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Tangential families and their envelopes
Let us consider a mapping f : R2 → R2 of the plane R2, whose coordinates are denoted by ξ and t. If ∂tf vanishes nowhere, then f defines the 1-parameter family of the plane curves parameterized by fξ := f (ξ, ·) and indexed by ξ (indicating the curve in the family). The curves in the family may have double points. The mapping f is called a parameterization of the family.
Definition. The 1-parameter family parameterized by f is a tangential family if the partial derivatives ∂ξ f and ∂tf are parallel non zero vectors at every point (ξ, t = 0), and the image of the mapping ξ 7→f (ξ, 0) is a curve, called the support of the family. In other terms, a family of plane curves, tangent to γ, is a tangential family whenever it can be parameterized by the tangency points of the family curves with the family support. Definition. The graph of the tangential family parameterized by f is the surface Φ := {(q, p) : q = f (ξ, 0), p = f (ξ, t), ξ, t ∈ R} ⊂ γ × R2 .
Let us consider the two natural projections of the family graph on the support and on the plane, respectively π1 : (q, p) 7→q and π2 : (q, p) 7→p.
Remark. The first projection π1 is a fibration; the images by π2 of its fibers are the curves of the family.
Definition. The criminant set (or envelope in the source in Thom’s notations) of a tangen-tial family is the critical set of the projection π2 of its graph to the plane. The envelope of the family is the apparent contour of its graph in the plane (i.e., the critical value set of π2). Remark. The envelope of a tangential family is the critical value set of any of its parame-terizations. In particular, the support of a tangential family belongs to its envelope. Let us end this section with some examples.
Example 1.2.1. The family of the tangent lines to the parabola y = x2 is parameterized by f (ξ, t) := (ξ + t, ξ2 + 2ξt) .
The criminant set is {t = 0}, so the envelope of the tangent lines coincides with the support parabola.
Example 1.2.2. Let us consider in the plane {x, y} the tangential family of support y = 0, defined by the graphs y = Pξ (x) of the polynomials Pξ (x) := (x − ξ)2(x − 2ξ) .
Tangential family germs of small codimension
In this section we study the A -orbits in the set X of small codimension. In order to do this, we introduce first some definitions.
Definition. Given a tangential family germ, the fiber π2−1 (0, 0) defines a unique character-istic direction in the tangent plane to the family graph at the origin, that we call the vertical direction.
Let f be a mapping from the plane to the plane, which germ at the origin f belongs to X0. A branch passing through the origin of the criminant set of the tangential family germ f is any irreducible component of the restriction of the f -critical set to an arbitrary small neighborhood of the origin. Lemma 1.2.1. Given any tangential family, the branch of its criminant set, whose projection in the plane is the support of the family, is non vertical. An example of the statement of Lemma 1.2.1 is shown in figure 1.5. The proof is given in section 1.4.1.
Tangential deformations and envelope stability
In this section we introduce the definition of tangential deformation and we state our main result. In some situations, as for instance in the study of geodesic tangential family evolution under small perturbations of the metric, it would be natural to perturb a tangential family only among tangential families.
Definition. A p-parameter tangential deformation of a tangential family germ f ∈ X0 is a p-parameter family of mappings {Fλ : R2 → R2 , λ ∈ Rp} .
such that the germ at the origin of F0 is f and Fλ parameterizes a tangential family whenever |λ| is small enough. For example, the translation of the origin is a 2-parameter tangential deformation. Note that the support of a tangential family does not change, up to diffeomorphisms, under tan-gential deformations of the family.
Projective tangential families
A (local) projective structure is the structure given, at any point of the plane near the origin, by a second order differential equation (see [4]). The graphs of its solutions are, by definition, the straight lines for this structure. For instance, the standard projective structure in the Euclidean plane is defined by the equation y00 = 0. A projective structure is said to be flat if there exists a change of local coordinates bringing it to the standard projective structure.
Remark. Let us fix a flat projective structure. The envelope of the straight lines tangent to a curve, having an inflection point, contains the support curve and the straight line, tangent to it at the inflection point. In particular, the envelope is not geometric.
Let us consider a curve γ in the plane R2, endowed with a projective structure. The straight lines tangent to γ define a tangential family, that we call the projective tangential family of support γ. Corollary. Let us consider the germ of the projective tangential family of support γ at a point P ∈ γ. If P is an ordinary point of γ, the family envelope is the germ of γ at P . If P is a simple inflection point of γ, then the family envelope has two branches passing through P (one of which is the support); these two branches have generically a second order self-tangency at P : the curve germs for which the claim does not hold are contained in the union of two codimension 1 submanifolds in the space of the curve germs having an inflection point at P .
Remark. This Corollary provides a criterion distinguishing non flat projective structures. Indeed, suppose that there exists a curve, having a simple inflection point, such that the envelope of its tangent straight lines is geometric. Then the projective structure is not flat. Consider a projective tangential family of support γ. A deformation of the projective structure induces a tangential deformation of the projective family. In this setting, Theorem 1.2.2 reads as follows.
Corollary. The second order self-tangency of the envelope of any projective tangential family germ is stable under small perturbations of the projective structure.
Geodesic tangential families
Let γ be a curve in a Riemannian surface. The geodesics tangent to γ form a tangential family, that we call the geodesic tangential family of support γ.
