Geometric realizations of homotopic paths over curved surfaces


Peters J. F., Alfano R., Smith P., Tozzi A., VERGİLİ T.

FILOMAT, vol.38, no.3, pp.793-802, 2024 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 3
  • Publication Date: 2024
  • Doi Number: 10.2298/fil2403793p
  • Journal Name: FILOMAT
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.793-802
  • Karadeniz Technical University Affiliated: Yes

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 simplyconnected 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.