Lipschitz-Killing curvatures and polar images

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Lipschitz-Killing curvatures and polar images

Nicolas Dutertre

To cite this version:

Nicolas Dutertre. Lipschitz-Killing curvatures and polar images. Advances in Geometry, De Gruyter,

2019, 19 (2), pp.205-230. �10.1515/advgeom-2018-0019�. �hal-01239927�

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LIPSCHITZ-KILLING CURVATURES AND POLAR IMAGES

NICOLAS DUTERTRE

Abstract. We relate the Lipschitz-Killing measures of a definable setX⊂Rⁿ in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds ofRⁿ, such results were established by Langevin and Shifrin [17].

Then we give infinitesimal versions of these results. As a corollary, we obtain a relation between the polar invariants of Comte and Merle [7] and the densities of generic polar images.

1. Introduction

Let M ⊂ Rⁿ be a smooth compact submanifold of dimension dM. To each x ∈ M, we can associate a sequence of curvatures K0(x), . . . , KdM(x) called the Lipschitz-Killing-Weyl curvatures. Let us recall their definition. We denote by SNxM the unit sphere ofNxM, the normal space toM at x. For i∈ {0, . . . , dM}, thei-th Lipschitz-Killing-Weyl curvatureKi(x) is defined by

Ki(x) = Z

S_NxM

σi(IIx,v)dv,

where IIx,v is the second fundamental form on M at x in the direction v and σi(IIx,v) is thei-th elementary symmetric function of its eigenvalues. The second fundamental formIIx,v is defined on TxM as follows:

IIx,v(W1, W2) =−h∇W1W, W2i,

forW1 andW2in TxM, where∇ is the covariant differentiation inRⁿ andW is a local extension ofv normal to M. Note that Ki= 0 if iis odd. These curvatures are important objects: the integralsR

MKi(x)dxappear in Weyl’s tube formula [34]

and the integralR

MKdM(x)dxin the Gauss-Bonnet formula [1, 11].

In [17], Langevin and Shifrin explained how to compute the integralsR

MKi(x)dx, which have a differential geometric definition, by methods of differential topology.

Let us explain briefly their work. Let q be an even integer in {0, . . . , d_M}, let G^d_n^M^−q+1be the Grassmann manifold of (dM −q+ 1)-dimensional linear spaces in Rⁿ and for P ∈ G^d_n^M^−q+1, let π_P^M : M → P be the restriction to M of the orthogonal projection on P. Generically the discriminant ∆^M_P of π^M_P, also called the polar image ofπ_P^M, is almost everywhere a (dM−q)-dimensional submanifold of P. With each regular pointy in ∆^M_P, we can associate an integerα(π^M_P , y), which is very roughly speaking a kind of Morse index.

Mathematics Subject Classification (2010) : 14B05, 53C65, 58K05.

Keywords: Lipschitz-Killing curvatures, stratified Morse theory, polar varieties, polar images, o-minimal sets, generic projections, fold points.

1

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Langevin and Shifrin showed that Z

M

Kq(x)dx= cst 1 g^dn^M^−q+1

Z

G^dMn ^−q+1

Z

∆^M_P

α(π_P^M, y)dydP,

where g^d_n^M^−q+1 is the volume of G^d_n^M^−q+1 and “cst” means a universal constant that depends only on dM −q and n. For q = dM, this formula is exactly the exchange formula or exchange principle (already proved in [16]) which relates the integralR

MKdM(x)dx to the Morse indices of critical points of generic orthogonal projections onto lines.

Our first aim is to extend Langevin and Shifrin’s results to a large class of non-smooth objects, namely the class of definable sets in an o-minimal structure.

Definable sets are a generalization of semi-algebraic sets and global subanalytic sets, we refer the reader to classical references [33, 32, 8, 21] for basic definitions and results on this topics. The study of the geometric properties of these objects was initiated by Fu [12], who developed integral geometry for compact subanalytic sets. Using the technology of the normal cycle, he associated with every compact subanalytic setX ofRⁿ a sequence of curvature measures

Λ0(X,−), . . . ,Λn(X,−),

called the Lipschitz-Killing measures. In [5] (see also [2, 3]), Br¨ocker and Kuppe gave a geometric characterization of these measures using stratified Morse theory, in the more general setting of definable sets.

Let us describe now how we adapt the technics of Langevin and Shifrin to the singular definable case. We consider a compact definable setX ⊂Rⁿ and assume that it is equipped with a finite definable Whitney stratification {Sa}a∈A. Let S be a stratum of X of dimensiondS < n and let U be an open subset of X. As in the smooth case, forq∈ {0, . . . , dS}andP generic inG^q+1_n , the discriminant ∆^S∩U_P of the restriction toS∩U of the orthogonal projection onP is almost everywhere a smooth hypersurface inP, and to almost ally in ∆^S∩U_P , we can assign an index α(π^S_P, y), defined by means of stratified Morse theory. Then we set (see Definition 3.17)

Lq(X, S, U) = cst Z

G^q+1n

Z

∆^S∩U_P

α(π^S_P, y)dydP.

FordS < q≤n, we setLq(X, S, U) = 0. IfdS =n, we setLq(X, S, U) = 0 ifq < n and Ln(X, S, U) = vol(S∩U). We define the polar lengths of X (see Definition 3.18) by

Lq(X, U) =X

a∈A

Lq(X, Sa, U) forq∈ {0, . . . , n}.

