Lecture
Let us consider the natural generalization of Whitney's theorem on the fold and
the cusp from the preceding section, for mappings of spaces of arbitrary (but equal) dimension:
f(x, y): Rⁿ⁺¹ → Rⁿ⁺¹, y = (y₁,…,yₙ). (10.1)
As before, we shall consider the germ of mapping (10.1) at the origin and assume that f(0) = 0. Suppose that the origin
0 is a critical point of corank 1, i.e. rk f'(0) = n.
Then (see Problem 8.1) the germ of mapping (10.1) is CK-equivalent to
z = F(x, y), y₁ = y₁, …, yₙ = yₙ, (10.2)
where F(x, y): Rⁿ⁺¹ → R is some smooth function, F(0) = 0.
Consider the sequence of singularities (types of critical points) defined by the following two conditions:
1. The function F has finite multiplicity μ in the variable x, i.e.
∂F/∂x = … = ∂^{μ—1}F/∂x^{μ—1} = 0, ∂^μF/∂x^μ ≠ 0.
2. The differentials (or, equivalently, the gradients)
d(∂F/∂x), …, d(∂^{μ—1}F/∂x^{μ—1})
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at the point 0 are linearly independent.
Critical points of mapping (10.2) satisfying the two conditions given above are called Morin singularities.
REMARK 10.1. By virtue of condition (10.3), in formula (10.4) one may take the differentials (gradients) not with respect to all n + 1 variables,
but only with respect to the collection y₁,…,yₙ. Taking this into account, it is easy to see that
both conditions are compatible only for μ ≤ n + 1. Further, of course, this
inequality will always be assumed to hold.
REMARK 10.2. Under the two conditions listed above, the set Σ of critical points of the mapping f of the form (10.2),
given by the equation Fₓ(x, y) = 0, is a smooth hypersurface in the source space. However, the set of critical
values f(Σ) is a smooth hypersurface in the target space only for μ = 1, while for μ > 1
it has non-regular points (see Problem 11.7).
Let us compute the codimensions of Morin singularities for different μ, as well as
the codimensions of the degenerations not falling into this sequence. For μ = 1 the codimension of a Morin singularity equals 1, and the collection (10.4) is empty,
so for μ = 1 there are no other critical points.
For μ = 2 the codimension of a Morin singularity equals 2, and the collection (10.4) consists
of a single vector, so the codimension of the set of points not satisfying the second condition equals 2 + n (all the derivatives Fₓₓ,…,Fₓyₙ must vanish). Similarly, for arbitrary
μ, the codimension of a Morin singularity equals μ, and using the corank product formula (4.3), in which one must take m = μ—1, r = μ—2, one obtains that the codimension of the set of points not satisfying the second
condition equals
μ + (n—r)(m—r) = μ + (n—(μ—2))(μ—1—(μ—2)) = 2 + n.
Thus, Morin singularities include all types of critical points of germs of mappings of corank 1 up to and including codimension n, and, consequently, cover the entire list of stable germs of mappings Rⁿ → Rⁿ.
The task of this section is to obtain the CK-normal forms of germs of mappings with Morin singularities. It is easy to notice that for
n = 1, Morin singularities are the fold (μ = 1) and the cusp (μ = 2), and
¹Bernard Morin (1931–2018) — French mathematician. The results presented
in this section were published by him in a paper, a Russian translation of which appears in
the collection: Singularities of Differentiable Mappings (Moscow: Mir, 1968).
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the corresponding normal forms are given in Theorem 9.1 (Whitney). The generalization of this theorem to the case of arbitrary n and μ ≤ n+1 is
as follows.
THEOREM 10.1 (Morin). The germ of a smooth mapping (10.1) at a Morin critical point of multiplicity μ is CK-equivalent to the germ
μ—1
z = xᵘ⁺¹ + Σ yᵢxⁱ, y₁, …, yₙ. (10.5)
i=1
PROOF of Theorem 10.1 for μ = 1 repeats word for word the proof of Whitney's theorem for the fold. The proof for μ ≥ 2
follows the same scheme as the proof of Whitney's theorem for the cusp. Let us outline the main points of the corresponding argument.
