Homotopy classes in Sobolev spaces and energy minimizing maps

Let M and N be compact Riemannian manifolds. The energy of a lipschitz map f:M -+N is fM \Df | 2 (where \Df(x)\ = £ | d / / c b i | 2 i£xu...,zm are normal coordinates for M at x). Mappings for which the first variation of energy vanishes are called harmonic. The identity map from M to M is always harmonic, but it may be homotopic to mappings of less energy. For instance, the identity map on S is homotopic to mappings of arbitrarily small energy (namely, conformai maps that pull points from the North Pole toward the South Pole). That suggests the question: For which manifolds M is the identity map homotopic to maps of arbitrarily small energy? In this paper we give the simple answer: Those M such that TTI(M) and ^{M) are both trivial. More generally, we consider energy functionals like $(ƒ) = fM \Df\ p