Lecture
Consider the germ of a smooth mapping
(у): М+М
of a plane to a plane, or of a two-dimensional surface to another two-dimensional surface, given by the functions
2=/(т,у), ш=Л (ту), Р(0=Л=0, (9-1)
at the critical point — the origin 0. By definition of a critical point, the Jacobi matrix of this mapping
=” ^ (9.2) в п
is degenerate at the point ().
In general, the critical (singular) points of the mapping (9.1) are determined by the condition 4её Л; = 0, which defines on the preimage plane a set 5, consisting of the union of two subsets:
бо = {18 Л =0}, 51 = {гв Л; = 1}.
The codimension of 60 equals 4 (all four entries of the matrix vanish,
„+, this gives four independent equations), while the codimension of 51 equals 1 (a single equation 4еф /; = 0). This is a trivial illustration of the product formula for coranks (4.3).
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9.1. The fold and the cusp
Let us begin with the study of germs of mappings at stable critical points, i.e. those having codimension 1 or 2. This excludes
from consideration the set 50, and from now on we shall consider the germ of the mapping (9.1) at a point 0 belonging to the set 51 С 5.
Obviously, the germ of the mapping (9.1) is then ®-equivalent to
2 =Е(т,у), шШ=У, (9.3)
where Р(т,у) is some smooth function, ЁР\(0) = 0 (see problem 8.1).
The set 5 of critical points of the germ of the mapping (9.3) is given by the equation
Е, (т, у) = 0. (9.4)
At stable critical points the regularity condition holds:
meaning that the point 0 is not a critical point of the function
Е, (т, у), and in its neighborhood the set 5 is a smooth
curve without singularities. Condition (9.5) persists under all sufficiently small perturbations of the function Р, but breaks down at critical points of codimension 3, since two more equalities are added to the equation of critical points РЁ, = 0:
Р,. = О and Ёуу = 0.
REMARK 9.1. From the above it is clear that there are two types of stable critical points of germs of mappings (9.3), which have
codimension 1 and 2 respectively and are defined by the conditions
® Ё.(0) =0, Ё,.(0) #0,
®е Р.(0) =0, Р,.(0) =0, Еъи(0) = 0, Ехк (0) = 0.
These types of critical points are called, respectively, the fold
(English: /014) and the cusp (English: сирз or реа). We shall show below that
each of these types corresponds to exactly one СК-normal form
of the germ of a mapping, and thus folds and cusps are the only stable singularities of mappings of a plane to a plane.
For a mapping in general position, almost all critical points are folds. They fill the curve 5 almost entirely; the exception is the isolated cusp points located on 5 —
they occur where 5 intersects the curve given by the equation
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Е,.(т, у) = 0. If we slightly «perturb» our mapping, then the curve
5 is slightly deformed and the cusp points on it also shift,
but do not disappear. Critical points that are neither folds nor
cusps are unstable and can be destroyed by an arbitrarily small perturbation of the mapping. This is easily explained geometrically.
All such points are intersections of three distinct curves in the
plane: Ру = 0, Е; = О and Ру = 0, or Вухх = 0, i.e. they have codimension 3. By means of an arbitrarily small perturbation of the function Р
the triple intersection of the curves can always be destroyed, turning it
into three pairwise intersections (see fig. 9.1).
Fig. 9.1 Left: a transversal intersection of two curves is stable
under small perturbations. Right: an intersection of three curves
is unstable and splits into three pairwise intersections
In defining the fold and the cusp we used special
coordinates in the preimage №, namely, ones in which one of the functions defining the mapping coincides with one of the coordinates. It is often
more convenient to define the fold and the cusp invariantly, i.e. in a purely
geometric way, without using any coordinate systems. Let us do this
by means of two geometric objects.
The first is the set of critical points of the mapping. In our
case this is the smooth curve 5. The second is the field of directions (the
field of kernels of the differential) А = Кег 4, defined at the points of the curve 5.
