Geodesic
The theory of curves in differential geometry from the viewpoint of the theory of functions of a real variable
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1165-1181 
Yu. G. Reshetnyak. The theory of curves in differential geometry from the viewpoint of the theory of functions of a real variable. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1165-1181. http://geodesic.mathdoc.fr/item/RM_2005_60_6_a9/
@article{RM_2005_60_6_a9,
     author = {Yu. G. Reshetnyak},
     title = {The theory of curves in differential geometry from the viewpoint of the theory of functions of a~real variable},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {1165--1181},
     year = {2005},
     volume = {60},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2005_60_6_a9/}
}
TY  - JOUR
AU  - Yu. G. Reshetnyak
TI  - The theory of curves in differential geometry from the viewpoint of the theory of functions of a real variable
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2005
SP  - 1165
EP  - 1181
VL  - 60
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/RM_2005_60_6_a9/
LA  - en
ID  - RM_2005_60_6_a9
ER  - 
%0 Journal Article
%A Yu. G. Reshetnyak
%T The theory of curves in differential geometry from the viewpoint of the theory of functions of a real variable
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2005
%P 1165-1181
%V 60
%N 6
%U http://geodesic.mathdoc.fr/item/RM_2005_60_6_a9/
%G en
%F RM_2005_60_6_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper contains a brief survey of the theory of curves constructed by A. D. Aleksandrov and the author. The basic notions of the theory are defined, the statements of some of its most important results are given, and the methods of study applied in this theory are described. The principal methods are: the method based on using the notions of the tangent indicatrix of a curve in $\mathbb R^n$ and the development of a spherical curve, and the method based on integral-geometric relations of the type found by Fáry and Milnor. The integral-geometric method for solving problems in the theory of curves was proposed and realized by the author.

[1] A. D. Aleksandrov, “Teoriya krivykh na osnove priblizheniya lomanymi”, Nauchnaya sessiya LGU, Tezisy dokladov. Sektsiya matem. nauk, LGU, L., 1946

[2] A. D. Aleksandrov, “Teoriya krivykh na osnove priblizheniya lomanymi”, UMN, 2:3 (1947), 182–184

[3] A. Ya. Yusupov, Teoriya krivykh na osnove priblizheniya lomanymi, Diss. ... kand. fiz.-matem. nauk, Ped. in-t im. A. I. Gertsena, L., 1949

[4] I. Fáry, “Sur la courbure totale d'une courbe gauche faisant un nœud”, Bull. Soc. Math. France, 77 (1949), 128–138 | MR | Zbl

[5] J. Milnor, “On the total curvature of knots”, Ann. of Math. (2), 52 (1950), 248–257 | DOI | MR | Zbl

[6] I. Fáry, “Sur certaines inégalités géométriques”, Acta. Sci. Math. Szeged., 12. Part A (1950), 117–124 | MR | Zbl

[7] V. A. Zalgaller, “Variatsii krivykh vdol fiksirovannykh napravlenii”, Izv. AN SSSR. Ser. matem., 15:5 (1951), 463–476 | MR

[8] Yu. G. Reshetnyak, O dline i povorote krivoi i o ploschadi poverkhnosti, Diss. ... kand. fiz.-matem. nauk, LGU, L., 1954

[9] Yu. G. Reshetnyak, “Integralno-geometricheskii metod v teorii krivykh”, Trudy 3-go Vsesoyuznogo matematicheskogo s'ezda, T. 1: Sekts. dokl. (Moskva, 1956), Izd-vo AN SSSR, M., 1956, 164

[10] “Yu. G. Reshetnyak”, Vestn. Leningr. un-ta. Ser. matem., mekh, astron., 1957, no. 13, vyp. 3, 22–26

[11] Yu. G. Reshetnyak, “O ponyatii pod'ema neregulyarnogo puti v rassloennom mnogoobrazii i ego prilozheniyakh”, Sib. matem. zhurn., 16:3 (1975), 588–598 | MR | Zbl

