This is a continuation of our previous paper . In the class of hyperbolic manifolds in the sense of S. Kobayashi , we obtained in  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  that two Siegel domains of the first kind are biholomorphically equivalent if and only if they are linearly equivalent.
"Diffeomorphisms between Siegel domains of the first kind preserving the holomorphic automorphism groups and applications." Kodai Math. J. 36 (2) 299 - 312, June 2013. https://doi.org/10.2996/kmj/1372337520