In Theorem 3.20, we establish a connection between the Lipschitz-Killing measures and the polar lengths, namely we prove that for any open subsetU ofX and for q∈ {0, . . . , n},

Λq(X, U) =Lq(X, U).

Our second goal is to give infinitesimal versions of the previous equalities. We consider (X,0) ⊂ (Rⁿ,0) a germ of a closed definable set equipped with a finite definable Whitney stratification. We define localized versions of the polar lengths that we denote byL^loc_k (X,0),k= 0, . . . , n. Roughly speaking, these localized polar

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lengths are mean-values over Grassmann manifolds of weighted sums of densities of polar images. We prove that (see Theorem 4.7)

ǫ→0lim

Λ(X, X∩Bǫ)

bkǫ^k =L^loc_k (X,0),

fork∈ {0, . . . , n}. As a consequence, we obtain in Corollary 4.8 a relation between theL^loc_k (X,0)’s and the polar invariants of Comte and Merle [7]. This implies that the L^loc_k (X,0)’s are continuous along the strata of a Verdier stratification of X. This result should be related to Teissier’s famous result on the constancy of polar multiplicities along the strata of a Whitney stratification of a complex analytic set [29].

In complex analytic and algebraic geometry, polar and relative polar varieties and their relations with curvatures, characteristic classes and equisingularity prob- lems have been widely studied by many authors since the 70’s. We cannot give here a complete list of all the interesting papers published on these subjects and we apol- ogize for this. The study of real polar varieties and real equisingularity, suggested by Trotman [30], was started by Comte in [6]. It was continued in [7], [31] and [25].

We hope that the present paper will provide new interesting developments in this theory.

Throughout the paper, we will use the following notations and conventions (some of them have already appeared in this introduction):

• sk is the volume of unit sphereS^k of dimensionk andbk is the volume of the unit ballB^k of dimensionk,

• β(n, k) =^Γ(^k+1_Γ(²1^)Γ(^n−k+1² ⁾

2)Γ(ⁿ⁺¹₂ ) where Γ is the Euler function,

• for k ∈ {0, . . . , n}, G^k_n is the Grassmann manifold of k-dimension linear spaces in Rⁿ equipped with the O(n)-invariant density (see for instance [27], p.200),g_n^k is its volume,

• A^k_n is the affine grassmanian ofk-dimensional affine spaces inRⁿ,

• if P is a linear subspace of Rⁿ, SP is the unit sphere in P, G^k_P is the Grassmann manifold of k-dimensional linear spaces in P, A^k_P is the affine grassmanian ofk-dimensional affine spaces inP,P^⊥is the orthogonal space to P,πP : Rⁿ →P is the orthogonal projection on P and for any subset A⊂Rⁿ,π^A_P :A→P is its restriction toA,

• forv∈Rⁿ, the functionv^∗:Rⁿ→Ris defined byv^∗(y) =hv, yi,

• in Rⁿ, Bǫ(x) is the closed ball of radiusǫ centered at xand Sǫ(x) is the sphere of radiusǫcentered atx, ifx= 0, we simply writeBǫ andSǫ,

• ifX ⊂Rⁿ, Sing(X) is the singular set ofX,X is its topological closure, ˚X its topological interior, Fr(X) =X\X its frontier,

• when it makes sense, vol(X) means the volume of the set X and dX its dimension,

• if v1, . . . , vk are vectors in Rⁿ, [v1, . . . , vk] is the linear space spanned by v1, . . . , vk,

• a universal constant that we do not want to specify will be denoted by

“cst”,

• the word “smooth” means of classC³ at least,

• iff(x, y) =f(x1, . . . , xn, y1, . . . , ym) is a smooth function,∇f is its gradient and∇_xf is its gradient with respect to the variablesx1, . . . , xn.

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The paper is organized as follows. Section 2 contains a summary of results on stratified mappings, stratified More theory and the definitions of the Lipschitz- Killing measures and the polar invariants. In Section 3, we study generic polar varieties and images and prove the relation between the Lipschitz-Killing measures and the polar lengths. Section 4 deals with the local situation and contains infinitesimal versions of the results of Section 3.

The author thanks David Mond and Terry Gaffney for interesting discussions on double points of fold singularities.

2. Stratified mappings, stratified Morse theory and Lipschitz-Killing curvatures

In this section, we present different mathematical objects, notions and results that we will use in the next sections.

2.1. Stratified mappings. In this subsection, we recall well-known facts on mappings defined on stratified sets and on their critical points and values.

LetX⊂Rⁿbe a compact definable set equipped with a finite definable Whitney stratificationS ={Sa}a∈A. The fact that such a stratification exists is due to Loi [20]. Recently Nguyen, Trivedi and Trotman [24] gave another proof of this result.

Let f : Rⁿ → R^k be a smooth definable mapping and let f|X : X → R^k be its restriction toX. We assume further thatf is a submersion in an open neighborhood ofX.

A pointxinX is a (stratified) critical point off|X ifxis a critical point off|S, whereS is the stratum that containsx. This means that

dim(TxS+Txf⁻¹(f(x)))< n, or equivalently

dim(TxS∩Txf⁻¹(f(x)))> dS−k.

Note that ifdS< k, then all the points ofS are critical. Ifxis not a critical point off|X, we say thatxis a regular point off|X. We denote by Σ^S_f the set of critical points off|S and we set Σ^X_f =∪a∈AΣ^S_f^a.

Lemma 2.1. The setΣ^X_f is a compact definable set of X.

