Divisibility of multilinear mappings with applications to affine differential geometry |
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Authors: | Paweł Witowicz |
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Institution: | 1. Department of Mathematics and Applied Physics , Rzeszow University of Technology , Powstancow Warszawy 8, Rzeszow 35-959, Poland witowicz@prz.edu.pl |
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Abstract: | For real finite-dimensional vector spaces V, W we call a bilinear symmetric mapping h?:?V?×?V?→?W non-degenerate if the components of h with respect to a certain basis are linearly independent and non-degenerate. We say that a symmetric trilinear mapping C?:?V?×?V?×?V?→?W is divisible by h if there exists a linear form α such that C(v,?v,?v)?=?α(v)h(v,?v) for every v?∈?V. In affine differential geometry of affine immersions h is the second fundamental form and C – the cubic form of the immersion. The immersion has pointwise planar normal sections if h(v,?v)?∧?C(v,?v,?v)?=?0 for every tangent vector v. We prove that it implies that C is divisible by h if h is non-degenerate and the codimension is greater than two. For immersions with Wiehe's or Sasaki's choice of transversal bundles divisibility of C by h implies vanishing of C. |
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Keywords: | bilinear mapping trilinear mapping affine immersion transversal bundle normal section |
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