Critical points of the Gauss map and the exponential tangent map

Abstract

Given M an immersed k-dimensional submanifold of Rk+n and CM its unit normal bundle, we study the critical points of the generalized Gauss map ψ : CM → Rk+n, ψ(m, Um) = Um with Um ∈ CmM and m ∈ M. Usually, the image of these points is called the parabolic set. The case k = 1, 2 is analyzed in detail. Finally, we analyze the relation between the parabolic set and the critical points of the exponential tangent map Ω : TM → Rk+n, ψ(m, Ym) = m + Ym with Ym ∈ TmM, in the case k = n.