STRATIFICATION THEORY FROM THE WEIGHTED POINT OF VIEW

TOSHIZUMI FUKUI AND LAURENTIU PAUNESCU

To the memory of Professor Nobuo Sasakura

Stratification theory is a fundamental tool in constructing topological trivialization for families of varieties or maps. The idea was first presented by R.Thom [21] (for further development see for instance [16] and [7], and also a good survey can be found in [6]). At that time the purpose was mainly to show that, in some suitable set up, topological stable maps are dense (successfully proved). Using the existence of some good tube systems for a regular stratification, they showed the existence of a vector field whose integration gives a topological trivialization.

This approach was good enough for the study of topological stability, but unfortunately, in our opinion, the expression was not explicit enough to show that some given family is topologically trivial.

In this paper, we investigate stratification theory in terms of the defining equations of strata and maps (without tube systems), offering a concrete approach for solving the above problem.

The key notion in stratification theory is the regularity between strata (nowadays many regularity conditions are known, see [1, 2, 22]). In §1, we present a criterion for Verdier’s (w)- regularity conditions and Kuo’s ratio test condition in terms of the defining equations of the strata. Next we give an explicit construction of a vector field for topological trivialization. The key step in our construction is the use of a new projection formula (Lemma 1.4) allowing us to treat at the same time the non-complete intersection case. In this approach, it is possible to consider a weighted version of (w)-regularity condition and Kuo’s ratio test condition, and we do this in §2. We show that these conditions imply the integrability of the vector fields constructed by the method in §1. In §3, using the regularity conditions defined in §2, we prove the isotopy lemmas. Actually, using a partition of unity, we patch up together the vector fields constructed in §2.

With appropriate modifications one can use the same techniques in the complex analytic case to obtain analogous results. Because the modifications are standard, we concentrate ourselves only on the real case.

1. Regularity conditions

1.1. A criterion for (w)-regularity. Let (x, y) = (x_{1}, ..., x_{n}, y_{1}, ..., y_{m}) denote a system of coordinates of R^{n+m} and X, Y submanifolds of R^{n+m}. For notational convenience we also use x_{n+s} = y_{s}. We assume that

Y = {(x, y) ∈ R : x_{1} = ⋯ = x_{n} = 0}.

Let π_{P} denote the orthogonal projection of R^{n+m} to the normal space of X at P ∈ X. Then, following [22], we say X is (w)-regular over Y at 0 ∈ Y, if for any unit vector v tangent to Y |π_{P} (v)| ≲ |x| at P ∈ X near 0. Here A ≲ B means there is some positive constant C with A ≤ CB. Of course we may restrict v to the members of a basis of the tangent space of Y at 0.

We next assume that X is some open set in the regular locus of the variety defined as the zero locus of some C^{2}-functions F_{1}(x, y), ..., F_{p}(x, y) near 0, i.e., setting F := (F_{1}, ..., F_{p}), the Jacobi matrix of F has rank k on X near 0, where k ≤ p is the codimension of X in R^{n+m}. We note that the normal space of X is generated by the gradients of the functions F_{j} (j = 1, ..., p) at each P ∈ X near 0.

Let j_{1}, ..., j_{k} be integers with 1 ≤ j_{1} < ⋯ < j_{k} ≤ p. We set J = {j_{1}, ..., j_{k}}, F_{J} = (F_{j_{1}}, ..., F_{j_{k}}) and

dF_{J} = dF_{j_{1}} ∧ ⋯ ∧ dF_{j_{k}}, where dF_{j} = ∑_{i=1}^{n+m} \frac{∂F_{j}}{∂x_{i}} dx_{i},

d_{x}F_{J} = d_{x}F_{j_{1}} ∧ ⋯ ∧ d_{x}F_{j_{k}}, where d_{x}F_{j} = ∑_{i=1}^{n} \frac{∂F_{j}}{∂x_{i}} dx_{i},

and we define d^{x}F_{J} by dF_{J} = d_{x}F_{J} + d^{x}F_{J}.

For I ⊂ \{1, ..., n\}, S ⊂ \{1, ..., m\}, J ⊂ \{1, ..., p\} with #I + #S = #J = k, we set \frac{∂F_{J}}{∂(x_{I},y_{S})} the Jacobian of (F_{J}) with respect to the variables x_{i} (i ∈ I) and y_{s} (s ∈ S). If S = ∅, we simply denote it by \frac{∂F_{J}}{∂x_{I}}. We then define ∥d_{x}F ∥, ∥d^{x}F ∥ by the following formulae:

∥d_{x}F ∥^{2} = ∑_{J} ∥d_{x}F_{J}∥^{2} where ∥d_{x}F_{J}∥^{2} = ∑_{I} |\frac{∂F_{J}}{∂x_{I}} |^{2}

∥d^{x}F ∥^{2} = ∑_{J} ∥d^{x}F_{J}∥^{2} where ∥d^{x}F_{J}∥^{2} = ∑_{I,S:S≠∅} |\frac{∂F_{J}}{∂(x_{I}, y_{S})}|^{2}

Let |x| denote the function defined by |x|^{2} = ∑_{i=1}^{n} |x_{i}|^{2}. For a matrix M we have used |M| as the absolute value of its determinant det(M) if M is a square matrix or 0 otherwise.

Theorem 1.1. The following conditions are equivalent:

(i) X is (w)-regular over Y at 0. (ii) ∥d^{x}F ∥ ≲ |x| ∥d_{x}F ∥ holds on X near 0. (iii) For any C^{1}-functions φ_{j} (j = 1, ..., p) near 0, and s = 1, ..., m,

|∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} | ≲ |x| sup \{|∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} | : i = 1, ..., n\} holds on X near 0.

(iv) For J ⊂ {1, ..., , p}, I = {i_{1}, ..., i_{k−1}} ⊂ {1, ..., n} with 1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n, s = 1, ..., m,

|\frac{∂F_{J}}{∂(x_{I}, y_{s})}| ≲ |x| ∥d_{x}F ∥ holds on X near 0.

The condition (iii) is inspired by several conditions appeared in T.Gaffney’s paper [8]. The first author thanks Leslie Wilson for informing him about the existence of that paper. It is also possible to obtain a similar result for Kuo’s ratio test condition and we present it in §1.5

We next state some sufficient conditions for (w)-regularity. For j = 1, ..., p, we set

∥d_{x}F^{[j]}∥^{2} = ∑_{J} ∑_{i_{1}<···<i_{k−1}}|\frac{∂F_{J}^{[j]}}{∂x_{\{i_{1},...,i_{k−1}\}}} |^{2}, and h_{j} = \frac{∥d_{x}F ∥}{∥d_{x}F^{[j]}∥},

where

F_{J}^{[j]} = \begin{cases}(F_{j_{1}}, ..., F̂_{j_{a}}, ..., F_{j_{k}}) & if j ∈ J = \{j_{1}, ..., j_{k}\}, j = j_{a}, \\ (0, ..., 0) ((k − 1)-tuple) & if j ∉ J. \end{cases}

Here, is the notation indicating that we omit the letter (or the portion) to which is attached.

Corollary 1.2. X is (w)-regular over Y at 0, if for j = 1, ..., p the following inequalities hold on X near 0,

|\frac{∂F_{j}}{∂y_{s}} | ≲ |x| h_{j}, s = 1, ..., m (or, equivalently, ∥d^{x}F_{j}∥ ≲ |x|\frac{∥d_{x}F ∥}{∥d_{x}F^{[j]}∥}) .

Corollary 1.3. X is (w)-regular over Y at 0, if the following inequalities hold on X near 0,

∥d^{x}F_{j}∥ ≲ |x| ∥d_{x}F_{j}∥, ∥d_{x}F_{j}∥ ≲ h_{j} = \frac{∥d_{x}F ∥}{∥d_{x}F^{[j]}∥}, for j = 1, ..., p.

By (iii) in Theorem 1.1, the first inequality here is a necessary condition for (w)-regularity. Note that when p = k = 1, this is just Teissier’s c-condition (see [19, 20]).

1.2. Linear algebra. We present here some lemmas in linear algebra which are needed later on. Let V denote a real vector space with dimension n, and V^{∗} denote the dual space of V. Let e_{1}, ..., e_{n} denote a basis of V, and e_{1}^{∗}, ..., e_{n}^{∗} the dual basis of V^{∗} defined by e_{i}^{∗}(e_{j}) = δ_{ij}. We consider a positive symmetric bilinear form of V:

⟨ , ⟩ : V × V → R.

For an element v ∈ V, v^{∨} denotes the linear functional of V defined by v^{∨}(w) = ⟨w, v⟩. This induces an identification between V and V^{∗} by v ↦ v^{∨}. Then, we have e_{i}^{∨} = ∑_{i=1}^{n} g_{ij}e_{j}^{∗} where g_{ij} = ⟨e_{i}, e_{j}⟩. Thus, e_{i}^{∗} = ∑_{j=1}^{n} g^{ij}e_{j}^{∨} where (g^{ij}) denotes the inverse matrix of (g_{ij}).

