Geodesic
The shortest polygonal chains in the Heisenberg group
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2024), pp. 81-87 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We describe the shortest polygonal chains that connect two points on the first Heisenberg group with the sub-Riemannian structure. The shortest polygonal chain connecting two points with a fixed number of links either is a straight line or consists of segments of the same length such that the projections of their endpoints are inscribed in a circle. The analytical description is obtained for the spheres of the quasimetric generated by the shortest polygonal chains with 3 links.
Keywords: Heisenberg group, shortest path.
Mots-clés : polygonal chain 
@article{IVM_2024_11_a6,
     author = {S. G. Basalaev},
     title = {The shortest polygonal chains in the {Heisenberg} group},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {81--87},
     year = {2024},
     number = {11},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2024_11_a6/}
}
TY  - JOUR
AU  - S. G. Basalaev
TI  - The shortest polygonal chains in the Heisenberg group
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2024
SP  - 81
EP  - 87
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/IVM_2024_11_a6/
LA  - ru
ID  - IVM_2024_11_a6
ER  - 
%0 Journal Article
%A S. G. Basalaev
%T The shortest polygonal chains in the Heisenberg group
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2024
%P 81-87
%N 11
%U http://geodesic.mathdoc.fr/item/IVM_2024_11_a6/
%G ru
%F IVM_2024_11_a6
S. G. Basalaev. The shortest polygonal chains in the Heisenberg group. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2024), pp. 81-87. http://geodesic.mathdoc.fr/item/IVM_2024_11_a6/

[1] Greshnov A.V., “Optimal horizontal joinability on the Engel group”, Atti Accad. Naz. Lincei Classe Sci. Fis. Mat. Natur., 32:3 (2021), 535–547 | DOI | MR | Zbl

[2] Greshnov A.V., Zhukov R.I., “Gorizontalnaya soedinimost na kanonicheskoi $3$-stupenchatoi gruppe Karno s gorizontalnym raspredeleniem koranga $2$”, Sib. matem. zhurn., 62:4 (2021), 736-746 | DOI | Zbl

[3] Greshnov A.V., “Metod Agracheva-Barilari-Boskaina i otsenki chisla zvenev gorizontalnykh lomanykh, soedinyayuschikh tochki v kanonicheskoi gruppe Karno $G_{3,3}$”, Tr. MIAN, 321, no. 1, 2023, 108–117 | DOI | Zbl

[4] Zhukov R.I., Greshnov A.V., “Gorizontalnaya soedinimost na $5$-mernoi $2$-stupenchatoi gruppe Karno s gorizontalnym raspredeleniem korazmernosti $2$”, Algebra i logika, 62:2 (2023), 205–218 | DOI | Zbl

[5] Bonfiglioli A., Lanconelli E., Uguzzoni F., Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Springer Monographs in Mathematics, Springer Berlin, Heidelberg, 2007 | DOI | MR | Zbl

[6] Hall B.C., Lie Groups, Lie Algebras, and Representations. An Elementary Introduction, GTM, 222, Springer, Cham, 2015 | MR | Zbl

[7] Rashevskii P.K., “O soedinimosti lyubykh dvukh tochek vpolne negolonomnogo prostranstva dopustimoi liniei”, Uchen. zap. ped. in-ta im. Libknekhta, Ser. fiz.-matem., 1938, no. 2, 83–94

[8] Chow W.L., “Über systeme von linearen partiallen differentialgleichungen erster ordnung”, Mathematische Annalen, 117 (1939), 98-105 | DOI | MR | Zbl

[9] Courant R., Robbins H., What is Mathematics? An Elementary Approach to Ideas and Methods, Oxford Univ. Press, Oxford, 1941 | MR

[10] Niven I., Maxima and Minima Without Calculus, Dolc. Math. Exp. (6), Cambridge Univ. Press, Cambridge, 1981 | MR

[11] Gaveau B., “Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents”, Acta Math., 139 (1977), 95-153 | DOI | MR | Zbl

[12] Berestovskii V.N., “Geodezicheskie negolonomnykh levoinvariantnykh vnutrennikh metrik na gruppe Geizenberga i izoperimetriksy ploskosti Minkovskogo”, Sib. matem. zhurn., 35:1 (1994), 3-11 | MR | Zbl