Harmonic map
This article is about harmonic maps between Riemannian manifolds. For harmonic functions, see harmonic function.
A (smooth) map φ:M→N between Riemannian manifolds M and N is called harmonic if it is a critical point of the Dirichlet energy functional
This functional E will be defined precisely below—one way of understanding it is to imagine that M is made of rubber and N made of marble (their shapes given by their respective metrics), and that the map φ:M→N prescribes how one "applies" the rubber onto the marble: E(φ) then represents the total amount of elastic potential energyresulting from tension in the rubber. In these terms, φ is a harmonic map if the rubber, when "released" but still constrained to stay everywhere in contact with the marble, already finds itself in a position of equilibrium and therefore does not "snap" into a different shape.
Harmonic maps are the 'least expanding' maps in orthogonal directions.
Existence of harmonic maps from a complete Riemannian manifold to a complete Riemannian manifold of non-positive sectional curvature was proved by Eells & Sampson (1964).
Given Riemannian manifolds (M,g), (N,h) and φ as above, the energy density of φ at a point x in M is defined as[edit]Mathematical definition
where the 
is the squared norm of the differential of φ, with respect to the induced metric on the bundle T*M⊗φ−1TN. The total energy of φ is given by integrating the density over M
where dvg denotes the measure on M induced by its metric. This generalizes the classical Dirichlet energy.
The energy density can be written more explicitly as
Using the Einstein summation convention, in local coordinates the right hand side of this equality reads
If M is compact, then φ is called a harmonic map if it is a critical point of the energy functional E. This definition is extended to the case where M is not compact by requiring the restriction of φ to every compact domain to be harmonic, or, more typically, requiring that φ be a critical point of the energy functional in the Sobolev spaceH1,2(M,N).
Equivalently, the map φ is harmonic if it satisfies the Euler-Lagrange equations associated to the functional E. These equations read
where ∇ is the connection on the vector bundle T*M⊗φ−1(TN) induced by the Levi-Civita connections on M and N. The quantity τ(φ) is a section of the bundle φ−1(TN) known as the tension field of φ. In terms of the physical analogy, it corresponds to the direction in which the "rubber" manifold M will tend to move in N in seeking the energy-minimizing configuration.
