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VERONESE SURFACE

Giuseppe Veronese (1854-1917): Italian mathematician.
See also Wikipedia, BERGER, page 47. |

Cartesian parametrization: , with . |

The *Veronese surface* is the image of the quotient
of the 2-dimensional sphere by the antipodal relation (in other words,
the real projective plane),
by the map: .

Since this function is injective, the Veronese surface is a surface (i.e. a 2-manifold) without singular points embedded in (since it is included in the hyperplane of ) and homeomorphic to the real projective plane.

The "projection"
defines a homeomorphism of the Veronese surface onto its image, which is
therefore an *embedding* of the real projective plane in .

However, all the "projections" of this type of the surface
in , called
Steiner surfaces, have singular
points:

For example, the "projection"
maps the Veronese surface onto the Roman
surface, and the projection
maps it onto the cross-cap.

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© Robert FERRÉOL
2017