Lecture
In this section we shall study the critical (singular) points
of smooth mappings of the plane into three-dimensional space:
f: (z,w,u) (11.1)
According to the general definition given earlier, critical points
are those points (x, y) of the plane at which the rank of the Jacobi
matrix is smaller than its maximal value, i.e. smaller than 2.
The codimension of the set {rk Jf = 0} equals 6, since this requires
the vanishing of all entries of the 2×3 matrix Jf. At first
glance it may seem that the set {rk Jf = 1} has codimension 3, since this requires the vanishing of all the second-order minors
of the matrix Jf: Δ1, Δ2, Δ3. However, the corank product formula (4.3) shows that the codimension of this degeneracy equals 2.
The reason is that these minors are functionally dependent. Namely, all three germs Δi lie in the ideal (in the ring of germs of
smooth functions) generated by two of them (see example 4.1). This
can formally be written as
(Δ1, Δ2, Δ3) = (Δ1, Δ2). (11.3)
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Then the set of critical points of the mapping (11.1) is given by two equations:
Δ1(x,y) = 0, Δ2(x,y) = 0. (11.4)
From now on we shall always consider germs of mappings (11.1)
at the origin 0, assuming that f(0) = 0 and that the condition
rk Jf(0) = 1 holds.
11.1. Mappings of general type
Consider the three mappings:
(x,y) ↦ (z,w), (x,y) ↦ (z,u), (x,y) ↦ (w,u), (11.5)
obtained by projecting (11.1) onto the coordinate planes of the target space. From the condition rk Jf(0) = 1 it follows that for a suitable choice of local coordinates on the plane (x, y) one of these coincides identically with one of the coordinates (z, w, u) of the target space (see problem 8.1). For definiteness, let w = y.
Further, it is easy to see that if at least two of the mappings (11.5)
fail to be folds, this entails a degeneracy of codimension ≥ 4. Thus, in suitable local coordinates
(in the target and in the source) the germ of the mapping (11.1) at a critical point
of codimension ≤ 3 has the form
z = x², w = y, u = F(x, y) (11.6)
with some smooth function F(x,y) such that F(0) = Fx(0) = 0.
The Jacobi matrix of the mapping (11.6) equals
[2x 0 ]
[0 1 ]
[Fx Fy]
and its second-order minors are 2x, −Fx, 2xFy. Hence,
the set of critical points of the mapping (11.6) is defined by the two
equations
x = 0, Fx(x, y) = 0,
from which the vanishing of the third minor follows automatically.
Stability holds at those points where the curve Fx(x, y) = 0 meets the line x = 0 transversally, which is expressed by the inequality
Fxy ≠ 0. These are critical points of codimension 2, the smallest possible codimension for critical points of mappings of this type. Such
(and only such) critical points are stable.
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11.1.1. Codimension 2. The Whitney umbrella
THEOREM 11.1. The germ of the mapping (11.1) at a stable critical point is C∞-equivalent to
z = x², w = y, u = xy. (11.7)
Proof. As explained above, consider the germ of
the mapping (11.6) with a function F(x, y) such that F(0) = Fx(0) = 0 and
Fxy(0) ≠ 0. By lemma 5.3 we may write F in the form
F(x, y) = xy·φ(x², y) + x²ψ(x², y),
and from the conditions F(0) = Fx(0) = 0, Fxy(0) ≠ 0 it follows that φ(0) = 0 and
ψ(0) ≠ 0. Making the substitution u ↦ u − ψ(x², u) in the target space, we reduce the germ of our
mapping to the form
z = x², w = y, u = xyφ(x², y) (11.8)
with a function φ satisfying φ(0) = 0, φy(0) ≠ 0. It is easy to see that the pair of substitutions in the source plane and in the target space
x ↦ x/√φ(x², y), u ↦ u
brings the germ of this mapping to the form (11.7). ∎
The image of the plane (x, y) under the mapping (11.7) is
a self-intersecting surface, shown in Fig. 11.1. It is called the Whitney umbrella (Whitney umbrella) or sometimes the Cayley umbrella (Cayley umbrella); in the English-language literature the name cross-cap is also frequently used. This same surface (with one extra half-line x ≤ 0, y = u = 0, the "handle of the umbrella," added) can also be given by the algebraic equation u²x − z² = 0 [i.e. wx − z² type relation].
