# Finding the maps inside of the Nambo-Goto action

Let's sps that $$\phi:M \to N$$ is a diffeomorphism where $$M$$ and $$N$$ are two separate manifolds. Let $$f$$ be a function s.t. $$f:M \to R$$ where $$R$$ is the set of all real numbers. The composition of our diffeomorphism mapping and the function $$f$$ defined on $$N$$ follows as $$f [ \phi]:N \to R$$. This can be defined as our pullback on $$f$$ by $$\phi$$ denoted $$\phi^*f$$. Now consider the Nambu-Goto action describing the propagation of a 1-brane/string through the ambient spacetime $$S=-T\int_Ed^2E\sqrt{-\det h_{\mu\nu}}$$ $$=-T \int_Ed^2E\sqrt{(\dot X \cdot X')^2-\dot X^2 X'^2}$$ where $$\dot X$$ is the derivative w.r.t $$\tau$$ and $$X'$$ is the derivative w.r.t $$\sigma$$. How do are the functions $$X$$ found such that we can compute the action for a certain case?

In the Nambu-Goto action the worldsheet metric tensor $$h=X^{\ast}G~\in~ \Gamma\left( {\rm Sym}^2(T^{\ast}E)\right)$$ is the pullback of the target space metric tensor $$G~\in~ \Gamma\left( {\rm Sym}^2(T^{\ast}N)\right)$$ via the string map $$E\stackrel{X}{\longrightarrow} N.$$