Farthest points on flat surfaces. (English) Zbl 1407.53074
A recent result by the authors states that on most (in the sense of Baire categories) Alexandrov surfaces most points have a unique farthest point. Notable exceptions are flat tori and flat Klein bottles. An elementary treatment of the farthest point map for these surfaces is the topic of this paper.
Denote by \(F_p\) the set of all farthest points of \(p\) and by \(F^n_p\) the set of all farthest points of \(p\) which are joined to \(p\) be exactly \(n\) segments. If the flat torus is obtained by identifying opposite sides of a parallelogram, then \(\# F_p = \# F_p^3 = 2\) in general and \(\# F_p = \# F_p^4 = 1\) precisely for rectangles.
The case of a flat Klein bottle, obtained by suitable identification of opposite sides in a rectangle, is more complicated. Results not only depend on the side lengths but also on the point’s distance to the main geodesics. In all cases one has \(\# F_p = \# F_p^n = k\) for some \(n \in \{3, 4\}\) and some \(k \in \{1, 2\}\).
Denote by \(F_p\) the set of all farthest points of \(p\) and by \(F^n_p\) the set of all farthest points of \(p\) which are joined to \(p\) be exactly \(n\) segments. If the flat torus is obtained by identifying opposite sides of a parallelogram, then \(\# F_p = \# F_p^3 = 2\) in general and \(\# F_p = \# F_p^4 = 1\) precisely for rectangles.
The case of a flat Klein bottle, obtained by suitable identification of opposite sides in a rectangle, is more complicated. Results not only depend on the side lengths but also on the point’s distance to the main geodesics. In all cases one has \(\# F_p = \# F_p^n = k\) for some \(n \in \{3, 4\}\) and some \(k \in \{1, 2\}\).
Reviewer: Hans-Peter Schröcker (Innsbruck)
MSC:
| 53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
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