0000000000024068
AUTHOR
Maria Del Carmen Romero Fuster
showing 8 related works from this author
The horospherical Gauss-Bonnet type theorem in hyperbolic space
2006
We introduce the notion horospherical curvatures of hypersurfaces in hyperbolic space and show that totally umbilic hypersurfaces with vanishing cur- vatures are only horospheres. We also show that the Gauss-Bonnet type theorem holds for the horospherical Gauss-Kronecker curvature of a closed orientable even dimensional hypersurface in hyperbolic space. + (i1) by using the model in Minkowski space. We introduced the notion of hyperbolic Gauss indicatrices slightly modified the definition of hyperbolic Gauss maps. The notion of hyperbolic indicatrices is independent of the choice of the model of hyperbolic space. Using the hyperbolic Gauss indicatrix, we defined the principal hyperbolic curv…
Inflection points and topology of surfaces in 4-space
2000
We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincare-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.
Umbilicity of space-like submanifolds of Minkowski space
2004
We study some properties of space-like submanifolds in Minkowski n-space, whose points are all umbilic with respect to some normal field. As a consequence of these and some results contained in a paper by Asperti and Dajczer, we obtain that being ν-umbilic with respect to a parallel light-like normal field implies conformal flatness for submanifolds of dimension n − 2 ≥ 3. In the case of surfaces, we relate the umbilicity condition to that of total semi-umbilicity (degeneracy of the curvature ellipse at every point). Moreover, if the considered normal field is parallel, we show that it is everywhere time-like, space-like or light-like if and only if the surface is included in a hyperbolic 3…
Flat lightlike hypersurfaces in Lorentz–Minkowski 4-space
2009
Abstract The lightlike hypersurfaces in Lorentz–Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.
Global properties of codimension two spacelike submanifolds in Minkowski space
2009
Abstract We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplanes and hyperquadrics in order to get some global information on them. As a consequence we obtain new versions of Carathéodory's and Loewner's conjectures on spacelike surfaces in 4-dimensional Minkowski space and 4-flattenings theorems for closed spacelike curves in 3-dimensional Minkowski space.
Curvature locus and principal configurations of submanifolds of Euclidean space
2017
We study relations between the properties of the curvature loci of a submanifold M in Euclidean space and the behaviour of the principal configurations of M, in particular the existence of umbilic and quasiumbilic fields. We pay special attention to the case of submanifolds with vanishing normal curvature. We also characterize local convexity in terms of the curvature locus position in the normal space.
The geometry of surfaces in 4-space from a contact viewpoint
1995
We study the geometry of the surfaces embedded in ℝ4 through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any generic convexly embedded 2-sphere in ℝ4 has inflection points.
A Global View on Generic Geometry
2018
We describe how the study of the singularities of height and distance squared functions on submanifolds of Euclidean space, combined with adequate topological and geometrical tools, shows to be useful to obtain global geometrical properties. We illustrate this with several results concerning closed curves and surfaces immersed in \(\mathbb {R}^n\) for \(n=3,4, 5\).