Pappus type theorems for motions along a submanifold

We study the volumes volume(D) of a domain D and volume(C) of a hypersurface C obtained by a motion along a submanifold P of a space form M^n(lambda). We show: (a) volume(D) depends only on the second fundamental form of P, whereas volume(C) depends on all the ith fundamental forms of P, (b) when the domain that we move D0 has its q -centre of mass on P, volume(D) does not depend on the mean curvature of P, (c) when D0 is q-symmetric, volume(D) depends only on the intrinsic curvature tensor of P; and (d) if the image of P by the ln of the motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra of SO(n-q-d), and C is closed, then volume(C) does not depend on the ith fundamental forms of P for i(greater than)2 if and only if the hypersurface that we move is a revolution hypersurface (of the geodesic (n-q)-plane orthogonal to P) around a d-dimensional