数学上，在黎曼流形M和N之间的一个（光滑）映射，称为调和映射，如果这个映射是狄利克雷能量泛函

的一个临界点。

试想像M是橡胶做的，N是大理石做的，形状由其度量决定，而映射φ:M→N给出把橡胶「贴附」在大理石上的方式。E(φ)就表示因橡胶的张力产生的弹性位能。用这个比喻，φ称为调和映射，如果把橡胶「松开」，但仍限制要处处与大理石接触时，那么橡胶已经在平衡的位置，所以不会「缩回」到另一个形状。

从完备黎曼流形到非正截面曲率的完备黎曼流形存在调和映射，这个结果是Eells & Sampson (1964)证出。

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 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.