Curvature measures of subanalytic sets.

Since the Euler characteristics of these sets are bounded (in fact, by 0.2.2, the masses of the subanalytic family of currents

!Mathematical Expression Omitted^

are bounded) the dominated convergence theorem applies to give the desired formula (2.2.2a) for X = !f.sup.-1^(0), Y = !g.sup.-1^(0).

3. Uniqueness of the (Co-) Normal Cycle. Let X !subset of^ !E.sup.n^ be compact. We give here a natural criterion to determine the normal cycle N(X) !is an element of^ !I.sub.n-1^ (S!E.sup.n^) if it exists. This criterion is Introduction. The study of curvature is fundamental to geometry. While curvature is usually thought of as a characteristic of a smooth manifold, classical studies of Steiner, Minkowski and others showed that certain types of curvature have meaning even in the presence of somewhat severe singularities. It is natural to try to extend the constructions of these curvatures to more general situations and to seek further analogies with the smooth case. While some progress has been made since mathematical antiquity (notably by Blaschke !B1^, Federer !Fe1^ and Banchoff !Ba^), a comparatively recent observation, made independently by P. Wintgen and M. Zahle, has significantly illuminated the problem. In this paper we would like to focus the light somewhat and to use it to probe the curvature properties of the rather large class of subanalytic sets.

Some History. The beginnings of our problem are pleasantly concrete. Let K !subset^ !E.sup.n^ be a compact convex set, and suppose first that the boundary of K is smooth. It is an exercise in elementary differential geometry to see that the n-dimensional volumes of the tubular neighborhoods !K.sub.r^ := {x !is element of^ !E.sup.n^: dist (x, K) !is less than or equal to^ r} are given by a polynomial:

(I1) Vol (!K.sub.r^) = !summation of^ !c.sub.i^(K)!r.sup.n - i^ where i=0 to n;

here

(I2) !c.sub.j^(K) = const !integral of^ !!Sigma^.sub.i^(K)!r.sup.n-i^ with the limit of bdry K;

where !!Sigma^.sub.i^ = the !(n - 1 - i).sup.th^ the elementary symmetric function and the !k.sub.j^ are the principal curvatures. (In fact (I1) holds for any compact smooth submanifold, for small r !is greater than^ 0, with appropriate integrals (I2).) Remarkably the formula (I1) remains valid even when K is no longer smooth. This permits us to define the "curvature integrals" (I2) via (I1), even though the resulting quantities (measures) may not be absolutely continuous with respect to (n - 1)-dimensional surface area.

The formula (I1) admits a generalization due to Blaschke, the principal kinematic formula, which in its original form states that if K, L !subset^ !E.sup.n^ are compact and convex then

(I3) !integral of^ !Chi^(K !intersection^ !Gamma^L)d!Gamma^ with the limit of G = !summation over i+j=n.^ !const.sub.ij^(K)!c.sub.j^(L).

Here G is the n-dimensional euclidean group and d!Gamma^ is its invariant measure. (Observe that the Euler characteristic !Chi^(K !intersection^ !Gamma^L) = 0 or 1 depending on whether K !intersection^ !Gamma^L is empty or not.) To recover (I1), simply evaluate (I3) with L a ball of radius r.

Federer was able to establish (I1) and (I3) for a class larger than the convex sets, namely the "sets with positive reach". However, the types of singularities occurring in these sets are essentially the same as in the convex case: where a convex set may be characterized as one for which all principal curvatures are, in a generalized sense, positive, sets with positive reach admit a parallel description as those for which all principal curvatures are bounded below. Banchoff proceeded in a different direction by studying the curvature of polyhedra, which was a restriction of a different kind. However it is noteworthy that he was able to show that the Gauss curvature of a polyhedron is an intrinsic invariant, independent of the embedding. Such a conclusion was not attained by Federer. An outstanding omission from this line of work is the study of the most natural singular spaces of all, namely the real analytic varieties with singularities.

The Normal Cycle of Wintgen and Zahle. The curvature integrals (I2) may be described in a more invariant way. Let M !subset^ !E.sup.n^ be a smooth compact submanifold, and let N(M) !subset^ !E.sup.n^ x !S.sup.n - 1^ be the manifold of unit normals to M. Then N(M) has dimension (n - 1), and inherits a natural orientation from !E.sup.n^ (in fact N(M) is a copy of the tube {x: dist (x, M) = r}, r small). It turns out that there are universal differential (n - 1)-forms !!Kappa^.sub.0^, . . ., !!Kappa^.sub.n - 1^ on the unit tangent bundle !E.sup.n^ x !S.sup.n - 1^ such that

(I4) !c.sub.i^(M) = !integral of^ !!Kappa^.sub.i^ with the limit of N(M), i = 0, . . ., n - 1.

Zahle observed that if M has positive reach, then (I4) is valid for some lipschitz submanifold N(M) !subset^ !E.sup.n^ x !S.sup.n - 1^. In any case, N(M) has no boundary; it was dubbed the normal cycle by Wintgen !Wi^. It is natural to think of N(M) as an integral current in the sense of Federer and Fleming.

The primary questions in this line now become: given a singular set X !subset^ !E.sup.n^, when does a normal cycle N(X) exist? How can it be characterized? How can it be constructed? This paper is primarily devoted to the resolution of these questions for subanalytic sets. We are able also to give a satisfactory characterization of the normal cycle of an arbitrary compact subset of !E.sup.n^ (Theorem 3.2). Moreover, in the course of the discussion of the subanalytic case we also describe a construction valid for a more general class that contains, for example, the sets with positive reach. Still, this construction lacks formal beauty and seems far from definitive, so we have emphasized instead the subanalytic aspects.

Plan of the Paper. Section 0 contains generalities which may be unfamiliar to some readers.

In section 1 we show how to associate a conormal cycle to the zero-sets of certain nonnegative functions, among them the subanalytic ones. At this stage the construction depends on the function, but we show in section 3 that this is an illusion. The crucial fact here is the Gauss-Bonnet theorem 1.5/1.8, which enables us to fulfill the hypothesis of the uniqueness theorem 3.2. Most of this is straightforward, using the notion of Monge-Ampere functions from !Fu2^.

In section 2 we prove the kinematic formula for subanalytic subsets of !E.sup.n^ (or at least a preliminary version of it). This is important for the uniqueness result of section 3, as the constructions involved here give the appropriate "localization" of the Gauss-Bonnet theorem. The idea is to reduce to the formal bundle-theoretic constructions of !Fu4^.

Section 3 is the heart of the paper, where we give a very general uniqueness result and apply it to show that the subanalytic constructions of section 1 give unambiguous results. Here again we use the apparatus of !Fu4^, a theorem of !Fu3^, and at a crucial point a simple argument of !Fu1^. The relevant theorem of !Fu3^ states roughly that the normal cycle is determined by the values it gives for the Gauss curvature measure. The work in this section is to show that via the formalism of !Fu4^ we may express the latter in terms of indices of height functions, in the manner of !Ch-La^ or !A-W^ or !Ba^.

In section 4 we give the simple proofs of some basic properties of the normal cycle, all easy consequences of sections 1 and 3.

