A NOTE ABOUT BIFURCATION POINTS

ZBIGNIEW JELONEK

Abstract. Let X ⊂ k^{n} be a smooth affine variety of dimension n − r and let f = (f_{1}, ..., f_{m}) : X → ℂ^{m} be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we give a direct, short proof of the last statement.

1. Introduction.

Let X be a smooth affine variety over k = ℝ or k = ℂ of dimension n − r and let f : X → k^{m} be a polynomial dominant mapping. It is well-known, that the mapping f is a locally trivial fibration outside a bifurcation set B(f), which has a measure 0.

Let us recall that in general the set B(f) is bigger than K_{0}(f) - the set of critical values of f. It contains also the set B_{∞}(f) of bifurcations points at infinity. Briefly speaking the set B_{∞}(f) consists of points at which f is not a locally trivial fibration at infinity (i.e., outside a compact set). To control the set B_{∞}(f) one can use the set of asymptotic critical values at infinity of f (see [5]):

K_{∞}(f) = {y ∈ k^{m} : there is a sequence x_{l} → ∞ s.t. f(x_{l}) → y and ∥x_{l}∥ν(res_{T_{x_{l}}X}df(x_{l})) → 0},

where we consider on X the induced Euclidean metric and ν is the function defined by Rabier (see Definition 2.1 below). Let K(f) = K_{0}(f) ∪ K_{∞}(f) be the set of generalized critical values of f.

In [2] (see also [1] and [3]) we have proved that the set K(f) is a proper algebraic subset of ℂ^{m} (or proper semi-algebraic in the real case) which can be computed effectively (we can also estimate the degree of an algebraic hypersurface which contains K(f)). Moreover it can be proved (e.g., by a general Fibration Theorem of Rabier [5]), that B(f) ⊂ K(f). These two facts together allows us to construct effectively an Zariski open dense subset U ⊂ k^{m} over which the mapping f is a locally trivial fibration.

In this note I prove directly that B(f) ⊂ K(f) in the case when X ⊂ k^{n} is a smooth submanifold and f : X → k^{m} is a smooth mapping. This fact follows from a very general Theorem of Rabier (see [5]), but it is so important (e.g., in the study of polynomial mappings), that (as I believe) it is worth to have a simple direct proof of it in a special case of a submanifolds of an Euclidean space.

2. About Rabier function ν

In this section we give several equivalent expressions for ν. A part of these results is also contained (with slightly different proofs) in [2].

1991 Mathematics Subject Classification. 14 D 06, 14 Q 20, 51N10, 51N20, 15A04.

Key words and phrases. smmoth mappings, polynomial mapping, fibration, bifurcation points.

The author is partially supported by the KBN grant 2 PO3A 017 22.

Let X ≅ k^{n}, Y ≅ k^{m} be finite dimensional vector spaces (over k). Let us denote by L(X, Y) the set of linear mappings from X to Y and by Σ(X, Y) ⊂ L(X, Y) the set of non-surjective mappings. Let us recall the following ([5]): Definition 2.1. Let A ∈ L(X, Y). Set

ν(A) = inf_{||ϕ||=1}||A^{∗}(ϕ)||,

where A^{∗} : L(Y^{∗}, X^{∗}) is adjoint operator and ϕ ∈ Y^{∗}.

In [3] the following characterization of ν is given: ν(A) = dist(A, Σ) = inf_{B∈Σ}||A − B||. Moreover, we have following useful characterization ([5] and [3]): Proposition 2.1. Let A ∈ L(X, Y). Then

a) ν(A) = sup{r > 0 : B(0, r) ⊂ A(B(0, 1)), where B(0, r) = {x ∈ X : ||x|| ≤ r}.

b) if A ∈ GL(X, Y) then ν(A) = ||A^{−1}||^{−1}. Proposition 2.2. Let A = (A_{1}, ..., A_{m}) ∈ L(X, Y) and let \overline{A_{i}} = grad A_{i}. Let

κ(A) = min_{1≤i≤m}dist(\overline{A_{i}}, < (\overline{A_{j}})_{j≠i} >),

be the Kuo number of A. Then ν(A) ≤ κ(A) ≤ \sqrt{m}ν(A).

