Search results for "Motion along a submanifold"

Pappus type theorems for motions along a submanifold

2004

Abstract 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 λ . We show: (a) volume( D ) depends only on the second fundamental form of  P , whereas volume( C ) depends on all the i th fundamental forms of  P , (b) when the domain that we move D 0 has its q -centre of mass on  P , volume( D ) does not depend on the mean curvature of  P , (c) when D 0 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 …