Pappus type theorems for hypersurfaces in a space form |
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Authors: | M Carmen Domingo-Juan Ximo Gual Vicente Miquel |
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Institution: | (1) Departamento de Economía Financiera y Matemática, Universidad de Valencia, Valencia, Spain;(2) Departamento de Matemáticas, Universitat Jaume I, Castellón, Spain;(3) Departamento de Geometría y Topología, Universidad de Valencia, Burjasot (Valencia), Spain |
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Abstract: | 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).
Work partially supported by a DGES Grant No. PB97-1425 and a AGIGV Grant No. GR0052. |
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Keywords: | |
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