Proof. It is enough to prove that it is closed. Let (xm)m∈Nbe a sequence of points in Σ^X_f that tends to x. We can assume that (xm)m∈N is included in Σ^S_f. If x belongs to S then x belongs to Σ^S_f ⊂ Σ^X_f. If x belongs to S^′ ⊂ Fr(S) then, by Whitney condition (a), there exists a vector space T of dimension dS such that dim(T+Txf⁻¹(f(x)))< nand such that TxS^′⊂T. Thereforex∈Σ^S_f^′. Lety∈R^k. We say that y is a critical value off|X iff_|X⁻¹(y) contains a critical point off|X. Otherwise we say thaty is a regular value off|X. Note that if yis a regular value off|X, thenf⁻¹(y)∩X is a Whitney stratified set (see for instance [26]).

Lemma 2.2. The set of regular values off|X is an open definable and dense subset of R^k.

Proof. This set is equal toR^k\f(Σ^X_f). By Bertini-Sard’s theorem (see [4]), each set f(Σ^S_f) is a definable set of dimension less or equal to k−1. Hence f(Σ^X_f) is a compact definable set of dimension less or equal tok−1.

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2.2. Stratified Morse theory. Let M be a smooth riemannian manifold. Let X be a Whitney-stratified set ofM. Let xbe a point in X and let S(x) be the stratum that containsx. A generalized tangent space atxis a limit of a sequence of tangent spaces (TymS1)m∈N, whereS1is a stratum distinct fromS(x) such that x∈S1 and (ym)m∈Nis a sequence of points inS1 tending tox.

A Morse functionf :X →Ris the restriction of a smooth function ˜f :M →R such that the following conditions hold:

(1) f is proper and the critical values off are distinct i.e., ifxandy are two distinct critical points off thenf(x)6=f(y),

(2) for each stratumS ofX, the critical points off|S are non-degenerate, (3) at each critical point x, the differential Df˜(x) does not annihilate any

generalized tangent space atx.

Letf :X →Rbe a Morse function, S a stratum of X andxa critical point of f|S. Let us writec=f(x). The local Morse data forf atxis the pair

B_ǫ^M(x)∩f⁻¹([c−δ, c+δ]), B_ǫ^M(x)∩f⁻¹(c−δ) ,

where 0< δ≪ǫ≪1 andB^M_ǫ (x) is the dM-dimensional closed ball centered atx and of radiusǫinM. This definition is justified by the following property (see [13]

or [14]). There exists an open subsetAin R⁺×R⁺ such that

(1) the closureAofA inR² contains an interval ]0, ǫ[, ǫ >0, such that for all a ∈]0, ǫ[, the set {b ∈R⁺ | (a, b)∈ A} contains an open interval ]0, δ(a)[

withδ(a)>0,

(2) for all (ǫ, δ)∈A, the above pairs of spaces are homeomorphic.

The local Morse data are Morse data in the sense thatf⁻¹(]−∞, c+δ]) is homeomorphic to the space one gets by attachingB_ǫ^M(x)∩f⁻¹([c−δ, c+δ]) atf⁻¹(]−∞, c−δ]) alongB_ǫ^M(x)∩f⁻¹(c−δ) (see [13], I 3.5).

Ifxbelongs to a stratum of dimensiondX, the local Morse data at xare homeomorphic to the classical Morse data (B^λ×B^d^X^−λ, ∂B^λ×B^d^X^−λ) whereλis the Morse index off atx.

Ifxbelongs to a zero-dimensional stratum thenB_ǫ^M(x)∩f⁻¹([c−δ, c+δ]) has the structure of a cone (see [13], I 3.11).

If xlies in a stratumS with 0< dS < dX, then one can consider the classical Morse data off_|S at x. We will call them tangential Morse data and denote them by (Ptg, Qtg). One may choose a normal slice ofN atx, that is a closed submanifold of M of dimension dM −dS, which intersectsS in xtransversally. We define the normal Morse data (Pnor, Qnor) at x to be the local Morse data of f_|X∩N at x.

Goresky and Mac-Pherson proved that

• the normal Morse data are well defined, that is to say they are independent of the Riemannian metric and the choice of the normal slice ([13], I 3.6),

• the local Morse data (P, Q) off atxare the product of the tangential and the normal Morse data ([13], I 3.7):

(P, Q)≃(Ptg×Pnor, Ptg×Qnor∪Pnor×Qtg).

This implies thatP =B_ǫ^M(x)∩f⁻¹([c−δ, c+δ]) has the structure of a cone.

Definition 2.3. Let x∈X be a critical point of the Morse function f : X →R.

Let (P, Q) be the local Morse data of f at x. The Euler-Poincar´e characteristic

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χ(P, Q) = 1−χ(Q) is called the stratified Morse index off atxand is denoted by ind(f, X, x). Ifx∈X is not a critical point off, we set ind(f, X, x) = 0.

If (Ptg, Qtg) and (Pnor, Qnor) are the tangential and normal Morse data, then one has

χ(P, Q) =χ(Ptg, Qtg)·χ(Pnor, Qnor).

Thus we can write

ind(f, X, x) = indtg(f, X, x)·indnor(f, X, x),

where indtg(f, X, x) is called the tangential Morse index and indnor(f, X, x) the normal Morse index. Note that if x belongs to a zero-dimensional stratum then indtg(f, X, x) = 1. Ifxbelongs a stratum of dimensiondX, then indtg(f, X, x) = (−1)^λand indnor(f, X, x) = 1, whereλis the Morse index off atx.

