On affine geometry of purely real submanifolds (Q1380847)

The author studies \(n\)-dimensional submanifolds of \( {\mathbb{R}}^{2n} \) from the viewpoint of complex affine geometry. \({\mathbb{R}}^{2n}\) is identified with \({\mathbb{C}}^n\) and the immersion \(f : M^n \to {\mathbb{C}}^n\) to be considered is assumed to be purely real. The spaces \(f_{*} (TM)\) and \(J f_{*} (TM)\), \(J\) the canonical complex structure of \({\mathbb{C}}^n\), are then transversal everywhere. The structure group of the ambient space is the (real representation of the) general complex affine group \(GA (n,{\mathbb{C}})\), a subgroup of the general real affine group \(GA (2n, {\mathbb{R}})\). The sum \(f_{*} (TM) \oplus J f_{*} (TM)\) immediately gives rise to an induced connection \(\nabla\) on \(M\) and a second fundamental form \(h\). In terms of these objects, theorems of Bonnet and Cartan-Norden type are established. Several examples of such immersions are considered with respect to affine properties such as flatness, local symmetry, etc. The method is a combination of moving frames and base free equations.