Pappus type theorems for hypersurfaces in a space form
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 formMλn of sectional curvature λ. LetP0 be a totally geodesic hypersurface ofMλn throughc(0) and orthogonal toc(t). LetC0 be a hypersurface ofP0. LetC be the hypersurface ofMλn obtained by a motion ofC0 alongc(t). We shall denote it byCPorCFif 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 ofC0, then volume(C) ≥ volume(C),P),and the equality holds whenC0 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).
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