Title: Path-like integrals of lenght on surfaces of constant curvature,
Abstract: We naturally associate a measurable space of paths to a couple of orthogonal
vector fields over a surface and we integrate the length function over it. This
integral is interpreted as a natural continuous generalization of indirect
influences on finite graphs and can be thought as a tool to capture geometric
information of the surface. As a byproduct we calculate volumes in different
examples of spaces of paths.