In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows: $$E(f):=\frac{1}{2}\int_M\|df\|^2dvol_g\qquad f:(M,g)\to(N,h).$$

In many Books such as *Calculus of Variations and Harmonic Maps-Hajime Urakawa*, used the covariant derivation as a map $\Gamma(f^{-1}TN)\to \Gamma(f^{-1}TN)$ and in another Reference such as *Graduate thesise of A.A. joshi* the covariant derivation have been considered as a map $\Gamma(TM^*\otimes f^{-1}TN)\to \Gamma(TM^∗ \otimes TM^∗\otimes f^{-1}TN)$.

why this two consideration is equivalent?

**Update:**

Another application of harmonic maps is the study of the topology of domain manifold. In this purpose how can I get the some topological properties of the domain manifold?

Thanks.

exactlythe passages that define the covariant derivation as a map $\Gamma(f^{-1}TN)$ to itself? Is that supposed the be the action of $\nabla_X$ for some vector field $X$? $\endgroup$ – Willie Wong May 23 '16 at 18:47