It is well known that this bilinear form ⟨ , ⟩ induces bilinear forms on the exterior products ⋀^{k} V, ⋀^{k} V^{∗}. This is defined in the following way. The set {e_{i_{1}} ∧ ⋯ ∧ e_{i_{k}} | i_{1} < ⋯ < i_{k}} gives a basis of ⋀^{k} V, and the bilinear form is defined by

⟨e_{i_{1}} ∧ ⋯ ∧ e_{i_{k}}, e_{j_{1}} ∧ ⋯ ∧ e_{j_{k}}⟩ = g_{i_{1}j_{1}} ⋯ g_{i_{k}j_{k}}.

Similarly, the set {e_{i_{1}}^{∗} ∧ ⋯ ∧ e_{i_{k}}^{∗} | i_{1} < ⋯ < i_{k}} gives a basis of ⋀^{k} V^{∗}, and the bilinear form is defined by

⟨e_{i_{1}}^{∗} ∧ ⋯ ∧ e_{i_{k}}^{∗}, e_{j_{1}}^{∗} ∧ ⋯ ∧ e_{j_{k}}^{∗}⟩ = g^{i_{1}j_{1}} ⋯ g^{i_{k}j_{k}}.

Let a^{j} = ∑_{i=1}^{n} a_{i}^{j}e_{i} (j = 1, ..., k) denote vectors in V, and b^{j} = ∑_{i=1}^{n} b_{i}^{j}e_{i}^{∗} (j = 1, ..., k) covectors in V^{∗}. Under the identification (⋀^{k} V)^{∗} = ⋀^{k} V^{∗}, we have ( by [17], theorem 9, p.78),

(b^{1} ∧ ⋯ ∧ b^{k})(a^{1} ∧ ⋯ ∧ a^{k}) = det(b^{j}(a^{i}))_{1≤i,j≤k}.

Let a^{j} = ∑_{i=1}^{n} a_{i}^{j}e_{i} (j = 1, ..., p) denote vectors in V, and W the linear span of a^{1}, ..., a^{p}, assumed to be of dimension k ≤ p. We set W^{⊥} = {v ∈ V | ⟨v, w⟩ = 0, ∀w ∈ W }. Let ϕ_{j_{1},..,j_{k}} (1 ≤ j_{1} < ⋯ < j_{k} ≤ p) denote some constants such that the denominator is not zero. For v = (v_{1}, ..., v_{n}) ∈ V, we set

π(v) = ∑_{i=1}^{n} \frac{∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}}((a^{j_{1}})^{∨} ∧ ⋯ ∧ (a^{j_{k}})^{∨} ∧ e_{i}^{∗})(a^{j_{1}} ∧ ⋯ ∧ a^{j_{k}} ∧ v)}{∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}}((a^{j_{1}})^{∨} ∧ ⋯ ∧ (a^{j_{k}})^{∨})(a^{j_{1}} ∧ ⋯ ∧ a^{j_{k}})}e_{i}.

Lemma 1.4. The image of π is W^{⊥}, and π : V → W^{⊥} is the orthogonal projection with respect to ⟨ , ⟩.

Since we could not find this lemma in literature, we present here a complete proof of it.

Proof. Obviously π is a linear map. Since π(a^{j}) = 0, we have π(W) = 0. Therefore it is enough to see that π(v) = v for any v ∈ W^{⊥}. For v ∈ W^{⊥} and j = 1, ..., p, we have (a^{j})^{∨}(v) = ⟨a^{j}, v⟩ = 0. Thus we obtain

∑_{i} ((a^{j_{1}})^{∨}∧⋯∧(a^{j_{k}})^{∨}∧e_{i}^{∗})(a^{j_{1}}∧⋯∧a^{j_{k}} ∧v)e_{i} = ∑_{i} det (\begin{matrix}(a^{j_{r}})^{∨}(a^{j_{q}}) & (a^{j_{r}})^{∨}(v) \\ e_{i}^{∗}(a^{j_{q}}) & e_{i}^{∗}(v) \end{matrix})_{1≤r,q≤k} e_{i}

= det (\begin{matrix}(a^{j_{r}})^{∨}(a^{j_{q}}) & 0 \\ a^{j_{q}} & v \end{matrix})_{1≤r,q≤k} = ((a^{j_{1}})^{∨} ∧ ⋯ ∧ (a^{j_{k}})^{∨})(a^{j_{1}} ∧ ⋯ ∧ a^{j_{k}})v

Then we obtain

∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}} ∑_{i} ((a^{j_{1}})^{∨} ∧ ⋯ ∧ (a^{j_{k}})^{∨} ∧ e_{i}^{∗})(a^{j_{1}} ∧ ⋯ ∧ a^{j_{k}} ∧ v)e_{i}

= ∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}}((a^{j_{1}})^{∨} ∧ ⋯ ∧ (a^{j_{k}})^{∨})(a^{j_{1}} ∧ ⋯ ∧ a^{j_{k}})v.

Thus we obtain π(v) = v for v ∈ W^{⊥}.

Note that

( ∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}}((a^{j_{1}})^{∨} ∧ ⋯ ∧ (a^{j_{k}})^{∨})(a^{j_{1}} ∧ ⋯ ∧ a^{j_{k}})) π(v)

= ∑_{i=1}^{n} ∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}}((a^{j_{1}})^{∨} ∧ ⋯ ∧ (a^{j_{k}})^{∨} ∧ e_{i}^{∗})(a^{j_{1}} ∧ ⋯ ∧ a^{j_{k}} ∧ v)e_{i}

for any constants ϕ_{j_{1},...,j_{k}}.

Since v = ∑_{i=1}^{n}(e_{i})^{∗}(v)e_{i}, if we set ϕ_{j_{1},...,j_{k}} = 1, we get the following formula

π(v) = \frac{∑_{j_{1}<···<j_{k}} det (\begin{matrix}⟨a^{j_{r}}, a^{j_{q}}⟩ & ⟨a^{j_{r}}, v⟩ \\ a^{j_{r}} & v \end{matrix})}{∑_{j_{1}<···<j_{k}} det(⟨a^{j_{r}}, a^{j_{q}}⟩)_{1≤r,q≤k}}.

Remark 1.5. Let V be the tangent space T_{x}M of a manifold M at a point x. Then V^{∗} is its cotangent space T_{x}^{∗}M. Let ⟨ , ⟩ be a Riemannian metric. Let x_{1}, ..., x_{n} be a local coordinate system around x, \frac{∂}{∂x_{1}} , ..., \frac{∂}{∂x_{n}} the usual basis of T_{x}M, dx_{1}, ..., dx_{n} the usual (dual) basis of T_{x}^{∗}M.

Let F_{1}, ..., F_{p} be C^{1}-functions and assume the rank of the Jacobian of (F_{1}, ..., F_{p}) is rank k on X ⊂ ⋂_{j=1}^{p} F_{j}^{−1}(0), and ϕ_{j_{1},...,j_{k}} (1 ≤ j_{1} < ⋯ < j_{k} ≤ p) positive functions on the zero set of (F_{1}, ..., F_{p}). Applying the previous lemma with a^{j} = gradF_{j}, where gradF_{j} is the gradient vector of F_{j} i.e. a vector with dF_{j} = (gradF_{j})^{∨}, we obtain that the orthogonal projection of v ∈ T_{x}M to the tangent space T_{x}X is expressed by the following form.

(1.1) v ↦ ∑_{i} \frac{∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}}(dF_{j_{1}} ∧ ⋯ ∧ dF_{j_{k}} ∧ dx_{i})(gradF_{j_{1}} ∧ ⋯ ∧ gradF_{j_{k}} ∧ v)}{∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}}∥dF_{j_{1}} ∧ ⋯ ∧ dF_{j_{k}}∥^{2}} \frac{∂}{∂x_{i}} .

1.3. Proof of Theorem 1.1.

Lemma 1.6. X is (w)-regular over Y at 0 iff the following inequalities hold on X near 0 for each J ⊂ {1, ..., p}, #J = k:

(a)  |⟨dF_{J} ∧ dx_{i}, dF_{J} ∧ dy_{s}⟩| ≲ |x| ∥dF_{J}∥^{2} for 1 ≤ i ≤ n; 1 ≤ s ≤ m
(b)  |⟨dF_{J} ∧ dy_{i}, dF_{J} ∧ dy_{s}⟩| ≲ |x| ∥dF_{J}∥^{2} for 1 ≤ i < s ≤ m
(c)  ∥dF_{J}∥^{2} − ∥dF_{J} ∧ dy_{s}∥^{2} ≲ |x| ∥dF_{J}∥^{2} for 1 ≤ s ≤ m

Proof. This is a consequence of the definition of (w)-regularity and (1.1). We can see this taking particular functions ϕ_{j_{1},...,j_{k}} in the expression (1.1). Namely, one can take ϕ_{j_{1},...,j_{k}} = 1, if J = {j_{1}, ..., j_{k}}; 0, otherwise.

Proof of (ii) ⟹ (i). Since the left hand side of (c) does not contain the terms containing \frac{∂F_{j}}{∂y_{s}} for 1 ≤ j ≤ p, and ∥dF_{J}∥^{2} = ∥d_{x}F_{J}∥^{2} + ∥d^{x}F_{J}∥^{2}, (c) is equivalent to the following inequality.

∥d^{x}F ∥^{2} ≲ |x| ∥d_{x}F ∥^{2}.

