Veronese map


The Veronese map of degree 2 is a mapping from to the space of symmetric matrices defined by the formula:
Note that for any.
In particular, the restriction of to the unit sphere factors through the projective space, which defines the Veronese embedding of. The image of the Veronese embedding is called the Veronese submanifold, and for it is known as the Veronese surface.

Properties

  • The matrices in the image of the Veronese embedding correspond to projections onto one-dimensional subspaces in. They can be described by the equations:
  • :
  • The Veronese embedding induces a Riemannian metric, where denotes the canonical metric on.
  • The Veronese embedding maps each geodesic in to a circle with radius.
  • *In particular, all the normal curvatures of the image are equal to.
  • The Veronese manifold is extrinsically symmetric, meaning that reflection in any of its normal spaces maps the manifold onto itself.

Variations and generalizations

Analogous Veronese embeddings are constructed for complex and quaternionic projective spaces, as well as for the Cayley plane.