The fundamental theorem in affine hypersurface theory
A relative normalization of a strongly convex immersion \(x: M\to {\mathcal A}\) of an n-dimensional differentiable manifold M into real affine space \({\mathcal A}\) of dimension \(n+1\geq 3\) induces, via the structure equations for x and the conormal immersion, two linear connections \(1\Gamma\) and \(2\Gamma\) on M. (Their arithmetic mean is the Levi-Civita connection of the relative metric.) Continuing and extending earlier work of Blaschke, Norden, P. A. and A. P. Shirokov, the author presents a systematic study of the fundamental theorems (in the sense of Bonnet's theorem in Euclidean surface theory) for hypersurfaces in affine space based on the use of these two connections. Special uniqueness and existence results are obtained for the centroaffine and equiaffine normalizations. For example, the following uniqueness theorems are proved. If \(X,\bar X: M\to {\mathcal A}\) are locally strongly convex differentiable hypersurface immersions with centroaffine normalizations and if \(1\Gamma =\) \(2\Gamma\), then X(M) and \(\bar X(M)\) are equivalent modulo a regular centroaffine transformation of \({\mathcal A}\). For equiaffine normalizations a similar result holds if M is closed.
scientific article; zbMATH DE number 4044628