Veronese map

The Veronese map of degree 2 is a mapping from ${\displaystyle \mathbb {R} ^{n+1}}$ to the space of symmetric matrices ${\displaystyle (n+1){\times }(n+1)}$ defined by the formula:^[1]

${\displaystyle V\colon (x_{0},\dots ,x_{n})\to {\begin{pmatrix}x_{0}\cdot x_{0}&x_{0}\cdot x_{1}&\dots &x_{0}\cdot x_{n}\\x_{1}\cdot x_{0}&x_{1}\cdot x_{1}&\dots &x_{1}\cdot x_{n}\\\vdots &\vdots &\ddots &\vdots \\x_{n}\cdot x_{0}&x_{n}\cdot x_{1}&\dots &x_{n}\cdot x_{n}\end{pmatrix}}.}$

Note that ${\displaystyle V(x)=V(-x)}$ for any ${\displaystyle x\in \mathbb {R} ^{n+1}}$.

In particular, the restriction of ${\displaystyle V}$ to the unit sphere ${\displaystyle \mathbb {S} ^{n}}$ factors through the projective space ${\displaystyle \mathbb {R} \mathrm {P} ^{n}}$, which defines Veronese embedding of ${\displaystyle \mathbb {R} \mathrm {P} ^{n}}$. The image of the Veronese embedding is called the Veronese submanifold, and for ${\displaystyle n=2}$ 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 ${\displaystyle \mathbb {R} ^{n+1}}$. They can be described by the equations:

  ${\displaystyle A^{T}=A,\quad \mathrm {tr} \,A=1,\quad A^{2}=A.}$

In other words, the matrices in the image of ${\displaystyle \mathbb {R} \mathrm {P} ^{n}}$ have unit trace and unit norm. Specifically, the following is true:

• The image lies in an affine space of dimension ${\displaystyle n+{\tfrac {n\cdot (n+1)}{2}}}$.
• The image lies on an ${\displaystyle (n-1+{\tfrac {n\cdot (n+1)}{2}})}$-sphere with radius ${\displaystyle r_{n}={\sqrt {1-{\tfrac {1}{n+1}}}}}$.
  • Moreover, the image forms a minimal submanifold in this sphere.

• The Veronese embedding induces a Riemannian metric ${\displaystyle 2\cdot g}$, where ${\displaystyle g}$ denotes the canonical metric on ${\displaystyle \mathbb {R} \mathrm {P} ^{n-1}}$.
• The Veronese embedding maps each geodesic in ${\displaystyle \mathbb {R} \mathrm {P} ^{n-1}}$ to a circle with radius ${\displaystyle {\tfrac {1}{\sqrt {2}}}}$.
  • In particular, all the normal curvatures of the image are equal to ${\displaystyle {\sqrt {2}}}$.
• 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.