  We combine symmetry structures of ordinary (parallel directions) and dual
(transversal directions) coordinates to construct the Dirac-Born-Infeld (DBI)
theory. The ordinary coordinates are associated with the Neumann boundary
conditions and the dual coordinates are associated with the Dirichlet boundary
conditions. Gauge fields become scalar fields by exchanging the ordinary and
dual coordinates. A gauge transformation of a generalized metric is governed by
the generalized Lie derivative. The gauge transformation of the massless closed
string theory gives the $C$-bracket, but the gauge transformation of the open
string theory gives the $F$-bracket. The $F$-bracket with the strong
constraints is different from the Courant bracket by an exact one-form. This
exact one-form should come from the one-form gauge field. Based on symmetry
point of view, we deduce a suitable action with a non-zero $H$-flux at the
low-energy level. From an equation of motion of the scalar dilaton, it defines
a generalized scalar curvature. Finally, we construct a double sigma model with
a boundary term and show that this model with constraints is classically
equivalent to the ordinary sigma model.