Finally, in section 5, we show that the top (Gauss) curvature measure is in some sense an intrinsic invariant of subanalytic spaces, i.e. that a subanalytic isometry f: X !right arrow^ Y takes the curvature measure of X to that of Y. Here we use a "graph construction" similar to that in !Fu5^.

After completing this work, we became aware that Kashiwara and Schapira have also constructed, using completely different methods, a lagrangian cycle for a general subanalytic set X (cf. !K-S^, Chapter 9). In fact their construction gives essentially the same result as our !Mathematical Expression Omitted^, introduced in section 3. This follows from the uniqueness theorem 3.2 and an apparently simple computation on the Kashiwara-Schapira cycle, which was kindly performed by Kashiwara in a personal communication. A short presentation of the equivalence proof is presented at the end of section 4.

Acknowledgements. I am happy to thank Indiana University and the University of Trento, Italy for their kind hospitality as this work was completed. Thanks are due also to Adam Parusinski, who pointed out an error in an earlier version of section 4.6. Finally, I would also like to express my gratitude to the Mathematics Department of the University of Georgia, particularly Ray Kunze, Clint McCrory and Ted Shifrin, for their support and encouragement over several years as the theory presented here slowly took shape.

0. Background.

0.0. Geometric Measure Theory. The foundation of our work is the theory of integral currents developed in Chapter 4 of !Fe2^; on the whole we retain the notation and terminology of that work (but cf. also !Si^). We adopt also the following conventions:

0.0.1. If U is a measurable subset of an oriented manifold M of dimension m, then !U^ denotes the m-dimensional current given by integration over U. If M = R then we will usually dispense with the brackets, identifying (for instance) the interval (a, b) with the current (f dx) !approaches^ !integral of^f dx between limits b and a.

0.0.2. Let U !subset^ M be open, and let f: U !right arrow^ N be locally lipschitzian. Let T be a rectifiable current in M, with !Mathematical Expression Omitted^. We will need to consider expressions of the form !f.sub.#^T. Such expressions should be taken literally, i.e. for any differential form !Phi^ !is an element of^ !and^*N we have !f.sub.#^T(f*!Phi^) = T(f*!Phi^). That is, !f.sub.#^T is a well-defined current in N iff the indicated integral converges absolutely for every such !Phi^. Alternatively we may define

!Mathematical Expression Omitted^

where the !U.sub.!Epsilon^ !subset^!subset^ U and exhaust U.

0.0.3. If T is a current, U a set and !Phi^ a differential form then we put

!Mathematical Expression Omitted^.

0.1. Approximate Gradients of Lipschitz Functions. Let M be a !C.sup.!infinity^ manifold and f: M !right arrow^ R a locally lipschitz function. Clarke !C1^ has defined the generalized differential D*f(x) at any point x !is an element of^ M as

D*f (x) = convex hull{t !is an element of^ !T*.sub.x^M: there is a sequence !x.sub.i^ !right arrow^ x such that f is differentiable at each !x.sub.i^, and t = lim df(!x.sub.i^) where i !approaches^ !infinity^};

if M has a riemannian metric then we put Df(x) !subset^ !T.sub.x^M to be the image of D*f(x) under the induced diffeomorphism T*M !right arrow^ TM. (For an alternative characterization of D*f, cf. !Fu2^, 2.2.) We say that x is a critical point of f iff 0 !is an element of^ D*f(x); otherwise x is a regular point. Likewise, a value r !is an element of^ R is a critical value of f iff there exists a critical point x of f such that f(x) = r; otherwise r is a regular value. Since D*f is obviously upper semicontinuous, it follows that the set of critical points is a closed subset of M.

These ideas have been used implicitly by !Gr-Sh^ and !Grm^ to construct f-increasing flows near any regular point of f. Given an open set U !subset^ M, we say that a !C.sup.!infinity^^-vector field V on U is an approximate gradient field forf on U iff

(V(x), t) !is greater than^ 0

for every t !is an element of^ D*f(x) and x !is an element of^ U.

PROPOSITION 0.1.1. If f, M are as above, then there exists an approximate gradient field for f on the set regf of regular points of f. If !Phi^ : W !right arrow^ M is the maximal flow of V, where W !subset^ regf x R, then for each compact set K !subset^ regf there exists !Delta^ !is greater than^ 0 such that

d/dt!where^t=0 f(!Phi^(x,t)) !is greater than^ !Delta^.

Proof. Using a partition of unity, it is enough to establish this for M = U !subset^ !R.sup.n^. Convolving with a suitable bump function !Psi^, we may take

V := grad (f * !Psi^).

0.1.2. We also need the following simple generalization of the Poincare-Hopf theorem.

PROPOSITION. Suppose X !subset^ M is compact with

X = !f.sup.-1^(-!infinity^, r^

where f is lipschitzian and r is a regular value of f. Suppose that V is a vector field on M, having only nondegenerate zeros inside X, and such that the restriction of V to some neighborhood of !f.sup.-1^(r) is an approximate gradient field there. Then

!Chi^(X) = !Sigma^{index(V,x): x !is an element of^ X, V(x) = 0}

02. Subanalytic Sets, Maps and Currents. Let M be a real analytic manifold. For the definition of the subanalytic subsets of M, cf. !Bi-Mi^. If N is a second analytic manifold and X !subset^ M is a subanalytic subset, then a map f: X !right arrow^ N is said to be subanalytic iff f is continuous and graph f is a subanalytic subset of M x N. Note that this definition depends on how X is embedded in M; thus we will at times abuse notation and say in the circumstances above that "f: M !right arrow^ N is a subanalytic map".

A subanalytic current in M, of dimension k, is an integer-multiplicity rectifiable current T !is an element of^ !R.sub.k^(M) such that there are (i) a locally finite collection {!V.sub.!Alpha^^} of subanalytic subsets of M, with each !V.sub.!Alpha^^ a smooth oriented submanifold of dimension k, and (ii) integers {!n.sub.!Alpha^^}, such that

T = !Sigma^!n.sub.!Alpha^^!!V.sub.!Alpha^^.

(cf. !Ha 1^).

PROPOSITION 0.2.1. If T is a subanalytic current in M, then so is !Delta^T. In particular, T is an integral current.

Proof. cf. !Ha 1^, 4.2.

PRINCIPLE OF SUBANALYTIC FINITUDE 0.2.2. Let S !subset^ M be compact and subanalytic, and let f: S !right arrow^ N be a subanalytic map. Then

sup(!H.sup.k^(!f.sup.-1^(p)): p !is an element of^ N, dim!f.sup.-1^(p) !is less than or equal to^ k) !is less than^ !infinity^

for each k.

Proof cf. !Ha 2^, Section 3, or !Bi-Mi^, Theorem 3.14.

PROPOSITION 0.2.3. Let f: M !right arrow^ N be a subanalytic map with domain U, where U !subset^ M is open. Suppose that f is locally lipschitzian and that !f.sup.-1^(K) is relatively compact in M whenever K !is subset or equal to^ N is compact. Let T be a subanalytic current such that

!Mathematical Expression Omitted^.

Then the expression !f.sub.#^T defines a subanalytic current in N with

spt(!Delta^!f.sub.#^T - !f.sub.#^!Delta^T) !subset^ !intersection^ {f(U - V); V !subset^!subset^ U).

