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On Pappus-Type Theorems on the Volume in Space Forms

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Abstract

Let c be a curve in a n-dimensional space form M λ n, let P t bea totally geodesic hypersurface of M λ n orthogonal to c at c(t),and let D0 be a domain in P 0. If D is thedomain in M λ n obtained by a ‘motion’ of D0 alongc and D t is the domain in P t obtained by the motion ofD0 from 0 to t, we show that the n-volume ofD depends only on the length of the curve c, its first curvature, themodified (n − 1)-volume of D0 and the moment of D t with respect to the totally geodesic hypersurface of P t orthogonal to the normal vector f 2(t) of c. As a consequence, if c(0) is the center of mass of D0, then the n-volume ofD is the product of the modified (n − 1)-volume of D0 and the length of c. We get an analogous theorem for ahypersurface of M λ n obtained by ‘parallel motion’ of ahypersurface of P 0 along c.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, U.S.A.

    Alfred Gray

  2. Departamento de Geometrí a y Topología, Universidad de Valencia, Burjasot, Valencia, Spain

    Vicente Miquel

Authors
  1. Alfred Gray
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  2. Vicente Miquel
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Gray, A., Miquel, V. On Pappus-Type Theorems on the Volume in Space Forms. Annals of Global Analysis and Geometry 18, 241–254 (2000). https://doi.org/10.1023/A:1006765422116

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