Geodesic
Infinitesimal and global rigidity and inflexibility of surfaces of revolution with flattening at the poles
Sbornik. Mathematics, Tome 204 (2013) no. 10, pp. 1516-1547 Cet article a éte moissonné depuis la source Math-Net.Ru

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The subject of this article is one of the most important questions of classical geometry: the theory of bendings and infinitesimal bendings of surfaces. These questions are studied for surfaces of revolution and, unlike previous well-known works, we make only minimal smoothness assumptions (the class $C^1$) in the initial part of our study. In this class we prove local existence and uniqueness theorems for infinitesimal bendings. We then consider the analytic class and establish simple criteria for rigidity and inflexibility of compact surfaces. These criteria depend on the values of certain integer characteristics related to the order of flattening of the surface at its poles. We also show that in the nonanalytic situation there exist nonrigid surfaces with any given order of flattening at the poles. Bibliography: 22 titles.
Keywords: pole of a surface of revolution, order of flattening, infinitesimal bending, harmonic number, rigidity. 
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I. Kh. Sabitov. Infinitesimal and global rigidity and inflexibility of surfaces of revolution with flattening at the poles. Sbornik. Mathematics, Tome 204 (2013) no. 10, pp. 1516-1547. http://geodesic.mathdoc.fr/item/SM_2013_204_10_a3/

[1] N. V. Efimov, “Kachestvennye voprosy teorii deformatsii poverkhnostei «v malom»”, Tr. MIAN SSSR, 30, Izd-vo AN SSSR, M.–L., 1949, 3–128 | MR | Zbl

[2] N. V. Efimov, “O zhestkosti v malom”, Dokl. AN SSSR, 60:5 (1948), 761–764 | MR | Zbl

[3] I. Ivanova-Karatopraklieva, I. Kh. Sabitov, “Surface deformation. I”, J. Math. Sci., 70:2 (1994), 1685–1716 | DOI | MR | Zbl | Zbl

[4] S. B. Klimentov, I. Kh. Sabitov, Z. D. Usmanov, “Deformatsiya poverkhnostei “v malom”: ot N. V. Efimova do sovremennykh issledovanii”, Matematika, Vyp. 2, Sovremennye problemy matematiki i mekhaniki, 6, Izd-vo Mosk. un-a, M., 2011, 34–48

[5] I. Kh. Sabitov, “A study of the rigidity and inflexibility of analytic surfaces of revolution with flattening at the pole”, Moscow Univ. Math. Bull., 41:5 (1986), 33–41 | MR | Zbl

[6] I. Ivanova-Karatopraklieva, I. Kh. Sabitov, “Second-order infinitesimal bendings of surfaces of rotation with densification at a pole”, Math. Notes, 45:1 (1989), 19–24 | MR | Zbl

[7] S. E. Chon-Vossen, “Unstarre geschlossene Flächen”, Math. Ann., 102:1 (1930), 10–29 | DOI | MR | MR

[8] N. V. Efimov, “Nekotorye predlozheniya o zhëstkosti i neizgibaemosti”, UMN, 7:5(51) (1952), 215–224 | MR | Zbl

[9] I. Kh. Sabitov, “Local theory of bendings of surfaces”, Geometry, v. III, Encyclopaedia Math. Sci., 48, Springer-Verlag, Berlin, 1992, 179–250 | DOI | MR | MR | Zbl | Zbl

[10] I. Kh. Sabitov, “On the relations between infinitesimal bendings of different orders”, J. Math. Sci., 72:4 (1994), 3237–3241 | DOI | MR | Zbl | Zbl

[11] R. Connelly, H. Servatius, “Higher-order rigidity – What is the proper definition?”, Discrete Comput. Geom., 11:1 (1994), 193–200 | DOI | MR | Zbl

[12] I. Kh. Sabitov, “The rigidity of “corrugated” surfaces of revolution”, Math. Notes, 14:4 (1973), 854–857 | DOI | MR | Zbl | Zbl

[13] I. Kh. Sabitov, “Possible generalizations of the Minagawa–Rado lemma on the rigidity of a surface of revolution with a fixed parallel”, Math. Notes, 19:1 (1976), 74–79 | DOI | MR | Zbl | Zbl

[14] A. V. Pogorelov, Extrinsic geometry of convex surfaces, Transl. Math. Monogr., 35, Amer. Math. Soc., Providence, R.I., 1973 | MR | MR | Zbl | Zbl

[15] J. L. Kazdan, F. W. Warner, “Curvature functions for open 2-manifolds”, Ann. of Math. (2), 99 (1974), 203–219 | DOI | MR | Zbl | Zbl

[16] A. O. Gel'fond, Solving equations in integers, Mir, Moscow, 1981 | MR | MR | Zbl | Zbl

[17] T. Nagell, Introduction to number theory, Wiley, New York, 1951 | MR | Zbl

[18] A. Yu. Nesterenko, “Determination of integer solutions of a system of simultaneous Pell equations”, Math. Notes, 86:4 (2009), 556–566 | DOI | DOI | MR | Zbl

[19] A. D. Milka, “O tochkakh otnositelnoi nezhestkosti vypukloi poverkhnosti vrascheniya”, Ukr. geom. sb., 1 (1965), 65–74 | MR | Zbl

[20] N. V. Efimov, “Kachestvennye voprosy teorii deformatsii poverkhnostei”, UMN, 3:2 (1948), 47–158 | MR | Zbl

[21] I. H. Sabitov, “On infinitesimal bendings of troughs of revolution. I”, Math. USSR-Sb., 27:1 (1975), 103–117 | DOI | MR | Zbl | Zbl

[22] I. H. Sabitov, “On infinitesimal bendings of troughs of revolution. II”, Math. USSR-Sb., 28:1 (1976), 41–48 | DOI | MR | Zbl | Zbl