  When considering flows in biological membranes, they are usually treated as
flat, though more often than not, they are curved surfaces, even extremely
curved, as in the case of the endoplasmic reticulum. Here, we study the
topological effects of curvature on flows in membranes. Focusing on a system of
many point vortical defects, we are able to cast the viscous dynamics of the
defects in terms of a geometric Hamiltonian. In contrast to the planar
situation, the flows generate additional defects of positive index. For the
simpler situation of two vortices, we analytically predict the location of
these stagnation points. At the low curvature limit, the dynamics resemble that
of vortices in an ideal fluid, but considerable deviations occur at high
curvatures. The geometric formulation allows us to construct the
spatio-temporal evolution of streamline topology of the flows resulting from
hydrodynamic interactions between the vortices. The streamlines reveal novel
dynamical bifurcations leading to spontaneous defect-pair creation and fusion.
Further, we find that membrane curvature mediates defect binding and imparts a
global rotation to the many-vortex system, with the individual vortices still
interacting locally.
