---
res:
bibo_abstract:
- The goal of this article is to introduce the reader to the theory of intrinsic
geometry of convex surfaces. We illustrate the power of the tools by proving a
theorem on convex surfaces containing an arbitrarily long closed simple geodesic.
Let us remind ourselves that a curve in a surface is called geodesic if every
sufficiently short arc of the curve is length minimizing; if, in addition, it
has no self-intersections, we call it simple geodesic. A tetrahedron with equal
opposite edges is called isosceles. The axiomatic method of Alexandrov geometry
allows us to work with the metrics of convex surfaces directly, without approximating
it first by a smooth or polyhedral metric. Such approximations destroy the closed
geodesics on the surface; therefore it is difficult (if at all possible) to apply
approximations in the proof of our theorem. On the other hand, a proof in the
smooth or polyhedral case usually admits a translation into Alexandrovâ€™s language;
such translation makes the result more general. In fact, our proof resembles a
translation of the proof given by Protasov. Note that the main theorem implies
in particular that a smooth convex surface does not have arbitrarily long simple
closed geodesics. However we do not know a proof of this corollary that is essentially
simpler than the one presented below.@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: Arseniy
foaf_name: Akopyan, Arseniy
foaf_surname: Akopyan
foaf_workInfoHomepage: http://www.librecat.org/personId=430D2C90-F248-11E8-B48F-1D18A9856A87
- foaf_Person:
foaf_givenName: Anton
foaf_name: Petrunin, Anton
foaf_surname: Petrunin
bibo_doi: 10.1007/s00283-018-9795-5
bibo_issue: '3'
bibo_volume: 40
dct_date: 2018^xs_gYear
dct_language: eng
dct_publisher: Springer@
dct_title: Long geodesics on convex surfaces@
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