Rigidity theorem for harmonic maps from Riemannian manifold to Grassmannian manifold

We first study the Grassmannian manifoldG_n (R^n+p)as a submanifold in Euclidean space ⁿ (R^n+p). Then we give a local expression for each map from Riemannian manifoldM toGⁿ (R^n+p) ⁿ (R^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG_n (R^n+p). As a consequence of this formula we get a rigidity theorem.