The following theorem relates the Euler characteristic ofX to the indices of the critical points of a Morse function.

Theorem 2.4. LetX ⊂M be a compact Whitney-stratified set and letf :X →R be a Morse function. One has

χ(X) = X

x∈X

ind(f, X, x).

2.3. Lipschitz-Killing measures of definable sets. In this subsection, we present the Lipschitz-Killing measures of a definable set in an o-minimal structure. We describe Br¨ocker and Kuppe’s approach [5].

LetX⊂Rⁿbe a compact definable set equipped with a finite definable Whitney stratificationS ={Sa}a∈A.

Let us fix a stratumS. For k∈ {0, . . . , dS}, letλ^S_k :S→Rbe defined by λ^S_k(x) = 1

sn−k−1

Z

S_TxS⊥

indnor(v^∗, X, x)σdS−k(IIx,v)dv,

where IIx,v is the second fundamental form onS in the direction of v and where σdS−k(IIx,v) is the (dS −k)-th elementary symmetric function of its eigenvalues.

The index indnor(v^∗, X, x) is defined as follows:

indnor(v^∗, X, x) = 1−χ

X∩Nx∩Bǫ(x)∩ {v^∗=v^∗(x)−δ}

,

where 0< δ≪ǫ≪1 andNxis a normal (definable) slice toSatxinRⁿ. Since we work in the definable setting, this index is well-defined thanks to Hardt’s theorem [15, 8]. Furthermore whenv_|X^∗ has a stratified Morse critical point atx, it coincides with the normal Morse index at xof a function f :Rⁿ →R such thatf|X has a stratified Morse critical point atxand∇f(x) =v. Fork∈ {dS+ 1, . . . , n}, we set λ^S_k(x) = 0.

IfS has dimension nthen for allx∈S, we putλ^S₀(x) =· · ·=λ^S_n−1(x) = 0 and λ^S_n(x) = 1. IfS has dimension 0 then we set

λ^S₀(x) = 1 sn−1

Z

Sⁿ⁻¹

indnor(v^∗, X, x)dv, andλ^S_k(x) = 0 ifk >0.

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Definition 2.5. For every Borel setU ⊂Xand for everyk∈ {0, . . . , n}, we define Λk(X, U) by

Λk(X, U) =X

a∈A

Z

Sa∩U

λ^S_k^a(x)dx.

These measures Λk(X,−) are called the Lipschitz-Killing measures ofX. Note that for any Borel setU ofX, we have

ΛdX+1(X, U) =· · ·= Λn(X, U) = 0,

and ΛdX(X, U) =LdX(U), whereLdX is thedX-th dimensional Lebesgue measure inRⁿ. IfX is smooth then fork∈ {0, . . . , dX}, Λk(X, U) is equal to

1 sn−k−1

Z

U

KdX−k(x)dx.

As in the smooth case, the measure Λ0(X,−) satisfies an exchange formula (see [5]).

Proposition 2.6. For every Borel set U ⊂X, we have Λ0(X, U) = 1

sn−1

Z

Sⁿ⁻¹

X

x∈X

ind(v^∗, X, x)dv.

In the next section, we will generalize this exchange formula to the other measures Λk(X,−),k≥1.

The Lipschitz-Killing measures satisfy the kinematic formula (see [12, 5, 2]). We will need a particular case of this formula, namely the linear kinematic formula.

Proposition 2.7. Let U be an open subset of X. For k∈ {0, . . . , n}, we have Λn−k(X, U) = cst

Z

A^k_n

Λ0(X∩E, X∩E∩U)dE.

Proof. See [5], Corollary 8.5.

In [9], we gave a localized version of this equality. Let (X,0) ⊂(Rⁿ,0) be the germ of a closed definable set. Let H ∈ G^n−k_n , k ∈ {1, . . . , n}, and let v be an element inS_H^⊥. Forδ >0, we denote byHv,δ the (n−k)-dimensional affine space H+δv and we set

β0(H, v) = lim

ǫ→0lim

δ→0Λ0(Hδ,v∩X, Hδ,v∩X∩Bǫ).

Then we set

β0(H) = 1 sk−1

Z

S_H⊥

β0(H, v)dv.

Theorem 2.8. For k∈ {1, . . . , n}, we have

ǫ→0lim

Λk(X, X∩Bǫ) bkǫ^k = 1

g^n−kn

Z

G^n−kn

β0(H)dH.

Proof. See [9], Theorem 5.5.

In [9], we also established a relation between the limits limǫ→0Λk(X,X∩Bǫ) bkǫ^k and the polar invariants introduced by Comte and Merle in [7]. These polar invariants are real versions of the vanishing Euler characteristics of Lˆe and Teissier (see [18],

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Proposition 6.1.8 or [19], (3.1.2)). They can be defined as follows. Let H ∈G^n−k_n , k∈ {1, . . . , n}, and letv be an element inS_H^⊥. We set

α0(H, v) = lim

ǫ→0lim

δ→0χ(Hδ,v∩X∩Bǫ), α0(H) = 1

sk−1

Z

S_H⊥

α0(H, v)dv, and then

σk(X,0) = 1 gn^n−k

Z

G^n−kn

α0(H)dH.

Moreover, we putσ0(X,0) = 1.

Theorem 2.9. For k∈ {0, . . . , n−1}, we have limǫ→0

Λk(X, X∩Bǫ)

bkǫ^k =σk(X,0)−σk+1(X,0).

