This is a continuation of our previous paper 4. In the class of hyperbolic manifolds in the sense of S. Kobayashi 3, we obtained in 4 an intrinsic characterization of bounded symmetric domains in Cn from the viewpoint of the holomorphic automorphism group. In connection with this, we give in this paper a structure theorem on diffeomorphisms between Siegel domains of the first kind that preserve the holomorphic automorphism groups. As an application, we obtain a well-known fact 2 that two Siegel domains of the first kind are biholomorphically equivalent if and only if they are linearly equivalent.
児玉 et al. (Sat,) studied this question.