The Geometry of $$C^1$$ C 1 Regular Curves in Sphere with Constrained Curvature
- PDF / 421,751 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 60 Downloads / 189 Views
The Geometry of C 1 Regular Curves in Sphere with Constrained Curvature Cong Zhou1 Received: 15 April 2020 / Accepted: 3 September 2020 © Mathematica Josephina, Inc. 2020
Abstract In this article, we study C 1 regular curves in the 2-sphere that start and end at given points with given directions, whose tangent vectors are Lipschitz continuous, and their a.e. existing geodesic curvatures have essentially bounds in an open interval. Especially, we show that a C 1 regular curve is such a curve if and only if the infimum of its lower curvature and the supremum of its upper curvature are constrained in the same interval. Keywords Geometry · Topology · Curves · Curvature Mathematics Subject Classification 53A04 · 53C42
1 Introduction In this article, we study the geometric properties of a C 1 regular curve in the unit 2-sphere S2 with its “geodesic curvature” constrained in an open interval. This work originates in our research on the topology of C r regular curves in S2 , r ≥ 1. Initially in 1956, Smale [12] showed that the space of C r (r ≥ 1) regular closed curves on S2 , has two connected components. Later, Little [4] proved that there are six connected components for the space of C r , r ≥ 2, regular closed curves in S2 with nonvanishing geodesic curvature. After that, Shapiro and Khesin [3] studied the more general case, that is, the space of all smooth regular locally convex curves in S2 which start and end at given points with given directions (not necessarily closed). Further in [5,6] and [7], Saldanha did research on the higher homotopy properties of the space of Shapiro and Khesin. Recently, in 2013, Saldanha and Zühlke [9] extended Little’s result to the space of C r , r ≥ 2, regular closed curves with geodesic curvature constrained in an
B 1
Cong Zhou [email protected] Departamento de Geometria, Instituto de Matemática e Estatística, Campus do Gragoatá, Niterói, Rio de Janeiro 24210-201, Brazil
123
C. Zhou
open interval −∞ ≤ κ1 < κ2 ≤ +∞. In [13], we studied the space Lκκ21 ( P, Q) of C 1 regular curves in S2 that start and end at given points with given directions, whose lower and upper curvatures are restricted in an open interval (see the definitions of lower, upper curvatures and Lκκ21 ( P, Q) in Sect. 2), and proved the existence of a nontrivial map F : Sn 1 → Lκκ21 ( P, Q), where the dimension n 1 is linked to the maximum number of arcs of angle π in critical curves (see Definition 6 in [13] for critical curves and Theorems 7, 14 in [13] for details). We refer the readers to the articles [3–12], and references therein for more knowledge on this subject. Motivated by our work in [13], we are interested in understanding the set of C 1 regular curves in S2 whose lower and upper curvatures are restricted in an open interval (κ1 , κ2 ), −∞ < κ1 < κ2 < +∞. In this article, we distinguish which of its subsets is just the set of C 1 regular curves whose tangent vectors are Lipschitz continuous, and their a.e. existing geodesic curvatures have essentially bounds in (κ1 , κ2 ). Now we state our resul
Data Loading...
