  Vectors fields defined on surfaces constitute relevant and useful
representations but are rarely used. One reason might be that comparing vector
fields across two surfaces of the same genus is not trivial: it requires to
transport the vector fields from the original surfaces onto a common domain. In
this paper, we propose a framework to achieve this task by mapping the vector
fields onto a common space, using some notions of differential geometry. The
proposed framework enables the computation of statistics on vector fields. We
demonstrate its interest in practice with an application on real data with a
quantitative assessment of the reproducibility of curvature directions that
describe the complex geometry of cortical folding patterns. The proposed
framework is general and can be applied to different types of vector fields and
surfaces, allowing for a large number of high potential applications in medical
imaging.
