Centroaffine first order invariants of surfaces in \(\mathbb{R}^ 4\)
The centroaffine differential geometry of oriented surfaces \(M^2\) in \(\mathbb{R}^4 \setminus \{0\}\) is studied using the method of moving frames developed by E. Cartan. Following this method, the author introduces two semiconformal structures on the surface: one is called the affine semiconformal structure (it is related to the Burstin-Mayer metric introduced in the study of nondegenerate equiaffine surfaces in \(\mathbb{R}^4\), see a.o. [\textit{K. Nomizu} and \textit{L. Vrancken}, Int. J. Math. 4, No. 1, 127-165 (1993; Zbl 0810.53006)]) and the other one is called the \(h^3\)-semiconformal structure which is obtained via the Gauss equation. Then, using these structures, a centroaffine metric is introduced and the surfaces are divided into several classes depending on properties of the centroaffine metric and the \(h^3\)-semiconformal structure. Finally, examples are given of surfaces belonging to different classes.
scientific article; zbMATH DE number 785205