Configurations of curves and geodesics on surfaces. (English) Zbl 1035.53053

Hass, Joel; Scott, Peter

Hass, Joel (ed.) et al., Proceedings of the Kirbyfest, Berkeley, CA, USA, June 22–26, 1998. Warwick: University of Warwick, Institute of Mathematics. Geom. Topol. Monogr. 2, 201-213 (1999).

Let \(f\) and \(g\) be general position immersions of a manifold \(M\) into the interior of a manifold \(N\). The functions \(f\) and \(g\) are said to have the same configuration if there is a regular homotopy from \(f\) to \(g\) through general position immersions. This defines an equivalence class on general position immersions and an equivalence class is called a configuration. In this paper, the authors are interested in cases when \(\dim(M)\) is 1 or 2 and \(\dim(N)\) is 2 or 3. They investigate the question of how many configurations a given homotopy class can have. For primitive curves on a surface, they show that the number is finite if one restricts to immersions with the least possible number of double points, but little can be said for curves with excess intersections.
They also investigate the possible configurations of closed geodesics on a surface equipped with a hyperbolic metric. It is known that geodesics in a hyperbolic metric minimise the number of double points in their homotopy class. M. Neumann-Coto [Algebr. Geom. Topol. 1, 349–368 (2001; Zbl 0991.53024)] has shown that any curve configuration (not necessarily connected) which minimises the number of double points is realised by shortest geodesics in some metric. The authors construct examples which show that some configurations cannot be realised by closed geodesics in a hyperbolic metric.
For the entire collection see [Zbl 0939.00055].

Reviewer: Donal Hurley (Cork)

MSC:

53C22	Geodesics in global differential geometry
57R70	Critical points and critical submanifolds in differential topology
57M50	General geometric structures on low-dimensional manifolds

Keywords:

configurations; double points; intersections

Citations:

Zbl 0991.53024