Symplectisation as a functor between appropriate categories

Question

Let $(M,\xi)$ be a transversally orientable contact manifold, that is, there exists a form $\alpha \in \Omega^1(M)$ such that $\xi = \ker \alpha$. Then we can associate to $(M,\xi)$ its symplectisation $(\mathbb{R} \times M,d(e^t\alpha))$, a symplectic manifold. I wondered, if there is a categorical setting for this process. I mean, naively, we could consider symplectisation as a map on objects $$S \colon \mathsf{TOCont} \to \mathsf{Symp}$$ where $\mathsf{TOCont}$ denotes the category of transversally orientable contact manifolds as objects and maps $F \in C^\infty(M,\widetilde{M})$ such that there exists a nowhere vanishing function $f \in C^\infty(M)$ with $F^* \widetilde{\alpha} = f\alpha$ as morphisms $F \colon (M,\xi = \ker \alpha) \to (\widetilde{M},\widetilde{\xi} = \ker \widetilde{\alpha})$. Likewise, $\mathsf{Symp}$ denotes the category with objects symplectic manifolds and morphisms $F \colon (M,\omega) \to (\widetilde{M},\widetilde{\omega})$ such that $F \in C^\infty(M,\widetilde{M})$ with $F^*\widetilde{\omega} = \omega$.

Now the problem I am facing is the following: I would define $S$ on morphisms $$S(F) \colon (\mathbb{R} \times M,d(e^t\alpha)) \to (\mathbb{R} \times \widetilde{M},d(e^t\widetilde{\alpha}))$$ by $$S(F) := \operatorname{id}_{\mathbb{R}} \times F.$$ But then, if $F^* \widetilde{\alpha} = f\alpha$, we compute $$S(F)^* d(e^t\widetilde{\alpha}) = d(e^tf\alpha),$$ that is, $S(F)$ is not a morphism in $\mathsf{Symp}$. If $f > 0$, we could use the definition $$S(F)(t,x) := (t - \log(f(x)),F(x))$$ and things would work out fine. However, this would impose a restriction on orientation.

I think everything boils down to the fact that if $(M,\xi = \ker \alpha)$ is a contact manifold, then also $\xi = \ker f\alpha$ for every nowhere vanishing smooth function $f$. But I guess the symplectisations are not symplectomorphic in general in this case, that is, a single t.o. contact manifolds admits different non-symplectomorphic symplectisations. Is that right? Do you have any idea how to turn symplectisation into a functor between appropriate categories?

Answer 1

first of all I think your $S(F)$ can be modified into \begin{align*} S(F)(t,x)=(t-\log(|f(x)|), F(x)) \end{align*} since $f$ is non-vanishing, this is always smooth. Nevertheless, there is a more conceptual way to see the symplectization: the symplectization $S$ is a functor from contact manifolds into homogeneous symplectic manifolds. The latter is the category of pairs $(P,\omega)$ consisting of a $\mathbb{R}^\times$-principal bundle $P$ and a symplectic structure $\omega\in \Omega^2(P)$, such that \begin{align*} h_r^*\omega=r\omega \end{align*} for the principal action $h\colon \mathbb{R}^\times\times P\to P$. The morphisms are equivariant symplectomorphisms. This functor is even an equivalence of categories and does not work just for co-orientable contact structures. Everything what I said is (more or less) done in Remarks on Contact and Jacobi Geometry (Bruce, Grabowska, Grabowski 2015).

HD

Answer 2

The right notion of the category $\mathsf{TOCont}$ is not transversally orientable, but rather transversally oriented, that is, we choose an orientation of the hyperplane distribution and the $1$-form compatible with this orientation. Then everything works well.