Unveiling topological modes on curved surfaces

In this paper, we investigate topological modes of different physical systems defined on arbitrary two-dimensional spatial curved surfaces. We consider the shallow water equations, inhomogeneous Maxwell's equations, and the Jackiw-Rebbi model and show how the topological protection mechanism responds to the presence of curvature in different situations. We show the existence of a line gap in the considered models and study the condition on the curve, which can host topological modes.