Recall that at each point р Е 5 the mapping } : № -» М defines a linear map (a linear operator) of the corresponding
tangent spaces
4, : ТЬМ > ТМ. (9.6)
If coordinates are given in М and ЛМ, then in the tangent spaces ТМ
and ТьМ the natural bases are defined, in which the matrix of the linear map (9.6) is the matrix /+(р). Since, according to the assumptions made, гв .Л;(р) = 1 at every point р Е 5 from a neighborhood of
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zero, the linear map (9.6) has a kernel — a one-dimensional subspace of the tangent space 7,М№, which we denote
А(р) = Кег 4».
Thus, at each point р Е 5 a direction Х(р) is defined,
while at points р 9 5 the mapping 4}, defines no direction at all, since at such points its kernel consists of the zero vector alone.
In the situation described, one says that a field of directions is defined on the set 5. In the general-position case the direction А’(р) is transversal to the curve 5 (i.e. it intersects it at a nonzero angle), but at
certain points Д’(р) may be tangent to 5. Now we need to introduce
the notion of the order of tangency of the curve 5 and the field Х.
REMARK 9.2. If we are dealing with two smooth curves
у ф(т), у 4 (т), (0) — 4(0),
then the order of their tangency at the point 0 is the number р — the multiplicity of the function (т) — 4(2) at the point 01. If we needed to define
the order of tangency of the curve 5 and a direction field Х', defined in a whole
neighborhood of the point 0 in the plane (т, у), it could be defined
as the order of tangency of the curve 5 and the integral curve of the field ДА passing through the point 0 under consideration. But now we are dealing with
a field А’ that is defined only at the points of the curve 5 itself, and therefore has no integral curves at all, while the idea of extending it to
points outside 5 leads to ambiguity.
For this reason we use another definition, one that may be new
to the reader. Let Ё# be a parameter giving a regular parametrization р(Ё) of the curve 5 (regularity meaning that р’(#) = 0).
Consider @({) the angle between the tangent to 5 and the direction
Д(р) at the point р(®. Then tangency of the curve 5 and the field А at the point # = 0
is equivalent to the condition ©(0) = 0, and the order of tangency is defined as the order of the zero of the function @(® at the point $ = 0, i.e. its multiplicity д,
defined by formula (2.1). For example, tangency of zero order
(transversal intersection) is given by the condition @(0) 5 0, tangency of
first order by the conditions @(0) =0 and а"(0) = 0, etc.
1'An equivalent condition: the functions ф and ф have the same р-jet at the point 0,
but different (р + 1)-jets.
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PROBLEM 9.1. Show that the order of tangency is a geometric
invariant, i.e. a quantity that does not depend either on the choice of the parameter
$ on the curve 5, or on the choice of coordinates (5,у) in the plane. At the same time
the angle а(К itself is not an invariant, but only a semi-invariant, i.e. the quantity а
changes under changes of coordinates, but if а(р) = 0 at some point
ре 5, then this vanishing is preserved under all changes of the parameter & and
any choice of coordinates in the plane.
EXAMPLE 9.1. Fig. 9.2 shows the curve of critical points
5 and the field of kernels of the differential Х for the mappings (9.3) with the functions
Е = т? and Е = 23 + 19. In the first case (shown on the left) the field А’
is transversal to 5 = {х = 0} at all points. In the second case (on the right)
as the parameter # on the curve 5 = {у = —3=22 } one can take the
coordinate т itself. Then the angle а(х) between the parabola 5 and the horizontal
direction field А’ equals агсбап(—65). The field А’ is tangent to 5 at the origin
of coordinates, and this tangency is of first order.
у У
_ х х
Fig. 9.2 The curve 5 (heavy line) and the field А” for the mappings (9.3) for
the functions Е = 1? (left) and Р =? + ту (right)
PROBLEM 9.2. Show that the fold and the cusp are defined by the following conditions, equivalent to the conditions of remark 9.1:
® the curve © is regular and transversal to А”.
