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On analogues of Bäcklund theorem in affine differential geometry of surfaces. (English) Zbl 1444.53014
The main aim of the paper is a version of the Bäcklund theorem in affine differential geometry. The classical Bäcklund theorem for surfaces \(f, \hat f:M \rightarrow \mathbb R^3\) \((\dim M=2\)) in Euclidean space uses concepts of Euclidean metric as distance and angle to characterize \(f\) and \(\hat f\) as surfaces of (the same) constant negative Gaussian curvature.
In the affine space \(\mathbb R^3\), the main ingredient is a volume form which is parallel with respect to the standard connection in \(\mathbb R^3\). Denoting by \(\xi\) and \(\hat\xi\) equiaffine transversal vector fields (not necessarily the Blaschke normal fields) on \(f\) and \(\hat f\), respectively, a number of conditions implies that the affine fundamental forms on \(f\) and \(\hat f\) are conformal to each other and the connections induced by \(\xi\) and \(\hat\xi\) are locally symmetric. It turns out that this result corresponds in special cases to the situation in the classical Bäcklund theorem or in the Bäcklund theorem for surfaces in Minkowski space.
Furthermore, a different proof of a theorem due to S.-S. Chern and C.-L. Terng [Rocky Mt. J. Math. 10, 105–124 (1980; Zbl 0407.53002)] concerning affine minimal surfaces \(f, \hat f\) is given (in this theorem, the affine normals are required to be parallel).
53A15 Affine differential geometry
53B05 Linear and affine connections
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
Full Text: DOI Euclid