By Cauchy-Schwarz, (b) comes from (c). Thus we have that X is (w)-regular over Y at 0 iff the inequalities (a) and (c) hold on X near 0.

We now assume (ii). We then have ∥d^{x}F ∥^{2} ≲ |x|^{2} ∥d_{x}F ∥^{2} on X near 0. Since we may assume that |x| is very small, (c) trivially holds. By Cauchy-Schwarz inequality, this implies (a). This completes the proof.

Proof of (i) ⟹ (iii). We first remark the following equality:

\begin{aligned}∑_{i≠n+s} ∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} ⟨dF_{J} ∧ dx_{i}, dF_{J} ∧ dy_{s}⟩ & = ⟨dF_{J} ∧ ∑_{j=1}^{p} φ_{j} (dF_{j} − \frac{∂F_{j}}{∂y_{s}} dy_{s}) , dF_{J} ∧ dy_{s}⟩ \\ & = − ∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} ∥dF_{J} ∧ dy_{s}∥^{2}. \end{aligned}

Then, by (a) and (b), we have

\begin{aligned}|∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} | ∥dF_{J} ∧ dy_{s}∥^{2} & ≤ ∑_{i≠n+s} |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} | |⟨dF_{J} ∧ dx_{i}, dF_{J} ∧ dy_{s}⟩| \\ & ≲ ∑_{i≠n+s} |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} | |x| ∥dF ∥^{2} . \end{aligned}

Then by (c) we have

\begin{aligned}|∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} | ∥dF_{J}∥^{2} & ≤ |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} | (∥dF_{J} ∧ dy_{s}∥^{2} + |x| ∥dF ∥^{2}) \\ & ≲ ∑_{i=1}^{n+m} |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} | |x| ∥dF ∥^{2} . \end{aligned}

Summing up for all J and s we get the desired inequality, which completes the proof.

In §2 we shall prove the generalized weighted version (Theorem 2.1) of the equivalence (ii) ⟺ (iii) ⟺ (iv), and this will complete the proof of our theorem.

Proof of Corollary 1.2. We define d_{ℓ} by the following formula: d_{ℓ}^{2} = ∑_{J} ∑_{I,S:#S=ℓ} |\frac{∂F_{J}}{∂(x_{I},y_{S})} |^{2} .

By Cauchy-Schwarz inequality, the inequality in Corollary 1.2 implies d_{1} ≲ |x|d_{0}, which implies (iv) in Theorem 1.1.

Proof of Corollary 1.3. Obvious. Use Corollary 1.2.

1.4. Kuo’s vector. Let U be a neighborhood of 0, and Y = {x_{1} = ⋯ = x_{n} = 0}. We assume that X is the regular locus of the zero locus of C^{2}-functions F_{j} (j = 1, ..., p), and assume the codimension is k.

Let v be a tangent vector to Y. The purpose of this subsection is to give an explicit construction of a tangent vector ξ to X which is an extension of v and dp(ξ) = v where p is the natural projection defined by (x, y) ↦ y.

We first remark that any multiple of the orthogonal projection of v = ∑_{s=1}^{m} c_{s} \frac{∂}{∂y_{s}} to the tangent space of X may not have this property (except the case m = 1).

For s = 1, ..., m, we use the following notation:

d^{(s)}F_{j} = ∑_{i=1}^{n} \frac{∂F_{j}}{∂x_{i}} dx_{i} + \frac{∂F_{j}}{∂y_{s}} dy_{s}, and d^{(s)}F_{J} = d^{(s)}F_{j_{1}} ∧ ⋯ ∧ d^{(s)}F_{j_{k}}.

Let us consider the orthogonal projection of \frac{∂}{∂y_{s}} to the tangent space of X in the space defined by y_{1} = ⋯ = y_{s−1} = y_{s+1} = ⋯ = y_{m} = 0.

This is expressed by

∑_{i=1}^{n} \frac{∑_{J} ϕ_{J}⟨d^{(s)}F_{J} ∧ dx_{i}, d^{(s)}F_{J} ∧ dy_{s}⟩}{∑_{J} ϕ_{J}⟨d^{(s)}F_{J}, d^{(s)}F_{J}⟩} \frac{∂}{∂x_{i}} + \frac{∑_{J} ϕ_{J}⟨d^{(s)}F_{J} ∧ dy_{s}, d^{(s)}F_{J} ∧ dy_{s}⟩}{∑_{J} ϕ_{J}⟨d^{(s)}F_{J}, d^{(s)}F_{J}⟩} \frac{∂}{∂y_{s}} ,

where ϕ_{J} are any positive function on X. Its multiple whose \frac{∂}{∂y_{s}} -component is 1, is expressed by the following formula:

∑_{i=1}^{n} \frac{∑_{J} ϕ_{J}(d^{(s)}F_{J} ∧ dx_{i})(grad_{x}F_{J} ∧ \frac{∂}{∂y_{s}})}{∑_{J} ϕ_{J}∥d_{x}F_{J}∥^{2}} \frac{∂}{∂x_{i}} + \frac{∂}{∂y_{s}} ,

where grad_{x}F_{J} denotes the wedge product of the gradients in coordinate x_{1}, ..., x_{n}.

Let v = ∑_{j} c_{s} \frac{∂}{∂y_{s}} be a unit vector tangent to Y. Then the above construction implies the following vector ξ is tangent to X:

ξ = ∑_{i=1}^{n} \frac{∑_{j=1}^{m} c_{s} ∑_{J} ϕ_{J}(d^{(s)}F_{J} ∧ dx_{i})(grad_{x}F_{J} ∧ \frac{∂}{∂y_{s}})}{∑_{J} ϕ_{J}∥d_{x}F_{J}∥^{2}} \frac{∂}{∂x_{i}} + v.

Since grad_{x}F_{j} has zero \frac{∂}{∂y_{s}} components, we thus have

ξ = ∑_{i=1}^{n} \frac{∑_{J} ϕ_{J}(dF_{J} ∧ dx_{i})(grad_{x}F_{J} ∧ v)}{∑_{J} ϕ_{J}∥d_{x}F_{J}∥^{2}} \frac{∂}{∂x_{i}} + v.

We call this ξ a Kuo’s vector, because T.-C.Kuo expressed this vector in such an explicit form in [15] when m = p = 1, (see also [11, 12, 14]). Here we remark that the \frac{∂}{∂x_{i}} -component of ξ (say ξ_{i}) is

ξ_{i} = ∑_{s=1}^{m} c_{s} \frac{∑_{J,I:#I=k−1} ϕ_{J} \frac{∂F_{J}}{∂(x_{I},x_{i})} \frac{∂F_{J}}{∂(x_{I},y_{s})}}{∑_{J,I} ϕ_{J}|\frac{∂F_{J}}{∂x_{I}} |^{2}} .

We assume that X is (w)-regular over Y at 0. Then the inequality (ii) of Theorem 1.1 implies

|\frac{∑_{J}⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ v^{∨}⟩}{∑_{J} ∥d_{x}F_{J}∥^{2}} | ≲ |x| for i = 1, ..., n on X near 0.

This implies that ξ satisfies (stratified) Lipschitz condition on (X, Y) near 0.

1.5. Ratio test conditions. We use the same notation as in Theorem 1.1. We define |(x, y)| and |y| by

|(x, y)|^{2} = ∑_{i=1}^{n} |x_{i}|^{2} + ∑_{j=1}^{m} |y_{j}|^{2}, and |y|^{2} = ∑_{s=1}^{m} |y_{s}|^{2}.

We say that X is (r)-regular over Y at 0, if for any unit vector v tangent to Y

|π_{P} (v)| |(x, y)| = o(|x|) when P = (x, y) → 0, P ∈ X.

Here “A = o(B) when P → 0” means lim_{P →0} |A/B| = 0. This was defined in [12].

Theorem 1.7. The following conditions are equivalent:

(i) X is (r)-regular over Y at 0. (ii) ∥d^{x}F ∥ |(x, y)| = o(|x| ∥d_{x}F ∥) when (x, y) → 0 on X. (iii) For any C^{1}-function φ_{j} (j = 1, ..., p) near 0 and s = 1, ..., m,

|∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} | |(x, y)| = o(|x| ∥d_{x}F ∥) when (x, y) → 0 on X.

(iv) For J ⊂ {1, ..., , p}, I = {i_{1}, ..., i_{k−1}} ⊂ {1, ..., n} with 1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n, s = 1, ..., m,

|\frac{∂F_{J}}{∂(x_{I}, y_{s})}| |(x, y)| = o(|x| ∥d_{x}F ∥) when (x, y) → 0 on X.

Proof. This is proved in exactly the same way as Theorem 1.1 (we omit the details).

Proposition 1.8. When X is (r)-regular over Y at 0, the flow of Kuo’s vector field ξ is unique.

Proof. We first remark that ξ does not depend on the choice of ϕ_{J}, and we set ϕ_{J} ≡ 1. If X is (r)-regular over Y at 0, then the inequalities obtained by replacing |(x, y)| by |y| from the inequalities in (ii), (iii), (iv) of Theorem 1.7 hold. This shows that the Kuo’s vector ξ satisfies Nagumo’s criterion (Corollary 6.1 on page 32 of [9]) and we have uniqueness of the flow if m = 1. Assume that m > 1. Let C be the integral curve of v. Then the flow of ξ is in π^{−1}(C) and we can reduce the problem to the case m = 1 and we are done.

