Geodesic
Isometric immersions and embeddings of locally Euclidean metrics in~$\mathbb R^2$
Izvestiya. Mathematics , Tome 63 (1999) no. 6, pp. 1203-1220.

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

This paper deals with the problem of isometric immersions and embeddings of two-dimensional locally Euclidean metrics in the Euclidean plane. We find explicit formulae for the immersions of metrics defined on a simply connected domain and a number of sufficient conditions for the existence of isometric embeddings. In the case when the domain is multiply connected we find necessary conditions for the existence of isometric immersions and classify the cases when the metric admits no isometric immersion in the Euclidean plane. 
@article{IM2_1999_63_6_a4,
     author = {I. Kh. Sabitov},
     title = {Isometric immersions and embeddings of locally {Euclidean} metrics in~$\mathbb R^2$},
     journal = {Izvestiya. Mathematics },
     pages = {1203--1220},
     publisher = {mathdoc},
     volume = {63},
     number = {6},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1999_63_6_a4/}
}
TY  - JOUR
AU  - I. Kh. Sabitov
TI  - Isometric immersions and embeddings of locally Euclidean metrics in~$\mathbb R^2$
JO  - Izvestiya. Mathematics 
PY  - 1999
SP  - 1203
EP  - 1220
VL  - 63
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1999_63_6_a4/
LA  - en
ID  - IM2_1999_63_6_a4
ER  - 
%0 Journal Article
%A I. Kh. Sabitov
%T Isometric immersions and embeddings of locally Euclidean metrics in~$\mathbb R^2$
%J Izvestiya. Mathematics 
%D 1999
%P 1203-1220
%V 63
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1999_63_6_a4/
%G en
%F IM2_1999_63_6_a4
I. Kh. Sabitov. Isometric immersions and embeddings of locally Euclidean metrics in~$\mathbb R^2$. Izvestiya. Mathematics , Tome 63 (1999) no. 6, pp. 1203-1220. http://geodesic.mathdoc.fr/item/IM2_1999_63_6_a4/

[1] Avkhadiev F. G., Aksentev L. G., “Osnovnye rezultaty v dostatochnykh usloviyakh odnolistnosti analiticheskikh funktsii”, UMN, 30:4 (1975), 3–60 | MR | Zbl

[2] Darboux G., Léçons sur la théorie gén erale des surfaces, V. 3, Paris, 1894

[3] Hartman Ph., Wintner A., “Gaussian curvature and local embedding”, Amer. J. of Math., 73:4 (1951), 876–882 | DOI | MR

[4] Sabitov I. Kh., “K voprosu o gladkosti izometrii”, Sib. matem. zhurn., 34:4 (1993), 169–176 | MR | Zbl

[5] Sabitov I. Kh., “Izometricheskie pogruzheniya i vlozheniya lokalno evklidovykh metrik v ${\mathbb R}^2$”, Tr. seminara po vektornomu i tenzornomu analizu, 23 (1988), 147–156 | MR | Zbl

[6] Sabitov I. Kh., “Lokalno evklidovy metriki: kharakteristicheskie priznaki i pogruzheniya i vlozheniya v ${\mathbb R}^2$ i ${\mathbb R}^3$”, Simpozium po geometrii v tselom i osnovaniyam teorii otnositelnosti, Tez. dokladov, Novosibirsk, 1982, 97–98

[7] Vekua I. N., Obobschennye analiticheskie funktsii, Fizmatgiz, M., 1959 | MR

[8] Aleksandrov A. D., Vnutrennyaya geometriya vypuklykh poverkhnostei, Gostekhizdat, M.-L., 1946

[9] Goluzin G. M., Geometricheskaya teoriya funktsii, Nauka, M., 1966 | MR

[10] Kaplan W., “Close-to-convex schlicht functions”, Michigan math. J., 1:2 (1952), 169–185 | DOI | MR | Zbl

[11] Titus C. J., “A theory of normal curves and some applications”, Pacif. J. of Math., 10:3 (1960), 1083–1096 | MR | Zbl

[12] Biernacki M., “Sur les fonctions univalents”, Matematica (Cluj), 12 (1936), 49–64 | Zbl

[13] Bazilevich I. E., “Oblasti nachalnykh koeffitsientov ogranichennykh odnolistnykh funktsii $n$-kratnoi simmetrii”, Matem. sb., 43:4 (1957), 409–428 | MR

[14] Reshetnyak Yu. G., “Izotermicheskie koordinaty v mnogoobraziyakh ogranichennoi krivizny. I; II”, Sib. matem. zhurn., 1:1 (1960), 88–116 | MR | Zbl

[15] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977 | MR