Proof. Let !Gamma^ !subset^ M x N be the closure of the graph of f. By the hypothesis, the restriction to !Gamma^ of !!Pi^.sub.N^ is proper. There is clearly a unique subanalytic current !Mathematical Expression Omitted^ supported on !Gamma^ such that !!Pi^.sub.M#^T = T. Then !Mathematical Expression Omitted^, the result follows easily.

0.3. Invariant Forms on !SE.sup.n^. Let G be the group of oriented euclidean motions of !E.sup.m^. The exterior algebra of G-invariant differential forms on the unit tangent bundle S!E.sub.m^ is completely described in !Fu4^, Lemma 0.4.3, as generated by the canonical 1-form !Alpha^, its differential d!Alpha^ = !Omega^, and the forms !!Kappa^.sub.0^, . . ., !!Kappa^.sub.m - 1^ of degree m - 1. These last elements may also be described as follows.

Let exp: !TE.sup.m^ !approxiamtely equals^ !E.sup.m^ x !E.sup.m^ !right arrow^ !E.sup.m^ be the exponential map

exp (x, y) := x + y

Pulling back the volume form, we get

exp* (!dx.sub.1^ !and^ . . . !and^ !dx.sub.m^) = d(!x.sub.1 + !y.sub.1^) !and^ . . . !and^ d(!x.sub.m^ + !y.sub.m^)

= !Sigma^!Epsilon^(I,J)!dx.sub.I) !and^ !dy.sub.J^

where the sum is taken over all partitions of {1, . . ., m} into subsets I, J and !Epsiloi^(I,J) = !plus or minus^ 1. Let v: !E.sup.m^ - {0} !right arrow^ !S.sup.m - 1^ be the normalization map. Using the G-invariance of !dx.sub.1^ !and^ . . . !and^ !dx.sub.m^ we may write

exp* (!dx.sub.1^ !and^ . . . !and^ !dx.sub.m^) = (!!Rho^.sup.-1^!Alpha^ + d!Rho^) !and^ !summation of^ !!Rho^.sup.m - i - 1^!!Kappa^.sub.i^ where i=0 to m - 1

where p(x, y) = !absolue value of^ y and the !!Kappa^.sub.i^ are pull-backs via v of some (m - 1)-forms on !SE.sup.m^. In particular !!Kappa^.sub.0^ is the volume form of !S.sup.m - 1^, and

(0.3a) !!Kappa^.sub.i^ !is an element of^ (!!and^.sup.i^ !E.sup.m^) !and^ (!!and^.sup.m - i - 1^ !S.sup.m - 1^).

Let !!Lambda^.sub.0^, . . . !!Lambda^.sub.n - 1^ be the corresponding forms on !SE.sup.n^, and !!Kappa^.sub.0^, . . ., !!Kappa^.sub.m+n-1^ those on !SE.sup.m+n^ = S(!E.sup.m^ x !E.sup.n^). We would like to express the !!Kappa^.sub.r^ in terms of the !!Kappa^.sub.s^ and !!Lambda^.sub.t^. Let !!Xi^.sub.1^, . . ., !!Xi^.sub.n^; !!Eta^.sub.1^, . . ., !!Eta^.sub.n^ be the usual coordinates of !TE.sup.n^. Then

!Mathematical Expression Omitted^

with the obvious notation. The singular forms !!Kappa^.sub.s^, !!Lambda^.sub.t^ may be thought of as forms on !SE.sup.m+n^, pull-backs of the !!Kappa^.sub.s^, !!Lambda^.sub.t^ under the singular maps

(x, !Xi^; y, !Eta^) !right arrow^ (x, !Xi^/!absolute value of^ !Xi^)

(x, !Xi^; y, !Eta^) !right arrow^ (y, !Eta^/!absolute value of^ !Eta^).

Thus if we restrict to {R = 1} we obtain

!Mathematical Expression Omitted^

Taking interior product with the radial vector field !Delta^/!Delta^R = !!Rho^.sub.1^ !Delta^/!Delta^!!Rho^.sub.1^ + !!Rho^.sub.2^ !Delta^/!Delta^!!Rho^.sub.2^,

!Mathematical Expression Omitted^

By (0.3a) it follows that

!Mathematical Expression Omitted^,

r = 0, . . ., m + n - 1.

0.4. Notational Conventions. The closure of a set A in a topological space is written closA, and its interior by intA. Its frontier is the set fronA := closA - A.

We will denote projections M !right arrow^ N by !!Pi^.sub.N^, or, when M is the total space and N the base space of a bundle !Beta^, by !!Pi^.sub.!Beta^^. This convention is superseded by any other explicit labeling scheme; these will be confined to localities within the text.

We denote the k-dimensional measure of the unit sphere in !E.sup.k+1^ by

!Mathematical Expression Omitted^

where !Gamma^ is the Euler gamma function.

1. The Conormal Cycle.

1.0. Throughout this section we let M be an oriented smooth manifold of dimension m. Let T*M be its cotangent bundle, with canonical 1-form !Alpha^ and symplectic 2-form !Omega^ = d!Alpha^. Let z : M !right arrow^ T*M be the zero-section, and let !Mathematical Expression Omitted^ denote the deleted cotangent bundle. Let S*M denote the cotangent ray space !Mathematical Expression Omitted^, where !Xi^ !is similar to^ !Eta^ iff !Xi^ = t!Eta^ for some t !is greater than^ 0. Thus S*M is an !S.sup.m-1^ bundle over M carrying a natural contact structure; any choice of smooth Riemannian metric ! ^ on M determines a diffeomorphism S*M !tautomer^ {!Xi^ !is an element of^ T*M: !absolute value of^ !Xi^ = 1} and we will identify these spaces as convenience dictates.

Our object here is to construct, for certain singular subsets X !subset of^ M, currents N*(X) !is an element of^ !I.sub.m-1^(S*M) fulfilling the role of the usual manifold of unit conormals to a smooth submanifold. The subsets X under consideration are the zero-sets of nonnegative Monge-Ampere functions f : M !approaches^ !0, !infinity^), satisfying some nondegeneracy condition at zero. The construction given here depends a priori on the choice of the function f, and therefore we denote the current for the moment by N*(f, 0); however we will show in section 3 that under some additional assumptions the dependence is only on X = !f.sup.-1^(0), so we may revert to the notation N*(X).

In case M is real analytic, all closed subanalytic subsets X !subset of^ M belong to the class above. However this case requires some special handling, as we will see.

1.1. Auras. We recall from !Fu2^ that a locally lipschitz function f : M !right arrow^ R is called Monge-Ampere if there exists a current T !is an element of^ !I.sub.m-1^(T*M) satisfying the properties:

(1.1a) !Delta^T = 0;

!Mathematical Expression Omitted^;

(1.1c) !!Pi^.sub.M^ !where^ spt T is proper; and

(1.1d) given any compactly supported smooth m-form !Theta^ on M and any smooth function !Phi^ : T*M !right arrow^ R we have

T(!Phi^ !center dot^ !!Pi^.sub.M^*!Theta^) = !integral of^ (!Phi^ !Omicron^ df) !center dot^ !Theta^ !with the limit of M^

By the main theorem of !Fu2^, this current T is unique and will be denoted !df^, the differential current of f. In case f is !C.sub.2^, T is the oriented graph of df. By Theorem 2.2, op. cit., we have

(1.1e) spt !df^ !subset of^ graph D*f.

