Orbifolds from a metric viewpoint
- PDF / 351,444 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 20 Downloads / 198 Views
Orbifolds from a metric viewpoint Christian Lange1 Received: 4 July 2019 / Accepted: 5 March 2020 © Springer Nature B.V. 2020
Abstract We characterize Riemannian orbifolds and their coverings in terms of metric geometry. In particular, we show that the metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and that the natural projection is an orbifold covering. Keywords Riemannian orbifolds · Submetries · Alexandrov spaces · Covering maps Mathematics Subject Classification (2010) 57R18 · 14H30 · 51F99
1 Introduction Orbifolds were introduced by Satake in the 50s under the name of V-manifolds [21,22] and rediscovered by Thurston in the 70s, when also the term “orbifold” was chosen, cf. [6]. Thurston moreover defined the notion of coverings of orbifolds and showed that the usual theory of coverings and fundamental groups works in the setting of orbifolds [24]. This theory was later generalized to the setting of groupoids by Haefliger [9,10] (see also [2], and [19,20] for the relation between groupoids and orbifolds). Orbifolds can be endowed with a Riemannian metric. In this form they for instance arise as quotients of isometric Lie group actions on Riemannian manifolds [17] or as Gromov–Hausdorff limits [8]. Our purpose is to characterize such Riemannian orbifolds and their coverings in terms of metric geometry. Besides being interesting on its own right we describe some applications of this perspective. Recall that a length space (or intrinsic metric space) is a metric space in which the distance between any pair of points can be realized as the infimum of the lengths of all rectifiable paths connecting these points [5]. The following definition of a Riemannian orbifold was proposed to us by Alexander Lytchak. Definition 1.1 A Riemannian orbifold of dimension n is a length space O such that for each point x ∈ O there exist an open neighborhood U of x in O and a connected Riemannian
The author was partially supported by the DFG funded Project SFB/TRR 191.
B 1
Christian Lange [email protected] Mathematisches Institut der Universität zu Köln, Weyertal 86-90, 50931 Cologne, Germany
123
Geometriae Dedicata
manifold M of dimension n together with a finite group G of isometries of M such that U and M/G are isometric with respect to the induced length metrics. Here M is endowed with the induced length metric and M/G with the corresponding quotient metric, which measures the distance between orbits in M. In Sect. 2 we explain in which sense this definition is equivalent to the original definition of a Riemannian orbifold. To obtain a metric notion of orbifold coverings we follow an idea by Lytchak to view submetries with discrete fibers as branched coverings [14]. We denote the closed balls in a metric space X as Br (x) and the open balls as Ur (x). A map p : X → Y between metric spaces is called submetry if p(Br (x)) = Br ( p(x)) holds for all x ∈ X and all r ≥ 0. It is known that a submetry between Riemannian manifolds is a Riemannian submersion [1
Data Loading...
