A characterization of the complex paraboloid
The authors study umbilical affine definite surfaces in \(\mathbb{R}^4\) with respect to the Nomizu-Vrancken normalization. They give an interesting inequality on the affine Gauss curvature, and using this inequality they prove that any umbilical affine definite surface in \(\mathbb{R}^4\) with complete Burstin-Mayer metric is affine equivalent to the complex paraboloid.
affine umbilical surfaces in \(\mathbb{R}^ 4\)
scientific article; zbMATH DE number 785217