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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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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DOI: https://doi.org/10.1023/A:1006765422116