Definition 1.1.1. Let X !subset of^ M be compact. An aura for X is a proper, nonnegative Monge-Ampere function f : M !approaches^ !0, !infinity^) such that !f.sup.-1^(0) = X.

An aura f for X is nondegenerate iff its generalized differential is bounded away from zero near X, i.e. iff there is a neighborhood U of X such that

!Mathematical Expression Omitted^.

Example 1.1.2. If M is riemannian and X !subset of^ M has positive reach in the sense of !Fe2^, then by !Bg^, and !Fu2^, Prop. 3.1, the function !Mathematical Expression Omitted^ is a nondegenerate aura for X. If M is analytic and X is a general closed subanalytic set, then dist(x,X) is an aura which may be degenerate; however, this problem is circumvented in 1.6 ff below.

PROPOSITION 1.2. If X admits a nondegenerate aura then bdry X is (m - 1)-rectifiable.

Proof. Let f be a nondegenerate aura for X. By the Clarke inverse function theorem (!Cl^, Theorem 7.11), the level sets !f.sup.-1^(r) are lipschitz submanifolds of M for all sufficiently small r !is greater than^ 0. Any approximate gradient flow for f then induces a lipschitz mapping of such !f.sup.-1^(r) onto bdry X.

1.3. Let !Mathematical Expression Omitted^ be the quotient map. Given a choice of Riemannian metric ! ^ for M we have a diffeomorphism

!Mathematical Expression Omitted^

such that !Upsilon^ !Omicron^ !Mu^ = !!Pi^.sub.S*M^ and !absolute value of^ !Mu^(!Xi^,t) !is equivalent to^ t. Let us fix such a choice for the rest of this section, though we emphasize that the choice is entirely inessential.

PROPOSITION. Let f be a nondegenerate aura for X, and U be as in (1.1f) above.

Put

!Mathematical Expression Omitted^;

clearly spt !Mathematical Expression Omitted^. Then

!Mathematical Expression Omitted^,

!Mathematical Expression Omitted^,

and

(1.3c) !!Upsilon^.sub.#^!T.sub.f^ !is an element of^ !I.sub.m - 1^(S*M).

Furthermore there is !Epsilon^ !greater than^ 0 such that

!Mathematical Expression Omitted^.

Proof. By !Fe2^, 4.3.2, we have

!Mathematical Expression Omitted^

by !Fe2^, 4.3.2(1). Since !Mathematical Expression Omitted^, we have also

!Mathematical Expression Omitted^.

To prove (1.3c), put

!Mathematical Expression Omitted^,

and observe that by !Fu3^

!Mathematical Expression Omitted^

since f !is equivalent to^ 0 on spt S, and that for almost all sufficiently small r !is greater than^ 0

!Mathematical Expression Omitted^

since f is a nondegenerate aura, by 1.1e. From (1.3e) it follows that for any V !subset of^ T*M, we have

!Mathematical Expression Omitted^,

and therefore the tangent planes of the rectifiable current !Mathematical Expression Omitted^ are planes of dimension m which annihilate the contact form !Alpha^ of S*M, as well as its differential d!Alpha^. It is an elementary fact of contact geometry that such planes do not exist in a (2m - 1)-dimensional contact manifold, so actually !Mathematical Expression Omitted^. Thus we have for almost all sufficiently small r !is greater than^ 0

!Mathematical Expression Omitted^,

so !!Upsilon^.sub.#^!T.sub.f^ !is an element of^ !I.sub.n-1^(S*M) by (1.3f).

Now we prove (1.3d). By (1.3g) we have

!!Mu^.sub.r#^!!Upsilon^.sub.#^!T.sub.f^ = <S, ! ^, r>

for almost all r !is less than^ !Epsilon^, where !!Mu^.sub.r^ : S*M !right arrow^ T*M is multiplication by r. Thus for a.e r, s, 0 !is less than^ r !is less than^ s !is less than or equal to^ !Epsilon^,

!Mathematical Expression Omitted^.

In view of (1.3b) and the identity !Mu^*!Omega^ = dt !and^ !Alpha^ + t!Omega^, the current !!Mu^.sub.#^(!!Upsilon^.sub.#^!T.sup.f^ X (r, s)) is Lagrangian. Furthermore both sides of (1.3d) are supported in !Mathematical Expression Omitted^, which has measure zero by Proposition 1.2. Thus (1.3h) and the uniqueness theorem 2.0 of !Fu2^ imply that (1.3d) holds with 0, !Epsilon^ replaced by r, s; now let r !down arrow^ 0, s !up arrow^ !Epsilon^.

Definitions 1.4. Given f, !T.sub.f^ and U in 1.3, we put

N*(f, r) := !!Upsilon^.sub.#^<!df^, !Pi^*f, r>

for a.e. small r !is greater than^ 0, and

!Mathematical Expression Omitted^.

It is also convenient to define

!Mathematical Expression Omitted^.

(The reader is warned that our use of !right arrow^ has nothing to do with that of !Fe2^, 4.1.7.)

If M has a riemannian structure, we put

N(f, r), N(f, 0), !Mathematical Expression Omitted^

to be the images of the corresponding starred currents under the natural diffeomorphisms S*M !right arrow^ SM = unit tangent bundle of M (resp. T*M !right arrow^ TM).

Remarks 1.4.1. a) If X,f are as in example 1.1.2 then N(f, 0) is the unit normal bundle of !Za^. The current !Mathematical Expression Omitted^ is given by integration over !!union^.sub.x^ Nor(X, x), where Nor(X, x) is defined as in !Fe1^. In this case, these two currents are particularly simple in that each is given by integration over an oriented lipschitz submanifold of (respectively) SM or TM.

b) Given a current T living in the total space E of a sphere bundle, let us define the corresponding conical current in the corresponding vector bundle by

!Mathematical Expression Omitted^

where !Mu^(!Xi^, t) = t!Xi^. Then we may write

!Mathematical Expression Omitted^.

c) Given X,f as in 1.3 and a !C.sup.1,1^ diffeomorphism !Phi^ : M!prime^ !right arrow^ M, the composition f !Omicron^ !Phi^ is a nondegenerate aura for !!Phi^.sup.-1^(X), with

!(!Phi^*).sub.#^N*(f, r) = N*(f !Omicron^ !Phi^, r), r !is greater than or equal to^ 0.

Here !Phi^* : S*M !right arrow^ S*M!prime^ is induced by !Phi^* : T*M !right arrow^ T*M!prime^.

d) Let us call an m-dimensional current T !is an element of^ !D.sub.m-1^(S*M) legendrian if !Mathematical Expression Omitted^. Then N*(f, 0) is legendrian, by (1.3b).

1.5. Now we prove the analogue of the classical Chern-Gauss-Bonnet theorem (!Ch^), where we use N(f, 0) to compute the integral of geodesic curvature. Let M be given a riemannian structure, with Chern-Gauss-Bonnet form !Omega^ !is an element of^ !!Lambda^.sup.m^(M) and geodesic curvature form !Phi^ !is an element of^ !!Lambda^^.sup.m-1^(SM). Thus d!Phi^ = -!Omega^, and !Phi^ restricts to a volume form (of volume one) on each fiber !S.sub.p^M !is congruent to^ !S.sup.m-1^.

