Search results for "Mathematics::Differential Geometry"
showing 10 items of 209 documents
Malliavin calculus of Bismut type without probability
2007
We translate in semigroup theory Bismut's way of the Malliavin calculus.
Geometric contacts of surfaces immersed in Rn, n⩾5
2009
Abstract We study the extrinsic geometry of surfaces immersed in R n , n ⩾ 5 , by analyzing their contacts with different standard geometrical models, such as hyperplanes and hyperspheres. We investigate the relation between different types of contact and the properties of the curvature ellipses at each point. In particular, we focalize our attention on the hyperspheres having contacts of corank two with the surface. This leads in a natural way to the concept of umbilical focus and umbilic curvature.
Principal configurations and umbilicity of submanifolds in $\mathbb R^N$
2004
We consider the principal configurations associated to smooth vector fields $\nu$ normal to a manifold $M$ immersed into a euclidean space and give conditions on the number of principal directions shared by a set of $k$ normal vector fields in order to guaranty the umbilicity of $M$ with respect to some normal field $\nu$. Provided that the umbilic curvature is constant, this will imply that $M$ is hyperspherical. We deduce some results concerning binormal fields and asymptotic directions for manifolds of codimension 2. Moreover, in the case of a surface $M$ in $\mathbb R^N$, we conclude that if $N>4$, it is always possible to find some normal field with respect to which $M$ is umbilic and …
Blending Planes and Canal Surfaces Using Dupin Cyclides
2011
We develop two different new algorithms of G1-blending between planes and canal surfaces using Dupin cyclides. It is a generalization of existing algorithms that blend revolution surfaces and planes using a plane called construction plane. Spatial constraints were necessary to do that. Our work consist in building three spheres to determine the Dupin cyclide of the blending. The first algorithm is based on one of the definitions of Dupin cyclides taking into account three spheres of the same family enveloping the cyclide. The second one uses only geometric properties of Dupin cyclide. The blending is fixed by a circle of curvature onto the canal surface. Thanks to this one, we can determine…
Curves as measured foliation on noncompact surfaces
1993
In the present work, that regards the Thurston's theory, we prove that, if we choose a closed curve, how we wish, on a noncompact surface, it is always possible to construct a particular masured foliation that has the choosed curve like a leaf; we also prove this foliation has a remarkable property that makes very easy to mesure all homotopy classes of closed curves of our surface. To prove this statement we need some Propositions and some Lemma that we also demonstre.
Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points
2021
Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. A Dupin cyclide can be defined as the envelope of a one-parameter family of oriented spheres, in two different ways. R. Martin is the first author who thought to use these surfaces in CAD/CAM and geometric modeling. The Minkowski-Lorentz space is a generalization of the space-time used in Einstein’s theory, equipped of the non-degenerate indefinite quadratic form $$Q_{M} ( \vec{u} ) = x^{2} + y^{2} + z^{2} - c^{2} t^{2}$$ where (x, y, z) are the spacial components of the vector $$ \vec{u}$$ and t is the time component of $$ \vec{u}$$ and c is the constant of the spee…
Spacelike energy of timelike unit vector fields on a Lorentzian manifold
2004
On a Lorentzian manifold, we define a new functional on the space of unit timelike vector fields given by the L2 norm of the restriction of the covariant derivative of the vector field to its orthogonal complement. This spacelike energy is related with the energy of the vector field as a map on the tangent bundle endowed with the Kaluza–Klein metric, but it is more adapted to the situation. We compute the first and second variation of the functional and we exhibit several examples of critical points on cosmological models as generalized Robertson–Walker spaces and Godel universe, on Einstein and contact manifolds and on Lorentzian Berger’s spheres. For these critical points we have also stu…
A historical account on characterizations ofC1-manifolds in Euclidean spaces by tangent cones
2014
Abstract A historical account on characterizations of C 1 -manifolds in Euclidean spaces by tangent cones is provided. Old characterizations of smooth manifold (by tangent cones), due to Valiron (1926, 1927) and Severi (1929, 1934) are recovered; modern characterizations, due to Gluck (1966, 1968) and Tierno (1997) are restated. All these results are consequences of the Four-cones coincidence theorem due to [1] .
Kähler manifolds with split tangent bundle
2006
( Varietes kahleriennes a fibre tangent scinde). - On etudie dans cet article les varietes kahleriennes compactes dont le fibre tangent se decompose en somme directe de sous-fibres. En particulier, on montre que si le fibre tangent se decompose en somme directe de sous-fibres en droites, alors la variete est uniformisee par un produit de courbes. Les methodes sont issues de la theorie des feuilletages de (co)dimension 1.
Volume-convergent sequences of Haken 3-manifolds
2003
Abstract Let M be a closed orientable 3-manifold and let Vol(M) denote its Gromov simplicial volume. This paper is devoted to the study of sequences of non-zero degree maps f i :M→N i to Haken manifolds. We prove that any sequence of Haken manifolds (Ni,fi), satisfying limi→∞deg(fi)×Vol(Ni)=Vol(M) is finite up to homeomorphism. As an application, we deduce from this fact that any closed orientable 3-manifold with zero Gromov simplicial volume and in particular any graph manifold dominates at most finitely many Haken 3-manifolds. To cite this article: P. Derbez, C. R. Acad. Sci. Paris, Ser. I 336 (2003).