2. Weighted versions of (w)-regularity

In the previous section we present (w) regularity in terms of defining equations in some coordinate. This allow us to consider a weighted version of (w)-regularity condition. We present it in this section.

2.1. A weighted version of (w)-regularity. Let (x, y) = (x_{1}, ..., x_{n}, y_{1}, ..., y_{m}) denote a system of coordinates of a neighborhood U of 0 in R^{n+m} and X, Y disjoint submanifolds of U. For notational convenience we put x_{n+s} = y_{s}. We assume that

Y = {(x, y) ∈ U : x_{1} = ⋯ = x_{n} = 0}. (2.1)

We fix a weight w = (w_{1}, ..., w_{n}), and consider the function defined by

∥x∥ = ∥x∥_{w} := (|x_{1}|^{\frac{2w}{w_{1}}} + ⋯ + |x_{n}|^{\frac{2w}{wn}} )^{\frac{1}{2w}} , where w = w_{1}w_{2} ⋯ w_{n}. (2.2)

We next assume that X is some open set in the regular locus of the variety defined as the zero locus of C^{2}-functions F_{1}(x, y), ..., F_{p}(x, y) near 0, i.e., setting F := (F_{1}, ..., F_{p}), the Jacobi matrix of F has rank k on X near 0, where k is the codimension of X in R^{n+m}. We note that the normal space of X is generated by the gradients of functions F_{j} (j = 1, ..., p) at each P ∈ X near 0.

We define ∥d_{x}F ∥_{w}, ∥d^{x}F ∥_{w}, D_{w}(ℓ) by the following formula:

(2.3)  ∥d_{x}F ∥_{w}^{2} = ∑_{J} ∥d_{x}F_{J}∥_{w}^{2} where ∥d_{x}F_{J}∥_{w}^{2} = ∑_{I} (∥x∥_{w}^{w_{I}} |\frac{∂F_{J}}{∂x_{I}} |)^{2}

(2.4)  ∥d^{x}F ∥_{w}^{2} = ∑_{J} ∥d^{x}F_{J}∥_{w}^{2} where ∥d^{x}F_{J}∥_{w}^{2} = ∑_{I,S:S≠∅} (∥x∥_{w}^{w_{I}} |\frac{∂F_{J}}{∂(x_{I}, y_{S})}|)^{2}

(2.5)  D_{w}(ℓ)^{2} = ∑_{J} ∑_{I,S:#S=ℓ} (∥x∥_{w}^{w_{I}} |\frac{∂F_{J}}{∂(x_{I}, y_{S})}|)^{2} . Here w_{I} = ∑_{i∈I} w_{i}.

We consider the singular metric of R^{n+m} defined by

⟨∥x∥_{w}^{w_{i}} \frac{∂}{∂x_{i}} , ∥x∥_{w}^{w_{j}} \frac{∂}{∂x_{j}} ⟩ = δ_{ij} := \begin{cases}1 & if i = j, \\ 0 & if i ≠ j. \end{cases} (2.6)

Here, we understand that w_{n+1} = ⋯ = w_{n+m} = 0. We first remark that

⟨dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{k}}, dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{k}}⟩ = ∥x∥_{w}^{2(w_{i_{1}}+···+w_{i_{k}})}

and ⟨dF, dF ⟩ = ∥dF ∥_{w}^{2}, ⟨d_{x}F, d_{x}F ⟩ = ∥d_{x}F ∥_{w}^{2}, etc.

Theorem 2.1. The following conditions are equivalent:

(i) D_{w}(m) ≲ D_{w}(m − 1) ≲ ⋯ ≲ D_{w}(1) ≲ D_{w}(0) holds on X near 0. (ii) ∥d^{x}F ∥_{w} ≲ ∥d_{x}F ∥_{w} holds on X near 0. (iii) For any C^{1}-functions φ_{j} (j = 1, ..., p) near 0, and s = 1, ..., m,

|∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} | ≲ ∑_{i=1}^{n} ∥x∥_{w}^{w_{i}} |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} | holds on X near 0.

(iv) For J ⊂ {1, …, p}, I = {i_{1}, ..., i_{k−1}} ⊂ {1, …, n} with 1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n, s = 1, ..., m,

∥x∥_{w}^{w_{i_{1}}+···+w_{i_{k−1}}} |\frac{∂F_{J}}{∂(x_{I}, y_{s})}| ≲ ∥d_{x}F ∥_{w} holds on X near 0.

(v) For J ⊂ {1, …, p}, i = 1, ..., n, s = 1, ..., m,

|⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ dy_{s}⟩| ≲ ∥x∥_{w}^{w_{i}}∥d_{x}F_{J}∥_{w}^{2} holds on X near 0.

(vi) For some positive C^{1}-functions ϕ_{J} on X with J ⊂ {1, …, p}, i = 1, ..., n, s = 1, ..., m,

|∑_{J} ϕ_{J}⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ dy_{s}⟩| ≲ ∥x∥_{w}^{w_{i}} ∑_{J} ϕ_{J}∥d_{x}F_{J}∥_{w}^{2} holds on X near 0.

We say that X is weighted (w)-regular over Y at 0 with respect to w (or w-(w)-regular for short), if one of the above equivalent conditions holds in some coordinate system (x, y) with (2.1). If w_{1} = ⋯ = w_{n} = 1, these coincide with the usual (w)-regular condition. However, it is not immediate to see that (ii) is the same condition as in the homogeneous case.

We also state the weighted version of Theorem 1.3.

Corollary 2.2. X is w-(w)-regular over Y at 0, if the following inequalities hold on X near 0:

∥d^{x}F_{j}∥_{w} ≲ ∥d_{x}F_{j}∥_{w}, ∥d_{x}F_{j}∥_{w} ≲ \frac{∥d_{x}F ∥_{w}}{∥d_{x}F^{[j]}∥_{w}} , for j = 1, ..., p.

Proof. The proof is similar to that of Corollary 1.3, and we omit it.

2.2. Proof of Theorem 2.1. Since implications (i) ⟹ (ii) ⟹ (iv), (v) ⟹ (vi) are clear, it is enough to see (iv) ⟹ (iii) ⟹ (i), (iii) ⟹ (v), and (vi) ⟹ (iii).

Proof of (iii) ⟹ (i). We assume (iii). By the inequality in (iii), we have

|\frac{∂F_{J}}{∂(x_{I}, y_{s})}| ≲ ∑_{i=1}^{n} ∥x∥_{w}^{w_{i}} |\frac{∂F_{J}}{∂(x_{I}, x_{i})}| (2.7)

and D_{w}(1) ≲ D_{w}(0) = ∥d_{x}F ∥_{w}. Similarly we also have

|\frac{∂F_{J}}{∂(x_{I}, y_{s_{1}}, y_{s_{2}})}| ≲ ∑_{i=1}^{n} ∥x∥_{w}^{w_{i}} |\frac{∂F_{J}}{∂(x_{I}, y_{s_{1}}, x_{i})}|

and, using (2.7), we obtain D_{w}(2) ≲ D_{w}(1) on X near 0. In a similar way we obtain that D_{w}(ℓ) ≲ D_{w}(ℓ − 1) (ℓ = 2, ..., m), and this completes the proof.

Proof of (iii) ⟹ (v). We assume (iii). Using (iii) in a similar way to the previous proof, we have

|⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ dy_{s}⟩| = |∑_{I} ∥x∥_{w}^{2(w_{I}+w_{i})} \frac{∂F_{J}}{∂(x_{I}, x_{i})} \frac{∂F_{J}}{∂(x_{I}, y_{s})}|

≲ ∥x∥_{w}^{w_{i}} ∑_{ℓ=1}^{n} ∑_{I} ∥x∥_{w}^{2w_{I}+w_{ℓ}+w_{i}} |\frac{∂F_{J}}{∂(x_{I}, x_{i})}| |\frac{∂F_{J}}{∂(x_{I}, x_{ℓ})}| ≤ ∥x∥_{w}^{w_{i}}∥d_{x}F_{J}∥_{w}^{2}

and we are done.

Proof of (vi) ⟹ (iii). Since

∑_{i=1}^{n} \frac{∂F_{j}}{∂x_{i}} ⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ dy_{s}⟩ = − ∑_{t=1}^{m} \frac{∂F_{j}}{∂y_{t}} ⟨dF_{J} ∧ dy_{t}, d_{x}F_{J} ∧ dy_{s}⟩ = −\frac{∂F_{j}}{∂y_{s}} ∥dF_{J}∥_{w}^{2},

we obtain

∑_{j=1}^{p} φ_{j} ∑_{i=1}^{n} \frac{∂F_{j}}{∂x_{i}} ∑_{J} ϕ_{J}⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ dy_{s}⟩ = − ∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} ∑_{J} ϕ_{J}∥dF_{J}∥_{w}^{2}.