Furthermore, we have

ǫ→0lim

Λn(X, X∩Bǫ)

bnǫⁿ =σn(X,0).

Proof. See [9], Theorem 5.6.

3. Curvature measures and volume of polar images

In this section, we relate the curvature measures of definable sets to the volumes of polar images of generic projections. We start recalling the definition of polar varieties and polar images.

Definition 3.1. Let P ∈ G^k_n, k = 1, . . . , n, and let M ⊂ Rⁿ be a smooth submanifold. The polar variety (or polar set) Σ^M_P is the set of critical points ofπ^M_P , i.e.,

Σ^M_P =

x∈M |dim(TxM∩P^⊥)≥dM−k+ 1 , ifk≤dM. Ifk > dM, we set Σ^M_P =M.

The polar image ofπ^M_P is the set ∆^M_P defined by ∆^M_P =π^M_P (Σ^M_P ).

LetX ⊂Rⁿ be a compact definable set in an o-minimal structure. We equip it with a definable Whitney stratificationS. The following statements describe the structure of Σ^S_P and ∆^S_P for a stratumS ofX.

The next two lemmas can be proved using the machinery of modular submanifolds of multi-jet spaces developed by Mather and the fact the Thom-Boardman manifolds are modular (see [22], Theorem 2). The proofs we present below are more elementary and enable us to stay in the definable setting.

Lemma 3.2. Let S be a stratum of X such that dS < n. If k ≤ dS + 1 then for almost all P ∈G^k_n,Σ^S_P is either empty or a(k−1)-dimensional definable set.

Moreover the set

Σ^′_P^S =

x∈Σ^S_P | dim(TxS∩P^⊥)≥dS−k+ 2 ,

is a definable subset of Σ^S_P of dimension at mostk−2 andΣ^S_P\Σ^′_P^S is smooth.

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Proof. We treat first the casek≤dS. LetV ⊂(Rⁿ)^k be the open subset consisting of thek-tuples (v1, . . . , vk) such that (v1, . . . , vk) has rankk. LetMbe the following definable set:

M =n

(x, v1, . . . , vk)∈Rⁿ× V |x∈S and dim(TxS∩P^⊥)≥ds−k+ 1 whereP = [v1, . . . , vk]o

. The setM is the finite disjoint union of the setsMi,i= 1, . . . , k, where Mi=n

(x, v1, . . . , vk)∈Rⁿ× V | x∈S and dim(TxS∩P^⊥) =ds−k+i whereP = [v1, . . . , vk]o

. The set M1 is a definable manifold of dimension nk+k−1. To see this, we can assume that locallyS is given by

S={x∈Rⁿ |f1(x) =· · ·=fcS(x) = 0},

where cS =n−dS andf1, . . . , fcS are smooth definable functions. From now on, we setV = (v1, . . . , vk). Then (x, V) is inM1 if and only if the matrix







∂f1

∂x1(x) · · · _∂x^∂f¹

n(x) ... . .. ...

∂f_cS

∂x1 (x) · · · ^∂f_∂x^cS

n(x) v¹₁ · · · vⁿ₁

... . .. ... v¹_k · · · vⁿ_k





 ,

has rankcS+k−1, where we use the notationvi= (v¹_i, . . . , vⁿ_i)∈Rⁿ. It is an easy exercise of linear algebra to see that we can assume that the matrix







∂f1

∂x1(x) · · · _∂x^∂f¹_n(x) ... . .. ...

∂f_cS

∂x1 (x) · · · ^∂f_∂x^cS

n(x) v¹₁ · · · vⁿ₁

... . .. ... v_k−1¹ · · · vⁿ_k−1





 ,

has rankcS+k−1 and so that the minor

∂f1

∂x1(x) · · · _∂x^∂f¹

cS+k−1(x) ... . .. ...

∂f_cS

∂x1 (x) · · · _∂x^∂f^cS

cS+k−1(x) v₁¹ · · · v₁^c^S^+k−1

... . .. ... v_k−1¹ · · · v_k−1^c^S^+k−1

,

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does not vanish. Therefore the matrix







∂f1

∂x1(x) · · · _∂x^∂f¹_n(x) ... . .. ...

∂f_cS

∂x1 (x) · · · ^∂f_∂x^cS

n(x) v¹₁ · · · vⁿ₁

... . .. ... v_k−1¹ · · · vⁿ_k−1





 ,

has rankcS+k−1 if and only if the following minors:

mj =

∂f1

∂x1(x) · · · _∂x^∂f¹

cS+k−1(x) ^∂f_∂x¹_j(x) ... . .. ... ...