We say that ν(A) and κ(A) are equivalent and write ν(A) ∼ κ(A). The sign X ∼ Y means that there are positive constants C_{1}, C_{2} such that C_{1}X ≤ Y ≤ C_{2}X. Definition 2.2. Let A ∈ L(X, Y) and let H ⊂ X be a linear subspace. We set

ν(A, H) = ν(res_{H}A), κ(A, H) = κ(res_{H}A),

where res_{H}A denotes the restriction of A to H.

From Proposition 2.2 we get immediately: Corollary 2.1. Let A ∈ L(X, Y) and let H ⊂ X be a linear subspace. Then

ν(A, H) ∼ κ(A, H).

We have the following: Proposition 2.3. Let A = (A_{1}, ..., A_{m}) ∈ L(X, Y) and let H ⊂ X be a linear subspace. Assume that H is given by a system of linear equations B_{j} = 0, j = 1, ..., r. Then

κ(A, H) = min_{1≤i≤m}dist(\overline{A_{i}}, < (\overline{A_{j}})_{j≠i}; (\overline{B_{j}})_{j=1,...,r} >),

where \overline{A_{i}} = grad A_{i} and \overline{B_{j}} = grad B_{j}.

Proof. Indeed, every vector \overline{A_{i}} we can write as a_{i} + b_{i}, where a_{i} is orthogonal to the subspace B =< \overline{B_{j}})_{j=1,...,r} > (what means that a_{i} ∈ H) and b_{i} ∈ B. Hence

dist(\overline{A_{i}}, < (\overline{A_{j}})_{j≠i}; (\overline{B_{j}})_{j=1,...,r} >) = dist(a_{i}, < (a_{j})_{j≠i} >)

and since grad(res_{H}A_{i}) = a_{i} the proof is finished. □

We need also: Definition 2.3. Let A ∈ L(X, Y) (where n ≥ m + r) and let H ⊂ X be a linear subspace given by system of independent linear equation B_{i} = ∑ b_{ij}x_{ij}, i = 1, ..., r. Let a = [a_{ij}] be the matrix of A. Let c = [c_{kl}] be a ((m+r)×n) matrix given by rows A_{1}, ..., A_{m}; B_{1}, ..., B_{r} (we identify A_{i} = ∑ a_{ij}x_{j} with the vector (a_{j1}, ..., a_{jn}), similarly for B_{j}). Let M_{I}, where I = (i_{1}, ..., i_{m+r}), denote a ((m + r) × (m + r)) minor of c given by columns indexed by I. Let M_{J}(j) denote a ((m + r − 1) × (m + r − 1) minor given by columns indexed by J and by deleting the jth row , where 1 ≤ j ≤ m. Then by the generalized Gaffney function of A with respect to a linear subspace H, we mean the number

g(A, H) = \frac{(∑_{I} |M_{I}|^{2})^{1/2}}{(∑_{J,1≤j≤m} |M_{J}(j)|^{2})^{1/2}} .

(If this number is not defined we put g(A, H) = 0.) Remark 2.1. It is easy to see that g(A, H) depends on A and H only.

We have: Proposition 2.4. Let A ∈ L(X, Y) (where n ≥ m) and let H ⊂ X be a linear subspace. Then g(A, H) ∼ κ(A, H).

Proof. By basic properties of the Gram determinant (see e.g., [4]) we have that

dist(\overline{A_{i}}, < (\overline{A_{j}})_{j≠i}, \overline{B_{j}})_{j∈1,...,r\}} >) = \frac{G(\{\overline{A_{i}}\}_{i∈\{1,...,m\}}, \{\overline{B_{j}}\})_{j∈\{1,...,r\}})^{1/2}}{G(\{\overline{A_{j}}\}_{j≠i}, \{\overline{B_{j}}\}_{j∈\{1,...,r\}})^{1/2}} = \frac{(∑_{I} |M_{I}|^{2})^{1/2}}{(∑_{J} |M_{J}(i)|^{2})^{1/2}} .

