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On rigidity of affine surfaces

Author: Barbara Opozda
Journal: Proc. Amer. Math. Soc. 124 (1996), 2175-2184
MSC (1991): Primary 53A15; Secondary 53B05
MathSciNet review: 1363435
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Abstract: Rigidity of nondegenerate Blaschke surfaces in $ \mathbf {R}^{3}$ is studied. The rigidity criteria are given in terms of $\nabla R$, where $R$ is the curvature of the Blaschke connection $\nabla $. If the rank of $\nabla R$ is 2, then the surface is rigid. If $\nabla R=0$, it is nonrigid. In the case where the rank of $\nabla R$ is 1 there are both rigid and nonrigid surfaces. This case is discussed for various types of surfaces.

References [Enhancements On Off] (What's this?)

  • [C] E. Cartan, Sur la connexion affine des surfaces, C. R. Acad. Sci. Paris 178 (1924), 292-295.
  • [DNV] Franki Dillen, Katsumi Nomizu, and Luc Vranken, Conjugate connections and Radon’s theorem in affine differential geometry, Monatsh. Math. 109 (1990), no. 3, 221–235. MR 1058409, 10.1007/BF01297762
  • [NO] Katsumi Nomizu and Barbara Opozda, Integral formulas for affine surfaces and rigidity theorems of Cohn-Vossen type, Geometry and topology of submanifolds, IV (Leuven, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 133–142. MR 1185721
  • [NS] K. Nomizu, T. Sasaki, Affine Differential Geometry, Cambridge University Press, 1994. CMP 95:06
  • [O1] Barbara Opozda, Locally symmetric connections on surfaces, Results Math. 20 (1991), no. 3-4, 725–743. Affine differential geometry (Oberwolfach, 1991). MR 1145305, 10.1007/BF03323207
  • [O2] B. Opozda, A class of projectively flat surfaces, Math. Z. 219 (1995), 77-92. CMP 95:15
  • [OS] B. Opozda, T. Sasaki, Surfaces whose affine normal images are curves, Kyushu J. Math. 49 (1995), 1-10.
  • [S] Udo Simon, Global uniqueness for ovaloids in Euclidean and affine differential geometry, Tohoku Math. J. (2) 44 (1992), no. 3, 327–334. MR 1176075, 10.2748/tmj/1178227299
  • [Sl] W. Slebodzinski, Sur quelques problèmes de la théorie des surfaces de l'espace affine, Prace Mat. Fiz. 46 (1939), 291-345.

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Additional Information

Barbara Opozda
Affiliation: Instytut Matematyki, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

Keywords: Blaschke surface, metric compatible with connection
Received by editor(s): May 31, 1994
Additional Notes: The research was supported by the Kambara Fund of Kobe University and the KBN grant 2P30103004.
Communicated by: Christopher Croke
Article copyright: © Copyright 1996 American Mathematical Society