∂f_cS

∂x1(x) · · · _∂x^∂f^cS

cS+k−1(x) ^∂f_∂x^cS_j v v¹₁ · · · v₁^c^S^+k−1 v₁^j

... . .. ... ... v¹_k · · · v_k^c^S^+k−1 v_k^j

,

j = cS +k, . . . , n, vanish. Hence M1 is defined locally by cS +dS −k+ 1 = n−k+ 1 equations. Looking at the partial derivatives of the mj’s with respect to the variables v_k^j, j =cS+k, . . . , n, it easy to see that the gradient vectors of the functions definingM1are linearly independent, because the above minor of size cS+k−1 is not zero. HenceM1 is a submanifold of dimensionnk+k−1. Now let us consider the projection π1 : M1 → V,(x, V) 7→V. Bertini-Sard’s theorem implies that the setDπ1 of critical values ofπ1 is a definable set of dimension less thannk. Hence for allV ∈ V \Dπ1,π⁻¹₁ (V) is a smooth definable set of dimension k−1 (possibly empty). Butπ⁻¹₁ (V) is exactly the set

x∈Σ^S_P | dim(TxS∩P^⊥) =dS−k+ 1 ,

where V = (v1, . . . , vk) and P = [v1, . . . , vk]. Similarly we can show that the sets Mi,i≥2, are definable manifolds of dimension strictly less thannk+k−1 and we can consider the projectionsπi:Mi→ V,(x, V)7→V. As above, for allV ∈ V \D_π_i, whereDπi is the discriminant ofπi, the set

x∈Σ^S_P |dim(TxS∩P^⊥) =dS−k+i ,

is a smooth definable set of dimension less or equal tok−2. Let us denote byDthe union of the discriminantsDπi. Then for allV inV \D, Σ^S_P is a (k−1)-dimensional definable set, Σ^′_P^Sis a definable subset of Σ^S_P of dimension at mostk−2 and Σ^S_P\Σ^′_P^S is smooth, whereV = (v1, . . . , vk) andP= [v1, . . . , vk].

The same proof works ifk=dS+ 1.

When k ≤ dS, let Σ^S,0_P be set of fold points ofπ^S_P and when k = dS + 1, let Σ^S,0_P be the set of regular points ofπ^S_P. The following lemma gives a more precise description of Σ^S_P.

Lemma 3.3. Let S be a stratum of X such that dS < n. If k≤dS+ 1 then for almost allP ∈G^k_n,Σ^S,0_P is a smooth definable set of dimensionk−1andΣ^S_P admits the following decomposition:

Σ^S_P = Σ^S,0_P ⊔Z_P^S,

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whereZ_P^S is a closed definable set of dimension strictly less thank−1.

Proof. Let us treat first the casek≤dS. We recall thatxis a fold point of π_P^S if dim(TxS∩P^⊥) =dS−k+ 1 and dim(TxΣ^S_P∩P^⊥) = 0. We keep the notations of the previous lemma. LetN be the following definable set:

N =n

(x, v1, . . . , vk)∈Rⁿ×(V \D)|x∈S, dim(TxS∩P^⊥) =ds−k+ 1 and dim(TxΣ^S_P∩P^⊥)≥1 whereP = [v1, . . . , vk]o

.

The setN is the finite union of the setsNi,i= 1, . . . , k−1, where

Ni=n

(x, v1, . . . , vk)∈Rⁿ×(V \D)|x∈S, dim(TxS∩P^⊥) =ds−k+ 1 and dim(TxΣ^S_P∩P^⊥) =iwhereP = [v1, . . . , vk]o

.

The set N1 is a definable submanifold of dimension nk+k−2. To see this, let (x, V) be a point inM1. Since dim(TxS∩P^⊥) =dS−k+ 1, by the proof of Lemma 3.2 we can assume that around (x, V), M1 is defined by the vanishing of smooth definable functions f1, . . . , fcS andmcS+k, . . . , mn, wheref1, . . . , fcS depend onx, mcS+k, . . . , mn depend onxandV and where

rank(∇f1(x), . . . ,∇fcS(x), v1, . . . , vk−1) =cS+k−1.

Furthermore sinceV /∈Dπ1, Σ^S_P is defined locally atxby the vanishing off1, . . . , fcS

andmcS+k(−, V), . . . , mn(−, V) and the following gradient vectors:

∇f1, . . . ,∇f_c_S,∇_xmcS+k(−, V), . . . ,∇_xmn(−, V),

are linearly independent. So we see that (x, V) belongs to N1 if and only if the matrix







∂f1

∂x1(x) · · · _∂x^∂f¹_n(x) ... . .. ...

∂f_cS

∂x1(x) · · · ^∂f_∂x^cS

n(x)

∂m_cS+k

∂x1 (x) · · · ^∂m_∂x^cS_n^+k(x) ... . .. ...

∂mn

∂x1(x) · · · ^∂m_∂x_nⁿ(x) v¹₁ · · · vⁿ₁

... . .. ... v¹_k · · · vⁿ_k





 ,

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has rank n−1. Since rank(∇f1(x), . . . ,∇fcS(x), v1, . . . , vk−1) = cS +k−1, this condition is equivalent to the fact that the matrix

C=







∂f1

∂x1(x) · · · _∂x^∂f¹

n(x) ... . .. ...

∂f_cS

∂x1(x) · · · ^∂f_∂x^cS_n(x)

∂m_cS+k

∂x1 (x) · · · ^∂m_∂x^cS^+k

n (x) ... . .. ...

∂mn

∂x1(x) · · · ^∂m_∂xⁿ

n(x) v¹₁ · · · vⁿ₁

... . .. ... v¹_k−1 · · · vⁿ_k−1





 ,

has rankn−1. As in Lemma 3.2, we can assume that the minor

∂f1

∂x1(x) · · · _∂x^∂f¹

n−1(x) ... . .. ...

∂f_cS

∂x1(x) · · · _∂x^∂f^cS

n−1(x)

∂m_cS+k

∂x1 (x) · · · ^∂m_∂x^cS^+k

n−1 (x) ... . .. ...

