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Cubic form theorem for affine immersions. (English) Zbl 0657.53007

The authors extend the notion of affine immersion presented previously [Math. Z. 195, 165-178 (1987; Zbl 0629.53012)], to the case of higher codimension. In this case one has the immersion \(f: (M^ n,\nabla)\to (M^{n+p},{\tilde \nabla})\), where \(\nabla\), \({\tilde \nabla}\) are torsion-free affine connections. The main conclusions are drawn under the additional hypothesis that \({\tilde \nabla}\) be projectively flat and that the “osculating dimension” \(:=\dim \{{\tilde \nabla}_ Xf_*Y:\) X, Y vector fields in \(M\}\) be equal to \(n+1\). Some generalizations of the classical theorem of Pick-Berwald, regarding vanishing of the affine cubic form, are proven, to include the case of degenerate hypersurfaces.
Reviewer: S.Gigena

MSC:

53A15 Affine differential geometry

Citations:

Zbl 0629.53012
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References:

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