Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 
 

 

On the total curvature of minimizing geodesics on convex surfaces


Authors: N. Lebedeva and A. Petrunin
Original publication: Algebra i Analiz, tom 29 (2017), nomer 1.
Journal: St. Petersburg Math. J. 29 (2018), 139-153
MSC (2010): Primary 53C21
DOI: https://doi.org/10.1090/spmj/1485
Published electronically: December 27, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A universal upper bound is given for the total curvature of a minimizing geodesic on a convex surface in the Euclidean space.


References [Enhancements On Off] (What's this?)

  • 1. P. K. Agarwal, S. Har-Peled, M. Sharir, and K. R. Varadarajan, Approximating shortest paths on a convex polytope in three dimensions, J. ACM 44 (1997), no. 4, 567-584. MR 1481315
  • 2. J. Pach, Folding and turning along geodesics in a convex surface, Geombinatorics 7 (1997), no. 2, 61-65. MR 1487759
  • 3. I. Bárány, K. Kuperberg, and T. Zamfirescu, Total curvature and spiralling shortest paths, Discrete Comput. Geom. 30 (2003), no. 2, 167-176. MR 2007957
  • 4. J. Liberman, Geodesic lines on convex surfaces, Dokl. Acad. Nauk SSSR 32 (1941), no. 5, 310-313. (Russian) MR 0010994
  • 5. V. V. Usov, The length of the spherical image of a geodesic on a convex surface, Sibirsk. Mat. Zh. 17 (1976), no. 1, 233-236; English transl., Sib. Math. J. 17 (1976), no. 1, 185-188. MR 0405316
  • 6. I. D. Berg, An estimate on the total curvature of a geodesic in Euclidean $ 3$-space-with-boundary, Geom. Dedicata 13 (1982), no. 1, 1-6. MR 679213
  • 7. A. V. Pogorelov, The extrinsic geometry of convex surfaces, Nauka, Moscow, 1969; English transl., Transl. Math. Monogr., vol. 35, Amer. Math. Soc., Providence, RI, 1973. MR 0244909
  • 8. V. A. Zalgaller, The question of the spherical representation of a geodesic, Ukrain. Geom. Sb. 10 (1971), 12-18. (Russian) MR 0298604
  • 9. A. D. Milka, A shortest path with nonrectifiable spherical representation. I, Ukrain. Geom. Sb. 16 (1974), 35-52. (Russian) MR 0385761
  • 10. V. V. Usov, The three-dimensional swerve of curves on convex surfaces, Sibirsk. Mat. Zh. 17 (1976), no. 6, 1427-1430; English transl., Sib. Math. J. 17, no. 6, 1043-1045. MR 0442862
  • 11. A. Petrunin, Applications of quasigeodesics and gradient curves, Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 203-219. MR 1452875

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 53C21

Retrieve articles in all journals with MSC (2010): 53C21


Additional Information

N. Lebedeva
Affiliation: Department of Mathematics and Mechanics St. Petersburg State University Universitetskii pr. 28 198504 St. Petersburg Russia; St. Petersburg Branch Steklov Institute of Mathematics Russian Academy of Sciences Fontanka 27 191023 St. Petersburg Russia
Email: lebed@pdmi.ras.ru

A. Petrunin
Affiliation: Department of Mathematics Pennsylvania State University University Park, PA 16802 USA
Email: petrunin@math.psu.edu

DOI: https://doi.org/10.1090/spmj/1485
Keywords: Geodesic, curvature, Liberman's lemma, development, tongue
Received by editor(s): May 20, 2016
Published electronically: December 27, 2017
Additional Notes: N. Lebedeva was partially supported by RFBR grant 14-01-00062. A. Petrunin was partially supported by NSF grant DMS 1309340
Dedicated: Dedicated to Yu. D. Burago on the occasion of his 80th birthday
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society