  This paper concerns the motion of a relativistic string in a curved
space-time. As a general framework, we first analyze relativistic string
equations, i.e., the basic equations for the motion of a one-dimensional
extended object in a curved enveloping space-time $(\mathscr{N}, \tilde g)$,
which is a general Lorentzian manifold, and then investigate some interesting
properties enjoyed by these equations. Based on this, under suitable
assumptions we prove the global existence of smooth solutions of the Cauchy
problem for relativistic string equations in the curved space-time
$(\mathscr{N}, \tilde g)$. In particular, we consider the motion of a
relativistic string in the Ori's space-time, and give a sufficient and
necessary condition guaranteeing the global existence of smooth solutions of
the Cauchy problem for relativistic string equations in the Ori's space-time.