11.1.2. Singularities of codimension 3
Let us now consider the germ of the mapping (11.8) with a function φ satisfying the conditions
φ(0) = φy(0) = 0, φx(0) ≠ 0, φyy(0) ≠ 0. (11.9)
THEOREM 11.2. The germ of the mapping (11.8) with a function φ satisfying conditions (11.9) is C∞-equivalent to one of the two
germs
z = x², w = y, u = x(x² ± y²). (11.10)
Fig. 11.1 The Whitney umbrella
PROOF.
Using the division theorem (theorem 2.1) we may represent the germ of the function φ(x², y) in the form
φ(x², y) = c(x², y)(y² + a1(x²)y + a2(x²)), c(0) ≠ 0, (11.11)
where both a1, a2 are smooth
functions,
and from (11.9) it follows that a1(x²) = x²·c1(x²), where both ci are smooth
and
a2(0) ≠ 0. Setting b(x², y) = |x²·c1(x²)·y + a2(x²)|, we obtain
c(x², y)·(y² + a1(x²)y + a2(x²)) = ± c(x², y)(y ± b(x², y))²,
where
the sign ± before b(x², y)² agrees with the sign of the quantity a2(0). Making the change
of variable y ↦ y − b(x², y) in the target, where the sign "±" agrees with the previous one, we bring the germ of
our mapping to the form
z = x², w = y, u = x²c(x², y)y², c(0) > 0. (11.12)
Then in the source and in the target we make, respectively, the changes of variables
x ↦ x/√c(x², y), y ↦ y√c(x², y),
as a result of which we bring the germ of the mapping (11.12) to the form
z = x², w = y, u = x·B(x², y)(x² + y²) (11.13)
with some
smooth function B such that B(0) > 0. The change x ↦ x/∜B(x², y) brings the germ of the mapping (11.13) to the form (11.10). ∎
The singularities (11.10) are described in the paper [38], where they are called
"Mond's umbrellas" (or similar), in honor of the authors of preceding
works in this direction.
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PROBLEM 11.1. Draw the images of the plane (x, y) in three-dimensional
space (z, w, u) under both mappings (11.10). Can the germs of the mappings (11.10) with different signs be C∞-equivalent?
(To answer this question it is useful to pay attention to the geometric characteristics of the corresponding surfaces.)
The following generalization of theorem 11.2 holds. Consider the germ of the mapping (11.8), where instead of (11.9) the more general
condition holds:
with some integer n ≥ 1 (this is the multiplicity of the function φ in y at the point 0).
PROBLEM 11.2. Prove that such a germ is C∞-equivalent to one of the
germs
z = x², w = y, u = x(x² ± yⁿ), (11.15)
where for even n the sign "±" can be replaced by plus, while for
odd n it cannot.
11.2. Frontal mappings
In this section we consider germs of mappings (11.1) of a special type which have infinite codimension of degeneracy in the space of germs of smooth mappings R² → R³. They are defined by the following condition: all the second-order minors of the
Jacobi matrix Jf lie in the principal ideal generated by one of them.
This minor we shall also call the principal one and denote it by Δ. Then relation (11.3) turns into
(Δ1, Δ2, Δ3) = (Δ), (11.16)
and the set of critical points Σ of our mapping is given not by
two equations, as usual, but by only one equation Δ(x, y) = 0.
DEFINITION 11.1. Mappings satisfying condition
(11.16), whose set of critical points is nowhere dense in the source plane, are called
frontal.
Let A be a metric space, and let a subset σ ⊂ A be called nowhere dense in A if for any open
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REMARK 11.1. In this definition the main role is played by
condition (11.16). The requirement that the set Σ be nowhere dense is quite mild and holds almost always (in particular,
if Σ is a curve on the plane). In the singularity literature the
definition of frontal mappings usually used is a different but equivalent one; see, e.g., the paper [25] (lemma 2.3).
