Is it possible to define contact manifolds as manifolds with a G-structure?

Question

Many geometries (Riemannian, symplectic, complex, Kähler, Calabi-Yau) can be defined as categories of G-structures on manifolds with the first integrability condition (zeroing of torsion of G-structure), and structure-preserving morphisms are defined naturally if $G \to GL_n$ is monomorphism. Is it possible to define contact geometry in such terms?

Answer 1

A contact structure on $M^{2n+1}$ defines a $G$-structure (actually, it defines more than one, but there is a 'minimal' $G$-structure that is preserved by all contact transformations, and that is the one that people usually consider). Conversely, a $G$-structure on $M^{2n+1}$ comes from a contact structure provided that its intrinsic torsion satisfies the appropriate identities.

For example, a contact structure on $M$ is defined by a $2n$-plane field $D\subset TM$ (satisfying some nondegeneracy property). So if $G_0\subset\mathrm{GL}(2n{+}1,\mathbb{R})$ is the linear subgroup that preserves the subspace $\mathbb{R}^{2n}\subset\mathbb{R}^{2n{+}1}$, then a contact structure defines a $G_0$-structure on $M$ and, conversely, that $G_0$-structure contains the information needed to recover $D$. In that sense, the $G_0$-structure and the $2n$-plane field $D$ are equivalent information.

However, you will note that not every $G_0$-structure on $M$ defines a contact structure, because the corresponding hyperplane field defined by $G_0$ might be integrable, say.

Meanwhile, via the Lie bracket, a contact structure $D\subset TM$ defines a linear bundle map $B:\Lambda^2(D)\to TM/D$ that is non-degenerate. Correspondingly, there is a subgroup $G_1\subset G_0$ that is the subgroup that preserves a non-degenerate mapping $b:\Lambda^2(\mathbb{R}^{2n})\to\mathbb{R}^{2n+1}/\mathbb{R}^{2n}$. Hence, a contact structure $D$ defines a canonical $G_1$-structure on $M$ in the obvious way. Conversely, a $G_1$-structure on $M$ whose associated induced mapping $B:\Lambda^2(D)\to TM/D$ coincides with the one induced by the Lie bracket on $M$ contains all the information in the contact structure and nothing else; automorphisms of the $G_1$-structure are exactly the contact transformations. (The condition that the $B$ associated to the $G_1$-structure coincide with the one induced by the Lie bracket is the analog of 'integrability' in this case. However, this is not the same as the $G_1$-structure being 'torsion-free'.)

Moreover, there is no proper subgroup $G_2\subset G_1$ for which contact transformations of $M$ preserve some $G_2$-structure on $M$. In this sense, the $G_1$-structure defined above is the 'minimal' $G$-structure defined by a contact structure.

For $n$ odd, the only proper subgroup $G\subset\mathrm{GL}(2n{+}1,\mathbb{R})$ such that the contact transformations of $\mathbb{R}^{2n{+}1}$ preserve a torsion-free $G$-structure is the index $2$ subgroup $G=\mathrm{GL}^+(2n{+}1,\mathbb{R})$ while, for $n$ even, there is no proper subgroup with this property.