Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric

The symplectic Stiefel manifold, denoted by Sp(2p,2n), is the set of linear symplectic maps between the standard symplectic spaces ℝ2p and ℝ2n. When p=n, it reduces to the well-known set of 2n× 2n symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space ℝ2n× 2p. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel manifold. We obtain the expression of the Riemannian gradient with respect to the Euclidean metric, which then used in optimization algorithms. Numerical experiments on the nearest symplectic matrix problem and the symplectic eigenvalue problem illustrate the effectiveness of Euclidean-based algorithms.