Thus g(A, H) ≤ κ(A, H). On the other hand there is a number i_{0}, such that the sum ∑_{J} |M_{J}(i_{0})|^{2})^{1/2} is maximal. Since

(∑_{J,j} |M_{J}(j)|^{2})^{1/2} = (∑_{r}(∑_{J} |M_{J}(r)|^{2}))^{1/2} ≤ \sqrt{m}(∑_{J} |M_{J}(i_{0})|^{2})^{1/2},

we have

g(A, H) ≥ C \frac{(∑_{I} |M_{I}|^{2})^{1/2}}{(∑_{J} |M_{J}(i_{0})|^{2})^{1/2}} = Cdist(\overline{A}_{i_{0}}, < (\overline{A}_{j})_{j≠i_{0}}, \{\overline{B_{j}}\}_{j∈\{1,...,r\}} >) ≥ Cκ(A, H),

where C = 1/\sqrt{m}. □

At the end of this section we introduce another important function (see [1]): Definition 2.4. Let A ∈ L(k^{n}, k^{m}) (where n ≥ m) and let H ⊂ k^{n} be a linear subspace given by system of independent linear equation B_{i} = ∑ b_{ij}x_{ij}, i = 1, ..., r. Let a = [a_{ij}] be the matrix of A. Let c = [c_{kl}] be a ((m+r)×n) matrix given by rows A_{1}, ..., A_{m}; B_{1}, ..., B_{r}. Let M_{I}, where I = (i_{1}, ..., i_{m+r}), denote a ((m + r) × (m + r)) minor of c given by columns indexed by I. Let M_{J}(j) denote a ((m + r − 1) × (m + r − 1) minor given by columns indexed by J and by deleting the jth row A_{j}, where 1 ≤ j ≤ m. Then we define the number

g′(A, H) = max_{I}\{min_{\{J⊂I, 1≤j≤m\}} \frac{|M_{I}|}{|M_{J}(j)|}\},

(where we consider only numbers with M_{J}(j) ≠ 0, if all numbers M_{J}(j) are zero, we put g′(A, H) = 0).

We have: Proposition 2.5. We have g′(A, H) ∼ g(A, H).

Proof. First we prove that there is a constant C > 0 such that g′(A, H) ≤ Cg(A, H). Let us fixed index I = (i_{1}, ..., i_{m+r}) such that |M_{I}| ≠ 0 and consider numbers \frac{|M_{I}|}{|M_{J} (s)|}, where J ⊂ I and 1 ≤ s ≤ m. For simplicity we can assume that I = (1, ..., m + r). Let the subspace H be given by system of independent linear equation B_{i} = ∑ b_{ij}x_{ij}, i = 1, ..., r, and let a = [a_{ij}] be the matrix of A.

Consider a system of linear equations:

∑_{j=1}^{n} a_{1j}x_{j} = y_{1},
⋮  ⋮
∑_{j=1}^{n} a_{mj}x_{j} = y_{m},
∑_{j=1}^{n} b_{j1}x_{j} = 0,
⋮  ⋮
∑_{j=1}^{n} b_{rj}x_{j} = 0,
x_{m+r+1} = 0,
⋮  ⋮
x_{n}  = 0.

Using the Cramer rules we can solve this system. We have

x_{1}  = ∑_{k=1}^{m}(−1)^{1+k}y_{k}M_{1k}/M_{I},
⋮  ⋮
x_{n}  = ∑_{k=1}^{m}(−1)^{m+k}y_{k}M_{mk}/M_{I},
x_{m+r+1} =  0,
⋮  ⋮
x_{n}  =  0.

In particular we have ||x|| ≤ (max\frac{|M_{J} (i)|}{|M_{I}|})||y||. Consequently we see that the image of a unit ball in a subspace H′ = {x ∈ H : x_{m+r+1} = 0, ..., x_{n} = 0} by the mapping A contain a ball of radius min_{\{J⊂I, 1≤j≤m\}} \frac{|M_{I}|}{|M_{J} (j)|}. Now by Proposition 2.1 a), we have that

min_{\{J⊂I,1≤j≤m\}} \frac{|M_{I}|}{|M_{J} (j)|} ≤ ν(A, H′) ≤ ν(A, H). Finally we get

ν(A, H) ≥ max_{I}\{min_{\{J⊂I, 1≤j≤m\}} \frac{|M_{I}|}{|M_{J}(j)|}\} = g′(A, H).

In particular there is a constant C such that Cg(A, H) ≥ g′(A, H).