® the curve 5 is regular and has tangency of first order with the field А’.
HINT. Since the conditions given above are geometrically
invariant (do not depend on the choice of coordinate system), it suffices to
establish the equivalence of the first and second definitions of the fold and
the cusp for mappings of the form (9.3), which is verified trivially.
Now we can state and prove one of the classical results of singularity theory:
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THEOREM 9.1 (Whitney). The germ of any smooth mapping
(т, у): М > М at a fold point is С®-equivalent to the germ
2=42, Ш=у, (9.7)
and at a cusp point is СК-equivalent to the germ
д=13+1у, ш=у. (9.8)
Proof. Obviously it suffices to prove the theorem
for germs of mappings of the form (9.3), which we shall now do.
FOLD. By the condition ЁР,.(0) 5 0, by the implicit function theorem the curve 5, given by the equation Ё,(х, у) = 0, is the graph of some smooth function 1 = (у). Making the change of variable
тнт — ф(у), we (locally) turn this curve into the axis г = 0.
In the new coordinates (which, like the new function А`, we shall denote by the same letters as before) the derivative Ё,(т, у) vanishes
on the line 5 = 0. Using the representation (1.7), we obtain
Е(т,у) = В (у) + =*ф(т,у), Ву) =Е(0,у),
where (т,у) is a smooth function. From ЁЕ,.(0) # 0 it follows that ф(0) # 0.
Moreover, by means of the change 2 ++ —2 we can arrange that ‹р(0) > 0.
After this we make the change of variable
сн: ху (т, у),
which gives Е(т,у) = Ю (у) + 12, i.e. brings our mapping to the form
#2 =В (у) +1, ш=у.
Making the change 2 +} 2 — Ро (и), we obtain the normal form (9.7).
CUSP. STEP 1. Suppose that the function Р(т, у) satisfies the conditions
Е, (0) =0, Е..(0) =0, Е.,(0) 20, Ешь (0) #0. (9.9)
Let us show that then the germ of the mapping (9.3) is С®-equivalent to a germ of the same form with the function
Е(х, у) = (ху) (+3 +ту, (9.10)
where (т, у) and (у) are smooth functions, $ф(0) #2 0. For this let us represent
the function Р` in the form
Е(т,у) = Ю (у) + т9(т,у), В (у) = Е(0, у).
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From condition (9.9) it follows that
Making the change 2 ++} #— Ро(1), we obtain Ео(у) = 0, i.e. Ё = т9(х, у).
Let us apply the division theorem to the germ of the function 9(т,5). Since the multiplicity of 9(т,у) in the variable г at the point 0 equals 1, we obtain
9(т, у) = ф(т, у) (5 + а(у)т + В(у)), (9.11)
where а(у), 8(у), Ф(т,у) are smooth germs, (0) = В(0) = О and (0) = 0.
From the condition 9,(0) 52 0 it follows that 8'(0) 7 0. Hence we can make
a «right» change of variables у +? В(у) and simultaneously a «left» change 10 +» В(ч); this pair of changes preserves the relation чи = у. After
this let us write the (new) coefficient @(у,) in the form а«(у) = ф(уу with some smooth function 12. As a result our mapping takes
the form (9.3) with the function (9.10).
CUSP. STEP 2. According to what was proved above (problems 7.14 and 7.15),
the germ of any smooth function #(т, у) can be represented in the form
в(х, у) = а1(Е,у) + таз(Е, у) + 1? аз(Е, у) (9.12)
with suitable smooth germs а; (-, -}, where the function ЁР = Р(т,у)
is taken from formula (9.10).
