  We introduce generalized calibrations that take into account the gauge field
on the D-brane so that calibrated submanifolds minimize the Dirac-Born-Infeld
energy. We establish the calibration bound and show that the calibration form
is closed in a supersymmetric background with non-vanishing NS-NS 3-form H and
dilaton. We show that the calibration conditions are equivalent to the
existence of unbroken supersymmetry on the D-brane. We study the problem of
supersymmetric D-branes in the presence of non-vanishing H also from the
world-sheet approach and find exactly the same conditions. Finally, we show
that our notion of generalized calibrations is equivalent to the calibrations
introduced in the context of generalized Calabi-Yau geometry in
math.DG/0401221.