Thus, we obtain

\begin{aligned}|∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} | ∑_{J} ϕ_{J}∥d_{x}F_{J}∥_{w}^{2} & ≤ ∑_{i=1}^{n} |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} | |∑_{J} ϕ_{J} ⟨d^{(s)}F_{J} ∧ dx_{i}, d^{(s)}F_{J} ∧ dy_{s}⟩| \\ & ≲ ∑_{i=1}^{n} |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} | ∥x∥_{w}^{w_{i}} ∑_{J} ϕ_{J}∥d_{x}F_{J}∥_{w}^{2}. \end{aligned}

Dividing by ∥d_{x}F_{J}∥_{w}^{2}, we obtain (iii).

Proof of (iv) ⟹ (iii). (iv) implies D_{w}(1) ≲ ∥d_{x}F ∥_{w} on X near 0. Then, by Cauchy-Schwarz inequality, we have

|⟨d^{(s)}F_{J} ∧ dx_{i}, d^{(s)}F_{J} ∧ dy_{s}⟩| ≲ ∥x∥_{w}^{w_{i}}∥d_{x}F_{J}∥_{w}^{2} on X near 0.

We next remark the following equality:

∑_{i=1}^{n} \frac{∂F_{j}}{∂x_{i}} ⟨d^{(s)}F_{J} ∧ dy_{s}, d^{(s)}F_{J} ∧ dx_{i}⟩ = ⟨d^{(s)}F_{J} ∧ dy_{s}, d^{(s)}F_{J} ∧ (d^{(s)}F_{j} − \frac{∂F_{j}}{∂y_{s}} dy_{s})⟩

= −\frac{∂F_{j}}{∂y_{s}} ∥d_{x}F_{J}∥_{w}^{2}.

Thus, we obtain

\begin{aligned}|∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} | ∥d_{x}F_{J}∥_{w}^{2} & ≤ ∑_{i=1}^{n} |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} | |⟨d^{(s)}F_{J} ∧ dx_{i}, d^{(s)}F_{J} ∧ dy_{s}⟩| \\ & ≲ ∑_{i=1}^{n} |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} | ∥x∥_{w}^{w_{i}} ∥d_{x}F ∥_{w}^{2}. \end{aligned}

Summing up the inequalities, for all J and dividing by ∥d_{x}F ∥_{w}^{2}, we obtain (iii).

2.3. ODE problems. To construct a topological trivialization using the weighted (w)- condi- tion (or Kuo’s ratio test condition), we consider the following classical ODE problem

\frac{dx}{dt} = f(t, x), x(t_{0}) = x_{0} (2.8)

where x = (x_{1}, ..., x_{n}) If f(t, x) is continuous, it is well known that the solution x = x(t) exists (see [9], theorems 2.1, page 12). We fix a weight w = (w_{1}, ..., w_{n}) and consider the function ∥x∥ = ∥x∥_{w} defined by (2.1).

We say f(t, x) satisfies a weighted Lipschitz condition with respect to the weight w (or, w-Lipschitz, for short) if the following condition holds:

∥f(t, x) − f(t, \overline{x})∥_{w} ≲ ∥x − \overline{x}∥_{w} for any x, \overline{x} near x_{0}.

We say that f(t, x) is weighted Lipschitz , if it satisfies a weighted Lipschitz condition for some weight w.

Lemma 2.3. If f(t, x) is weighted Lipschitz near (t, x) = (t_{0}, x_{0}), then (2.8) admits at most one solution near t = t_{0}.

Proof. Let x(t), and \overline{x}(t) be two solutions of (3.1) with x(t_{0}) = \overline{x}(t_{0}) = x_{0}. Since

x_{i}(t) − x_{i}(t_{0}) = ∫_{t_{0}}^{t} f_{i}(t, x(s))ds, \overline{x}_{i}(t) − \overline{x}_{i}(t_{0}) = ∫_{t_{0}}^{t} f_{i}(t, \overline{x}(s))ds,

we have

x_{i}(t) − \overline{x}_{i}(t) = ∫_{t_{0}}^{t}\{f_{i}(s, x(s)) − f_{i}(s, \overline{x}(s))\}ds.

Thus

∑_{i=1}^{n} |x_{i}(t) − \overline{x}_{i}(t)|^{\frac{2w}{w_{i}}} = ∑_{i=1}^{n} |∫_{t_{0}}^{t}\{f_{i}(s, x(s)) − f_{i}(s, \overline{x}(s))\}ds|^{\frac{2w}{w_{i}}}

≤ ∑_{i=1}^{n} (∫_{t_{0}}^{t} |f_{i}(s, x(s)) − f_{i}(s, \overline{x}(s))|ds)^{\frac{2w}{w_{i}}}

≤ ∑_{i=1}^{n} |t − t_{0}|^{\frac{2w}{w_{i}} −1} ∫_{t_{0}}^{t} |f_{i}(s, x(s)) − f_{i}(s, \overline{x}(s))|^{\frac{2w}{w_{i}}} ds (by Hölder)

≤ ∑_{i=1}^{n} ∫_{t_{0}}^{t} |f_{i}(s, x(s)) − f_{i}(s, \overline{x}(s))|^{\frac{2w}{w_{i}}} ds (since |t − t_{0}| < 1).

We remark that the w-Lipschitz condition implies that there is C > 0 such that

∑_{i=1}^{n} |f_{i}(s, x) − f_{i}(s, \overline{x})|^{\frac{2w}{w_{i}}} ≤ C ∑_{i=1}^{n} |x_{i} − \overline{x}_{i}|^{\frac{2w}{w_{i}}} .

Setting φ(t) = ∑_{i=1}^{n} |x_{i}(t) − \overline{x}_{i}(t)|^{\frac{2w}{w_{i}}} we thus have

φ(t) ≤ ∫_{t_{0}}^{t} ∑_{i=1}^{n} |f_{i}(s, x) − f_{i}(s, \overline{x})|^{\frac{2w}{w_{i}}} ds = C ∫_{t_{0}}^{t} φ(s)ds

We here set Φ(t) = ∫_{t_{0}}^{t} φ(s)ds. Then we have \frac{dΦ}{dt} = φ(t) ≤ CΦ(t). Since

\frac{d}{dt} (e^{−C(t−t_{0})}Φ(t)) = e^{−C(t−t_{0})} (\frac{dΦ}{dt}(t) − CΦ(t)) ≤ 0,

and Φ(t) ≥ 0, Φ(t_{0}) = 0, we obtain Φ(t) = 0, and we are done.

We say f(t, x) satisfies a weighted Nagumo condition with respect to the weight w (or, w-Nagumo, for short) if the following condition holds:

∥f(t, x) − f(t, \overline{x})∥_{w} ≤ ∥\frac{x − \overline{x}}{t − t_{0}} ∥_{w} for any t near t_{0} and any x, \overline{x} near x_{0}.

We say that f(t, x) is weighted Nagumo , if it satisfies a weighted Nagumo condition for some weight w.

Lemma 2.4. If f(t, x) is weighted Nagumo near (t, x) = (t_{0}, x_{0}), then (2.8) admits at most one solution near t = t_{0}.

Proof. Let x(t), \overline{x}(t) be same as the proof of Lemma 2.3. We define φ(t) = (t − t_{0})∥(x − \overline{x})/(t − t_{0})∥_{w}^{2w} and as in the previous proof we get

φ(t) ≤ ∫_{t_{0}}^{t} \frac{φ(s)}{s − t_{0}} ds.

We set Φ(t) = ∫_{t_{0}}^{t} \frac{φ(s)}{s−t_{0}} ds. Then

\frac{d}{dt} (\frac{Φ(t)}{t − t_{0}} ) = \frac{φ(t) − Φ(t)}{(t − t_{0})^{2}} ≤ 0.

Since Φ(t) ≥ 0, Φ(t_{0}) = 0, we obtain Φ(t) = 0, and we are done.

Note that weighted Lipschitz trivially implies weighted Nagumo..

2.4. Kuo’s vector. We consider the Euclidean space R^{n+m} with the singular metric defined by (2.6). Let U be a neighborhood of 0, and Y = {x_{1} = ⋯ = x_{n} = 0}. In this section we assume that X is the regular locus of the zero locus of C^{2}-functions F_{j} (j = 1, ..., p), and assume the codimension is k. By elementary calculation, we can express the gradient vector field of the function F_{j} with respect to this singular metric is expressed as follows:

gradF_{j} = grad_{x}F_{j} + ∑_{s=1}^{m} \frac{∂F_{j}}{∂y_{s}} \frac{∂}{∂y_{s}} , where grad_{x}F_{j} = ∑_{i=1}^{n} ∥x∥_{w}^{2w_{i}} \frac{∂F_{j}}{∂x_{i}} \frac{∂}{∂x_{i}} .