THEOREM. Iff : M !approaches^ !0, !infinity^) is a nondegenerate aura for a compact set X !subset of^ M, then

!integral of^ !Omega^ + N(f, 0)(!Phi^) = !Chi^(X) !with the limit of X^,

where !Chi^ = Euler characteristic.

Proof. Let !Xi^ be a smooth unit vector field on U (taken as in (1.1f)) that is an approximate gradient field for f !where^ U. By the generalized Poincare-Hopf theorem 0.1.2 and the usual proof (!Ch^) of the Chern-Gauss-Bonnet theorem, we have

!Mathematical Expression Omitted^

since the flow of !Xi^ gives a deformation retraction of !f.sup.-1^!0, r^ onto X.

In view of (1.1e), (1.1d) and the hypothesis that !Xi^ is an approximate gradient field, we may construct a smooth homotopy H: (SM !where^ U) X !0, 1^ !right arrow^ SM !where^ U such that

!Mathematical Expression Omitted^

and !!Pi^.sub.M^H(!Theta^, t) = !!Pi^.sub.M^(!Theta^) for all !Theta^ !is an element of^ SM !where^ U and t !is an element of^ !0, 1^. This may be done, for instance, by taking

H(!Theta^, t) := !Upsilon^((1 - t)!Theta^ + t!Xi^ !Omicron^ !!Pi^.sub.M^(!Theta^))

whenever <!Theta^, !Xi^ !Omicron^ !!Pi^.sub.M^(!Theta^)> !is greater than or equal to^ 0 and extending smoothly to the rest of SM !where^ U. Thus we have for a.e. small r !is greater than^ 0

!Mathematical Expression Omitted^.

Applying this current to !Phi^, we obtain

!Mathematical Expression Omitted^

by the support theorem (!Fe2^, 4.1.20), since dim !!Pi^.sub.M#^!J.sub.r^ = m and spt !!Pi^.sub.M#^!J.sub.r^ is included in the (m - 1) dimensional set !f.sup.-1^(r). Thus we may replace the second term on the left-hand side of (1.5a) by N(f, r)(!Phi^). Letting r !approaches^ 0, the theorem follows.

1.6. Next we treat the subanalytic case, where in the most interesting examples the set X does not admit any nondegenerate aura in the sense above. On the other hand the subanalytic category has many special features which allow us to construct conormal cycles by approximation. Consider, for instance, the following simple fact. We assume here, and for the rest of section 1, that M is real analytic.

LEMMA. Let f : M !right arrow^ R be proper, locally lipschitz and subanalytic. Then the set of critical values {f(x) : 0 !is an element of^ D*f(x)} is locally finite in R.

Proof. Let S be a stratification of graph(f) !subset of^ M X R satisfying Whitney's condition a. Then the projections !!Pi^.sub.M^(S), S !is an element of^ S constitute a Whitney a-stratification of the manifold M. The condition a, applied to S, implies that for S !is an element of^ S, x !is an element of^ !!Pi^.sub.M^(S) and !Xi^ !is an element of^ D*f(x) we have

!Xi^ !where^ !T.sub.x^(!!Pi^.sub.M^(S)) = d(f !where^ !Pi^S)(x).

Thus 0 !is an element of^ D*f (x) iff d(f !where^ !Pi^S)(x) = 0. Since f is proper, the set of critical values !C.sub.s^ := {f(x): x !is an element of^ S, d(f !where^ !Pi^S)(x) = 0} is subanalytic in R for each S !is an element of^ S. Since f !where^ !Pi^S is !C.sup.!infinity^^, Sard's theorem implies that !C.sub.s^ has dimension zero. Now 0.2.2 implies that Cs is locally finite, and the lemma follows from the properness off and the local finiteness of S.

LEMMA 1.7. v : T*M !right arrow^ S*M is a semi-analytic map.

Proof. Choosing locally an analytic riemannian metric ! ^ on M, the graph of !Upsilon^ is realized as

!Mathematical Expression Omitted^,

which is clearly semianalytic in T*M X S*M.

COROLLARY. If f : M !right arrow^ R is lipschitz, subanalytic and proper, then the expression !Mathematical Expression Omitted^ defines a subanalytic integral current in S*M, with boundary supported in !Mathematical Expression Omitted^.

Proof. Proof is immediate from 0.2.3.

1.8. Now let X !subset of^ M be a compact and subanalytic, and let f : M !approaches^ !0, !infinity^) be a subanalytic aura for X. According to lemma 1.6, we may define N*(f, r) as in 1.4 for small r !is greater than^ 0. Using the corollary to Lemma 1.7 we may put

!Mathematical Expression Omitted^

where X and U !subset of^!subset of^ !f.sup.-1^!0, !r.sub.1^) and !r.sub.1^ is the smallest positive critical value of f. Finally, we define !Mathematical Expression Omitted^ by the first formula of (1.4c) (it seems likely that the second formula is valid also, but we have not checked this).

Remark 1.4.1(b) remains true if "!C.sup.1,1^" and "nondegenerate" are replaced by "analytic" and "subanalytic", respectively. If M has a riemannian metric (smooth, but not necessarily analytic) we may define N(f, r), !Mathematical Expression Omitted^ as in 1.4. The Chern-Gauss-Bonnet theorem remains valid. Let !Omega^, !Phi^ be as in Theorem 1.5.

THEOREM. With X, f, M as above, we have

!integral of^ !Omega^ + N(f, 0)(!Phi^) = !Chi^(X) !with the limit of X^.

Proof. Let !Xi^ be an approximate gradient field for f !where^ (U-X). Then the flow of !Xi^ defines deformation retractions of !f.sup.-1^!0, r^ onto !f.sup.-1^!0, s^ for all small r !is greater than^ s !is greater than^ 0, which shows that the inclusion X !subset of^ !f.sup.-1^!0, r^ induces an isomorphism in Cech cohomology. Since X is an ANR (in fact it is triangulable), Cech cohomology is the same as singular cohomology. Therefore, for small r !is greater than^ 0

!Mathematical Expression Omitted^

(by 1.5, since max {f - r, 0} is a nondegenerate aura for !f.sup.-1^!0, r^)

!approaches^ !integral of^ !Omega^ + N(f, 0)(!Phi^) !with the limit of X^.

as r !approaches^ 0.

1.9. A subanalytic family {!X.sup.r^}r !is an element of^ J in M is a family of subsets !X.sup.r^ !subset of^ M such that there is a subanalytic subset !Mathematical Expression Omitted^ with

!Mathematical Expression Omitted^

for all r !is an element of^ J !subset of^ R. In applications it is important to know the continuity properties of N*(!X.sup.r^), as a function of r, for such families {!X.sup.r^}. For this reason we prove at this stage the

THEOREM. Let !{!X.sup.r^}.sub.r!is an element of^!0,1^^ be a monotonically increasing subanalytic family of compact subsets of M with !X.sup.0^ = !!intersection^.sub.r!is greater than^0^!X.sup.r^. Then !X.sup.r^ has the homotopy type of !X.sup.0^ for all sufficiently small r !is greater than^ 0.

