Rigidity of the canonical isometric imbedding of the symplectic group \(Sp(n)\) (Q2467082)

The symplectic group \(\text{ Sp}(n)\) admits a natural embedding \(f_0\) into the real vector space \(M_{n,n}({\mathbb H})\) of \(n\times n\)-matrices over the skew field \({\mathbb H}\) of quaternions, which is an isometry if \(M_{n,n}({\mathbb H})\) is considered as an Euclidean space \({\mathbb R}^{4n^2}\) and \(\text{ Sp}(n)\) is equipped with a two-sided invariant metric. In a previous publication [Geom. Dedicata 71, No. 1, 75--82 (1998; Zbl 0921.53019)] the authors have shown that \(\text{ Sp}(n)\) does not even locally admit an immersion into \({\mathbb R}^{4n^2-1}\). The result of the paper under review says that for any isometric immersion \(f\) of a connected open set \(U\) in \(\text{ Sp}(n)\) into \({\mathbb R}^{4n^2}\) one has \(f=af_0\) on \(U\) where \(a\) is an Euclidean transformation of \({\mathbb R}^{4n^2}\).