Harmonic map

A (smooth) map ${\displaystyle \phi }$: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 ${\displaystyle \phi }$:M→N prescribes how one "applies" the rubber onto the marble: E(${\displaystyle \phi }$) then represents the total amount of elastic potential energy resulting 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.

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