This result states eells-sampson existence theorem for harmonic maps: given (M,g) be a closed smooth Riemannian manifold and let (N,h) be a compact smooth Riemannian manifold whose sectional curvature satisfies sec h 0. For every smooth map u 0: M.... It is useful in harmonic map theory and elliptic regularity, where variational identities, curvature estimates, and compactness arguments control geometric objects.