Equivalence theorems for complex affine hypersurfaces
The author studies under which conditions two complex affine hypersurface immersions \(f_1\), \(f_2 : (M^n, \nabla) \to \mathbb{C}^{n + 1}\) are affine equivalent, i.e. when does there exist a complex affine transformation \(A\) of \(\mathbb{C}^{n + 1}\) such that \(Af_1 = f_2\). These conditions are in terms of the affine connection and the complex-valued, complex-linear second fundamental form induced by the choice of any transversal vector field \(\xi_1\) resp. \(\xi_2\) fields being holomorphic or anti-holomorphic. Next she applies her results to the case when the induced affine shape operators are anti-complex. The key result here is that, if the induced affine shape operators are anti-complex, if the induced affine connections coincide, and if the curvature tensor of the induced affine connection is non zero at some point, then the two immersions are equivalent. As an application, she recovers Calabi's theorem stating that two Kähler hypersurfaces of \(\mathbb{C}^{n + 1}\) are unitarily equivalent provided their induced metrics coincide.
scientific article; zbMATH DE number 785219