Let us apply representation (9.12) to the function А(т,у) = 13, for convenience replacing the coefficients: а2 +} —а2 and аз +» Заз. As a result we obtain
13 =а1 (Е, у) — таз(Ё, у) + Зт?аз(Е,у). (9.13)
Representation (9.13) can be rewritten in the form
(2 —а(Е,у))3 +5 Еу)(е —а(Е,у)) = <(Е,у), (9.14)
where
а=аз, Б=а2 -— За”, е=а! -а5-а?. (9-15)
From representation (9.14) it is seen that the following pair of changes:
(т, у) => (5,9): Е=г-а(Е(т, у), у), У=ЫЬЕ(т, у), У), (9.16)
(2, ш) н+ (2,0): Е=с(2, 1), Ш=Ьа,щ) (9-17)
brings the germ of the mapping (9.3) with the function (9.10) to the normal
form (9.8).
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CUSP. STEP 3. To complete the proof of the theorem it remains to verify that both changes (9.16) and (9.17) are local
diffeomorphisms, i.e. that their Jacobi matrices at the origin are nondegenerate. For this let us show that
дал
ОР
и == (0) = 20) ду (©) = (9.18)
Indeed, comparing the coefficients of the monomial 1” on the left- and
right-hand sides of the identity (9.13), taking (9.10) into account, we immediately obtain
the first inequality in (9.18). Now let us look at the coefficient of
the monomial 2. On the left-hand side of the identity (9.13) it equals zero, while on the right —
it is composed of two terms, corresponding to the first and second terms
in (9.13) (the third term contains no monomial 2). Taking into account the first
inequality (9.18), from this we obtain the second condition.
Let us prove that the change (9.16) is a local diffeomorphism. Taking into account the expressions (9.15), as well as formulas (9.10) and (9.18), at
the origin we have
3
р да ду От 0) =1-— орт — (0) 72.10) =1 — 900) = = ЕЕ) —(0)2,(0) =
09 ОЬ 96) _ 042 — (0) = —(0)2,(0) + —(0 0
Hence it follows that the Jacobian of the change (9.16) at the origin equals 1.
A similar check for the change (9.17) is left to the reader. ||
REMARK 9.3. Theorem 9.1 was proved in 1955 by the American mathematician Hassler Whitney [42]. In proving the normal form for the cusp, Whitney did not use the Malgrange preparation
theorem (it appeared later), which made his proof substantially longer.
Let us picture how the fold and cusp mappings look geometrically, using the normal forms (9.7) and (9.8).
Since in both cases one coordinate in the image (namely ш) coincides with a coordinate in the preimage (namely у), both mappings can
be realized as the projection п of a smooth surface = in
the space (1,у,2) onto the plane (у,2) along the axis г.
DEFINITION 9.1. The set 5 of critical (singular) points
of the projection mapping т is called the criminant, and its projection л(5) is called the discriminant set or discriminant curve of this mapping.
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In the case of the fold (9.7) the surface - is a parabolic cylinder 5 = 12. The projection л is a two-sheeted
covering of the half-plane (у,2 > 0} branched along the axis 2 = 0.
The criminant is the axis у in the space (2, у, 2); the discriminant curve is
the axis у in the plane (у, 2); see fig. 9.3 (left).
In the case of the cusp (9.8) the surface .Я is the graph of the function 2 = 3 +жу.
The criminant is the parabola 322 + у = 0 on the surface „Я, and
the discriminant curve is the semicubical parabola
= 312, з= —2т3
in the plane (у,2). The cusp arises at the point where the criminant
is tangent to the field of kernels of the differential А’ = Кег а; this point is the origin
of coordinates; see fig. 9.3 (right).
Fig. 9.3 The fold and the cusp, arising when projecting
a surface onto a plane. The criminant (top) and the discriminant curve
(bottom) are shown by dashed lines
PROBLEM 9.3. Verify that the critical points of the following mappings (5,3/) ++ (2, 2) are stable under small
perturbations: 2 = 23, Ш=у; 2=ЕЗЖ, ШЕУ 2 = 22, ш=у?.
PROBLEM 9.4. Investigate the critical points of the mapping 2 = ту,
и = уи and its perturbations:
1) 2 = ту + =х”, = У,
2) 2= ту+ ЕТ, ШУ.
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