Rigidity of holomorphic maps between Shilov boundaries of type-I bounded symmetric domains

We introduce the concept of orthogonal structure on complex Grassmannians. Based on this structure, we define the notion of orthogonal mappings. This class of maps generalizes holomorphic maps between the Shilov boundaries of type-I bounded symmetric domains. By analyzing the geometric properties of these orthogonal mappings, we obtain rigidity results for such mappings. As an application, we establish a rigidity result for the holomorphic maps from the Shilov boundary of rank $1$ type I bounded symmetric domain $Ω_{1,s}$ (i.e. unit spheres) to the Shilov boundary of a higher-rank type-I domain $ Ω_{r',s'}$, where $s\ge 2$ and $2\le r'\le s'$. More specifically, we show that: (1) such maps are constant when $s'-r' < s-1$; (2) for $s-1 \le s'-r' < 2s-2$, they reduce to the standard linear embeddings after normalization by applying automorphisms on both sides. Our results are optimal and generalize the well-known optimal bounds for proper mappings between rank $1$ cases and CR maps between Shilov boundaries of higher-rank type I bounded symmetric domains.