Rigidity of the canonical isometric imbedding of the Hermitian symmetric space $Sp(n)/U(n)$

Abstract

In this paper we discuss the rigidity of the canonical isometric imbedding $\pmb{f}_0$ of the Hermitian symmetric space $Sp(n)/U(n)$ into the Lie algebra $\mathfrak{sp}(n)$. We will show that if $n \ge 2$, then $\pmb{f}_0$ is strongly rigid, i.e., for any isometric immersion $\pmb{f}_1$ of a connected open set $U$ of $Sp(n)/U(n)$ into $\mathfrak{sp}(n)$ there is a euclidean transformation $a$ of $\mathfrak{sp}(n)$ satisfying $\pmb{f}_1=a\pmb{f}_0$ on $U$.