The theorem of the area of a rectangular trihedral of the hyperbolic plane of positive curvature
Dalʹnevostočnyj matematičeskij žurnal, Tome 13 (2013) no. 1, pp. 127-147.

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On the hyperbolic plane $\widehat{H}$ of positive curvature the orthogonal hypercyclic coordinates system is introduced. Formulas for calculation of the areas of a rectangular trihedral, in particular with a parabolic hypotenuse, double rectangular trihedral and the simple 4-contours, being a cell in simple partitions of the plane $\widehat{H}$ are received.
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L. N. Romakina. The theorem of the area of a rectangular trihedral of the hyperbolic plane of positive curvature. Dalʹnevostočnyj matematičeskij žurnal, Tome 13 (2013) no. 1, pp. 127-147. http://geodesic.mathdoc.fr/item/DVMG_2013_13_1_a11/

[1] B. A. Rozenfeld, M. P. Zamakhovskii, Geometriya grupp Li. Simmetricheskie, parabolicheskie i periodicheskie prostranstva, MTsNMO, M., 2003 | MR

[2] B. A. Rozenfeld, Neevklidovy prostranstva, Nauka, M., 1969 | MR

[3] H. S. M. Coxeter, “A Geometrical Background for De Sitter's World”, The American Mathematical Monthly, Volume 50, No. 4, (April 1943), 217–228 | DOI | MR | Zbl

[4] de Sitter W, “On the Relativity of Inertia. Remarks Concerning Einstein’s Latest Hypothesis”, Proc. Royal Acad. Amsterdam [KNAW], Volume 19, Issue 2, 1917, 1217–1225

[5] K. Akutagawa, “On space-like hypersurfaces with constant mean curvature in the de Sitter space”, Math. Z., 196, (1987), 13–-19 | DOI | MR | Zbl

[6] S. Montiel, “An integral inequality for compact space-like hypersurfaces in a de Sitter space and application to the case of constant mean curvature”, Indiana Univ. Math. J., 37, (1988), 909–-917 | DOI | MR | Zbl

[7] Q. M. Cheng, “Complete space-like submanifolds in a de Sitter space with parallel mean curvature vector”, Math. Z., 206, (1991), 333–-339 | DOI | MR | Zbl

[8] Q. M. Cheng, “Hypersurfaces of a Lorentz space form”, Arch. Math., 63, (1994), 271–-281 | DOI | MR | Zbl

[9] Huili Liu, Guili Liu, “Weingarten rotation surfaces in 3-dimensional de Sitter space”, Journal of Geometry, Volume 79, Issue 1–2, (April 2004), 156–168 | MR | Zbl

[10] Takesi Fusho, Shyuichi Izumiya, “Lightlike surfaces of spacelike curves in de Sitter 3-space”, Journal of Geometry, Volume 88, Issue 1–2, (March 2008), 19–29 | MR | Zbl

[11] Roland Hefer, “Metric and Periodic Lines in de Sitter's World”, Journal of Geometry, Volume 90, Issue 1–2, (December 2008), 66–82 | MR | Zbl

[12] Yunhi Cho, “Trigonometry in extended hyperbolic space and extended de Sitter space”, Bull. Korean Math. Soc. 46, No. 6, DOI 10.4134/BKMS.2009.46.6.1099, (2009), 1099–-1133 | MR | Zbl

[13] Masaki Kasedou, “Singularities of lightcone Gauss images of spacelike hypersurfaces in de Sitter space”, Journal of Geometry, Volume 94, Issue 1–2, (September 2009), 107–121 | MR | Zbl

[14] Immanuel Asmus, “Duality between hyperbolic and de Sitter geometry”, Journal of Geometry, Volume 96, Issue 1-2, (December 2009), 11–40 | MR

[15] Oscar M. Perdomo, “Algebraic zero mean curvature hypersurfaces in de Sitter and anti de Sitter spaces”, Geometriae Dedicata, Volume 152, Issue 1, (June 2011), 183–196 | MR | Zbl

[16] Dan Yang, Zhonghua Hou, “Linear Weingarten spacelike submanifolds in de Sitter space”, Journal of Geometry, Volume 103, Issue 1, (April 2012), 177–190 | MR | Zbl

[17] Takami Sato, “Pseudo-spherical evolutes of curves on a spacelike surface in three dimensional Lorentz-Minkowski space”, Journal of Geometry, Volume 103, Issue 2, (August 2012), 319–331 | MR | Zbl

[18] L. N. Romakina, “Konechnye zamknutye 3(4)-kontury rasshirennoi giperbolicheskoi ploskosti”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 10:3 (2010), 14–26

[19] L. N. Romakina, “Konechnye zamknutye 5-kontury rasshirennoi giperbolicheskoi ploskosti”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 11:1 (2011), 38–49

[20] L. N. Romakina, Teoriya otnositelnosti, gravitatsiya i geometriya: Mezhd. konf. “Petrov 2010 Anniversary Sympozium on General Relativity and Gravitation”. Trudy. Kazan, 1–6 noyabrya 2010 g., Kazan: Kazan. un-t, 2010

[21] L. N. Romakina, Sovremennye problemy matematiki i mekhaniki. Tom 6, vypusk 3, Izd-vo Moskovskogo universiteta, M., 2011

[22] L. N. Romakina, “Analog mozaiki na giperbolicheskoi ploskosti polozhitelnoi krivizny”, Sb. nauch. tr. Mekhanika. Matematika, Izd-vo Sarat. un-ta, Saratov, 2010, 69–72

[23] Sb. Math., 203:9 (2012), 1310–1341 | DOI | DOI | MR | Zbl

[24] L. N. Romakina, “Ovalnye linii giperbolicheskoi ploskosti polozhitelnoi krivizny”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 12:3 (2012), 37–44

[25] L. N. Romakina, Trigonometriya giperbolicheskoi ploskosti polozhitelnoi krivizny, SGU im. N.G. Chernyshevskogo, Saratov, 2012

[26] L. N. Romakina, Geometrii koevklidovoi i kopsevdoevklidovoi ploskostei, OOO Izd-vo “Nauchnaya kniga”, Saratov, 2008

[27] L. N. Romakina, “Opredelenie luchei, otrezkov i kvaziotrezkov razlichnogo tipa pryamykh pri postroenii klassicheskikh neevklidovykh geometrii na modelyakh Keli-Kleina”, Mezhdun. konferentsiya «62-e Gertsenovskie chteniya»: Sb. nauch. tr., Izd-vo RGPU im.A.I. Gertsena, Sankt-Peterburg, 2009, 103–109