On some properties of the curvature and Ricci tensors in complex affine geometry
Let \(f:(M^n, \nabla) \to \mathbb{C}^{n + 1}\) be a complex affine hypersurface immersion and let \(\xi\) be any transversal vector field along \(f\). Then the transversal complex plane bundle \(\mathcal N\) spanned by \(\xi\) and \(J \xi\) induces a complex affine connection \(\nabla\) on \(M^n\), in the sense that \(\nabla J = 0\), where \(J\) is the complex structure on \(M^n\). The author defines for an abstract complex manifold with complex affine connection a complex Ricci tensor ric, which, in case the connection is holomorphic, is the complex trace of the curvature tensor \(R\). Next, the author assumes the induced connection to be holomorphic, or affine Kähler, and proves some properties of ric. In the third section, it is proved that a nondegenerate complex affine hypersurface \(M^n\), equipped with a complex transversal bundle is semi-symmetric (i.e. \(R \cdot R = 0\)) if and only if ric is symmetric (if \(n = 2)\) or ric is proportional to the complex second fundamental form (if \(n = 2\)). If \(f\) is degenerate (i.e. the second fundamental form is degenerate), then the result is no longer true. In the next section, it is proved that a nondegenerate complex affine hypersurface \(M^n\) is locally symmetric if and only if \(M^n\) is a quadric or flat. If \(f\) is degenerate, then the author proves a cylindrical decomposition theorem. The final section contains some instructive examples.
scientific article; zbMATH DE number 806225
scientific article