6533b853fe1ef96bd12ac11d
RESEARCH PRODUCT
Pappus type theorems for motions along a submanifold
Vicente MiquelM. Carmen Domingo-juansubject
Mean curvatureGeodesicVolumeSpace formParallel motionImage (category theory)Second fundamental formMathematical analysisSubmanifoldMotion along a submanifoldCombinatoricsHypersurfaceComputational Theory and MathematicsTubePappus formulaeLie algebraDomain (ring theory)Comparison theoremMathematics::Differential GeometryGeometry and TopologyAnalysisMathematicsdescription
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 is closed, then volume( C ) does not depend on the i th fundamental forms of P for i >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 geodesic plane.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2004-09-01 | Differential Geometry and its Applications |