Local Isometric Imbeddings of Symplectic Groups

We show that the canonical isometric imbedding of the symplectic group Sp(n) into R4n2 gives the least-dimensional isometric imbedding into the Euclidean space, even in the local standpoint. We prove this result by calculating the quantity pG determined by the curvature of Sp(n), which serves as an obstruction to the existence of local isometric imbeddings. We also exhibit the estimates on the value pG for the remaining compact classical simple Lie groups, and improve the previous results on the codimension of local isometric imbeddings of these groups.