Divisibility of multilinear mappings with applications to affine differential geometry

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.