On isomorphic classical diffeomorphism groups. III (Q1894601)

[For part II see the first author, J. Differ. Geom. 28, No. 1, 23-35 (1988; Zbl 0645.53019)]. In this note the main result is the following theorem: Let \((M_i, \alpha_i)\), \(i = 1, 2\), be two paracompact, connected, smooth manifolds of dimension \(2n + 1\) equipped with contact forms \(\alpha_i\). If \(G_{\alpha_1} (M_1) \to G_{\alpha_2} (M_2)\) is a group isomorphism, then there exists a unique diffeomorphism \(w : M_1 \to M_2\) such that for all \(h \in G_{\alpha_1} (M_1)\), \(\phi (h) = whw^{-1}\) and \(w^*\alpha_2 = \lambda \alpha_1\), for some nowhere vanishing function \(\lambda\) (i.e., the contact structures are determined by their automorphism groups).