https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincare ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.

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Singularities of hyperbolic Gauss maps

We show four Legendrian dualities between pseudo-spheres in Minkowski space as a basic theorem. We can apply such dualities for constructing extrinsic differential geometry of spacelike hypersurfaces in pseudo-spheres. In this paper we stick to spacelike hypersurfaces in the lightcone and establish an extrinsic differential geometry which we call the lightcone differential geometry.

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Legendrian dualities and spacelike hypersurfaces in the lightcone

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.

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Inflection points and topology of surfaces in 4-space

Recently we discovered a new geometry on submanifolds in

 hyperbolic $n$-space

 which is called {\it horospherical geometry}

 Unfortunately this geometry is not invariant under the hyperbolic motions (it is invariant under the canonical action of $SO(n)$), but it has quite interesting

 features

 For example, the flatness in this geometry is a hyperbolic invariant and the total curvatures are topological invariants In this paper, we investigate the {\it horospherical flat surfaces}

 (flat surfaces in the sense of horospherical geometry) in hyperbolic $3$-space

 Especially, we give a generic classification of singularities

 of such surfaces As a consequence, we can say that such a class of surfaces has

 quite a rich geometric structure

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Horospherical flat surfaces in Hyperbolic 3-space

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.

The geometry of surfaces in 4-space from a contact viewpoint

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.

Differential Geometry from a Singularity Theory Viewpoint

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 curvatures ¯ • § and the hyperbolic Gauss-Kronecker curvature K § h by exactly the same way as the definition of those of classical Gaussian dif- ferential geometry in Euclidean space. Totally umbilic hypersurfaces with respect to the above curvatures are equidistant hypersurfaces, hyperspheres or hyperhorospheres which are called model hypersurfaces in hyperbolic space. The hyperbolic Gauss-Kronecker curvature is a hyperbolic invariant which describes the contact of hypersurfaces with such model hypersurfaces. We remark that Kobayashi (13), (14) had already defined the notion of hyperbolic Gauss-Kronecker curvature under a dierent framework and studied some basic properties of it from the view point of the theory of Fourier transformations. In this paper we introduce the principal horospherical curvature e § (cf., x3). This new curvature is not a hyperbolic invariant but an SO(n)-invariant, where we consider the canonical SO(n)-subgroup in the group of hyperbolic motions. However, we can show that e § (p) is invariant under hyperbolic motions if and only if e § (p) = 0. We can

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The horospherical Gauss-Bonnet type theorem in hyperbolic space

/pdf/umbilicity-of-space-like-submanifolds-of-minkowski-space-3mhaopf9qp.pdf

Maria del Carmen Romero Fuster

Papers

The geometry of surfaces in 4-space from a contact viewpoint

Differential Geometry from a Singularity Theory Viewpoint

Inflection points and topology of surfaces in 4-space

The horospherical Gauss-Bonnet type theorem in hyperbolic space

Umbilicity of space-like submanifolds of Minkowski space