Three dimensional locally homogeneous nondegenerate centroaffine hypersurfaces with null Tchebychev vector field (Q2452337)

The author gives the following classification theorem. Theorem. Let \(M\) be a connected 3-dimensional locally homogeneous nondegenerate centroaffine hypersurface in \(\mathbb R^4\) with null Tchebychev vector field. Then \(M\) is centroaffinely equivalent to an open part of one of the following surfaces: (1) \(x_1^{\alpha_1}x_2^{\alpha_2}x_3^{\alpha_3}x_4^{\alpha_4}=1\), (2) \(\exp(-\alpha_1\arctan{{x_1}\over{x_2}})(x_1^2+x_2^2)^{\alpha_2}x_3^{\alpha_3}x_4^{\alpha_4}=1\), (3) \(\exp(-\alpha_1\arctan{{x_1}\over{x_2}}-\alpha_2\arctan{{x_3}\over{x_4}}) (x_1^2+x_2^2)^{\alpha_3}(x_3^{2}+x_4^2)^{\alpha_4}=1\), (4) \(x_4=-x_1(\alpha_1\log x_1+\alpha_2\log x_2+\alpha_3\log x_3)\), (5) \(x_4={{x_2^2}\over{2x_1}}-x_1(\alpha_1\log x_1+\alpha_3\log x_3)\), (6) \(x_4=x_3(\alpha_1\arctan{{x_1}\over{x_2}}+{\alpha_2\over 2}\log(x_1^2+x_2^2)+\alpha_3\log x_3)\), (7) \(x_4={x_2^2\over 2x_1}+\delta x_1^{\alpha_1}x_3^{\alpha_3}\), (8) \(x_4={x_2^2\over 2x_1}+\delta x_1^{\alpha_1}\exp{x_3\over x_1}\), (9) \(x_4={x_2^2\over 2x_1}-x_1(\alpha_1\log x_1+\alpha_3\log x_3)\), where \(\delta=\pm 1, \alpha_1,\alpha_2,\alpha_3,\alpha_4\) are constants and satisfy certain conditions.