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