Relating second order geometry of manifolds through projections and normal sections

We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in R6 R5) with surfaces R5 R4 ). For example, we show how generate a Roman surface by family ellipses different Steiner’s way. also study relations between and cases through projections. that there is commutative diagram projections which relates loci types manifolds, therefore, second order geometry all them related. In particular, define asymptotic directions for 1 R4.