Let $(M, D, J)$ be a strictly pseudoconvex hypersurface type CR manifold with $J$ integrable.
Let $D$ be the kernel of a $1$-form $\eta\_0$.
As known the automorphism group is defined to be
$$
CR = \{ \phi \in Diff(M): \phi^\*\eta\_0 = f\_\phi \eta\_0 \text{ with $f\_\phi$ nowhere vanishing and } \phi\_\* J = J \phi\_\* \}.
$$
My question is: does the function $f\_\phi$ have to be always positive?


It looks like one needs this when proving that the cone of "positive" CR fields (in the sense of Boyer&Galicki, i.e. st. $\eta\_0 (\xi)>0$) is $CR$-invariant. See the Sasakian geometry monograph chap 8 for reference.


thank you


David



Group of CR automorphisms
Yes, $f\_\phi$ is always positive, as long as you choose $\eta\_0$ properly. The reason is that you have assumed that the CR-structure is strictly pseudo-convex, which means that there is a choice of $\eta\_0$ (unique up to a positive multiple) such that ${\mathrm{d}}\eta\_0$ restricted to $D$ is a positive $(1,1)$-form (i.e., $\mathrm{d}\eta\_0(v,Jv)>0$ for all nonzero $v\in D$). Since the sign of $\eta\_0$ is determined by the CR-structure, any CR-automorphism must preserve $\eta\_0$ up to a positive multiple.


