The Veronese map of degree 2 is a mapping from
to the space of symmetric matrices
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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e66b518be0dc7b58cf87662e7ac9fac6898b20b5)
Note that
for any
.
In particular, the restriction of
to the unit sphere
factors through the projective space
, which defines Veronese embedding of
. The image of the Veronese embedding is called the Veronese submanifold, and for
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
. They can be described by the equations: ![{\displaystyle A^{T}=A,\quad \mathrm {tr} \,A=1,\quad A^{2}=A.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f30bdedd9168a712ecf56b4a91fd46d882961231)
- In other words, the matrices in the image of
have unit trace and unit norm. Specifically, the following is true: - The image lies in an affine space of dimension
. - The image lies on an
-sphere with radius
. - Moreover, the image forms a minimal submanifold in this sphere.
- 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.
Notes
- ^ Lectures on Discrete Geometry. Springer Science & Business Media. p. 244. ISBN 978-0-387-95374-8.
- ^ Hazewinkel, Michiel (31 January 1993). Encyclopaedia of Mathematics: Stochastic Approximation — Zygmund Class of Functions. Springer Science & Business Media. p. 416. ISBN 978-1-55608-008-1.
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.