The proof depends on the following more general lemma. Let us say that a family of subanalytic functions !{!f.sub.r^ : M !right arrow^ R}.sub.r!is an element of^J^ is subanalytic if there is a function !Mathematical Expression Omitted^ with subanalytic graph such that !Mathematical Expression Omitted^ for each r !is an element of^ J. Note that we do not require !Mathematical Expression Omitted^ to be continuous.

LEMMA 1.9.1. Let !{!f.sub.r^}.sub.r!is an element of^!0,1^^ be a monotonically decreasing subanalytic family of functions M !right arrow^ R, each of which is lipschitz, nonnegative and proper, and such that !f.sub.0^ = !sup.sub.r!is greater than^0^!f.sub.r^. Then for each sufficiently small !Epsilon^ !is greater than^ 0, the inclusion !Mathematical Expression Omitted^ is homotopy equivalence for all sufficiently small r !is greater than^ 0.

Proof. Put C := {(r,!f.sub.r^(x)) : 0 !is an element of^ D*!f.sub.r^(x)} !subset of^ !0, 1^ X !0, !infinity^). It is clear that C is a subanalytic subset. Since (by lemma 1.5) each set !Mathematical Expression Omitted^ is zero-dimensional, the set C is one-dimensional and therefore (by !Bi-Mi^, Theorem 6.1) a locally finite union of pieces of analytic curves. The same is true of clos C. Let !!Gamma^.sub.1^, . . ., !!Gamma^.sub.k^ be those curves of clos C intersecting !0, 1^ X {0}, and put

!Mathematical Expression Omitted^

Finally, put for r !is an element of^ !0, 1^

g(r) := inf{s: (r, s) !is an element of^ !!Gamma^.sub.1^ !union^ . . . !union^ !!Gamma^.sub.k^} !down arrow^ 0

as r !down arrow^ 0. We have for each r !is an element of^ !0, 1^

(g(r), !!Epsilon^.sub.0^) !intersection^ !f.sub.r^({x : 0 !is an element of^ D*!f.sub.r^(x)}) !down arrow^ 0.

Let !Epsilon^ !is an element of^ (0, !!Epsilon^.sub.0^) be fixed. We claim that for all sufficiently small r !is greater than^ s !is greater than^ 0, the inclusion !Mathematical Expression Omitted^ is a homotopy equivalence. To see this, let Vt be an approximate gradient field for !f.sub.t^, t !is an element of^ !0, 1^. Then the flow of !V.sub.t^ gives a deformation retraction of !Mathematical Expression Omitted^ onto !Mathematical Expression Omitted^; let !Mathematical Expression Omitted^ be the corresponding retraction/homotopy equivalence. If r !is greater than^ s !is greater than^ 0 are sufficiently small, then

!Mathematical Expression Omitted^,

and we have the corresponding commutative diagram of homotopy groups

!Mathematical Expression Omitted^

A diagram chase shows that the last inclusion is an isomorphism, as claimed.

If follows that !Mathematical Expression Omitted^ is a homotopy equivalence for all small s, and the lemma follows as in the first paragraph of the proof of 1.8.

1.2. Proof of 1.9. It follows from !Gra^ that M admits a !C.sup.!Omega^^ riemannian metric. By the lemma, it is enough to show that there is !Epsilon^ !is greater than^ 0 such that !f.sub.r^ := dist(!center dot^, !X.sub.r^) has no critical values in (0, !Epsilon^) for all sufficiently small r !is greater than^ 0. We argue by contradiction. By !Fu0^, the set of critical points of !f.sub.r^ near !X.sub.r^ is included in

{x !is an element of^ M: there are points !p.sub.0^ . . ., !p.sub.m^ !is an element of^ X and to and !t.sub.0^, . . ., !t.sub.m^ !is an element of^ !0, 1^ such that

!Mathematical Expression Omitted^

where !d.sub.!p.sub.i^^ = dist(!center dot^, !p.sub.i^). Consider the subanalytic subset

!Mathematical Expression Omitted^.

The negation of the assertion above is equivalent to the assertion that the projection !Pi^B = !!Pi^.sub.RXR^B of B onto the last two factors has (0, 0) as a limit point. If this is so then by the curve selection lemma, we may find a real analytic curve

!Gamma^ : !0, 1^ !approaches^ M X !X.sup.m+1^ X !R.sup.m^ X R X R

such that !Gamma^(0,1) !subset of^ B and !Pi^ !convolution^ !Gamma^(0) = (0,0). Writing

!Gamma^(r) = (

we compute

d/dr !Delta^(r) = d/dr !summation of^ !t.sub.i^(r) dist (x(r),!p.sub.i^(r)) where i = 0 to m

(since dist (x(r),!p.sub.i^(r)) !is equivalent to^ !Delta^(r))

= !summation of^ !t.sub.i^(r) d/dr !dist (x(r),!p.sub.i^(r))^

= !summation of^ !t.sub.i^(r) where i = 0 to m {<x!prime^(r), grad !d.sub.!p.sub.i^(r)^(x(r))> + <!p!prime^.sub.i^(r), grad !d.sub.x(r)^(!p.sub.i^(r))>}

= <x!prime^(r), !summation of^ !t.sub.i^(r) grad !d.sub.!p.sub.i^(r)^(!p.sub.i^(r)) where i = 0 to m> + !summation of^ <!t.sub.i^(r)!p!prime^.sub.i^(r), grad !d.sub.x(r)^(!p.sub.i^(r))> where i = 0 to m

= !summation of^ !t.sub.i^(r)<!p!prime^.sub.i^(r), grad !d.sub.x(r)^(!p.sub.i^(r))> where i = 0 to m

(by the definition of B).

On the other hand, since the !X.sub.r^ are increasing we have that -!p!prime^.sub.i^(r) !is an element of^ Tan(!X.sub.r^,!p.sub.i^(r)) for each i, r; since the definition of B implies that -grad !d.sub.x(r)^(!p.sub.i^(r)) belongs to the dual cone Nor(!X.sub.r^,!p.sub.i^(r)), the expression above is !less than or equal to^ 0 for every r. Thus !Delta^ !is less than or equal to^ 0, which is a contradiction.

Remark 1.9.3. For the main application of 1.9 (namely 4.3 below) it is sufficient to know the result for !X.sup.r^ !subset of^ !E.sup.n^. In particular the result of !Gra^ is superfluous to the main line of development here.

2. The Kinematic Formula.

2.0. We establish the principal kinematic formula for pairs of compact subanalytic subsets of !E.sup.n^. This is carried out by showing how the construction of !Fu2^ for the differential current of a sum of two nondegenerate auras is related to that of !Fu4^ for the normal cycle of a genetic intersection: the point is that the "connecting current" of the latter is just a fancy manifestation of vector addition.

Some of the constructions of the present section will be used in the next, where we prove that N*(f,0) depends only on !f.sup.-1^(0) for subanalytic auras f.

2.1. We recall some of the constructions of !Fu4^, simplifying the notation and interpreting the "connecting current" of section 4, op. cit., somewhat more concretely.

