Affine surfaces in \(\mathbb{R}^5\) with zero cubic form (Q5936469)

The authors classify affine surfaces with zero cubic form in \(\mathbb{R}^5\). They prove: NEWLINENEWLINENEWLINETheorem 1. Let \(f:M^2\to \mathbb{R}^5\) be a nondegenerate affine surface immersed in \(\mathbb{R}^5\) with zero cubic form and zero Ricci tensor. Then, locally, the image of \(M^2\) is the image under an affine motion of \((x_1,x_2, \frac 12 x_1^2,x_2x_2, \frac 12x_2^2)\). NEWLINENEWLINENEWLINETheorem 2. Let \(f:M^2\to \mathbb{R}^5\) be a nondegenerate affine surface immersed in \(\mathbb{R}^2\) with zero cubic form and Ricci tensor of rank 1. Then, locally, the image of \(M^2\) is the image under an affine motion of NEWLINE\[NEWLINE(- \tfrac 12 \tau c(2u), \tfrac 12 s(2u), -\tau\nu c(u), \nu s(u), \tfrac 12 \nu^2),NEWLINE\]NEWLINE where \(\tau=\pm 1\) and, when \(\tau=1\), then \(c(x)= \cos(x)\) and \(s(x)= \sin(x)\); when \(\tau=-1\), then \(c(x)= \cosh(x)\) and \(s(x)= \sinh(x)\). NEWLINENEWLINENEWLINETheorem 3. Let \(f:M^2\to \mathbb{R}^5\) be a nondegenerate affine surface immersed in \(\mathbb{R}^5\) with zero cubic form and zero Ricci tensor of rank 2. Then there is a (pseudo-) Riemannian metric on \(\mathbb{R}^5\) such that the image of \(M^2\) is an open subset of the Veronese surface in \(S^4\) or an open subset of the Lorentzian Veronese surface in \(S_2^4\).