Spaces of Curves with Constrained Curvature on Hyperbolic Surfaces

Author(s)

Saldanha, Nicolau C. ^[1] ; Zuhlke, Pedro ^[2]

Affiliation

^[1] Pontificia Univ Catolica Rio de Janeiro PUC RIO, Dept Matemat, Rua Marques Sao Vicente 225, BR-22453900 Gavea, RJ - Brazil

^[2] Univ Fed Santa Maria UFSM, Dept Matemat, Ave Roraima 1000, BR-97105900 Santa Maria, RS - Brazil

• Document type: Journal article
• Source: Indiana University Mathematics Journal; v. 69, n. 4, p. 1403-1443, 2020.
• Web of Science Citations: 0

Abstract

Let S be a hyperbolic surface. We investigate the topology of the space of all curves on S which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval (kappa(1), kappa(2)). Such a space falls into one of four qualitatively distinct classes, according to whether (kappa(1), kappa(2)) contains, overlaps, is disjoint from, or contained in the interval {[} -1, 1]. Its homotopy type is computed in the latter two cases. We also study the behavior of these spaces under covering maps when S is arbitrary (not necessarily hyperbolic nor orientable), and show that if S is compact, then they are always nonempty. (AU)

• FAPESP's process: 14/22556-3 - Geometry of isoparametric submanifolds of Hilbert space and topology of spaces of curves with bounded curvature on surfaces.
• Grantee: Pedro Paiva Zühlke D'Oliveira
• Support Opportunities: Scholarships in Brazil - Post-Doctoral