REMARK 11.2. Interest in frontal mappings is related to various applications, for example to the problem of propagation of a perturbation (a wave front) in a three-dimensional medium. We have already
touched on an analogous question in the two-dimensional case (section 6.2.4),
but the three-dimensional case turns out to be much harder; see [6, 5, 7],
as well as the paper [25] on singularities of frontal mappings, some results of which we have used.
In what follows we restrict ourselves to those frontal mappings for
which the regularity condition ∇Δ ≠ 0 holds, i.e.
|Δx| + |Δy| ≠ 0, (11.17)
so that the set of critical points Σ is
a smooth curve without singularities. Our next task is the C∞-classification of germs of frontal mappings at critical points up to and including
codimension 3 (here we mean the
codimension within the class of frontal mappings itself, and not within the class of all smooth mappings R² → R³).
11.2.1. Codimension 1. The cuspidal edge
First consider frontal mappings (11.1) for
which at least one of the germs (11.5) has a fold at the origin. Then in suitable local coordinates we
have the mapping (11.6) with a smooth function F(x, y) such that
F(0) = Fx(0) = 0.
As the principal minor we may take Δ = 2x, and the set
of critical points coincides with the line x = 0. Since Fx(x,y) vanishes on the line x = 0, we have a representation
Fx(x, y) = xg(x, y)
where the open subset U' ⊂ U contains no
point of the set Σ, i.e. U' ∩ Σ = ∅. (In this definition the open sets U and U' are often replaced by balls.) In our case A is the plane
of the variables (x, y) with the standard Euclidean metric.
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with some smooth function g (see problem 1.4). From this, using
lemma 5.3, we obtain
F(x, y) = xg(x, y) =
= x(φ(x², y) + xψ(x², y)) =
= xφ(x², y) + x²ψ(x², y). (11.18)
The substitution u ↦ u − φ(w, u) − x·ψ(x, u) in the target space brings the germ
of our mapping to the form
z = x², w = y, u = x³ψ(x², y). (11.19)
Critical points of codimension 1 arise under the assumption
that the condition ψ(0) ≠ 0 holds. In this case, in the target space
one can make the change of variable u ↦ u/∛ψ(x, u), which brings
the germ of our mapping to the form
z = x², w = y, u = x³. (11.20)
The corresponding surface — the image of the plane (x, y) in three-dimensional space (z, w, u) — is called the semicubical cuspidal edge (cuspidal edge). Here the edge itself is called the
line (in our coordinates, the axis w) consisting of the images of critical points, see fig. 11.2.
Fig. 11.2 The semicubical cuspidal edge
PROBLEM 11.3. Consider the cubic polynomials
P(t) = t³ + at + b·t + c
with real coefficients a, b, c. Each polynomial P(t) corresponds bijectively to a point in the space with Cartesian
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coordinates (a, b, c). What does the set of points look like that correspond to polynomials having multiple roots? What does the set of points look like that correspond to polynomials having a single root
of multiplicity three?
ANSWER. The first set is a surface in the space (a, b, c),
diffeomorphic to the semicubical cuspidal edge. The second set is the edge itself. Here one may use the results of section 6.2.3.
11.2.2. Codimension 2. The folded Whitney umbrella
Now consider the germ of the frontal mapping (11.19) in the
case where ψ(0) = 0. Critical points of codimension 2 arise under the assumption that the condition
ψ(0) = 0, ψy(0) ≠ 0 (11.21)
holds. In this case the substitution u ↦ u/∛ψ(x, u) used in the previous section no longer
works (the corresponding mapping is not even continuous). But one can make substitutions of variables in the source and in the target
respectively,
y ↦ ψ(x², y), w ↦ ψ(x², w).
After this our mapping takes the form
z = x², w = y, u = xy. (11.22)
The corresponding surface — the image of the plane (x, y) in three-dimensional space (z, w, u) — is called the folded Whitney umbrella
(folded Whitney umbrella, another name also used is
cuspidal cross-cap), see fig. 11.3.
