How many geometric structures on manifolds are there?

Question

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?

Answer 1

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.

Answer 2

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.