Pacific Journal of Mathematics

Infinitesimal rigidity of almost-convex oriented polyhedra of arbitrary Euler characteristic.

Edgar Kann

Article information

Source
Pacific J. Math., Volume 144, Number 1 (1990), 71-103.

Dates
First available in Project Euclid: 8 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.pjm/1102645827

Mathematical Reviews number (MathSciNet)
MR1056667

Zentralblatt MATH identifier
0718.52019

Subjects
Primary: 52C25: Rigidity and flexibility of structures [See also 70B15]

Citation

Kann, Edgar. Infinitesimal rigidity of almost-convex oriented polyhedra of arbitrary Euler characteristic. Pacific J. Math. 144 (1990), no. 1, 71--103. https://projecteuclid.org/euclid.pjm/1102645827


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References

  • [A] A. D. Alexandrov, Konvexe Polydeder, Akademie-Verlag, Berlin, 1958.
  • [AZ] A. D. Alexandrov and V. A. Zalgaller, Intrinsic Geometry of Surfaces, Trans- lations of Mathematical Monographs, Vol. 15, American Mathematical So- ciety, Providence, 1967.
  • [AR] L. Asimov and B. Roth, The rigidity of graphs, II, J. Math. Anal, and Appl., 68 (1979), 171-190.
  • [B] H. Busemann, Convex Surfaces, Interscience, New York, 1958.
  • [C 1] R. Connelly, A counterexample to the rigidity conjecturefor polyhedra, Inst. Hautes Etudes Sci. Publ. Math., 47 (1978), 333-338.
  • [C 2] R. Connelly, The rigidity of certain cabledframeworks and the second order rigidity of arbitrarily triangulated convex surfaces, Advances in Math., 37 (1980), 272-299.
  • [CW 1] H. Crapo and W. Whiteley, Statics offrameworks and motions ofpanel struc- tures, a projectivegeometric introduction, Structural Topology, 6 (1982), 43-
  • [CW 2] H. Crapo and W. Whiteley, Plane stresses and projected polyhedra, preprint, Champlain Regional College, 900 Riverside Drive, St. Lambert, Quebec, J4P-3B8, 1985.
  • [Da] G. Darboux, Theorie GeneraldesSurfaces, vol. Ill, Paris 1894, vol. IV, Paris, 1896.
  • [D] M. Dehn, Uber die Starrheit konvexer Polyeder, Math. Ann., 77 (1916), 466- 473.
  • [E] N. W. Efimov, Flchenverbiegung im Grossen mit einem Nachtrag von E. Rembs und K. P. Grotemeyer,Akademie-Verlag, Berlin, 1957.
  • [E i] N. W. Efimov, Qualitative problems of the theory of deformations of surfaces,Transl. Amer. Math. Soc, Series 1, Vol. 6: Differential Geometry and the Calculus of Variations, (1962), 274-423, (English translation of the first part of the above).
  • [F] A. Fogelsanger, Thegeneric rigidity of minimal cycles, Ph.D. Thesis,Cornell University, May 1988.
  • [G] H. Gluck,Almost all simply connected closedsurfaces are rigid, Geometric Topology, Lecture Notes in Math., Vol. 438, Springer-Verlag, Berlin, 1975, 225-239.
  • [K i] E. Kann, A new method for infinitesimal rigidity of surfaces with K > 0, J. Differential Geom., 4 (1970), 5-12.
  • [K 2] E. Kann,An elementary proof of a finite rigidityproblem by infinitesimal rigidity methods, Proc. Amer. Math. Soc, 60 (1976), 252-258.
  • [K3] E. Kann, Glidebending of general caps: An infinitesimal treatment, Proc. Amer. Math. Soc, 84 (1982), 247-255.
  • [Ko] D. Koutroufiotis, Monotopy of convex caps, Archiv der Mathematik,XXVII (1976), 657-662.
  • [L] S. Lefschetz, Introduction to Topology, Princeton University Press, Prince- ton, 1949.
  • [Ly] L. A. Lyusternik, Convex Figures and Polyhedra, D. C. Heath and Co., Boston, 1966.
  • [Mi] E. Moise, Elementary Geometryfrom an Advanced Standpoint, SecondEd., Addison-Wesley, Reading, 1974.,
  • [M2] E. Moise, Geometric Topology in Dimensions 2 and 3 , Springer-Verlag, 1977.
  • [MR] T. Minagawa and T. Rado, On the infinitesimal rigidity of surfaces, Osaka Math. J., 4(1952), 241-285.
  • [N] L. Nirenberg,Rigidity of a classof closedsurfaces, NonlinearProblems,The University of Wisconsin Press, 1963, 177-193.
  • [P] A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces, Transl. of Math. Monographs, Vol. 35, Amer. Math. Soc, Providence, 1973.
  • [Ra] W. J. M. Rankine,On the Application of Barycentric Perspective to theTrans- formation of Structures, Phil. Mag. Series, 4 26 (1863), 387-388.
  • [R] B. Roth, Rigid and flexible frameworks, Amer. Math. Monthly, 88 (1981), 6-21.
  • [Rl] B. Roth, Rigidity of Convex Surfaces, Chapter 4 in "The Geometry of Rigid Structures", H. Crapo and W. Whiteley eds. to appear.
  • [RW] B. Roth and W. Whiteley, Tensegrity frameworks, Trans. Amer. Math. Soc, 265(1981), 419-446.
  • [W 1] W. Whiteley, Motions, stressesand projectedpolyhedra, StructuralTopology, 7(1982), 13-38.
  • [W 2] W. Whiteley, Infinitesimally rigid polyhedra I: Statics of frameworks, Trans. Amer. Math. Soc, 285 (1984), 431-465.
  • [W 3] W. Whiteley, Rigidity and polarity I: Statics of sheet structures,Geom. Dedicata,22 (1987), 329-362.
  • [W4] W. Whiteley, Infinitesimally rigid polyhedra II: Modified spherical frameworks, Trans. Amer. Math. Soc, 306 (1988), 115-137.
  • [W5] W. Whiteley Infinitesimally Rigid Polyhedra IV: Motions of Hinged-Panel Manifolds, preprint Champlain Regional College, 900 Riverside Drive, St. Lambert, Quebec, J4P-3P2 1986.