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We generalize the notion of calibrated submanifolds to smooth maps and showthat the several examples of smooth maps appearing in the differential geometrybecome the examples of our situation. Moreover, we apply these notion to givethe lower bound to the energy of smooth maps in the given homotopy classbetween Riemannian manifolds, and consider the energy functional which isminimized by the identity maps on the Riemannian manifolds with specialholonomy groups.

Smooth maps minimizing the energy and the calibrated geometry