Using the same construction as §1.4, the vector field

ξ = ∑_{i=1}^{n} \frac{∑_{J} ϕ_{J}(dF_{J} ∧ dx_{i})(grad_{x}F_{J} ∧ v)}{∑_{J} ϕ_{J}∥d_{x}F_{J}∥_{w}^{2}} \frac{∂}{∂x_{i}} + v.

is tangent to X. Here we remark that the \frac{∂}{∂x_{i}} -component of ξ (say ξ_{i}) is

ξ_{i} = ∑_{s=1}^{m} c_{s} \frac{∑_{J} ϕ_{J}⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ dy_{s}⟩}{∑_{J} ϕ_{J}∥d_{x}F_{J}∥_{w}^{2}}

We assume that X is w-(w)-regular over Y at 0. Then the inequality in (vi) of Theorem 2.1 implies

|ξ_{i}| ≲ ∥x∥_{w}^{w_{i}} on X near 0. (2.9)

This implies the w-Lipschitz condition and the uniqueness of the flow of ξ on X ∪ Y by Lemma 2.3. Thus, we obtain the following:

Theorem 2.5. The conditions in Theorem 2.1 are equivalent to the following condition: (vii) The Kuo’s vector ξ satisfies (2.9) for any unit vector v = ∑_{s=1}^{m} c_{s} \frac{∂}{∂y_{s}}.

Remark 2.6. Assume that the closure of X is X ∪ Y. We consider an extension of ξ to the neighborhood U of 0 and denote it by ξ̃. If the denominator ∥d_{x}F ∥^{2} is not zero on U − Y, there are no problems to extend ξ on U. In this case, the integration of ξ gives a R-trivialization (if it is w-Lipschitz or w-Nagumo). In general, if we set

ξ̃ = \begin{cases}∑_{i=1}^{n} \frac{∑_{J} ϕ_{J} (dF_{J} ∧dx_{i})(grad_{x}F_{J} ∧v)}{∑_{J} ϕ_{J} ∥d_{x}F_{J} ∥_{w}^{2}+∑_{j=1}^{p} |F_{j}|^{2}} \frac{∂}{∂x_{i}} + v & on U − Y, \\ v & on Y, \end{cases}

this is the desired extension on U. We remark that ξ̃satisfies (w)-Lipschitz (or (w)-Nagumo), if ξ does so.

Thus, by theorem 2.1 of [9], page 94, the integration of ξ̃gives a family of homeomorphisms which trivialize the family (R^{n}, X_{y}, y) (y ∈ Y) near 0 where X_{y} = {x ∈ R^{n} : (x, y) ∈ X}. Actually in this way we obtain a K-trivialization of the family.

The notations R, K we use here are the standard ones. For the definition and more about these equivalence relations, consult the excellent survey [24].

Theorem 2.7. (Same notation as above) Suppose that X is w-(w)-regular over Y at 0 and that the singular set of the variety defined by F_{1}, ..., F_{p} is Y. Then the family (R^{n}, X_{y}, y) (y ∈ Y) near 0 is topologically trivial.

Theorem 2.8. (Same notation as above) We assume that F_{j} (j = 1, ..., p) are C^{ω}. We write for j = 1, ..., p the weighted Taylor series of F_{j}(x, y) by H_{j,d_{j}}(x, y) + H_{j,d_{j}+1}(x, y) + ⋯, where

H_{j,ℓ}(x, y) = ∑_{α=(α_{1},...,α_{n})} c_{j,ℓ,α}(y)x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}; w_{1}α_{1} + ⋯ w_{n}α_{n} = ℓ

is a weighted homogeneous ℓ-form in variables x with respect to the weight w = (w_{1}, ..., w_{n}). If the zero locus of (H_{1,d_{1}}(x, y), ..., H_{p,d_{p}}(x, y)) has codimension k and its singular set is Y, then the family (R^{n}, X_{y}, y) (y ∈ Y) is topologically trivial.

See [5], for similar topological triviality theorem.

Proof. It is enough to see the following condition which is (vi) in Theorem 2.1 with ϕ_{J} = ∥d_{x}F_{J}∥_{w}^{2d^{[J]}−2} where d_{J} = ∑_{j∈J} d_{j}, d^{[J]} = ∏_{J′:#J′=k,J′≠J} d_{J′}: There is C > 0 such that for i = 1, ..., n, s = 1, ..., m,

|∑_{J} ∥d_{x}F_{J}∥_{w}^{2d^{[J]}−2}⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ dy_{s}⟩| ≤ C∥x∥_{w}^{w_{i}} ∑_{J} ∥d_{x}F_{J}∥_{w}^{2d^{[J]}} (2.10)

holds on X near 0.

The weighted expansion of the right hand side of (2.10) with respect to the weight w is given by

H_{d}(x, y) + H_{d+1}(x, y) + ⋯,

where H_{ℓ}(x, y) is a weighted homogeneous ℓ-form in variables x (not necessarily polynomial). By our supposition the zero set of H_{d}(x, y) is in Y. We are also able to write the weighted expansion of the left hand side of (2.10) by

K_{d}(x, y) + K_{d+1}(x, y) + ⋯, (2.11) where K_{ℓ}(x, y) is a weighted homogeneous ℓ-form in variables x with respect to the weight w. Using these expressions, we see that H_{d} is not zero outside the origin, and this implies our inequality.

Using the method above, we can show the following

Proposition 2.9. Let V be a set defined by F_{1}(x) = ⋯ = F_{a}(x) = 0 in R^{n}. We consider a map f : V × R^{m} → R^{p}. We denote by f_{y} : V → R^{p} the map defined by f_{y}(x) = f(x, y) for x ∈ V. We also denote by the same letter f an extension of f to R^{n} × R^{m}. Assume that the jacobi matrix \frac{∂(F,f)}{∂x} is of constant rank (say k) on (V −\{0\})×R^{m}. If the following inequalities hold on V × R^{m} (resp. R^{n} × R^{m}) for s = 1, ..., m, i_{1}, ..., i_{k−1} with 1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n then the family f_{y}, y ∈ R^{m}, is C^{0}-K_{R(V)}-trivial (resp. C^{0}-R(V)-trivial).

∥x∥_{w}^{w_{i_{1}}+···+w_{i_{k−1}}} |\frac{∂(F_{J}, f_{J′})}{∂(x_{i_{1}}, ..., x_{i_{k−1}}, y_{s})}| ≲ ∥d_{x}(F, f)∥_{w}.

For other equivalent conditions to this inequality, see Theorem 2.1.

This application was very much inspired by Maria Ruas’s talk (related to J.N.Tomazella’s Ph D. Thesis) at International Symposium “Topology of Singularities” held in Kochi, 1998. Here K_{R(V)} = R(V).C. We used the standard notation from [24]. For the definition and more about R(V), see [4].

2.5. Weighted Nagumo regularity conditions. Using Lemma 2.4, it is possible to give a similar treatment to §2.4. We replace ∥x∥_{w} by ∥x/|y|∥_{w} in (2.3), (2.4), (2.5), and denote them by ∥d̃_{x}F ∥_{w}, ∥d̃^{x}F ∥_{w}, D̃_{w}(ℓ), respectively. We define a new singular metric by replacing ∥x∥_{w} by ∥x/|y|∥_{w} in (2.6).

Theorem 2.10. The following conditions are equivalent:

(i) For ℓ = 1, ..., m, D̃_{w}(ℓ) = o(D̃_{w}(ℓ − 1)) when (x, y) → 0, (x, y) ∈ X. (ii) ∥d̃^{x}F ∥_{w} = o(∥d̃_{x}F ∥_{w}), when (x, y) → 0, (x, y) ∈ X. (iii) For any C^{1}-functions φ_{j} (j = 1, ..., p) near 0, and s = 1, ..., m,

|∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂y_{s}} | = o (∑_{i=1}^{n} ∥\frac{x}{|y|}∥_{w}^{w_{i}} |∑_{j=1}^{p} φ_{j} \frac{∂F_{j}}{∂x_{i}} |) when (x, y) → 0, (x, y) ∈ X.

(iv) For J ⊂ {1, …, p}, I = {i_{1}, ..., i_{k−1}} ⊂ {1, …, n} with 1 ≤ i_{1} < ⋯ < i_{k−1} ≤ n, s = 1, ..., m,

∥\frac{x}{|y|}∥_{w}^{w_{i_{1}}+···+w_{i_{k−1}}} |\frac{∂F_{J}}{∂(x_{I}, y_{s})}| = o(∥d̃_{x}F ∥_{w}) when (x, y) → 0, 0 ∈ X.

(v) For J ⊂ {1, …, p}, i = 1, ..., n, s = 1, ..., m,

|⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ dy_{s}⟩| = o (∥\frac{x}{|y|}∥_{w}^{w_{i}} ∥d̃_{x}F_{J}∥_{w}^{2}) when (x, y) → 0, (x, y) ∈ X.

(vi) For some positive C^{1}-functions ϕ_{J} on X with J ⊂ {1, …, p}, i = 1, ..., n, s = 1, ..., m,

|∑_{J} ϕ_{J}⟨dF_{J} ∧ dx_{i}, d_{x}F_{J} ∧ dy_{s}⟩| = o (∥\frac{x}{|y|}∥_{w}^{w_{i}} ∑_{J} ϕ_{J}∥d̃_{x}F_{J}∥_{w}^{2}) when (x, y) → 0, (x, y) ∈ X.

Proof. The proof is similar to that of Theorem 2.1, and we omit the details.

We say that X is w-Nagumo regular over Y at 0 if one (thus any) of the equivalent condition above holds. By the same way as the proof of Proposition 1.8, we obtain that the Kuo’s vector ξ has a unique flow under w-Nagumo regularity.

