Geometric Realizations of Homotopic Paths over Curved Surfaces

Abstract

This paper introduces geometric realizations of homotopic paths over simply-connected surfaces with non-zero curvature as a means of comparing and measuring paths between antipodes with either a Feynman path integral or Woodhouse contour integral, resulting in a number of extensions of the Borsuk Ulam Theorem. All realizations of homotopic paths reside on a Riemannian surface $S$, which is simply-connected and has non-zero curvature at every point in $S$.
A fundamental result in this paper is that for any pair of antipodal surface points, a path can be found that begins and ends at the antipodal points. The realization of homotopic paths as arcs on a Riemannian surface leads to applications in Mathematical Physics in terms of Feynman path integrals on trajectory-of-particle curves and Woodhouse countour integrals for antipodal vectors on twistor curves. Another fundamental result in this paper is that the Feynman trajectory of a particle is a homotopic path geometrically realizable as a Lefschetz arc.