The Veronese map to the space of symmetric matrices defined by the formula:

.

factors through the projective space, which defines the Veronese embedding. The image of the Veronese embedding is called the Veronese submanifold it is known as the Veronese surface.

Properties

• . They can be described by the equations:

have unit trace and unit norm. Specifically, the following is true:

• The image lies in an affine space.

.

• Moreover, the image forms a minimal submanifold in this sphere.

  The Veronese embedding induces a Riemannian metric.

.

• In particular, all the normal curvatures.

• 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.

Notes

References

• Cecil, T. E.; Ryan, P. J. Tight and taut immersions of manifolds Res. Notes in Math., 107, 1985.

• K. Sakamoto, Planar geodesic immersions, Tohoku Math. J., 29 (1977), 25–56.