Equations for the motion of relativistic torus in the Minkowski space R1+n

Abstract

In this paper we derive the equations for the motion of relativistic torus in the Minkowski space R1+n (n⩾3). This kind of equations also describes the three-dimensional timelike extremal submanifolds in the Minkowski space R1+n. We show that these equations can be reduced to a (1+2)-dimensional quasilinear symmetric hyperbolic system and the system possesses some interesting properties such as nonstrict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, strong null condition, etc. We also find and prove an interesting fact that all plane wave solutions to these equations are lightlike extremal submanifolds and vice versa except for a type of special solution.