Harmonic map

A (smooth) map ϕ {displaystyle phi } :M→N between Riemannian manifolds M and N is called harmonic if it is a critical point of the Dirichlet energy functional A (smooth) map ϕ {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 ϕ {displaystyle phi } :M→N prescribes how one 'applies' the rubber onto the marble: E( ϕ {displaystyle phi } ) then represents the total amount of elastic potential energy resulting from tension in the rubber. In these terms, ϕ {displaystyle phi } 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.