Smooth Maps Minimizing the Energy and the Calibrated Geometry

We generalize the notion of calibrated submanifolds to smooth maps and show that several kinds of smooth maps appearing in the differential geometry are applicable to our situation. Moreover, we apply this notion to give the lower bound to some energy functionals of smooth maps in the given homotopy class between Riemannian manifolds and consider the energy functional which is minimized by the identity maps on the Riemannian manifolds with special holonomy groups.