3. Construction of vector fields for the isotopy lemmas

3.1. A vector field for the first isotopy lemma. Let (x, y) = (x_{1}, ..., x_{n}, y_{1}, ..., y_{m}) denote a system of coordinates of R^{n+m}. Let p : R^{n+m} → R^{m} denote the projection defined by p(x, y) = y. We set X_{0} = 0 × R^{m}. Let X_{1}, X_{2}, ..., X_{s} denote disjoint submanifolds of R^{n+m} satisfying the following conditions:

(i) The dimensions of any two (thus all) connected components of X_{k} are the same for k = 1, ..., s.

(ii) m < dim X_{1} < dim X_{2} < ⋯ < dim X_{s}. (iii) X_{j} is weighted (w)-regular (or, w-Nagumo regular) over X_{i} at each point of X_{i}, for

0 ≤ i < j ≤ s. (iv) The restriction of p to X_{0} ∪ X_{1} ∪ ⋯ ∪ X_{k} is proper for k = 1, ..., s. Then, we show that {X_{i,y}}_{0≤i≤s} (y ∈ R^{m}) is a topological trivial stratification (the first isotopy lemma), where X_{i,y} = {x ∈ R^{n} : (x, y) ∈ X_{i}} for i = 0, 1, ..., s. We construct such a trivialization following [18] (see chapter I) under the following supposition:

(v) There is a C^{2}-map F^{(k)} = (F_{1}^{(k)}, ..., F_{p_{k}}^{(k)}) : R^{n+m} → R^{p_{k}}, so that the closure of X_{k} is the zero locus of F^{(k)} and that X_{k} is its regular locus.

By the tubular neighborhood theorem, we have the following:

(1) There is a tubular neighborhood T_{i} of X_{i} and a submersion π_{i} : T_{i} → X_{i} for i = 0, 1, ...s so that π_{i}|_{X_{i}} is the identity and π_{0} = p|_{U_{0}}.

(2) π_{i} ∘ π_{j}(x) = π_{i}(x) for x ∈ T_{i} ∩ T_{j} for 0 ≤ i < j ≤ s. Locally we can use the construction in §2.4, understanding as a local system of coordinates some of components of F^{(i)} (x-coordinates) and π_{i} (y-coordinates).

Without (iv), we claim there are C^{1}-vector fields ξ_{i} (i = 0, 1, ..., s) on X_{i} so that {ξ_{i}}_{0≤i≤s} yields a continuous vector field on X_{0} ∪ X_{1} ∪ ⋯ ∪ X_{s} so that

(3) d(π_{i}|_{X_{j}})ξ_{j} = ξ_{i} on X_{j} ∩ T_{i} for 0 ≤ i < j ≤ s.

(4) {ξ_{i}, ξ_{j}} on {X_{i}, X_{j} ∩ T_{i}} satisfies (stratified) weighted Lipschitz (or, Nagumo) condition for 0 ≤ i < j ≤ s.

We show this fact by induction on s. We first assume s = 1. Because of partitions of unity, it is enough to see it locally and it has been already shown this fact in §2.4.

We assume that there are such ξ_{i}’s for i = 0, 1, ..., s − 1, and C^{1}-vector fields ξ_{is} (i = 0, 1, ..., s − 1) on X_{s}, such that for i = 0, 1, ..., s − 1, (5) d(π_{i}|_{U_{i}∩X_{s}})ξ_{is} = ξ_{i} where U_{i} is a small neighborhood of X_{i}.

(6) {ξ_{i}, ξ_{is}} on {X_{i}, X_{s}∩U_{i}} satisfies (stratified) weighted Lipschitz (or, Nagumo) condition. Shrinking U_{i}’s, if necessary, we have for 0 ≤ j ≤ i < s

(7) d(π_{j}|_{U_{j}∩X_{s}})ξ_{is} = ξ_{j}

(8) {ξ_{j}, ξ_{is}} on {X_{j}, X_{s}∩U_{s}} satisfies (stratified) weighted Lipschitz (or, Nagumo) condition ((8) follows from (v)).

We remark {U_{j} ∩X_{k}}_{0≤j<k} ∪{X_{k}} is an open covering and we take its refinement {V_{i}}_{0≤i≤s} defined by

V_{s−1} = U_{s−1} ∩ X_{s}
V_{s−2} = U_{s−2} ∩ X_{s} − (a small closed neighborhood of X_{s−1})
⋯
V_{1} = U_{1} ∩ X_{s} − (a small closed neighborhood of ⋃_{i=2}^{s−1} X_{i})

V_{0} = U_{0} ∩ X_{s} − (a small closed neighborhood of ⋃_{i=1}^{s−1} X_{i})

V_{s} = X_{s} − (a small closed neighborhood of ⋃_{i=0}^{s} X_{i}).

Let {ψ_{i}}_{0≤i≤s} be a partition of unity on X_{s} subordinate to {V_{i}}_{0≤i≤s}. Then ξ_{s} = ∑_{i=0}^{s−1} ψ_{i}ξ_{is} is the desired vector field on X_{s}.

Using (iv), we show the integration of {ξ_{i}}_{0≤i≤s} yields a family of homeomorphisms which trivialize the family (X_{i,y}, y) (y ∈ R^{m}).

3.2. A lemma in linear algebra. We use the notation in §1.2. Let A = (a_{i}^{j})_{i=1,...,n;j=1,...,p} be an n by p matrix and b = (b^{j})_{j=1,...,p} a column vector of dimension p. We set ã^{j} = (a_{1}^{j}, ..., a_{n}^{j}, −b^{j}) for j = 1, ..., p. Assume that rankA = k. We set, for i = 1, ..., n,

x_{i} = \frac{∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}}((ã^{j_{1}})^{∨} ∧ ⋯ ∧ (ã^{j_{k}})^{∨} ∧ e_{i}^{∗})(ã^{j_{1}} ∧ ⋯ ∧ ã^{j_{k}} ∧ e_{n+1})}{∑_{j_{1}<···<j_{k}} ϕ_{j_{1},...,j_{k}}∥a^{j_{1}} ∧ ⋯ ∧ a^{j_{k}}∥^{2}}. (3.1)

We assume that the denominator is not zero.

Lemma 3.1. If rank(A b) = k, then Ax = b where x = (x_{i})_{i=1,...,n}.

Proof. It is enough to see that

(A −b) (\begin{matrix}x \\ 1 \end{matrix}) = 0. (3.2)

Consider the subspace W generated by ã^{j}, j = 1, ..., p. Multiplying the orthogonal projection of e_{n+1} to W^{⊥} that the last component is 1, we get the lemma by Lemma 1.4.

Remark 3.2. When n = p = k, this reduces to G.Cramer’s formula.

3.3. A vector field for the second isotopy lemma. To obtain an analogous treatment for the second isotopy lemma, we should construct a vector filed trivializing a family of maps under some reasonable conditions. We present here a construction of such a vector field in the case of two strata. Then using patching and inductuion (in a resonable setup) one can proceed in constructing a vector field for a second isotopy lemma type result (like §3.1).

We consider the Euclidean spaces R^{n+m} with coordinates x = (x_{1}, ..., x_{n}), y = (y_{1}, ..., y_{m}), and R^{n′+m′} with coordinates x′ = (x′_{1}, ..., x′_{n′}), y′ = (y′_{1}, ..., y′_{m′}). Set

Y = {(x, y) ∈ R^{n+m} : x = 0}, Y′ = {(x′, y′) ∈ R^{n′+m′} : x′ = 0},
p :R^{n+m} → R^{m} the projection defined by p(x, y) = y, and
p′ :R^{n′+m′} → R^{m′} the projecton defined by p′(x′, y′) = y′.

Let X, X′ be submanifolds of R^{n+m}, R^{n′+m′}, respectively. We assume

(i) X is the regular locus of the zero locus of C^{2}-functions F_{j} (j = 1, ..., p). We assume the jacobi matrix of (F_{1}, ..., F_{p}) has rank k on X.

We now consider a C^{2}-map f : X ∪ Y → X′ ∪ Y′ so that (ii) f(Y) ⊂ Y′, and f|_{Y} is a submersion, (iii) f(X) ⊂ X′, and f|_{X} is a submersion, (iv) p′ ∘ f = f ∘ p on X near Y. Suppose there are vector fields v′, ξ′ and v on Y′, X′ and Y respectively so that

(v) d(p′|_{X′})ξ′ = v′, (vi) d(f|_{Y})v = v′. We want to construct a vector field ξ on X such that

(1) d(p|_{X})ξ = v,

(2) d(f|_{X})ξ = ξ′.

(3) ξ and v define a C^{0}-vector field on X ∪ Y.

