groups. Later, in 1953, M. Berger classified the possible irreducible holonomies.

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    $\begingroup$ It seems to me that de Rham and Berger's work is about holonomy groups, which is related to but not the same as the idea of a Lie group defining a geometry (which I interpret as defining the symmetries of a space). In particular, any Lie group is the group of symmetries of a space (namely itself), but Berger's theorem shows that only some Lie groups arise as a holonomy group. Another point is that Berger's theorem applies only to Riemannian manifolds and compact Lie groups, so it misses a lot of important and interesting geometric structures (such as the ones in Thurston's classification). $\endgroup$
    – Deane Yang
    Commented Oct 15, 2010 at 13:19
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    $\begingroup$ Whereas indeed, as Deane points out, this Berger theorem is about riemannian manifolds whose holonomy groups are therefore compact, his results can be extended. First, to the pseudo-riemannian situation: lorentzian holonomy groups (the indecomposable ones necessarily non-compact) have been classified recently and there is a partial classification for index 2. But Berger also worked on the classification of affine torsion-free connections which need not preserve a metric. This classification has also been completed by Merkulov and Schwachhöfer. $\endgroup$
    – José Figueroa-O'Farrill
    Commented Oct 15, 2010 at 15:20
  • $\begingroup$ The Berger classification is for irreducible nonsymmetric spaces. Symmetric spaces yield more generic holonomy groups. $\endgroup$
    – Steve Huntsman
    Commented Oct 15, 2010 at 19:34

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