Abstract
We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the singularities of the family of affine distance functions appear at points on the affine normal plane and the affine focal points correspond to degenerate singularities of this family. Moreover we show that if M is immersed in a locally strictly convex hypersurface, then the affine normal plane contains the affine normal vector to the hypersurface and conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.
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The first author has been partially supported by DGICYT Grant MTM2012-33073 and CAPES-PVE. The second and third authors have been partially supported by CAPES, FAPESP and CNPq.
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Nuño-Ballesteros, J.J., Saia, M.J. & Sánchez, L.F. Affine Focal Points for Locally Strictly Convex Surfaces in 4-Space. Results Math 71, 357–376 (2017). https://doi.org/10.1007/s00025-016-0606-z
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DOI: https://doi.org/10.1007/s00025-016-0606-z