Corollary. Let us consider the geodesic tangential family germ at a point P of its support γ. If P is an ordinary point of γ, then the envelope of the family germ is the germ of γ at P . If P is a simple inflection point of γ, then the envelope of the family germ has two branches passing through the point P (one of which is the family support). Generically, these two branches have a second order self-tangency at P : the curve germs γ for which the claim does not hold are contained in the union of two codimension 1 submanifolds in the space of all the curve germs having an inflection point at P . Perturbations of the metric of the Riemannian surface induce tangential deformations on geodesic tangential families. Corollary. The second order self-tangencies of envelopes of geodesic tangential family germs in a Riemannian surface are stable under small deformations of the metric. Corollary. The second order self-tangencies of envelopes of geodesic tangential family germs in a surface in the Euclidean three-space are stable under small deformations of the surface (equipped with the induced metric). We point out that the preceding Corollaries of Theorems 1.2.1 and 1.2.2 hold only in the local setting.
Preliminary results of Singularity Theory
The proof of Theorems 1.2.1 and 1.2.3 is based on the reduction of the prenormal form to a polynomial map germ and the computation of the corresponding A -miniversal deformation. In order to do this, first we recall some facts on Singularity Theory of smooth maps. For a complete presentation of this theory, we refer the reader to [11], [13] or [26].
As before, let us denote respectively by s, t and x, y the coordinates in the source and in the target space of a map germ f : (R2, 0) → (R2 , 0). Such a germ defines, by the formula f ∗g := g ◦ f , a homomorphism from the ring Ex,y of the function germs on the target to the ring Es,t of the function germs on the source. Hence we can consider every Es,t-module as an Ex,y-module via this homomorphism.
Let hf i ⊂ Es,t be the ideal generated by the components of f (for the structure of Es,t-module). Let us recall the well known Preparation Theorem of Mather-Malgrange-Weierstrass, in the case we deal with: the module Es,t is finitely generated as Ex,y -module if and only if the quotient space Es,t/hf i is a real vector space of finite dimension; moreover, a basis of the vector space provides a generator system of the module. From now on, we shall use the notation ms,t instead of m2 to point out that the variables of the considered germs are s and t.
Table of contents :
1 Stable tangential families and their envelopes
1.1 Introduction
1.2 Presentation of the results
1.2.1 Tangential families and their envelopes
1.2.2 Equivalences of tangential families
1.2.3 Tangential family germs of small codimension
1.2.4 Tangential deformations and envelope stability
1.3 Examples and applications
1.3.1 Projective tangential families
1.3.2 Geodesic tangential families
1.3.3 A global geodesic tangential family on the sphere
1.4 Proof of Theorems 1.2.1, 1.2.2 and 1.2.3
1.4.1 Prenormal forms of tangential families
1.4.2 Preliminary results of Singularity Theory
1.4.3 Proof of Theorem 1.2.1, 1.2.1 and 1.2.3
2 Simple tangential families and perestroikas of their envelopes
2.1 Introduction
2.2 Simple singularities of tangential families
2.2.1 Preliminary results on tangential families
2.2.2 Classication of tangential families
2.2.3 Simple tangential families
2.2.4 Miniversal tangential deformations
2.2.5 Adjacencies of tangential family singularities
2.3 Bifurcation diagrams of simple singularities
2.3.1 Bifurcation diagrams of singularities S1;n
2.3.2 Bifurcation diagram of the singularity S2;2
2.3.3 Bifurcations diagrams of the singularities Tn
2.4 Proofs
2.4.1 Prenormal forms of tangential families
2.4.2 Preliminary background of Singularity Theory
2.4.3 S1-type tangential families
2.4.4 S2-type tangential families
2.4.5 T-type tangential families
2.4.6 Proof of Theorem 2.2.3
3 Singularities of Legendrian graphs and singular tangential families
3.1 Introduction
3.2 Presentation of the results
3.2.1 Envelopes of tangential families
3.2.2 Legendrian graphs and their singularities
3.2.3 Stability of singularities of Legendrian graphs
3.2.4 Normal forms of Legendrian graph projections
3.2.5 Singular tangential families and their envelopes
3.3 Proofs
3.3.1 Parameterizations of Legendrian graphs
3.3.2 Proof of Theorem 3.2.2
3.3.3 Preliminary background of Singularity Theory
3.3.4 Proof of Theorem 3.2.4
3.3.5 Proof of Proposition 3.3.8
3.3.6 Proof of Theorems 3.2.5, 3.2.6 and 3.2.7
4 Minimax solutions to Hamilton-Jacobi equations
4.1 Introduction
4.2 Minimax values of functions
4.2.1 Preliminary results
4.2.2 Morse complex of generic functions
4.2.3 Incident, coupled and free critical points
4.2.4 Minimax critical values
4.3 Minimax solutions to Hamilton{Jacobi equations
4.3.1 Generating families of Lagrangian submanifolds
4.3.2 Geometric solutions
4.3.3 Minimax solutions
4.4 Characterization of minimax solutions
4.4.1 Preliminary notations
4.4.2 Admissible decompositions of wave fronts
4.4.3 Characterization of the minimax solution
4.4.4 Vanishing triangles
4.5 Generic singularities of minimax solutions
References