[12] I. F. Mainik, O monotonnykh krivykh i integralnoi geometrii v $n$-mernom evklidovom i sfericheskom prostranstvakh, Dep. v VINITI, No 2194-76, M., 1976

[13] I. F. Mainik, O neregulyarnykh krivykh v $n$-mernom evklidovom i sfericheskom prostranstvakh, Dep. v VINITI, No 3184-76, M., 1976

[14] I. F. Mainik, “O krivykh v reduktivnykh prostranstvakh”, Dokl. AN SSSR, 235:3 (1977), 531–533 | MR | Zbl

[15] I. F. Mainik, “Ob $r$-vpisannykh krivykh”, Sib. matem. zhurn., 20:1 (1979), 95–108 | MR | Zbl

[16] I. F. Mainik, “Ob integralnykh kriviznakh v $\mathbb R^n$ i $S^n$”, Simpozium po geometrii v tselom i osnovam teorii otnositelnosti, Tezisy dokladov, Nauka, Novosibirsk, 1982, 53–54

[17] I. F. Mainik, “Proektsionnyi priznak ravenstva krivykh”, Sib. matem. zhurn., 24:4 (1983), 100–103 | MR | Zbl

[18] I. F. Mainik, “Proektsionnyi priznak ravenstva prostranstvennykh krivykh”, Sib. matem. zhurn., 26:2 (1985), 207–209 | MR | Zbl

[19] I. F. Mainik, “Ob approksimatsii integralnykh krivizn krivykh v $R^n$ i $S^n$”, Sib. matem. zhurn., 28:5 (1987), 128–133 | MR | Zbl

[20] A. D. Aleksandrov, Yu. G. Reshetnyak, “Povorot krivoi v $n$-mernom evklidovom prostranstve”, Sib. matem. zhurn., 29:1 (1988), 3–22 | MR

[21] Yu. G. Reshetnyak, “Nekotorye primeneniya integralnoi geometrii k teorii krivykh konechnogo povorota”, Sib. matem. zhurn., 29:1 (1988), 141–150 | MR

[22] A. D. Alexandrov, Yu. G. Reshetnyak, General Theory of Irregular Curves, Kluwer, Dordrecht, 1989 | MR | Zbl

[23] I. F. Mainik, “K otsenke dliny krivoi spuska”, Sib. matem. zhurn., 33:4 (1992), 215–218 | MR

[24] I. F. Mainik, “K ustoichivosti v teoreme ob otsenke dliny krivoi spuska”, Sib. matem. zhurn., 38:4 (1997), 867–875 | MR | Zbl

[25] Yu. G. Reshetnyak, “O teorii neregulyarnykh krivykh v $n$-mernom evklidovom prostranstve”, Geometriya i prilozheniya, Trudy konferentsii, posvyaschennoi 70-letiyu V. A. Toponogova (Novosibirsk, 2000), Izd-vo IM SO RAN, Novosibirsk, 2001, 158–170 | MR | Zbl

[26] J. Radon, “Über Randwertaufgaben beim logarithmischen Potential”, Sitzber. Akad. Wiss. Wien., 128 (1919), 1123–1167

[27] I. M. Liberman, “Geodezicheskie linii na vypuklykh poverkhnostyakh”, Dokl. AN SSSR, 32:5 (1941), 310–313 | MR | Zbl

[28] A. V. Pogorelov, Vneshnyaya geometriya vypuklykh poverkhnostei, Nauka, M., 1969 | MR

[29] A. G. Khovanskii, S. Yu. Yakovenko, “Generalized Rolle theorem in ${\mathbb R}^n$ and ${\mathbb C}$”, J. Dynam. Control Systems, 2:1 (1996), 103–123 | DOI | MR | Zbl

[30] Yu. G. Reshetnyak, “Otsenka ugla obzora krivoi v metricheskom prostranstve nepolozhitelnoi krivizny”, Sib. matem. zhurn., 43:3 (2002), 694–701 | MR | Zbl