The second variation formula for exponentially harmonic maps

Abstract

We derive the formula in the title and deduce some consequences. For example we show that the identity map from any compact manifold to itself is always stable as an exponentially harmonic map. This is in sharp contrast to the harmonic or p -harmonic cases where many such identity maps are unstable. We also prove that an isometric and totally geodesic immersion of S m into S n is an unstable exponentially harmonic map if m ≠ n and is a stable exponentially harmonic map if m = n .