On the other hand, there exists I_{0}, such that the minor M_{I_{0}} has a maximal norm. Since

g(A, H) = \frac{(∑_{I} |M_{I}|^{2})^{1/2}}{(∑_{J,j} |M_{J}(j)|^{2})^{1/2}} ≤ ( \begin{matrix}n \\ m + r \end{matrix} )^{1/2} \frac{|M_{I_{0}}|}{(∑_{J,j} |M_{J}(j)|^{2})^{1/2}} ≤

≤ ( \begin{matrix}n \\ m + r \end{matrix} )^{1/2} \{min_{\{J⊂I_{0},1≤j≤m\}} \frac{|M_{I_{0}}|}{|M_{J}(j)|}\} ≤ ( \begin{matrix}n \\ m + r \end{matrix} )^{1/2} g′(A, H),

we have that there is a constant C′ > 0 such that g(A, H) ≤ C′g′(A, H). □

Corollary 2.2. We have g′(A, H) ∼ ν(A, H).

3. Main Result

In this section we give a short, direct proof of the fact B(f) ⊂ K(f) for smooth mapping f : X → k^{m}, where X is a smooth submanifold of k^{m} and f is a smooth mapping. Our approach is a modification of our proof from [1]. Let us recall the following basic definition: Definition 3.1. Let k = ℂ or k = ℝ and let X be a smooth submanifold of k^{n}. Let f : X → k^{m} be a k−smooth mapping. Then we define the set of generalized critical values K(f) = K_{0}(f) ∪ K_{∞}(f), where K_{0}(f) is the set of critical values of f and

K_{∞}(f) = {y ∈ k^{m} : there is a sequence x_{l} → ∞ s.t. f(x_{l}) → y and ∥x_{l}∥ν(df(x_{l}), T_{x_{l}}X) → 0}

is the set of critical values at infinity Remark 3.1. Note, that by virtue of results of section 2, in the place of the function ν above we can put also κ, g, or g′.

We have the following simple observation (see [1], [5]): Proposition 3.1. Let k = ℂ or k = ℝ and let X be a smooth affine variety over k. Let f : X → k^{m} be a k−smooth mapping. Then the set K(f) = K_{0}(f) ∪ K_{∞}(f) is closed.

We need also the following lemma, which is a slight modification of the lemma from [1]. For the sake of completeness we give a short proof. Lemma 3.1. Let U ⊂ k^{n} be an open set and V : U → k^{n} be a smooth mapping. Let y ∈ U and let

x′(t) = V (x), with x(0) = y,

be a differential equation. Let x(y, t), t ∈ [0, t_{0}) be a solution of this equation. Assume that for ||x(t)|| large enough, we have ||V (x(y, t))|| < M||x(y, t)||. Then this trajectory is bounded. In particular this trajectory either is defined for every t > 0 or it intersects the boundary ∂U of U.

Proof. By our assumption we have ||V (x(y, t))|| < M||x(y, t)|| for sufficiently large ||x(y, t)||, say for ||x(y, t)|| > R. Let r(t) = ||x(y, t)||^{2}. We show that the function r is bounded. Assume by contradiction that there is a sequence of points t_{i} → t_{0} such that r(t_{i}) → ∞ as t_{i} → t_{0}.

Let N be any regular value of r with N > R. The preimage r^{−1}(N) is at most countable set W = {w_{1}, w_{2}, ....}. For every w_{i} ∈ W two cases are possible:

a) r(t) ≤ N for every t ∈ [w_{i}, w_{i+1}),

b) r(t) ≥ N for every t ∈ [w_{i}, w_{i+1}), (if w_{i+1} is not defined then we put w_{i+1} = t_{0}). Of course only the case b) is interesting. Hence assume that b) holds. By the assumption for a number t ∈ [w_{i}, w_{i+1}) we have ||V (x(t))|| < M||x(t)||. Consequently for every such t, we have r′(t) ≤ 2||x||||V (x)|| < 2M||x||^{2} = 2Mr(t). It means that ln(r(t))′ ≤ 2M and finally r(t) ≤ r(w_{i})exp(2M(t − w_{i})) ≤ Nexp(2M(t − w_{1})). This means that the set {r(t), w_{1} ≤ t < t_{0}} is bounded, a contradiction. □

Now we give a short, direct proof of the fact that B(f) ⊂ K(f), which is a particular version of very general result of Rabier [5]. Theorem 3.1. Let k = ℂ or k = ℝ and let X ⊂ k^{n} be a smooth submanifold (i.e., X is smooth for k = ℝ or Stein for k = ℂ). Let f : X → k^{m} be a k−smooth mapping (i.e., f is smooth for k = ℝ or holomorphic for k = ℂ). Then

B(f) ⊂ K(f) = K_{0}(f) ∪ K_{∞}(f),

i.e., the mapping f is a locally trivial fibration outside the set K(f).

