Book contents
- Frontmatter
- Contents
- Preface
- 1 On a conjecture by A. Durfee
- 2 On normal embedding of complex algebraic surfaces
- 3 Local Euler obstruction, old and new, II
- 4 Branching of periodic orbits in reversible Hamiltonian systems
- 5 Topological invariance of the index of a binary differential equation
- 6 About the existence of Milnor fibrations
- 7 Counting hypersurfaces invariant by one-dimensional complex foliations
- 8 A note on topological contact equivalence
- 9 Bi-Lipschitz equivalence, integral closure and invariants
- 10 Solutions to PDEs and stratification conditions
- 11 Real integral closure and Milnor fibrations
- 12 Surfaces around closed principal curvature lines, an inverse problem
- 13 Euler characteristics and a typical values
- 14 Answer to a question of Zariski
- 15 Projections of timelike surfaces in the de Sitter space
- 16 Spacelike submanifolds of codimension at most two in de Sitter space
- 17 The geometry of Hopf and saddle-node bifurcations for waves of Hodgkin-Huxley type
- 18 Global classifications and graphs
- 19 Real analytic Milnor fibrations and a strong Łojasiewicz inequality
- 20 An estimate of the degree of ℒ-determinacy by the degree of A-determinacy for curve germs
- 21 Regularity of the transverse intersection of two regular stratifications
- 22 Pairs of foliations on surfaces
- 23 Bi-Lipschitz equisingularity
- 24 Gaffney's work on equisingularity
- 25 Singularities in algebraic data acquisition
16 - Spacelike submanifolds of codimension at most two in de Sitter space
Published online by Cambridge University Press: 07 September 2011
- Frontmatter
- Contents
- Preface
- 1 On a conjecture by A. Durfee
- 2 On normal embedding of complex algebraic surfaces
- 3 Local Euler obstruction, old and new, II
- 4 Branching of periodic orbits in reversible Hamiltonian systems
- 5 Topological invariance of the index of a binary differential equation
- 6 About the existence of Milnor fibrations
- 7 Counting hypersurfaces invariant by one-dimensional complex foliations
- 8 A note on topological contact equivalence
- 9 Bi-Lipschitz equivalence, integral closure and invariants
- 10 Solutions to PDEs and stratification conditions
- 11 Real integral closure and Milnor fibrations
- 12 Surfaces around closed principal curvature lines, an inverse problem
- 13 Euler characteristics and a typical values
- 14 Answer to a question of Zariski
- 15 Projections of timelike surfaces in the de Sitter space
- 16 Spacelike submanifolds of codimension at most two in de Sitter space
- 17 The geometry of Hopf and saddle-node bifurcations for waves of Hodgkin-Huxley type
- 18 Global classifications and graphs
- 19 Real analytic Milnor fibrations and a strong Łojasiewicz inequality
- 20 An estimate of the degree of ℒ-determinacy by the degree of A-determinacy for curve germs
- 21 Regularity of the transverse intersection of two regular stratifications
- 22 Pairs of foliations on surfaces
- 23 Bi-Lipschitz equisingularity
- 24 Gaffney's work on equisingularity
- 25 Singularities in algebraic data acquisition
Summary
Abstract
The aim of this paper is to provide a description of the main geometrical properties of spacelike submanifolds of codimension at most two in de Sitter space, that have been studied by the author with full details in other papers, as an application of the theory of Legendrian singularities. We analyze the geometrical meaning of the singularities of lightcone Gauss images, lightcone Gauss maps and lightlike hypersurfaces of generic spacelike surfaces in de Sitter 3-space and de Sitter 4-space.
Introduction
In this paper we consider de Sitter space, which is a Lorentzian space form with positive curvature defined by a pseudo n-sphere in Minkowski space. The spacelike curves in de Sitter 3-space are investigated in [5] and the lightlike surface of the spacelike curves are constructed from the Frenet-Serret type formula. In [9] the differential geometry of the timelike surfaces in de Sitter space are discussed, and the singularities of de Sitter Gauss images of timelike surfaces in de Sitter 3-space are classified. The principal, asymptotic and characteristic curves associated to the de Sitter Gauss maps are investigated in [10], and the contact of timelike surfaces with geodesic loci are investigated in [11]. In [12] we investigated the lightcone Gauss image of spacelike hypersurface in de Sitter space, which is the analogous tool in [6]. The singularities of the Gauss images correspond to the parabolic sets of spacelike hypersurfaces.
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- Real and Complex Singularities , pp. 211 - 228Publisher: Cambridge University PressPrint publication year: 2010