Dispersal and migration can be important drivers of species distributions. Because the paths followed by individuals of many species are curvilinear, spatial statistical models based on rectilinear coordinates systems would fail to predict population connectivity or the ecological consequences of migration or species invasions. I propose that we view migration/dispersal as if organisms were moving along curvilinear geometrical objects called smooth manifolds. In that view, the curvilinear pathways become the ‘shortest realised paths’ arising from the necessity to minimise mortality risks and energy costs. One can then define curvilinear coordinate systems on such manifolds. I describe a procedure to incorporate manifolds and define appropriate coordinate systems, with focus on trajectories (1D manifolds), as parts of mechanistic ecological models. I show how a statistical method, known as ‘manifold learning’, enables one to define the manifold and the appropriate coordinate systems needed to calculate population connectivity or study the effects of migrations (e.g. in aquatic invertebrates, fish, insects and birds). This approach may help in the design of networks of protected areas, in studying the consequences of invasion, range expansions, or transfer of parasites/diseases. Overall, a geometrical view to animal movement gives a novel perspective to the understanding of the ecological role of dispersal and migration.