A map u:M->N, between two compact Riemannian manifolds, is a harmonic map if it is a critical point for the energy functional integral_M ( left bracketing bar d u right bracketing bar )^2 d μ_M. The norm of the differential left bracketing bar d u right bracketing bar is given by the metric on M and N and d μ_M is the measure on M. Typically, the class of allowable maps lie in a fixed homotopy class of maps.

Bochner identity | calculus of variations | curvature | Euclidean space | Euler-Lagrange differential equation | geodesic | harmonic function | Hodge's theorem | homotopy class | Riemannian manifold | Riemann surface