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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

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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.

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