# How many geometric structures on manifolds are there?

Let $M$ be a smooth manifold, of dimension $n$. I know that there are many types of geometric structures on $M$. A large number of them are captured by the notion of a $G$-structure, which is a reduction of $M$'s frame bundle $F(M)$ to a Lie subgroup $G$ of $GL(n,\mathbb{R})$, the structure group of $F(M)$.

Kobayashi, in his text "Transformation Groups in Differential Geometry," is kind enough to provide a list of some $G$-structures. His list includes structures with which most people are acquainted, such as $O(n)$-structures (Riemannian metrics), $O(n)\times \mathbb{R}_{>0}$-structures (conformal structures), and $SL(n,\mathbb{R})$-structures (volume forms). Several important ones are missing, such as almost CR-structures, which in the case $n=2k+1$ are reductions of $F(M)$ to the subgroup $G_0$ of $GL(2k+1,\mathbb{R})$ with elements $$\begin{bmatrix} A & x\\ 0 & y \end{bmatrix},$$ where $A\in GL(k,\mathbb{C})$, $x\in \mathbb{R}^{2k}$, and $y\in \mathbb{R}^\times$.

Questions: Is there a more comprehensive list of $G$-structures somewhere? Are there "exotic" $G$-structures that have appeared in the literature?

• According to your definition, a list of $G$-structures is a list of subgroups of $GL(n)$. (In fact, one should also include at least various coverings of these subgroups, to accommodate for all sorts of $\mathrm{Spin}$-structures.) IMHO, a very exotic but popular structure is the $G_2$-structure on $7$-manifolds. – Alex Degtyarev Jan 29 '15 at 21:57
• I agree. However I don't imagine that every possible structure group reduction has been studied, nor do I imagine that they all are worthy of study. If $M$ is a $4$-manifold, then I can reduce its frame bundle to, say, $SO(3)\times 1$, or $SL(2,\mathbb{R})\times SO(2)$. Has anyone looked at these $G$-structures? – user41626 Jan 29 '15 at 22:12
• Well, some reductions of $SO(2n+1)$ to $SO(2n)$ are called contact structures and are indeed studied a lot. Although I do agree that not everything is studied/worth studying, but this does not seem to be relevant to your question title :) – Alex Degtyarev Jan 29 '15 at 23:10
• Also, I think that any reduction of $SO$ to a subgroup is basically a question about the holonomy group of the metric, and I think that such things are studied. – Alex Degtyarev Jan 29 '15 at 23:12
• You should probably make explicit what makes your question different from «what are the coverings of linear Lie groups?» which is what the comments are converging to. – Mariano Suárez-Álvarez Jan 29 '15 at 23:25

Any map whatsoever from a space $X$ to $BO(n)$ gives a notion of $X$-structure for $n$-manifolds given by a choice of lift (up to homotopy) of the classifying map $M \to BO(n)$ of the tangent bundle of such a manifold $M$ to $X$ (together with a choice of homotopy).

When $X$ is itself $BG$ for some Lie group $G$ and the map $BG \to BO(n)$ is induced by a map $G \to O(n)$ of Lie groups this reproduces a more classical flavor of structure, but this more general setting includes, for example, the case of string structures, where $X = BString(n)$ does not arise in the more classical way. This very general notion of structure on a manifold arises, for example, in Lurie's discussion of the cobordism hypothesis.

So one terrible answer to your question is: there is a notion of (tangential) structure on an $n$-manifold for every isomorphism class of $n$-dimensional vector bundle on some space.

• This is fantastic and is what I was hoping my question would produce. – user41626 Jan 30 '15 at 1:36

I don't have a comprehensive list and I'll mention some $G$ structures that seemingly were ignored.

A presimplectic structure on an even dimensional manifold $M$ is a $G$ structure, with $G=Sp(2n,\mathbb{R}).$

An almost complex structure on a vector bundle of rank $2n$ is a $G$ structure with $G= \bigg\{ \left( \begin{array}{ll} A & B \\ -B & A \end{array} \right) \,\ |\,\ A,B \in GL(n,\mathbb{R}) \bigg\}$

More exotic ones:

A nonvanising section $\sigma$ on a vector bundle $E \rightarrow M$ of rank $n$ can be viewd as a $G$ structure with $G= \bigg\{ \left( \begin{array}{ll} 1 & A \\ 0 & B \end{array} \right) \,\ |\,\ A \in M_{n-1,1}(\mathbb{R}) \,\ ,B \in M_{n-1}(\mathbb{R}) \bigg\},$

A parralelization of a plane bundle(for simplicity!) can be viewed as a $G$ structure with

$G= \bigg\{ \left( \begin{array}{ll} a & b \\ b & a \end{array} \right) \,\ |\,\ a^2-b^2 \neq 0 \bigg\}$

Moreover, since the structure group $G$ acts by conjugations on $GL(n,\mathbb{R}))$, one can calculate the invariant polynomial functions in the entries of a matrix and see if one can obtain cohomology invariants for the $G$ structure. A concrete example is the pfaffian of a matrix asociated to an $SO(n,\mathbb{R})$ stucture that yields the Euler form of a connection.