Abstract Purpose: This paper introduces discrete and continuous paths over simply-connected surfaces with non-zero curvature as means of comparing and measuring paths between antipodes with either a Feynman path integral
or Wodehouse contour integral, resulting in a number of extensions of the Borsuk Ulam Theorem.
Methods: All paths originate on a Riemannian surface S, which is simply�connected and has non-zero curvature at every point in S. A surface is a simply connected, provided every cross-cut divides the surface into disjoint regions. A cross cut is an arc that runs through the interior of the surface S
without self-intersections and joins one boundary to another.
Results: A fundamental result in this paper is that for any pair of antipodal surface points, a path that begins and ends at the antipodal points can be found. This result is extended to Feynman path integrals on trajectory-of particle curves and to N.M.J. Wodehouse countour integrals for antipodal
vectors on twistor curves. Another fundamental result in this paper is that the Feynman trajectory of a particle is realizable as a Lefschetz arc.