∂mn

∂x1(x) · · · _∂x^∂mⁿ

n−1(x) v¹₁ · · · v₁ⁿ⁻¹

... . .. ... v_k−2¹ · · · v_k−2ⁿ⁻¹

,

does not vanish. The setN1is defined locally by the vanishing of thefi’s, themj’s and the determinant ofC. Because the above (n−1)×(n−1)-minor is not zero, we see that the gradient vectors of the functions definingN1 have rankn−k+ 2 and N1 is a submanifold of dimensionnk+k−2. In the same way, the setsNi,i≥2, have dimension less thannk+k−2 andN is a definable set of dimension at most nk+k−2. Let us consider the projectionρ:N→ V \D,(x, V)7→V. LetD^′be the union of the discriminants of the restrictions ofρto theNi’s. It is a definable set of dimension strictly less thannk. Hence for allV ∈ V \(D∪D^′),ρ⁻¹(V) is a definable set of dimensionk−2 (possibly empty). Butρ⁻¹(V) is exactly Σ^S_P\(Σ^S,0_P ∪Σ^′_P^S), where V = (v1, . . . , vk) and P = [v1, . . . , vk]. We take Z_P^S = Σ^′_P^S∪ρ⁻¹(V). It is closed because Σ^′_P^S is closed andρ⁻¹(V)⊂ρ⁻¹(V)∪Σ^′_P^S.

Fork=dS+ 1, the statement is just a reformulation of Lemma 3.2.

Lemma 3.4. Let S be a stratum of X such that dS < n. If k≤dS+ 1 then for almost allP ∈G^k_n,∆^S_P is a definable set of dimensionk−1 (or empty).

Proof. The set ∆^S_P is definable as the projection of a definable set. Moreover dim(∆^S_P) ≤ dim(Σ^S_P) = k−1. The projection π^S_P_|Σ^S,0

P : Σ^S,0_P → ∆^S_P is a local diffeomorphism because for all x∈ Σ^S,0_P , TxΣ^S_P ∩P^⊥ ={0}. Letxbe a point in Σ^S,0_P . Then there exists a neighborhoodUx ofxincluded in Σ^S,0_P on whichπ^S_P_|Σ^S,0

P

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is a diffeomorphism and so dimπ^S_P(Ux) =k−1. But π^S_P(Ux) is included in ∆^S_P. Therefore the dimension of ∆^S_P is greater or equal tok−1.

LetD(π^S_P_|Σ^S,0

P ) be the set of double-points ofπ^S_P_|Σ^S,0

P , i.e., D(π_P^S_|Σ^S,0

P ) =n

x∈Σ^S,0_P | ∃y∈Σ^S,0_P withx6=y andπP(x) =πP(y)o . Lemma 3.5. Let S be a stratum of X such that dS < n. If k≤dS + 1 then for almost all P ∈ G^k_n, D(π^S_P_|Σ^S,0

P ) is a definable set of dimension at most k−2 (or empty).

Proof. The setD(π^S_P_|Σ^S,0

P ) is clearly definable. Let us treat first the case k ≤dS. We keep the notations of the previous lemmas. Let L be the following definable set:

L=n

(x, y, v1, . . . , vk)∈Rⁿ×Rⁿ×(V \D∪D^′)|x∈Σ^S,0_P , y∈Σ^S,0_P , x6=y andhv_j, xi=hv_j, yifor allj∈ {1, . . . , k}whereP = [v1, . . . , vk]o

. This set is a definable submanifold of dimension nk+k−2. Let us explain this assertion. As explained in Lemma 3.2, ifx∈Σ^S,0_P then there exist smooth definable functionsf1, . . . , fcS andmcS+k(−, V), . . . , mn(−, V), whereV = (v1, . . . , vk), such that in a neighborhood ofx, Σ^S,0_P is the set

{f1=· · ·=fcS =mcS+k(−, V) =· · ·=mn(−, V) = 0},

and the gradient vectors of these functions are linearly independent. Furthermore, sincexis a fold point,

rank(∇f1(x), . . . ,∇f_c_S(x),∇_xmcS+k(x, V), . . . ,∇_xmn(x, V), v1, . . . , vk) =n.

Similarly, there exist smooth definable functionsg1, . . . , gcS and ˜mcS+k(−, V), . . . ,

˜

mn(−, V) such that in a neighborhood ofy, Σ^S,0_P is the set

{g1=· · ·=gcS = ˜mcS+k(−, V) =· · ·= ˜mn(−, V) = 0},

and the gradient vectors of these functions are linearly independent. We also have that

rank(∇g1(y), . . . ,∇gcS(y),∇ym˜cS+k(y, V), . . . ,∇ym˜n(y, V), v1, . . . , vk) =n.

ThereforeLis locally given by the vanishing of the functions f1, . . . , fcS, mcS+k, . . . , mn, g1, . . . , gcS,m˜cS+k, . . . ,m˜n, and

hv1, x−yi, . . . ,hvk, x−yi.

Let us remark that thefi’s depend onx, themj’s onxandV, thegi’s onyand the

˜

mj’s ony and V. We will show that the gradient vectors of these functions have rank 2n−k+ 2. As in the proof of Lemma 3.2, we can assume thatvk belongs to [∇f1(x), . . . ,∇f_c_S(x), v1, . . . , vk−1]. Since x6=y, there isj ∈ {1, . . . , n} such that

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xj−yj6= 0. Let us consider the following matrixJ of size (2n−k+ 2)×(2n+ 1):

J =







∇f1(x) 0 0

.. .