Proof. It is well-known that we can assume that f can be extended to a k−smooth mapping \overline{f} on the whole of k^{n} (in real case it is an easy exercise, in complex it follows from the theory of Stein manifolds).

First assume that X is a global complete intersection, i.e. X = {b_{1} = 0, ..., b_{r} = 0} and rank {d_{x}b_{1}, ..., d_{x}b_{r}} = r for every x ∈ X.

Let a ∉ K(f). Without loss of generality we can assume that a = 0. We have that a ∉ K_{0}(f) and a ∉ K_{∞}(f). This implies that there are R > 0, ϵ > 0, η > 0, such that for every x ∈ X with ||x|| ≥ R and ||f(x)|| < η, we have

(3.1)  max_{I}\{min_{\{J⊂I\}}||x||\frac{|M_{I}|}{|M_{J}(j)|}\} > ϵ.

Moreover, there is ω > 0, such that for every x ∈ X with ||x|| ≤ R and ||f(x)|| < η, we have max_{I}|M_{I}(x)| ≥ ω.

Let U = {y ∈ k^{m} : ||y|| < η} and let Γ = f^{−1}(0). We show that f^{−1}(U) ≅ Γ × U and f is a projection Γ × U ∋ (γ, u) → u ∈ U. Indeed, let us define a set U_{I} = {x ∈ \overline{f}^{−1}(U) : if ||x|| ≥ R then min_{\{J⊂I\}}||x||\frac{|M_{I}|}{|M_{J} (j)|} ≥ ϵ, if ||x|| ≤ R then |M_{I}(x)| ≥ ω\}.

Further, let V_{I} = \{x ∈ \overline{f}^{−1}(U) : if ||x|| ≥ R then min_{\{J⊂I\}}||x||\frac{|M_{I}|}{|M_{J} (j)|} ≤ ϵ/2, if ||x|| ≤ R then |M_{I}(x)| ≤ ω/2}. The sets V_{I} and U_{I} are disjoint. Consequently there is a C^{∞} function δ_{I} : k^{n} → [0, 1], which is equal to 1 on U_{I} and to 0 on V_{I}. It is easy to see that sets H_{I} = {x : δ_{I} > 0} cover the set f^{−1}(U). Now take δ := ∑_{I} δ_{I} and let Δ_{I} = δ_{I}/δ.

Take y = (y_{1}, ..., y_{n}) ∈ U. Take the index I = (1, ..., m + r) and let consider a (formal) system of differential equations:

∑_{j=1}^{n} \frac{∂\overline{f}_{1}}{∂x_{j}} (x(t))x_{j}(t)′ = y_{1},
⋮  ⋮
∑_{j=1}^{n} \frac{∂\overline{f}_{m}}{∂x_{j}} (x(t))x_{j}(t)′ = y_{m},
∑_{j=1}^{n} \frac{∂b_{1}}{∂x_{j}} (x(t))x_{j}(t)′ = 0,
⋮  ⋮
∑_{j=1}^{n} \frac{∂b_{r}}{∂x_{j}} (x(t))x_{j}(t)′ = 0,
x_{m+r+1}(t)′  = 0,
⋮  ⋮
x_{n}(t)′  = 0.

Using the Cramer rules we can solve this system (at least in U_{I}). Let M_{J}(i) := M_{ki} for J = I \ {k}. We have

x_{1}(t)′  = ∑_{k=1}^{m}(−1)^{1+k}y_{k}M_{1k}/M_{I},
⋮  ⋮
x_{m+r}(t)′ = ∑_{k=1}^{m}(−1)^{m+k}y_{k}M_{mk}/M_{I},
x_{m+r+1}(t)′ =  0,
⋮  ⋮
x_{n}(t)′  =  0.

We can write this system shortly as

x(t)′ = V_{I}(y, x(t)).

By the Cramer rules, we have df(V_{I}(y, x)) = y and db(V_{I}(y, x)) = 0. In an analogous way we can define V_{I} for an arbitrary index I = (i_{1}, ..., i_{m}).

