Centroaffine Differential Geometry: Submanifolds of Codimension 2

Flat centroaffine surfaces with the degenerate second fundamental form and vanishing Pick invariant inR4

“…For a centroaffine immersion into the affine space, the position vector yields its first canonical transversal vector field. A standard method of choosing a second one was proposed in 1950 by Lopsic (see [10]). Reorganizing geometry of equi-centroaffine immersions of codimension two, Nomizu and Sasaki [11] took the prenormalized Blaschke normal field as the second canonical transversal vector field, and by this structure Furuhata [12] proved that an equi-centroaffine immersion is minimal if and only if the trace of the affine shape operator with respect to the prenormalized Blaschke normal field vanishes identically.…”

Projective Surfaces and Pre-Normalized Blaschke Immersions of Codimension Two

“…One transversal vector field is the radial vector field, that is, the position vector field of a surface, and the other is chosen to be a pre-normalized Blaschke normal vector field, which was defined in [6]. See [4,5,11,12] for other choices of transversal vector fields. Following [6], Furuhata [3] studied surfaces in R 4 with vanishing shape operator, which can be considered from a viewpoint of a certain variation problem.…”