A relation between the curvature ellipse and the curvature parabola

Abstract

At each point in an immersed surface in ℝ4 there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. Recently, at the singular point of a corank 1 singular surface in ℝ3, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in ℝ4 to ℝ3 in a tangent direction, corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where more degenerate singularities can appear. In this paper we relate the geometry of an immersed surface in ℝ4 at a certain point to the geometry of the projection of the surface to ℝ3 at the singular point. In particular we relate the curvature ellipse of the surface to the curvature parabola of its singular projection.

Relating second order geometry of manifolds through projections and normal sections

“…Proof: The proof follows as in the regular case. The only different consideration is that T p M 3 sing is a plane and if w ∈ T p M 3 sing is a non zero vector, then (dg q ) −1 (w) ⊂ T qM is a plane which contains the subset ker(dg q ), where g is the corank 1 map at q used in the initial construction and g(q) = p.…”

“…It is natural to expect certain relations between the curvature loci in each case. For example, in [3] we showed the relation between the curvature ellipse of M 2 reg ⊂ R 4 and the curvature parabola of the projection M 2 sing ⊂ R 3 and obtained some relations between their second order geometry. It is also known that in the previous case, the tangent direction is asymptotic if and only if the singularity of the projection is worse than a cross-cap ( [9,19]).…”

“…Let M 3 sing ⊂ R 5 be the projection under a tangent direction of M 3 reg ⊂ R 6 and p ∈ M 3 reg the point which is projected to p. Since the coefficients of the second fundamental form are the same for both manifolds, the height functions are the same. Therefore, by (3) in Proposition 5.3 and its regular counterpart (Theorem 2.2 in [12]), the binormal directions are the same and u ∈ T p M 3 reg is asymptotic if and only if u ∈ T qM is asymptotic.…”

“…the corank of the differential of the parametrisation at any point is at most 1. For singular corank 1 surfaces in R N , N = 3, 4, we can cite [3,4,5,16], and for singular corank 1 3-manifolds in R 5 , [6]. Here, the curvature locus is a parabola or a parabolic version of a Veronese surface.…”

“…For M 3 reg ⊂ R 6 projected to M 3 sing ⊂ R 5 , an analogous result to the equivalence between (i) and (ii) can be proved using results from [6]. However, the equivalence between (i) and (iii) for 3-manifolds is not true in general (see [6] for a partial result):…”