PROBLEM 10.1. Apply the PTM (preparation theorem for mappings) to the germ of mapping (10.1) given by the formula
μ—1
z = f(x, y) = xᵘ⁺¹ + Σ yᵢxⁱ,
i=1
y₁ = y₁, …, yₙ = yₙ,
(10.6)
where g(y) is an arbitrary smooth function, g(0) = 0.
This problem is a multidimensional generalization of Problem 7.14, and its solution is analogous. As a result we shall establish that the multiplicity of the germ of mapping (10.6) equals μ+1, and its local algebra Qf = R[[x]]/(f'ₓ)
is the algebra of truncated polynomials of degree below μ—1, with generators
e₁(x, y) = e₂(x, y) = x, …, e_{μ+1}(x, y) = xᵘ.
Hence, by the PTM, for any germ of a function h(x, y) there follows the
representation
h(x, y) = a₀(y)h(x, y) + … + aᵤ(y)xᵘ, (10.7)
where the function f(x, y) is taken from formula (10.6).
PROBLEM 10.2. Establish representation (10.7) for the function
μ
F(x, y) = Σ φₛ(x, y)(x + ψₛ(y)) + a₀(x, y), s=1
(10.8)
where φₛ(x, y) and ψₛ(y) are smooth functions, φₛ(0) ≠ 0 and ψₛ(0) = 0.
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This is a generalization of Problem 7.15, and its solution is entirely analogous.
The further arguments completely repeat the proof of Whitney's theorem for the cusp.
STEP 1. First of all, let us bring the germ of the function F(x, y) to the form
μ—1
F(x, y) = φ(x, y)(xᵘ⁺¹ + a(y)) + Σ x^{μ—1—i} bᵢ(y))
i=1
with some functions φ, a, b₁,…,b_{μ—1}, where φ(0) ≠ 0, a(0) = 0 and
all bᵢ(0) = 0. This is done by means of the division theorem and precisely the same techniques that were used at the first step in the proof of Whitney's theorem for the cusp.
From the second condition in the definition of Morin singularities it follows
that the differentials db₁,…,db_{μ—1} at zero are linearly independent. Consequently, one can make a «right» change of variables yᵢ ↦ bᵢ(y)
and simultaneously a «left» change of variables uᵢ ↦ bᵢ(u). As a result of this pair of substitutions, the germ of our mapping takes the form (10.2) with
the function F(x, y) having the form (10.8).
STEP 2. Let us write representation (10.7) for the function f(x, y) = xᵘ⁺¹.
Here, for convenience, we replace aᵢ by —aᵢ:
xᵘ⁺¹ + a(y)xᵘ + a₁(y)xᵘ⁻¹ + … + aᵤ(y) = 0. (10.9)
Set c = a/(μ+1). It is not hard to show that then there exist functions c₀,…,c_{μ—1}, which are polynomials in a₀,…,aᵤ, for which
the identity
(x+c)ᵘ⁺¹ + a(y)(x+c)ᵘ + (a₁+…)(x+c)ᵘ⁻¹ + …
+ (aᵤ+…)(x+c)⁰ = 0 (10.10)
holds. To prove this statement one must successively compare the coefficients of all the monomials x^μ, xᵘ⁻¹,…, x⁰ in formulas
(10.9) and (10.10). The equalities of coefficients of the monomials xᵘ, xᵘ⁻¹ hold automatically. From the equality for the monomial xᵘ⁻¹ it follows that…
Continuing further, we obtain that
cᵢ = aᵢ — Pᵢ(a₀,…,aᵢ), i = 1,…,μ—1,
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where Pᵢ are polynomials, whose exact form is not important to us.
From representation (10.10) it follows that the following pair of substitutions
x = x + c(F(x, y), y), (10.11)
y₁ = c₁(F(x, y), y), …, y_{μ—1} = c_{μ—1}(F(x, y), y),
z = —c₀(z, u), u₁ = b₁(z, u), …, u_{μ—1} = b_{μ—1}(z, u) (10.12)
brings the germ of our mapping to the normal form (10.5).
PROBLEM 10.3 (STEP 3). To complete the proof of Theorem 10.1 it remains to verify that both substitutions (10.11) and (10.12) are
local diffeomorphisms. Carry out the corresponding check yourself. |
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