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.

Caustics of convex curves

It is well-known that the focal set (i.e. the image of the caustic) of a given convex closed curve γ admits singular points. In this paper, we classify the diffeomorphic type of focal sets of convex curves which admit at most four cusps.

Embeddings of nonorientable surfaces with totally reducible focal set

In an earlier paper [5] we introduced the idea of an immersion f: Mm-ℝn with totally reducible focal set. Such an immersion has the property that, for all p ∈ M, the focal set with base p is a union of hyperplanes in the normal plane to f(M) at f(p). Trivially, this always holds if n = m + 1 so we only consider n > m + 1.

Generic isotopies of space curves

For a single space curve (that is, a smooth curve embedded in ℝ3) much geometrical information is contained in the dual and the focal set of the curve. These are both (singular) surfaces in ℝ3, the dual being a model of the set of all tangent planes to the curve, and the focal set being the locus of centres of spheres having at least 3-point contact with the curve. The local structures of the dual and the focal set are (for a generic curve) determined by viewing them as (respectively) the discriminant of a family derived from the height functions on the curve, and the bifurcation set of the family of distance-squared functions on the curve. For details of this see for example [6, pp. 123–8].

Focal Sets in Certain Riemannian Manifolds

In recent years the geometry of generic submanifolds of Euclidean space has been theobject of much study. Thorn hinted in [7] that the focal set of such a submanifold couldprofitably be studied by using the family of distance squared functions on thesubmanifold from points of the ambient space. For a generic submanifold the focal set isthe catastrophe or bifurcation set of this family. The key to obtaining results on thelocal structure of this focal set is a transversality theorem of Looijenga [5]; for analternative exposition see [8].

Distance-Genericity for Real Algebraic Hypersurfaces

One of the original applications of catastrophe theory envisaged by Thom was that of discussing the local structure of the focal set for a (generic) smooth submanifold M ⊆ Rn + 1. Thom conjectured that for a generic M there would be only finitely many local topological models, a result proved by Looijenga in [4]. The objective of this paper is to extend Looijenga's result from the smooth category to the algebraic category (in a sense explained below), at least in the case when M has codimension 1.Looijenga worked with the compactified family of distance-squared functions on M (defined below), thus including the family of height functions on M whose corresponding catastrophe theory yields the local structure of the focal set at infinity. For the family of height functions the appropriate genericity theorem in the smooth category was extended to the algebraic case in [1], so that the present paper can be viewed as a natural continuation of the first author's work in this direction.