Tension field

Summary

The tension field of a smooth map between Riemannian manifolds is a vector field along the map. It is the trace of the covariant derivative of the differential of the map. It generalizes the Laplacian of a function and the acceleration of a curve. Minus the tension field is the gradient of the energy, so a map is harmonic exactly when its tension field vanishes. The harmonic map heat flow moves a map in the direction of its tension field.