• 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,\quad \mathrm{tr}\,A=1,\quad A^2=A. :In other words, the matrices in the image of \R\mathrm{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\cdot(n+1)}2. :*The image lies on an (n-1+\tfrac {n\cdot(n+1)}2)-sphere with radius r_n= \sqrt{1-\tfrac1{n+1}}. :**Moreover, the image forms a
minimal submanifold in this sphere. • The Veronese embedding induces a
Riemannian metric 2\cdot g, where g denotes the canonical metric on \R\mathrm{P}^{n-1}. • The Veronese embedding maps each geodesic in \R\mathrm{P}^{n-1} to a circle with radius \tfrac1{\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==