Morphisms between Grassmannians, II

Denote by \({\mathbb {G}}(k,n)\) $\mathbb{G}(k,n)$ the Grassmannian of linear subspaces of dimension k in \({\mathbb {P}}^n\) ${\mathbb{P}}^n$ . We show that if \(\varphi :{\mathbb {G}}(l,n) \rightarrow {\mathbb {G}}(k,n)\) $\phi :\mathbb{G}(l,n)\to \mathbb{G}(k,n)$ is a nonconstant morphism and \(l \not =0,n-1\) $l\ne 0,n-1$ , then \(l=k\) $l=k$ or \(l=n-k-1\) $l=n-k-1$ and \(\varphi \) $\phi$ is an isomorphism.