Affine homogeneous surfaces in \(\mathbb{R}^ 3\) with vanishing Pick invariant
A surface \(M\) in \(\mathbb{R}^3\) is called a locally homogeneous surface if for any \(p\in M\) there exists a neighborhood \(U_p\) of \(p\) in \(M\) and a mapping \(\varphi\) from \(U_p\) to the affine transformation group in \(\mathbb{R}^3\), such that for any \(q\in U_p\) the affine transformation \(\varphi(q)\) maps \(p\) to \(q\) and \(U_p\) to \(M\). In this paper, the authors classify all nondegenerate locally homogeneous surfaces in \(\mathbb{R}^3\) with vanishing Pick invariant. They show that any such surface is affinely equivalent to an open part of one of the following surfaces: 1. a nondegenerate quadric; 2. \(Z=XY+ \ln X\); 3. \(Z=XY +X^b\), \(b\in\mathbb{R}\); 4. \(Z=XY +e^x\); 5. \(Z=XY -X\ln X\); 6. \(Z=XY -X^2\ln X\); 7. \(Z=YX^b\), \(b\in \mathbb{R}\); 8. \(Z=e^{-\alpha \arctan (Y/X)}\); 9. \(Z=X \ln Y\); 10. \(Z=e^XY\); 11. \(\tan Z=Y/X\).
non-degenerate surfaces in \(\mathbb{R}^ 3\)
scientific article; zbMATH DE number 1114164