∇f_cS(x) 0 0

∇_xm_cS+k(x, V) 0 ∗

.. .

∇_xmn(x, V) 0 ∗

0 ∇g₁(y) 0

.. .

0 ∇g_cS(y) 0

0 ∇_ym˜_cS(y, V) ∗

.. .

0 ∇_ym˜n(y, V) ∗

v₁ −v₁ 0

.. .

v_k−1 −v_k−1 0

v_k −v_k x_j−y_j





 ,

where 0= (0, . . . ,0) in Rⁿ. It is a submatrix of the matrix whose lines are the gradient vectors of thefi’s, themj’s, thegi’s, the ˜mj’s and thekfunctionshvi, x−yi.

Sincevk belongs to [∇f1(x), . . . ,∇fn−dS(x), v1, . . . , vk−1],J has the same rank as the matrix ˜J given by

J˜=







∇f1(x) 0 0

.. .

∇f_cS(x) 0 0

∇_xm_cS_+k(x, V) 0 ∗

.. .

∇_xmn(x, V) 0 ∗

0 ∇g₁(y) 0

.. .

0 ∇g_cS(y) 0

0 ∇_ym˜_cS(y, V) ∗

.. .

0 ∇_ym˜_n(y, V) ∗

v1 −v1 0

.. .

vk−1 −vk−1 0

0 w x_j−y_j





 ,

where w = (w1, . . . , wn) is a linear combination of the −vi’s and thus belongs to P. Let us denote the lines of ˜J by

F₁, . . . ,F_c_S,M_c_S_+k, . . . ,M_n,G₁, . . . ,G_c_S,M˜_c_S_+k, . . . ,M˜_n, and

V₁, . . . ,V_k,

and show that they are linearly independent. So let us suppose that X

i

αiF_i+X

j

βjM_j+X

i

γiG_i+X

j

δjM˜_j+X

i

ξiV_i= 0.

Since

rank(∇f1(x), . . . ,∇fcS(x),∇xmcS+k(x, V), . . . ,∇xmn(x, V), v1, . . . , vk−1) =n, we see that

α1=· · ·=αcS =βcS+k =· · ·=βn =ξ1=· · ·=ξk−1= 0.

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We are left with the equality X

i

γiG_i+X

j

δjM˜_j+ξkV_k= 0, which implies thatξkwbelongs to

[∇g1(y), . . . ,∇gcS(y),∇ym˜cS+k(y, V), . . . ,∇ym˜n(y, V)].

Buty is a fold point so dim(TyS∩P^⊥) =dS −k+ 1 and dim(TyΣ^S_P ∩P^⊥) = 0.

Therefore, dim(NyS∩P) = dim(NyΣ^S_P ∩P) = 1 andNyS∩P =NyΣ^S_P ∩P. So the vectorξkw belongs to [∇g1(y), . . . ,∇gcS(y)], andδcS+k =· · ·=δn = 0. Since xj−yj6= 0, we find that ξk = 0 and finally thatγ1 =· · ·=γcS = 0. We conclude that the gradient vectors of the functions which define L locally at (x, y, V) are linearly independent and that L has dimension nk+k−2. As in the previous statements, we see that for almost all (v1, . . . , vk) in V \D∪D^′, the set

n(x, y)∈Rⁿ×Rⁿ |x∈Σ^S,0_P , y∈Σ^S,0_P , x6=y

andπ_P^S(x) =π^S_P(y) whereP = [v1, . . . , vk]o , has dimension at most k−2. Therefore D(π_P^S_|Σ^S,0

P ) has dimension at mostk−2 because it is the image of this set by the projection (x, y)7→x.

Let us treat the casek=dS + 1. Here we recall that Σ^S,0_P is the set of regular point ofπ^S_P. LetQbe the following definable set:

Q=n

(x, y, v1, . . . , vk)∈Rⁿ×Rⁿ× V |x∈S, y∈S, x6=y

andhvj, xi=hvj, yifor allj∈ {1, . . . , k}whereP = [v1, . . . , vk]o . This set is a definable submanifold of dimensionnk+k−2. The proof of this fact is an in the previous case. Locally at a point (x, y, V),Qis given by the vanishing of functionsf1, . . . , fcS, g1, . . . , gcS and the functionshv1, x−yi, . . . ,hvk, x−yi. As before thefi’s depend only onxand thegi’s only ony. It is not difficult to see that the gradient vectors of these functions are linearly independent becausex−y6= 0.

Therefore for almost allV inV, the set

n(x, y)∈S×S |x6=y andπ_P^S(x) =π^S_P(y) whereP = [v1, . . . , vk]o , has dimension at mostk−2 and so hasD(π_P^S_|Σ^S,0

P ).

Remark 3.6. As noticed to the author by Terry Gaffney, we can also use Mather’s technology of modular submanifolds to prove the above lemma. Double points of fold singularities form a contact class and a Thom-Boardmann manifold. Hence by [22], pages 233 and 234, they form a modular submanifold. Therefore by Theorem 1 in [22], for almost allP ∈G^k_n, π_P^S is transverse with respect to this submodular manifold. Our proof avoids the notions of modular manifolds and contact classes and enables us to stay in the definable setting.

For each stratumSand eachP ∈G^k_n,k≤dS+ 1, let Σ^S_P,limbe the following set:

Σ^S_P,lim=

x∈S¯\S | ∃(x_m)m∈NinS such thatxm→x, TxmS→T

and dim(T∩P^⊥)≥dS−k+ 1 .