REMARK 11.3. The germ (11.22) cannot be brought to the form (11.20),
even using not diffeomorphisms but homeomorphisms (both in the
target and in the source). This is obvious from comparing the properties of the corresponding surfaces (figs. 11.2 and 11.3). The semicubical cuspidal edge,
has no self-intersections, while the folded Whitney umbrella does. The property
of having self-intersections is evidently preserved under diffeomorphisms
and even under homeomorphisms, i.e. it is topological.
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Fig. 11.3 The folded Whitney umbrella
11.2.3. Singularities of codimension 3
Consider the germ of the frontal mapping (11.19), for which
neither the condition ψ(0) = 0 nor the condition (11.21) holds. Critical points of codimension 3 arise under the assumption that
ψ(0) = 0, ψy(0) = 0, ψx(0) ≠ 0, ψyy(0) ≠ 0. (11.23)
By the
division theorem, exactly as in
(11.11), we represent the germ of the function ψ(x², y) in the
form
where both ci
are smooth functions, and
from conditions (11.23) it follows that a1(x²) = x²c1(x²),
after which, using substitutions and reasoning
fully analogous to those used in section 11.1.2,
we bring the germ of our mapping to a form which differs from (11.10) only by the exponent of x in the third equation:
z = x², w = y, u = x³(x² ± y²). (11.24)
PROBLEM 11.4. Draw the images of the plane (x, y) in three-dimensional space (z, w, u) under both mappings (11.24). Can the germs of the mappings (11.24) with different signs be C∞-equivalent?
Prove the following generalization of the result obtained:
PROBLEM 11.5. The germ of a frontal mapping of the form (11.19), for which
instead of (11.23) condition (11.14) holds, is C∞-equivalent
to one of the germs
z = x², w = y, u = x³(x² ± yⁿ⁺¹), (11.25)
where for even n the sign "±" can be replaced by plus.
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REMARK 11.4. The singularities (11.24), (11.25) and some other
singularities of frontal mappings of codimension greater than 3 have been studied in the paper [38].
11.2.4. Codimension 2. The swallowtail
Finally, consider the germ of the frontal mapping (11.1),
for which none of the germs of the mappings (11.5) is a
fold, but one of them is a cusp. Note that, unlike
mappings (11.1) of general type, in the class of frontal mappings
this type of degeneracy has codimension 2, not 4.
Without loss of generality we may assume that the cusp is the
first of the germs of the mappings (11.5), so in suitable local
coordinates we have the mapping
z = 2x³ + xy, w = y, u = F(x, y) (11.26)
with a smooth function F(x, y). The Jacobi matrix of the mapping (11.26) has the form
[6x²+y x ]
[0 1 ]
[Fx Fy]
and its minors are Δ1 = 6x² + y, Δ2 = −Fx, and the third lies in the ideal generated by
them, in accordance with the general formula (11.3). From the condition that the
mapping is frontal, it follows that from among Δ1, Δ2 one can choose a principal minor Δ, such that (11.16) holds. It is easy to
see that as the principal minor one may take Δ = 6x² + y.
Fx(x, y) vanishes on the set Σ of critical points of the mapping (11.26), given by the equation 6x² + y = 0. This can be written as the identity Fx(x, −6x²) = 0, holding for all x in a
neighborhood of zero, from which follow the equalities Fx(0) = Fxx(0) = 0.
Thus, the condition that for a frontal mapping (11.1) none
of the germs (11.5) is a fold follows automatically
from the assumption that one of them is a cusp. Hence,
in the class of frontal mappings this degeneracy has codimension 2.
REMARK 11.5. The attentive reader will note that unlike the normal form (9.8) for the cusp, in formula (11.26) the coefficient in front of the monomial x³ is doubled. Of course this is of no essential significance and is done only so that the normal form
(11.29) given below has integer coefficients.
