  We explore the properties of chiral superfluid thin films coating a curved
surface. Due to the vector nature of the order parameter, a geometric gauge
field emerges and leads to a number of observable effects such as anomalous
vortex-geometric interaction and curvature-induced mass/spin supercurrents. We
apply our theory to several well-known phases of chiral superfluid $\rm ^3 He$
and derive experimentally observable signatures. We further discuss the cases
of flexible geometries where a soft surface can adapt itself to compensate for
the strain from the chiral superfluid. The proposed interplay between geometry
and chiral superfluid order provides a fascinating avenue to control and
manipulate quantum states with strain.
