Differential Geometry from a Singularity Theory Viewpoint provides a new look at the fascinating and classical subject of the differential geometry of surfaces in Euclidean spaces. The book uses ...singularity theory to capture some key geometric features of surfaces. It describes the theory of contact and its link with the theory of caustics and wavefronts. It then uses the powerful techniques of these theories to deduce geometric information about surfaces embedded in 3, 4 and 5-dimensional Euclidean spaces. The book also includes recent work of the authors and their collaborators on the geometry of sub-manifolds in Minkowski spaces.
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.
Inflection points and topology of surfaces in 4-space Garcia, Ronaldo Alves; Mochida, Dirce Kiyomi Hayashida; Romero Fuster, Maria del Carmen ...
Transactions of the American Mathematical Society,
07/2000, Letnik:
352, Številka:
7
Journal Article
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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 Poincaré-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 Izumiya, Shyuichi; Pei, Donghe; del Carmen Romero Fuster, María
Proceedings of the Royal Society of Edinburgh. Section A. Mathematics,
04/2004, Letnik:
134, Številka:
2
Journal Article
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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-space, a de Sitter 3-space or a three-dimensional light cone, respectively. We also give characterizations of total semi-umbilicity for surfaces contained in hyperbolic 4-space, de Sitter 4-space and four-dimensional light cone.
We introduce the notion horospherical curvatures of hypersurfaces in hyperbolic space andshow that totally umbilic hypersurfaces with vanishing curvatures 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.
We associate a graph with weights in its vertices and edges to any stable map from a 3-manifold to
. These graphs are
-invariants from a global viewpoint. We study their properties and give a ...sufficient and necessary condition for a graph to be the graph of a stable map from a 3-sphere with handles to
. We also obtain a sufficient condition in the general case of a closed stably paralellizable 3-manifold in terms of its Heegaard genus.
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