On Pappus-type theorems on the volume in space forms (Q1586049)

A formula of Pappus states that the volume of the set generated by rotation of a plane domain \(D_0\) in \(\mathbb{R}^3\) around an axis, which does not meet \(D_0\) but lies in the same plane as \(D_0\), is the product of the are of \(D_0\) and the length of the circle \(c\) described by the center of mass of \(D_0\). An analogous formula holds for the area of a surface in \(\mathbb{R}^3\) generated by the rotation of a curve. It is also known that the volume of a tube around a curve \(c\) in \(\mathbb{R}^n\) only depends on the radius of the tube and on the length of \(c\). Thus, in each of the above cases, the expression for the considered volume or area does not involve the curvature of \(c\). In this paper the authors examine a more general situation, considering a curve \(c:[0,L] \to M^n_\lambda\), where \(M^n_\lambda\) denotes an \(n\)-dimensional space-form of sectional curvature \(\lambda\). For any \(t\in [0,L]\), let \(P_t\) be a totally geodesic hypersurface in \(M^n_\lambda\) orthogonal to \(c\) at \(c(t)\) and fix a domain \(D_0\) in \(P_0\). Denote by \(D_t\) the domain in \(P_t\) obtained by the motion of \(D_0\) from 0 to \(t\) and by \(D\) the solid obtained by the motion of \(D_0\) along \(c\). The authors prove an integral formula stating that the \(n\)-volume of \(D\) only depends on the length of \(c\), the first curvature of \(c\), the modified \(n-1\)-volume of \(D_0\) and the moment of \(\{D_t\}_{t\in [0,L]}\). For any \(t\), the moment of \(D_t\) is evaluated with respect to the totally geodesic hypersurface of \(P_t\) orthogonal to the normal vector \(f_2(t)\) of \(c\), where \((f_1(t), f_2(t), \dots, f_n(t))\) is a Frénet frame. In particular, if \(c(0)\) is the center of mass of \(D_0\), the \(n\)-volume of \(D\) is the product of the length of \(c\) and the modified \(m-1\)-volume of \(D_0\). Finally, the authors state an analogous theorem for the volume of a hypersurface of \(M^n_\lambda\) obtained by parallel motion of a hypersurface of \(P_0\) along \(c\).