On Pappus-Type Theorems on the Volume in Space Forms

Let c be a curve in a n-dimensional space form Mⁿ, let P_t bea totally geodesic hypersurface of Mⁿ orthogonal to c at c(t),and let D₀ be a domain in P₀. If D is thedomain in Mⁿ 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_torthogonal to the normal vector f₂(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 Mⁿ obtained by parallel motion of ahypersurface of P₀ along c.