If ϕ is a non-constant map of a closed Riemannian manifold M into a Euclidean space with arbitrary dimension, then the tension field τ(ϕ) and the energy E(ϕ) of ϕ satisfy the following optimal inequality:

∫M |τ(ϕ)|2 *1≥ 2 λ1 E(ϕ),

where λ1 is the first nonzero eigenvalue of the Laplacian of M (see [B.-Y. Chen, J.-M. Morvan and T. Nore, Energy, tension and finite type maps. Kodai Math. J. 9 (1986), no. 3, 406–418]). 

My question is that can we generalize this result to more general ambient spaces than Euclidean spaces?