Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Affine surfaces whose geodesics are planar curves

Author: Luc Vrancken
Journal: Proc. Amer. Math. Soc. 123 (1995), 3851-3854
MSC: Primary 53A15; Secondary 53B05
MathSciNet review: 1283565
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the geometry of nondegenerate affine surfaces $ {M^2}$ in $ {\mathbb{R}^4}$, with respect to the Burstin-Mayer, the Weise-Klingenberg and the equiaffine transversal plane bundle. A classification is obtained of the surfaces whose geodesies with respect to the induced connection are planar curves.

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

  • [BM] C. Burstin and W. Mayer, Die Geometrie zweifach ausgedehnter Mannigfaltigkeiten $ {F_2}$ im affinen Raum $ {R_4}$, Math. Z. 27 (1927), 373-407.
  • [DMVV] F. Dillen, G. Mys, L. Verstraelen, and L. Vrancken, The affine mean curvature vector for surfaces in $ {\mathbb{R}^4}$, Math. Nachr. 166 (1994), 155-165. MR 1273330 (95b:53013)
  • [K] W. Klingenberg, Zur affinen Differentialgeometrie, Teil I: Uber p-dimensionale Minimal-flächen und Sphären im n-dimensionalen Raum, Math. Z. 54 (1951), 65-80. MR 0049637 (14:206b)
  • [NV] K. Nomizu and L. Vrancken, A new equiaffine theory for surfaces in $ {\mathbb{R}^4}$, Internat. J. Math. 4 (1993), 127-165. MR 1209963 (94f:53014)
  • [SV] C. Scharlach and L. Vrancken, Affine transversal planes for surfaces in $ {\mathbb{R}^4}$, Geometry and Topology of Submanifolds V, World Scientific, Singapore, 1993, pp. 249-253. MR 1339979 (96k:53009)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53A15, 53B05

Retrieve articles in all journals with MSC: 53A15, 53B05

Additional Information

Keywords: Affine differential geometry, higher codimension, planar geodesies
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society