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From the condition Fx(x, −6x²) = 0 it follows that for any integer n ≥ 1
the function F can be represented in the form
Fx(x, y) = φ(x, y)(6x² + y)ⁿ = (6x² + y)ⁿ·Σ ak(y)xᵏ + (6x² + y)ⁿ⁺¹ψ(x, y),
where ai, ψ are smooth functions of their arguments. Integrating this last
equality with respect to x, we obtain
F(x, y) = a−1(y) + Σ ak(y)·(...) +
+ xⁿ⁺²φ(x, y) + xⁿ⁺³ψ(x, y) =
= a−1(y) + a0(y)(x³ + xy) + ½a1(y)(3x⁴ + x³y)+
+ A(x, y) + B(x, y),
where
A(x, y) = Σ ak(y)(...), k ≥ 2,
B(x, y) = xⁿ⁺²φ(x, y) + xⁿ⁺³ψ(x, y). (11.27)
From this it follows that the substitution u ↦ u − a−1(y) − a0(y)z brings
the germ of our mapping to the form (11.26) with the function
F(x, y) = 3a1(y)(3x⁴ + x³y) + A(x, y) + B(x, y).
Critical points of codimension 2 correspond to the condition
a1(0) ≠ 0. Then one can use the substitution u ↦ u/a1(y), which
brings the germ of our mapping to the form
z = 2x³ + xy, w = y, u = 3x⁴ + x³y + A(x, y) + B(x, y), (11.28)
where the functions A = 2A/a1(y), B = 2B/a1(y) have the form (11.27).
PROBLEM 11.6 (*). Show that the germ of the frontal mapping (11.28) is C∞-equivalent to a germ of the same form with functions
A = B = 0, i.e.
z = 2x³ + xy, w = y, u = 3x⁴ + x³y. (11.29)
The surface corresponding to the mapping (11.29) — the image of the plane (x, y) in three-dimensional space (z, w, u) — is shown on
fig. 11.4. It is called the swallowtail (swallowtail).
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Fig. 11.4 The swallowtail. Left: with sections by planes
w = const added. Right: view of the surface from another angle
PROBLEM 11.7. Let f : R³ → R³ be the germ of a mapping (10.5) with
n = 2. Prove that the set of critical values f(Σ) is a plane for μ = 1, a semicubical cuspidal edge for μ = 2, and a swallowtail
for μ = 3, and moreover in the latter two cases the reduction to the normal forms (11.20) and (11.29) respectively is achieved by means of a
linear change of scale in the target space.
11.3. Examples
11.3.1. Discriminant surfaces
of polynomials
Consider the cubic polynomials
P(t) = t³ + at + bt + c, (11.30)
depending on the real coefficients a, b, c. Each polynomial
P(t) corresponds to a point in the space with Cartesian coordinates
(a, b, c). Let us denote by Σ the set of points corresponding to polynomials having multiple real roots. What does this
set look like? To answer this question we need to write out the equalities
P(t) = P'(t) = 0
and consider them as equations in the variables b, c and the parameters a, t. As a result we obtain that the surface Σ is the
image of the frontal mapping (a, t) ↦ (a, b, c), where
b = −(3t² + 2at), c = 2t³ + at².
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Let us show that this is a semicubical cuspidal edge. For this we make the substitution on the right, t = x + a/3, a = y/3 [rescaled parameters], and the substitution on the left u = ... After
this our mapping takes the form (x, y) ↦ (a, b, c), where
w = y, b = 3(y² − x²), c = 2x³ − 3xy + y³ [rescaled].
Making in succession two substitutions on the left: z = w² − 3b, and then (denoting the new variables by the same letters) the substitution u = (c − 3zw − w³)/... .
In this way we bring the mapping obtained to the normal form (11.20). Note that all the substitutions of variables we made are
global, so we have reduced to the normal form not only the germ, but the
mapping as a whole.
REMARK 11.6. The semicubical cuspidal edge Σ divides
the space (a, b, c) into two open connected regions, in one of
which the polynomial P(t) has three real roots, and in the other —
one. At points of the surface Σ not lying on the edge itself, the polynomial
P(t) has one simple root and one root of multiplicity 2. At points of
the edge of the surface Σ, the polynomial P(t) has one root of multiplicity 3.
Consider now the quartic polynomials
P(t) = t⁴ + at² + bt + c
and denote by Σ the set of points in space (a, b, c) corresponding to polynomials P(t) having multiple roots.
