Let M and N be compact Riemannian manifolds and $f : M o N$ be a smooth map. Following J. Eells, f is exponentially harmonic if it represents a critical point of the exponential energy integral $$ E(f) = int_{M} exp(left$mid$ df ight$mid$^2) dM $$</TEX> where $(left df $mid$ ight$mid$^2$</TEX> is the energy density defined as $sum_{i=1}^{m} left$mid$ df(e_i) ight$mid$^2$</TEX>, m = dimM, for orthonormal frame $e_i$ of M. The Euler- Lagrange equation of the exponential energy functional E can be written $$ exp(left$mid$ df ight$mid$^2)( au(f) + df( ablaleft$mid$ df ight$mid$^2)) = 0 $$</TEX> where $ au(f)$ is the tension field along f. Hence, if the energy density is constant, every harmonic map is exponentially harmonic and vice versa.