Veronese map explained

The Veronese map of degree 2 is a mapping from

\R ^{n+1} to the space of symmetric matrices (n +1 ){ x }(n +1 ) defined by the formula:^[1] V\colon(x _{0,...,x} _{n)\to \begin{pmatrix} x} _{0 ⋅} x _{0&x} _{0 ⋅} x _{1&...&x} _{0 ⋅} x _{n \\ x} _{1 ⋅} x _{0&x} _{1 ⋅} x _{1&...&x} _{1 ⋅} x _{n \\ \vdots&\vdots&\ddots&\vdots \\ x} _{n ⋅} x _{0&x} _{n ⋅} x _{1&...&x} _{n ⋅} x _{n \end{pmatrix}.} Note that V(x) = V( - x) for any x\in\R ^{n+1} .

In particular, the restriction of

V to the unit sphere S ^{n} factors through the projective space \R P ^{n} , which defines Veronese embedding of \R P ^{n} . The image of the Veronese embedding is called the Veronese submanifold, and for n =2 it is known as the Veronese surface.^[2]

Properties

• The matrices in the image of the Veronese embedding correspond to projections onto one-dimensional subspaces in

\R ^{n+1} . They can be described by the equations:A ^{T=A,} tr A =1 , A ^{2=A.}

In other words, the matrices in the image of

\R P ^{n} have unit trace and unit norm. Specifically, the following is true:

• The image lies in an affine space of dimension

n + \tfrac {n ⋅ (n +1 )} 2 .

• The image lies on an

(n -1+ \tfrac {n ⋅ (n +1 )} 2 ) -sphere with radius r _{n=} \sqrt{ 1- \tfrac 1 {n +1 }} .

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

2 ⋅ g , where g denotes the canonical metric on \R P ^{n-1} .

• The Veronese embedding maps each geodesic in

\R P ^{n-1} to a circle with radius \tfrac 1 {\sqrt{ 2 }} .

• • In particular, all the normal curvatures of the image are equal to

\sqrt{ 2 } .

• 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.

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.

Notes and References

1. Book: Lectures on Discrete Geometry . Springer Science & Business Media . 978-0-387-95374-8 . 244 . en.
2. Book: Hazewinkel . Michiel . Encyclopaedia of Mathematics: Stochastic Approximation — Zygmund Class of Functions . 31 January 1993 . Springer Science & Business Media . 978-1-55608-008-1 . 416 . en.