PROBLEM 11.8. Prove that Σ is the swallowtail, dividing the space (a, b, c) into three open connected regions, in one
of which P(t) has four real roots, in another two, and in the
third, none. What are the multiplicities of the roots of P(t) at points of the cuspidal edges,
of the surface Σ, at points of its self-intersection line? at the vertex?
PROBLEM 11.9. What does the analogous set Σ look like for the polynomials P(t) = t⁵ + at³ + bt² + ct + d
in the 4-dimensional space of their coefficients?
11.3.2. Tangent surfaces to curves
In the space with coordinates (x, y, z) consider the curve
x = φ(t), y = ψ(t), z = η(t) (11.31)
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with smooth functions φ, ψ, η. We shall assume that the curve (11.31)
has a tangent line at every point, including critical ones. The tangent
to the curve at a critical point is defined as the limit of the tangent lines at non-critical points tending to it, if such a limit exists.
The tangent surface of the curve (11.31) is the ruled surface made up of all the lines tangent to it.
Consider the three curves
Γ123: x = t, y = t², z = t³,
Γ124: x = t, y = t², z = t⁴,
Γ234: x = t², y = t³, z = t⁴,
the first two of which have no critical points at all, while the third
has a critical point at zero and a tangent line there. Let us show that
the tangent surfaces of the curves Γ123, Γ124, Γ234 are diffeomorphic to
the semicubical cuspidal edge, the folded Whitney umbrella, and the swallowtail, respectively.
THE CURVE Γ123. The tangent surface is given by the equations
x = t + s, y = t² + 2ts, z = t³ + 3t²s, (11.32)
where the parameter t determines the point on the curve at which the tangent is taken, and s is the parameter along the tangent line itself. Consider the surface (11.32) as the image of the mapping
(t, s) ↦ (x, y, z) (11.33)
and let us show that this mapping is C∞-equivalent to (11.20).
Indeed, the substitution (t, s) ↦ (x, u) given by x = t + s in the source brings the mapping to the form
x = x, y = x² − 2xu + u², z = ... .
After the substitution y ↦ x² − y in the target space we obtain
x = x, y = u, z = ... .
Applying to the last mapping the substitution z ↦ (z − x³ + 3xy)/2,
we obtain the normal form (11.20). ∎
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THE CURVE Γ124. The tangent surface is given by the equations
x = t + s, y = t² + 2ts, z = t⁴ + 4t³s.
We reason as in the previous case, and show
that the corresponding mapping (11.33) is C∞-equivalent to (11.22).
Successive application of the substitutions (t, s) ↦ (x, u), y ↦ x² − y,
brings our mapping to the form
x = x, y = u, z = ... .
Finally, the substitution z ↦ (...)/8 brings the last mapping to the normal form (11.22). ∎
THE CURVE Γ234. The tangent surface is given by the equations
x = t² + 2ts, y = t³ + 3t²s, z = t⁴ + 4t³s,
here in computing the tangent vector we have cancelled the common factor t, which vanishes at the origin — the unique critical point of this curve. Let us show that the corresponding mapping (11.33) is C∞-equivalent to (11.29).
Successive application of the substitutions (t, s) ↦ (x, w), ..., and
z ↦ ... brings our mapping to the form
...
Finally, making a further pair of substitutions x ↦ −6x and u ↦ −6u, we bring
the last mapping to the normal form (11.29). ∎
PROBLEM 11.10. Let φ(t), ψ(t), η(t) be smooth functions not vanishing at t = 0. Show that the germs of the tangent
surfaces to the curves
x = t²φ(t), y = t³ψ(t), z = t⁴η(t),
x = t²φ(t), y = t³ψ(t), z = t⁵η(t),
x = t²φ(t), y = t³ψ(t), z = t⁴η(t)
at the origin are diffeomorphic to the same surfaces as the tangent
surfaces to the corresponding curves Γ123, Γ124, Γ234.
PROBLEM 11.11. For which space curves are the tangent surfaces the images of the mappings given by formulas (11.24) and (11.25)?
Comments