Geodesic
The space of Schwarz–Klein spherical triangles
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 16 (2020) no. 3, pp. 263-282 
Alexandre Eremenko; Andrei Gabrielov. The space of Schwarz–Klein spherical triangles. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 16 (2020) no. 3, pp. 263-282. http://geodesic.mathdoc.fr/item/JMAG_2020_16_3_a3/
@article{JMAG_2020_16_3_a3,
     author = {Alexandre Eremenko and Andrei Gabrielov},
     title = {The space of {Schwarz{\textendash}Klein} spherical triangles},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {263--282},
     year = {2020},
     volume = {16},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2020_16_3_a3/}
}
TY  - JOUR
AU  - Alexandre Eremenko
AU  - Andrei Gabrielov
TI  - The space of Schwarz–Klein spherical triangles
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2020
SP  - 263
EP  - 282
VL  - 16
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/JMAG_2020_16_3_a3/
LA  - en
ID  - JMAG_2020_16_3_a3
ER  - 
%0 Journal Article
%A Alexandre Eremenko
%A Andrei Gabrielov
%T The space of Schwarz–Klein spherical triangles
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2020
%P 263-282
%V 16
%N 3
%U http://geodesic.mathdoc.fr/item/JMAG_2020_16_3_a3/
%G en
%F JMAG_2020_16_3_a3

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

We describe the space of spherical triangles (in the sense of Schwarz and Klein). It is a smooth connected orientable $3$ manifold, homotopy equivalent to the $1$-skeleton of the cubic partition of the closed first octant in $\mathbb{R}^3$. The angles and sides are real analytic functions on this manifold which embed it to $\mathbb{R}^6$.

[1] G. Chisholm, Algebraisch-gruppentheoretische Untersuchingen zur shpärischen Trigonometrie, Inaugural-Dissertation, Göttingen, 1895 (German) | Zbl

[2] A. Eremenko, “Metrics of positive curvature with conic singularities on the sphere”, Proc. Amer. Math. Soc., 132:11 (2004), 3349–3355 | DOI | MR | Zbl

[3] A. Eremenko, A. Gabrielov, V. Tarasov, “Metrics with conic singularities and spherical polygons”, Illinois J. Math., 58 (2014), 739–755 | DOI | MR | Zbl

[4] A. Eremenko, A. Gabrielov, V. Tarasov, “Metrics with conic singularities and spherical quadrilaterals”, Conform. Geom. Dyn., 20 (2016), 128–175 | DOI | MR | Zbl

[5] A. Eremenko, A. Gabrielov, G. Mondello, D. Panov, Moduli spaces for Lamé functions and Abelian differentials of the second kind, arXiv: 2006.16837

[6] A. Eremenko, G. Mondello, D. Panov, Moduli of spherical tori with one conical point, arXiv: 2008.02772

[7] G. Mondello, D. Panov, “Spherical metrics with conical singularities on a 2-sphere: angle constraints”, Int. Math. Res. Not. IMRN, 2016, no. 16, 4937–4995 | DOI | MR | Zbl

[8] S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara, K. Yamada, “CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere”, Proc. Japan Acad., 87 (2011), 144–149 | MR | Zbl

[9] E.H. Spanier, Algebraic topology, Springer-Verlag, New York–Berlin, 1966 | MR

[10] E. Study, Sphärische Trigonometrie, orthogonale Substitutionen und elliptische Funktionen, S. Hirzel, Leipzig, 1893 (German)