Framed Curves, Ribbons, and Parallel Transport on the Sphere

Abstract

We consider curves γ:[0,1]→R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma : [0, 1]\\rightarrow {\\mathbb {R}}^3$$\\end{document} endowed with an adapted orthonormal frame r:[0,1]→SO(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r: [0, 1]\\rightarrow SO(3)$$\\end{document}. We wish to deform such framed curves (γ,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma , r)$$\\end{document} while preserving two contraints: a local constraint prescribing one of its ‘curvatures’ (i.e., off-diagonal elements of r′rT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r'r^T$$\\end{document}), and a global constraint prescribing the initial and terminal values of γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} and r. We proceed in two stages. First we deform the frame r in a way that is naturally compatible with the constraints on r, by interpreting the local constraint in terms of parallel transport on the sphere. This provides a link to the differential geometry of surfaces. The deformation of the base curve γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications: first, we characterize the boundary conditions on (γ,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma , r)$$\\end{document} that are accessible without violating the local constraint; then, we provide a short proof of a smooth approximation result for framed curves satisfying both the differential and the global constraints. Finally, we also apply these ideas to elastic ribbons with nonzero width.