Relating the second order geometry of manifolds in Euclidean spaces

We use normal sections to relate the curvature locus of regular (resp. singular
corank 1) 3-manifolds in R6 (resp. R5) with regular (resp. singular corank 1)
surfaces in R5 (resp. R4). For example we show how to generate a Roman
surface by a family of ellipses dierent to Steiner's way. Furthermore, we give
necessary conditions for the 2-jet of the parametrisation of a singular 3-manifold
to be in a certain orbit in terms of the topological types of the curvature loci of
the singular surfaces obtained as normal sections. We also study the relations
between the regular and singular cases through projections. We show there is
a commutative diagram of projections and normal sections which relates the
curvature loci of the dierent types of manifolds, and therefore, that the second
order geometry of all of them is related. In particular we dene asymptotic
directions for singular corank 1 3-manifolds in R5 and relate them to asymptotic
directions of regular 3-manifolds in R6 and singular corank 1 surfaces in R4.