Affine focal sets of codimension-2 submanifolds contained in hypersurfaces

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g-Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the (n + 1)-space, the immersion (ν, ν · (f − O)), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.

Equiaffine characterization of Lagrangian surfaces in ℝ4

For non-degenerate surfaces in ℝ4, a distinguished transversal bundle called affine normal plane bundle was proposed in [K. Nomizu and L. Vrancken, A new equiaffine theory for surfaces in ℝ4, Internat. J. Math. 4(1) (1993) 127–165]. Lagrangian surfaces have remarkable properties with respect to this normal bundle, like for example, the normal bundle being Lagrangian. In this paper, we characterize those surfaces which are Lagrangian with respect to some parallel symplectic form in ℝ4.