Rigidity of the canonical isometric imbedding of the symplectic group $Sp(n)$

Abstract

In this paper, we discuss the rigidity of $Sp(n)$ as a Riemannian submanifold of $M(n,n;\mathbb{H})$. We prove that the inclusion map $\pmb{f}_0$, which is called the canonical isometric imbedding of $Sp(n)$, is rigid in the following strongest sense: Any isometric immersion $\pmb{f}_1$ of a connected open set $U (\subset Sp(n))$ into $\pmb{R}^{4n^2}\,(\cong M(n,n;\mathbb{H}))$ coincides with $\pmb{f}_0$ up to a euclidean transformation of $\pmb{R}^{4n^2}$, i.e., there is a euclidean transformation $a$ of $\pmb{R}^{4n^2}$ satisfying $\pmb{f}_1=a\pmb{f}_0$ on $U$.