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Rigidity of affine hypersurfaces with rank 1 shape operator. (English) Zbl 1058.53009
Consider two positive definite hypersurface immersions \(f: M\to \mathbb{R}^{n+1}\), \(g: M\to \mathbb{R}^{n+1}\) which induce the same Blaschke connection \(\nabla\). The following question arises: Does there exist an affine map \(A\) of \(\mathbb{R}^{n+1}\), such that \(f= A\circ g\)?
If \(n=\dim M\geq 3\) and \(\dim(\text{im\,}R)\geq 2\) (\(R\) curvature tensor of \(\nabla\)) the answer is in the affirmative [L. Vrancken and K. Nomizu, Result. Math. 27, No. 1–2, 93–96 (1995; Zbl 0873.53011)]. When \(\dim(\text{im\,} R)= 0\) then \(M\) is an improper affine sphere and the answer is no.
In the present paper the author shows that in case \(n\geq 3\) and \(\dim(\text{im\,}R)= 1\) the answer is no again. Furthermore all such locally non-rigid hypersurfaces are described using the solutions of differential equations of Monge-Ampère type.
MSC:
53A15 Affine differential geometry
53C24 Rigidity results
53B05 Linear and affine connections
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[1] DOI: 10.1515/9783110870428
[2] K. Nomizu and B. Opozda, Geometry and Topology of Submanifolds IV (World Scientific, 1992) pp. 133–142. · Zbl 0838.53015
[3] Nomizu K., Affine Differential Geometry (1994) · Zbl 0834.53002
[4] DOI: 10.1007/BF03322272 · Zbl 0837.53011
[5] DOI: 10.2748/tmj/1178227299 · Zbl 0761.53008
[6] DOI: 10.2748/tmj/1113247798 · Zbl 1008.53014
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