Smooth maps minimizing the energy and the calibrated geometry

Abstract

We generalize the notion of calibrated submanifolds to smooth maps and show that the several examples of smooth maps appearing in the differential geometry become the examples of our situation. Moreover, we apply these notion to give the lower bound to the energy 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.