  The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set
of linear symplectic maps between the standard symplectic spaces
$\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the
well-known set of $2n\times 2n$ symplectic matrices. We study the Riemannian
geometry of this manifold viewed as a Riemannian submanifold of the Euclidean
space $\mathbb{R}^{2n\times 2p}$. 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.