(4) {v, ξ} satisfies (stratified) weighted Lipschitz (or, Nagumo) condition on {Y, X}. To describe such ξ, we write

ξ′ = ∑_{j=1}^{n′} ξ′_{j} \frac{∂}{∂x′_{j}} + v′ on X′, and ξ = ∑_{i=1}^{n} ξ_{i} \frac{∂}{∂x_{i}} + v on X

where ξ′_{j} (j = 1, ..., n′) are C^{1}-functions on X′, and ξ_{i} (i = 1, ..., n) are C^{1}-functions on X. It is enough to find ξ_{i}’s so that

(\begin{matrix}\frac{∂F_{s}}{∂x_{i}} & \frac{∂F_{s}}{∂y_{k}} \\ \frac{∂f_{j}}{∂x_{i}} & \frac{∂f_{j}}{∂y_{k}} \\ \frac{∂f′_{ℓ}}{∂x_{i}} & \frac{∂f′_{ℓ}}{∂y_{k}} \end{matrix} ) (\begin{matrix}ξ_{i} \\ v_{k} \end{matrix}) = (\begin{matrix}0 \\ ξ′_{j} \\ v′_{ℓ} \end{matrix}) (3.3)

where f_{j} = x′_{j} ∘ f (j = 1, ..., n′), f′_{ℓ} = y′_{ℓ} ∘ f (ℓ = 1, ..., m′), and v = ∑_{k=1}^{m} v_{k} \frac{∂}{∂y_{k}}.

Now we consider the regularity condition for f, namely, (vii) (p, f) : X → Y ×_{Y′} X′ is a submersion.

See Lemma (I.2.4) in [7] and [10] for the relation between (vii) and Thom’s (a_{f})-condition.

If we assume (vii), (3.3) has a solution. So, by (v), it is enough to find ξ_{i} (i = 1, ..., n) on X near Y so that

(\begin{matrix}\frac{∂F_{s}}{∂x_{i}} & \frac{∂F_{s}}{∂y_{k}} \\ \frac{∂f_{j}}{∂x_{i}} & \frac{∂f_{j}}{∂y_{k}} \end{matrix} ) (\begin{matrix}ξ_{i} \\ v_{k} \end{matrix}) = (\begin{matrix}0 \\ ξ′_{j} \end{matrix}) . (3.4)

Thus, it is enough to solve the following system on X near Y:

A(ξ_{i}) = b where A = (\frac{∂F_{s}}{∂x_{i}} ; \frac{∂f_{j}}{∂x_{i}} ) , b = b′ − b′′, b′ = (\begin{matrix}0 \\ ξ′_{j} \end{matrix}) , b′′ = (\frac{∂F_{s}}{∂x_{i}} ; \frac{∂f_{i}}{∂y_{k}} ) (v_{k}) (3.5)

Therefore, (vii) implies the existence of ξ_{i} satisfying (3.5) for any v_{k} and ξ′_{j}. In other words, (vii) implies

(3.6) rankA = rank(A b) = ℓ (constant) on X near 0.

By Lemma 3.1, ξ_{i} defined by (3.1) is a solution for (3.5).

Proposition 3.3. We assume (i)-(vii). We consider the singular metric defined by (2.6). ξ = ∑_{i} ξ_{i} \frac{∂}{∂x_{i}} + v is w-Lipschitz, if the following inequalities hold for s = 1, ..., m, i = 1, ..., n on X near 0:

(3.7) |∑_{J,J′} ϕ_{J,J′}(d_{x}F_{J} ∧ d^{(s)}f_{J′} ∧ dx_{i})(grad_{x}F_{J} ∧ grad_{x}f_{J′} ∧ \frac{∂}{∂y_{s}} )|

≲ ∥x∥_{w}^{w_{i}} |∑_{J,J′} ϕ_{J,J′}∥d_{x}F_{J} ∧ d_{x}f_{J′}∥_{w}^{2}|

(3.8) |∑_{J,J′} ϕ_{J,J′}(d_{x}F_{J} ∧ d̃f_{J′} ∧ dx_{i})(grad_{x}F_{J} ∧ grad_{x}f_{J′} ∧ \frac{∂}{∂t})|

≲ ∥x∥_{w}^{w_{i}} |∑_{J,J′} ϕ_{J,J′}∥d_{x}F_{J} ∧ d_{x}f_{J′}∥_{w}^{2}|

where d̃f_{J′} = ⋀_{j∈J′} d̃f_{j}, d̃f_{j} = d_{x}f_{j} + ξ′_{j}dt and t is a parameter, #J + #J′ = ℓ, and ϕ_{J,J′} are some C^{1}-functions so that the right hand side is nowhere zero on X.

Proof. Obvious.

Since the left hand side of (3.7) is

| ∑_{J,J′,I:I∋i} ϕ_{J,J′}∥x∥_{w}^{2w_{I}} \frac{∂(F_{J}, f_{J′})}{∂(x_{I−\{i\}}, y_{s})} \frac{∂(F_{J}, f_{J′})}{∂x_{I}} | ,

if ϕ_{J,J′} are non-negative, then (3.7) comes from

ϕ_{J,J′}^{1/2}∥x∥_{w}^{w_{I}−w_{i}} |\frac{∂(F_{J}, f_{J′})}{∂(x_{I−\{i\}}, y_{s})}| ≲ ∑_{J,J′;I} ϕ_{J,J′}^{1/2}∥x∥^{w_{I}} |\frac{∂(F_{J}, f_{J′})}{∂x_{I}} | on X near 0. (3.9)

Similarly, if ϕ_{J,J′} are non-negative, then (3.8) comes from

(3.10) ϕ_{J,J′}^{1/2}|ξ′_{j}|∥x∥_{w}^{w_{I}−w_{i}} |\frac{∂(F_{J}, f_{J′−\{j\}})}{∂(x_{I−\{i\}})}| ≲ ∑_{J,J′;I} ϕ_{J,J′}^{1/2}∥x∥_{w}^{w_{I}} |\frac{∂(F_{J}, f_{J′})}{∂x_{I}} | on X near 0.

Proposition 3.4. We assume (i)-(vii). If ξ′ satisfies the w′-Lipschitz condition, then (3.8) follows from the following inequality:

(3.11) ϕ_{J,J′}∥f∥_{w′}^{w′_{j}}∥x∥_{w}^{w_{I}−w_{i}} |\frac{∂(F_{J}, f_{J′−\{j\}})}{∂(x_{I−\{i\}})}| ≲ ∑_{J,J′;I} ϕ_{J,J′}∥x∥_{w}^{w_{I}} |\frac{∂(F_{J}, f_{J′})}{∂x_{I}} | on X near 0,

for any I, J, J′ with J′ ∋ j, I ∋ i. Here, ϕ_{J,J′} are some non-negative C^{1}-functions so that the right hand side is nowhere zero on X.

Proof. It follows that |ξ′_{j}| ≲ ∥f∥_{w′}^{w′_{j}} on X near 0 and (3.10).

In the propositions above we assume that J ⊂ {1, ..., p} and J′ ⊂ {1, ..., n′}.

Proposition 3.5. We assume (i)-(iv). We also assume that F and f are real analytic. Consider the weighted expansion of F and (f_{1}, ..., f_{n′}) with respect to the weight w. We assume that the weighted initial form of (f_{1}, ..., f_{n′}) has its degree a multiple of w′. We assume that the jacobi matrix of weighted initial form of (F_{1}, .., F_{p}, f_{1}, ..., f_{n′}) by the variables (x_{1}, ..., x_{n}) is of constant rank on X and that the closure of X is X ∪ Y. Then for any w′- Lipschitz vector field {ξ′, v′} on (X′, Y′) and any lift v on Y of v′ there is a vector field ξ on X with (1)–(3) so that {ξ, v} on (X, Y) is w-Lipschitz.

Proof. Same as that of Theorem 2.8, and we omit the details.

Example 3.6 (Example 2.5.10 in [6]). Let F : R^{3} → R be the function defined by

F (x, y, z) = y − x^{2k+1} − z^{2}x^{2ℓ+1}.

We consider the restriction of the map f : R^{3} ∋ (x, y, z) ↦ (y, z) ∈ R^{2} to F^{−1}(0). Set Y = {x = y = 0} and X = F^{−1}(0) − Y. Note that f|(X, Y) is always a Thom regular map. If k ≤ ℓ, one can see we can apply Proposition 3.5. However if k > ℓ, our assumption in Proposition 3.5 are not satisfied, and it is shown in [6] that actually \frac{∂}{∂z} does not admit a continuous lift by f on F^{−1}(0).

It is also possible to obtain a similar criterion for weighted Nagumo conditions as (3.7), (3.8), and we leave it to the reader.

We also remark that a pinch map does not satisfy Thom’s (a_{f})-condition, and so the use of (a_{f})-condition is somehow restrictive. It is clear that there are maps with pinching so that there are ξ′ and v′ satisfying the following conditions:

(i) (3.6).

(ii) The inequalities that imply the weighted Lipschitz condition for ξ (defined by using (3.1)).

Example 3.7. Consider the map f : R^{3} → R^{3} defined by

(x, y, t) ↦ (x′, y′, t′) = (x^{2}, x^{4}y + x^{3}t, t).

This map has pinching, i.e. the stratum {x = 0} is sent to t′-axis, and f is local diffeomorphism on the stratum {x ≠ 0}. Following our method one can find that the vector field |x′|^{3/2} \frac{∂}{∂y′} + \frac{∂}{∂t′} has a lift \frac{∂}{∂t} by f (note that for example \frac{∂}{∂t′} does not lift properly), and f_{t} : R^{2} → R^{2}, f_{t}(x, y) = (x^{2}, x^{4}y + x^{3}t) is topologically right-left trivial.

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Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubo, Urawa 338, Japan

School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia