action $$\begin{aligned}
    S = -\mu \int_\Sigma d^2 \zeta \ \sqrt{- \gamma }, 
\label{NG} \end{aligned}$$ where $\mu$ is the string tension and $\gamma $ is the determinant of the induced metric on $\Sigma$ given by $$\begin{aligned}
    \gamma_{ab} 
        = g_{\mu\nu}{\partial_a}x^\mu 
            {\partial_b}x^\nu,\end{aligned}$$ where $g_{\mu\nu}$ is the metric of ${\cal M}$.

Let us consider that the metric $g_{\mu\nu}$ of ${\cal M}$ possesses isometries generated by Killing vector fields. If $\Sigma$ is tangent to one of the Killing vector fields of ${\cal M}$, say $\xi$, we call the world surface $\Sigma$ a cohomogeneity-one string associated with the Killing vector $\xi$. The stationary string is one of the example of the cohomogeneity-one string, where the Killing vector is timelike.

The group action of isometry generated by $\xi$ on ${\cal M}$ defines the orbit of $\xi$. The metric on the orbit space is naturally introduced by the projection tensor with respect to $\xi$, $$\begin{aligned}
    h_{\mu\nu}:=g_{\mu\nu}-\xi_\mu\xi_\nu/f, 
\label{hab}
\end{aligned}$$ where $f:=\xi^\mu\xi_\mu$. The action for the cohomogeneity-one string associated with a Killing vector $\xi$ is reduced to the action for a curve ${\cal C}$ in the form[@FSZH; @IshiharaKozaki] $$\begin{aligned}
    S=\int_{{\cal C}} \sqrt{-f h_{\mu\nu}dx^\mu dx^\nu}. 
\label{line_action}
\end{aligned}$$ The metric $h_{\mu\nu}$ has Euclidean signature in the case of timelike Killing vector, $f<0$, 