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Lightcone dualities for hypersurfaces in the sphere. (English) Zbl 1307.53011

The unit sphere \(S^n\) can be naturally embedded into the lightcone and into the de Sitter space in the Minkowski space \(M^{n+2}\). In this context, hypersurfaces in \(S^n\) give rise to dual hypersurfaces using Legendrian dualities and lightcone dualities.
In previous works, curves in the unit 2-sphere and the unit 3-sphere were studied using Legendrian duality. For example, the evolute of a curve in the unit 2-sphere is the dual of the tangent indicatrix of the original curve.
In the present work, hypersurfaces in the unit \(n\)-sphere are studied using the theory of Legendrian singularities. The special case are curves in the unit 2-sphere. The geometric meaning of singularities of lightcone dual hypersurfaces is interpreted using the contact of hypersurfaces with parabolic spheres and parabolic hyperquadrics. As an application, the singularities of the lightcone dual hypersurfaces for surfaces in the unit 3-sphere are classified.

MSC:

53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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