2.1.1. Put !Mathematical Expression Omitted^ to be the euclidean group in n dimensions, equipped with its biinvariant measure d!Gamma^. Let B be the bundle

!Mathematical Expression Omitted^

with projection p(!Xi^,!Gamma^) := (!!Pi^.sub.!E.sup.n^^(!Xi^), !!Gamma^.sup.-1^!!Pi^.sub.!E.sup.n^^(!Xi^)), fiber !S.sup.n-1^ x SO(n), and group SO(n) x SO(n) acting on the left on !S.sup.n-1^ x SO(n) by

(2.1.1a) (!Alpha^,!Beta^)(!Theta^,!Gamma^) = (!Alpha^!Theta^, !Alpha^!Gamma^!!Beta^.sup.-1^)

This bundle is G x G equivariant, where G x G acts on S!E.sup.n^ x G by the formula (2.1.1a). Let P: S!E.sup.n^ x S!E.sup.n^ !approaches^ !E.sup.n^ x !E.sup.n^ be the projection, and !Mathematical Expression Omitted^ the induced bundle over S!E.sup.n^ x S!E.sup.n^, with total space E. Thus the G x G-equivariant diagram

!Mathematical Expression Omitted^

commutes.

A most important observation is that the group of !Mathematical Expression Omitted^ reduces to SO(n - 1) x SO(n - 1) in such a way that !Mathematical Expression Omitted^ remains G x G-equivariant (!Fu4^, lemma 2.2).

2.1.2. As a simple case of the current constructed in !Fu4^, section 4, we here construct an integral current connect !is an element of^ !I.sub.dimSO(n)+1^(!S.sup.n-1^ x SO(n)), invariant under the action (2.1.1a), as follows. Define H: SO(n) x !0,1^ !approaches^ !S.sup.n-1^ x SO(n) by

H(!Alpha^,s) := (!Nu^(s!Alpha^!e.sub.n^ + (1 - s)!e.sub.n^),!Alpha^)

where !Nu^: !E.sup.n-1^ - {0} !approaches^ !S.sup.n-1^ is the normalization map and !e.sub.n^ is the nth element of the standard basis of !E.sup.n^. Then, putting K := {!Alpha^: !Alpha^!e.sub.n^ = -!e.sub.n^},

connect := !H.sub.#^(!SO(n) - K^ x !0,1^),

where the orientation of SO(n) will be decided for our convenience later. (Recall (0.1.2) that the singularities of the map H do not obstruct the performance of the push-forward here; at most they introduce some boundary.) The boundary of connect decomposes as

!Delta^ connect = !bd.sub.1^ - !bd.sub.0^ + cut,

where

!bd.sub.i^ := !H.sub.i#^!SO(n)^, i = 0, 1

and

cut = !!S.sup.n-1^^ x !{!Alpha^ !is an element of^ SO(n): !Alpha^!e.sub.n^ = -!e.sub.n^}^

The "cone" over connect may be expressed as the image

!Mathematical Expression Omitted^,

where

!Mathematical Expression Omitted^

The invariance of connect is clear from the definition. Thus, given currents S, T !is an element of^ !I.sub.*^(S!E.sup.n^) we may form

!Mathematical Expression Omitted^,

as in !Fu4^, section 1.

2.2. We first prove the kinematic formula for pairs of nondegenerate auras satisfying a further measure-theoretic condition.

Definition. If f: M !approaches^ !0,!infinity^) is a nondegenerate aura, then we put

!Mathematical Expression Omitted^,

where the lim sup is in the sense of Hausdorff convergence of subsets. If M is Riemannian then nor(f, 0) will denote the corresponding set in SM.

THEOREM 2.2.1. Let f,g: !E.sup.n^ !approaches^ !0,!infinity^) be nondegenerate auras for subsets X,Y !subset of^ !E.sup.n^ respectively. Suppose that, of the two closed sets nor(f,0) and nor(g,0), one is (n - 1) rectifiable and the other has !H.sup.n^ measure zero. Then there is a closed set K !subset of^ G with d!Gamma^(K) = 0 such that whenever !Gamma^ !is an element of^ G - K the function f + g !convolution^ !!Gamma^.sup.-1^ is a nondegenerate aura for X !intersection^ !Gamma^Y. Furthermore we have for a.e. !Gamma^ !is an element of^ G the relation of normal cycles

!Mathematical Expression Omitted^

From this and the Gauss-Bonnet theorem 1.5 we obtain from !Fu4^, Theorem 2.3,

COROLLARY 2.2.2. Putting

!Mathematical Expression Omitted^

and

!!Phi^.sub.n^(h,0) := !L.sup.n^(!h.sup.-1^(0)),

we have

(2.2.2a) !integral of^ !Chi^(X !intersection^ !Gamma^Y)d!Gamma^ with the limit of G = !summation of^ !!Beta^.sub.ij^!!Phi^.sub.i^(f,0)!!Phi^.sub.j^(g,0) where i + j = n

where !!Beta^.sub.ij^ are constants depending only on n.

Remark. In fact, the measure d!Gamma^ and the curvature measures !!Phi^.sub.i^ may be normalized so that

!!Beta^.sub.ij^ = (i+j)]/i]j]

in (2.2.2a). (See also !Ni^.) It is worth noticing that Federer's treatment of the formula yields this result quite readily. Observe first that the tube formula for convex bodies K, or smooth compact submanifolds M, is an immediate consequence of the kinematic formula, taking the two objects under consideration to be A = K or M and !Mathematical Expression Omitted^, the closed ball of a varying radius r:

!Mathematical Expression Omitted^

(here the radius r must be restricted in case A = M). If we take in particular !Mathematical Expression Omitted^ then we obtain

!Mathematical Expression Omitted^.

It follows that if we normalize d!Gamma^ so that

d!Gamma^({!Gamma^: !absolute value of^ !Gamma^(0) !is less than or equal to^ 1}) = 1

and let !Mathematical Expression Omitted^ be the multiples of !!Phi^.sub.i^ so that

!Mathematical Expression Omitted^,

we then arrive at the desired formula

!Mathematical Expression Omitted^.

2.2.3. We prove the first assertion of 2.2.1 in the following strengthened form. Let f, g, X, Y as above.

LEMMA. There is a closed set K !subset of^ G of measure zero such that whenever !Mathematical Expression Omitted^, there is !Delta^ !is greater than^ 0 such that if !Gamma^ !is an element of^ U, (x,u) !is an element of^ nor (f,0) and (x,v) !is an element of^ nor(g !convolution^ !!Gamma^.sup.-1^, 0) = !Gamma^ nor(g,0), we have

!absolute value of^ u + v !is greater than^ !Delta^ (!absolute value of^ u + !absolute value of^ v).

In particular, for such !Gamma^ we may find !Delta^ !is greater than^ 0 and a neighborhood V of X !intersection^ !Gamma^Y in !E.sup.n^ such that if x !is an element of^ V - (X !intersection^ !Gamma^Y) and w !is an element of^ D(f + !Gamma^ !convolution^ !g.sup.-1^)(x) then !absolute value of^ w !is greater than^ !Delta^!prime^.

Proof. By the proof of !Fe2^, 3.2.22,

!H.sup.2n-1^ (nor(f,0) x nor(g,0)) = 0.

