Pappus type theorems for hypersurfaces in a space form
- 67 Downloads
In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following situation. Letc(t) be a curve in a space formM λ n of sectional curvature λ. LetP 0 be a totally geodesic hypersurface ofM λ n throughc(0) and orthogonal toc(t). LetC 0 be a hypersurface ofP 0. LetC be the hypersurface ofM λ n obtained by a motion ofC 0 alongc(t). We shall denote it byC PorC Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC 0, then volume(C) ≥ volume(C),P),and the equality holds whenC 0 is contained in a geodesic sphere or the motion corresponds to a curve contained in a hyperplane of the Lie algebraO(n−1) (whenn=3, the only motion with these properties is the parallel motion).
KeywordsSectional Curvature Space Form Parallel Motion Geodesic Sphere Frenet Frame
Unable to display preview. Download preview PDF.
- [BK]P. Buser and H. Karcher,Gromov’s almost flat manifolds, Astérisque81 (1981).Google Scholar
- [NB]J. J. Nuño-Ballesteros,Bitangency properties of generic closed curves in ℝ n,inReal and Complex Singularities (J. W. Bruce and F. Tari, eds.), Research Notes in Mathematics, Vol. 412, Chapman and Hall/CRC, Boca Raton, London, 2000, pp. 188–201.Google Scholar
- [Wa]C. T. C. Wall,Geometric properties of generic differentiable manifolds, inGeometry and Topology, Rio de Janeior, July 1976 (J. Palis and M. do Carmo, eds.), Lecture Notes in Mathematics597, Springer-Verlag, Berlin, 1977, pp. 707–774Google Scholar