Relating second order geometry of manifolds through projections and normal sections

Abstract

We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in R 6 (resp. R 5 ) with regular (resp. singular corank 1) surfaces in R 5 (resp. R 4 ). For example, we show how to generate a Roman surface by a family of ellipses different to Steiner's way. We also study the relations between the regular and singular cases through projections. We show that there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular, we define asymptotic directions for singular corank 1 3-manifolds in R 5 and relate them to asymptotic directions of regular 3-manifolds in R 6 and singular corank 1 surfaces in R 4 .

Geometry of surfaces in $$\mathbb R^5$$ through projections and normal sections

“…The 2-jet of the parametrization now has the form (0, 0, 1 2 (a 20 x 2 + 2a 11 xy + a 02 y 2 ), 1 2 (b 20 x 2 + 2b 11 xy + b 02 y 2 ), 0). The curvature ellipse ∆ 5 e can be parametrized by (2) η e (θ) = a 20 cos(θ) 2 + 2a 11 sin(θ) cos(θ) + a 02 sin(θ) 2 , b 20 cos(θ) 2 + 2b 11 sin(θ) cos(θ) + b 02 sin(θ) 2 , 0 . Now, by rotation in the tangent space we can take u to be (0, 1).…”

“…As the dimension and codimension of the immersed manifolds grow, deeper singularity theory concepts are needed. Besides this, the attention has recently changed to singular manifolds M k sing ⊂ R n , n > k ≥ 2, ( [3,4,15,23]) and the relation of their geometry with regular manifolds 2,6,19,20,21]).…”

“…This has been studied in [8], [16] and [22], however, our motivation is slightly different. In [2] a commutative diagram between regular and singular surfaces and 3-manifolds through projections and normal sections was established. This diagram induces a commutative diagram between the corresponding curvature loci.…”

“…This means that the second order geometry of all these manifolds is related. For example, in [2] a relation between the asymptotic directions at a point p ∈ M n reg ⊂ R 2n and at π u (p) ∈ M n sing ⊂ R 2n−1 , where π u is the orthogonal projection along the tangent direction u, is given. Namely, since the second fundamental form and the height functions coincide, a direction is asymptotic for p if and only if it is asymptotic for π u (p).…”

The axial curvature for corank 1 singular surfaces

“…The parameter y ∈ R ∪ [y ∞ ] corresponding to an asymptotic direction X ∈ T q R 2 is also called an asymptotic direction. The number of asymptotic directions is characterized by the topological type of the curvature parabola and when ∆ p is degenerate y ∞ is an asymptotic direction (see also [2] for an explanation):…”