  On the curved surfaces of living and nonliving materials, planar excitable
waves frequently exhibit directional change and subsequently undergo a
topological change; that is, a series of wave dynamics from fusion,
annihilation to splitting. Theoretical studies have shown that excitable planar
stable waves change their topology significantly depending on the initial
conditions on flat surfaces, whereas the directional-change of the waves occurs
based on the geometry of curved surfaces. However, it is not clear if the
geometry of curved surfaces induces this topological change. In this study, we
first show the curved surface geometry-induced topological changes in a planar
stable wave by numerically solving an excitable reaction-diffusion equation on
a bell-shaped surface. We determined two necessary conditions for inducing
topological change: the characteristic length of the curved surface (i.e.,
height of the bell-shaped structure) should be larger than the width of the
wave and than a threshold independent of the wave width. As for the geometrical
mechanism of the latter, we found that a bifurcation of the globally minimum
geodesics (i.e. minimal paths) on the curved surface leads to the topological
change. These conditions imply that wave topology changes can be predicted on
the basis of curved surfaces, whose structure is larger than the wave width.
