Search results for "Second fundamental form"

showing 4 items of 4 documents

Hadamard-type theorems for hypersurfaces in hyperbolic spaces

2006

Abstract We prove that a bounded, complete hypersurface in hyperbolic space with normal curvatures greater than −1 is diffeomorphic to a sphere. The completeness condition is relaxed when the normal curvatures are bounded away from −1. The diffeomorphism is constructed via the Gauss map of some parallel hypersurface. We also give bounds for the total curvature of this parallel hypersurface.

Pure mathematicsGauss mapMathematics::Dynamical SystemsMathematics::Complex VariablesHyperbolic spaceSecond fundamental formMathematical analysisCauchy–Hadamard theoremGauss–Kronecker curvatureSecond fundamental formHypersurfaceMathematics::Algebraic GeometryComputational Theory and MathematicsBounded functionHadamard theoremTotal curvatureDiffeomorphismGeometry and TopologyMathematics::Differential GeometryAnalysisConvex hypersurfaceMathematicsDifferential Geometry and its Applications
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ASYMPTOTIC CURVES ON SURFACES IN ℝ5

2008

We study asymptotic curves on generically immersed surfaces in ℝ5. We characterize asymptotic directions via the contact of the surface with flat objects (k-planes, k = 1 - 4), give the equation of the asymptotic curves in terms of the coefficients of the second fundamental form and study their generic local configurations.

Surface (mathematics)Asymptotic curveAsymptotic analysisApplied MathematicsGeneral MathematicsSecond fundamental formMathematical analysisGravitational singularityAsymptotic expansionMathematicsCommunications in Contemporary Mathematics
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Boundary reconstruction for the broken ray transform

2013

We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary. For determination of the values of the function itself we obtain the usual geodesic ray transform, but for derivatives this transform has to be weighted by powers of the second fundamental form. The problem studied here is related to Calder\'on's problem with partial data.

Mathematics - Differential GeometryDifferential Geometry (math.DG)GeodesicAstrophysics::High Energy Astrophysical PhenomenaGeneral MathematicsSecond fundamental formta111Mathematical analysisFOS: MathematicsBoundary (topology)Function (mathematics)53C65 78A05 (Primary) 35R30 58J32 (Secondary)MathematicsAnnales Academiae Scientiarum Fennicae Mathematica
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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 …

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 TopologyAnalysisMathematicsDifferential Geometry and its Applications
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