Schwarz Lemma at the Boundary on the Classical Domain of Type III

Let $\mathcal{R_\mathcal{\uppercase\expandafter{\romannumeral3}}}(n)$ be the classical domain of type $\mathcal{\uppercase\expandafter{\romannumeral3}}$ with $n\geq 2$. This article is devoted to a deep study of the Schwarz lemma on $\mathcal{R_\mathcal{\uppercase\expandafter{\romannumeral3}}}(n)$ via not only exploring the smooth boundary points of $\mathcal{R_\mathcal{\uppercase\expandafter{\romannumeral3}}}(n)$ but also proving the Schwarz lemma at the smooth boundary point for holomorphic self-mappings of $\mathcal{R_\mathcal{\uppercase\expandafter{\romannumeral3}}}(n)$.