  We have studied spinning superfluid $^4$He nanodroplets at zero temperature
using Density Functional theory. Due to the irrotational character of the
superfluid flow, the shapes of the spinning nanodroplets are very different
from those of a viscous normal fluid drop in steady rotation. We show that when
vortices are nucleated inside the superfluid droplets, their morphology, which
evolves from axisymmetric oblate to triaxial prolate to two-lobed shapes, is in
good agreement with experiments. The presence of vortex arrays confers to the
superfluid droplets the rigid-body behavior of a normal fluid in steady
rotation, and this is the ultimate reason of the surprising good agreement
between recent experiments and the classical models used for their description.