Since

A := {(!Xi^, !Eta^, !Gamma^) !is an element of^ nor(f,0) x nor(g,0) x G: !Gamma^!Eta^ = -!Xi^}

is a bundle over nor(f,0) x nor(g,0) with fiber SO(n - 1), it follows from the result just cited that

!H.sup.dimSO(n-1)+2n-1^(A) = 0

Since dimSO(n - 1) + 2n - 1 = dim G, it follows that the closed set K := !!Pi^.sub.G^(A) has measure zero in G. Now the lemma follows from the definition of the sets nor(h,0).

Observe that the first assertion of 2.2.1 follows from this lemma and the construction of the proof of Prop. 2.6, !Fu2^.

2.2.4. Proof of (2.2.1a). Using !Fe2^, 4.3.5 we may rewrite the construction of !Fu2^, Prop. 2.6 as follows. Put

Q: T!E.sup.n^ x T!E.sup.n^ x SO(n) !approaches^ T!E.sup.n^ x G

!Mathematical Expression Omitted^.

Here !!Beta^,z^ !is an element of^ G is defined to be the element

!!Beta^,z^(u) = !Beta^u + z.

Then we have for a.e !Gamma^ !is an element of^ G

(2.2.4a) !d(f + g !convolution^ !!Gamma^.sup.-1^)^ = !!Pi^.sub.T!E.sup.n#^ <!Q.sub.#^(!df^ x !dg^ x !SO(n)^), !!Pi^.sub.G^, !Gamma^>.

Since !Mathematical Expression Omitted^ for some t !is greater than or equal to^ 0} for h = f, g, it follows from Lemma 2.2.3 that for each !Mathematical Expression Omitted^ the map Q is proper when restricted to

!Mathematical Expression Omitted^.

Now by (2.2.4a), the flat continuity of push-forward by proper maps, and the flat continuity of slicing (c.f. !Fe2^ 4.3.2(2)) it follows that for a.e. !Gamma^ !is an element of^ U

!Mathematical Expression Omitted^

Writing as in (1.4d)

!Mathematical Expression Omitted^

for h = f, g, where Z = integration over the zero-section of T!E.sup.n^, we get after expanding the indicated products above

!Mathematical Expression Omitted^

Referring to (1.4c), it remains only to identify the last term with the of the last term of (2.2.1a). To emphasize the completely formal nature of this identification, we isolate this step in the following

LEMMA. Let S, T !is an element of^ !I.sub.*^(S!E.sup.n^), and let !Mathematical Expression Omitted^ be the corresponding conical currents. Then for a.e. !Gamma^ !is an element of^ G

!Mathematical Expression Omitted^

Proof. In fact, we prove the equality of currents in T!E.sup.n^ x G

!Mathematical Expression Omitted^.

Working locally, let !Sigma^, !Tau^ be local cross-sections of the principal SO(n - 1) bundle G !approaches^ S!E.sup.n^, with projection !Mathematical Expression Omitted^. Thus we may write by abuse of notation

!Sigma^(x,!Theta^) = !!Sigma^(!Theta^),x^, where !Sigma^(!Theta^)!e.sub.n^ = !Theta^,

and similarly for !Tau^. These maps induce a local trivialization !Phi^ of the bundle !Mathematical Expression Omitted^, characterized by the relation

!Mathematical Expression Omitted^

for (x, !Theta^) !is an element of^ domain(!Sigma^), (y,!Psi^) !is an element of^ domain(!Tau^), !Phi^ !is an element of^ !S.sup.n-1^ and !Alpha^ !is an element of^ SO(n) (cf. the proof of lemma 2.2 of !Fu4^). Suppose that the currents S and T are supported in the domains of !Sigma^ and !Tau^ respectively. Then

!Mathematical Expression Omitted^

where J: S!E.sup.n^ x S!E.sup.n^ x SO(n) x !0,1^ !approaches^ S!E.sup.n^ x S!E.sup.n^ x !S.sup.n-1^ x SO(n) is

J(!Xi^,!Eta^,!Alpha^,s) = (!Xi^,!Eta^,H(!Alpha^,s))

and H is defined in 2.1.2. Thus the left-hand side of (2.2.4b) may be written

!Mathematical Expression Omitted^

where A: S!E.sup.n^ x S!E.sup.n^ x SO(n) x !0,1^ x R !approaches^ T!E.sup.n^ x G is the map

A(!Xi^,!Eta^,!Alpha^,s,t) = (t!Sigma^(!Xi^) !center dot^ !Nu^(s!Alpha^!e.sub.n^ + (1 - s)!e.sub.n^),!Sigma^(!Xi^)!Alpha^!Tau^!(!Eta^).sup.-1^).

On the other hand, the right-hand side may be written (up to sign)

!Mathematical Expression Omitted^

where !Mathematical Expression Omitted^ is the map

!Mathematical Expression Omitted^.

But if we put

g: S!E.sup.n^ x S!E.sup.n^ x SO(n) x (0,!infinity^) x (0,!infinity^) !approaches^ S!E.sup.n^ x S!E.sup.n^ x SO(n) x (0,1^ x (0,!infinity^) to be the map

g(x,!Theta^; y,!Phi^; !Alpha^,u,v) = (x,!Theta^; y,!Phi^; !Sigma^(!Theta^)!Alpha^!Tau^!(!Psi^).sup.-1^, v/u + v, !absolute value of^ u!Theta^ + v!Sigma^(!Theta^)!Alpha^!Tau^!(!Psi^).sup.-1^!Psi^)

then g is a diffeomorphism, with inverse

!g.sup.-1^(x,!Theta^; y,!Psi^; !Beta^,s,t) = (x,!Theta^; y,!Psi^; !Sigma^!(!Theta^).sup.-1^!Beta^!Tau^(!Psi^), t(1 - s)/!(1 - s)!Theta^ + s!Beta^!Psi^^, st/!(1 - s)!Theta^ + s!Beta^!Psi^^).

Furthermore,

!Mathematical Expression Omitted^

and it follows that the currents of (2.2.4c,d) are equal up to sign.

2.3. We may now deduce the kinematic formula 2.2.2 for subanalytic auras f,g as follows. Put, for r,s !is greater than^ 0

!f.sub.r^ := max{f - r, 0}

!g.sub.s^ := max{g - s, 0}

Then N(!f.sub.r^,0) = N(f,r) and N(!g.sub.s^,0) = N(g,s) for all small r,s !is greater than^ 0, and furthermore the auras !f.sub.r^,!g.sub.s^ are nondegenerate for such r,s. Thus we may apply 2.2.2 to obtain

!integral of^ !Chi^(!f.sup.-1^!0,r^ !intersection^ !Gamma^!g.sup.-1^!0,s^)d!Gamma^ with the limit of G = !summation of !!Beta^.sub.ij^!!Phi^.sub.i^(!f.sub.r^,0)!!Phi^.sub.j^(!g.sub.s^,0) where i + j = n

for small r,s. But

!!Phi^.sub.i^(!f.sub.r^,0) !approaches^ !!Phi^.sub.i^ (f,0)

!!Phi^.sub.j^(!g.sub.s^,0) !approaches^ !!Phi^.sub.j^ (g,0)

as r,s !approaches^ 0, by 1.8, and by Theorem 1.9

!Chi^(!f.sup.-1^!0,r^ !intersection^ !Gamma^!g.sup.-1^!0,s^) !approaches^ !Chi^(!f.sup.-1^(0) !intersection^ !Gamma^!g.sup.-1^ (0)).
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