Now consider a vector field V (y, x) = ∑_{I} Δ_{I}V_{I}(y, x) in a domain \overline{f}^{−1}(U). By the construction, we have ||V (x)|| ≤ 2mη/ϵ||x|| for ||x|| ≥ R and x ∈ X. Let us consider the differential equation

(3.2)  x(t)′ = V (y, x(t)), x(0) = γ,

where γ ∈ Γ. Let us note that

df(V (y, x)) = df(∑_{I} Δ_{I}V_{I}(y, x)) = ∑_{I} df(Δ_{I}V_{I}(y, x)) = ∑_{I} Δ_{I}df(V_{I}(y, x)) = (∑_{I} Δ_{I})y = y.

Similarly db(V (x, y) = 0. Consequently, if x(t, y, γ) is a solution of system 3.2, then the trajectory is contained in X and yt = \overline{f}(x(t), y, γ) = f(x(t), y, γ). Since y ∈ U, we have that the trajectory x(t, y, γ), t ∈ [0, t_{0}) does not cross the border ∂f^{−1}(U) for every 0 ≤ t_{0} ≤ 1 + δ, for some δ > 0. Consequently by Lemma 3.1 the trajectory x(t, y, γ) is defined on the whole of [0, 1] and it is contained in X. Since f(x(t, y, γ)) = yt, we have that the phase flow x(t, y, γ), t ∈ [0, 1] transforms f^{−1}(0) = Γ into f^{−1}(y) (in fact, by the symmetry, it transforms Γ onto f^{−1}(y)). Let

Φ : Γ × U ∋ (γ, y) → x(1, y, γ) ∈ f^{−1}(U).

It is easy to see that Φ is a diffeomorphism. Thus 0 ∉ B(f).

In the general case we can choose a locally finite cover {U_{i}} of k^{n}, such that in each U_{i} the manifold X ∩ X_{i} is a complete intersection. Now we can construct vector fields V_{i} on U_{i} (construction is as above) and then glue them to one field V by a partition of unity subordinate to the cover {U_{i}}. The rest of the proof is the same as above. □

At the end of this note we give two simple examples. Example 3.1. Let us consider a Stein curve Γ = {(x, y) ∈ ℂ^{2} : exp(xy) = 2}. Let us consider the projection f : Γ ∋ (x, y) → y ∈ ℂ. Using generalized Gaffney function we see that

K_{0}(f) = f({(x, y) ∈ Γ : yexp(xy) = 0}) = ∅

and

K_{∞}(f) = {limf(x_{n}, y_{n}) = y_{n}; where (|x_{n}| + |y_{n}|) → ∞ and |y_{n}| → 0} = {0}.

Hence finally K(f) = {0} and indeed we can check directly that in this case B(f) = K(f) = {0} (in fact f is a topological covering outside 0). Note that the mapping f has no usual critical values. Example 3.2. Let us consider a smooth mapping

f : ℂ^{3} ∋ (x, y, z) → (xexp(z), yexp(z)) ∈ ℂ^{2}.

Using the function g′ we can easily computed that K(f) = {0}. But the function f is a global fibration of ℂ^{3} - in fact it gives a fibration

ℂ^{2} × ℂ ∋ ((x, y), z) → ((xexp(−z), yexp(−z), z) ∈ ℂ^{3}.

Thus in general B(f) ≠ K(f). Remark 3.2. In the case of Stein submanifolds in general it may happen that K(f) = ℂ^{m}. The latter equality is impossible for algebraic submanifolds and polynomial mappings - see [2], [3].

References

[1] Z. Jelonek, On the generalized critical values of a polynomial mapping, Manuscripta Mathematica, to appear.

[2] Z. Jelonek, K. Kurdyka, Quantitative Bertini-Sard theorem for smooth affine varieties, to appear.

[3] K. Kurdyka, P. Orro, S. Simon, Semialgebraic Sard Theorem for generalized critical values, Journal of Differential Geometry 56 (2000), 67-92.

[4] A. Mostowski, M. Stark, Elementy Algebry Wyższej, PWN, Warszawa, (1968), (in polish).

[5] P. J. Rabier, Ehresmann’s fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Annals of Math., 146 (1997), 647-691.

(Z. Jelonek) Instytut Matematyczny, Polska Akademia Nauk, Św. Tomasza 30, 31-027 Kraków, Poland & Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

E-mail address: najelone@cyf-kr.edu.pl