Uniqueness of curvature measures in pseudo-Riemannian geometry

The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a K\"unneth-type formula for Lipschitz-Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

An important example is given by the intrinsic volumes. If V is a Euclidean vector space of dimension n, K ∈ K(V ), then, as observed by Steiner [35], the volume of the r-tube K r := K + rB around K is a polynomial in r: vol K r = n k=0 µ k (K)ω n−k r n−k .
Here ω n−k is the volume of the (n − k)-dimensional unit ball. The coefficient µ k (K) is called k-th intrinsic volume. If ι : V → W is an isometric embedding of Euclidean vector spaces, then µ W k (ι(K)) = µ V k (K) for all K ∈ K(V ). In particular, µ k is invariant under translations and rotations. Conversely, if µ is a continuous (with respect to the Hausdorff metric on K(V )), translation-and rotation-invariant realvalued valuation, then µ is a linear combination of intrinsic volumes by a famous theorem of Hadwiger.
Hadwiger's theorem has inspired a lot of research. To mention just a few of the numerous results, we refer the reader to [1,8,9,10,14,15] for versions for subgroups of the orthogonal group, to [6,27,29,33,36,37] for valuations taking values in some abelian semigroups, and to [30,31] for semi-continuous valuations.
A differential geometric version of Steiner's formula was found by Weyl [38]. Instead of taking a compact convex body, he considered a compact submanifold M (possibly with boundary) of a Euclidean space and showed that the volume of an r-tube is a polynomial for small enough r. Moreover, the coefficients only depend on the intrinsic geometry of the submanifold, and not on the embedding. We refer to this as Weyl's principle.
For both formulas, the Steiner and the Weyl formula, local versions exist, where one looks only at those points in the r-tube such that the foot point on K or M belongs to a given Borel subset of V . The coefficients Λ k , k = 0, . . . , n are then valuations with values in the space of signed measures on V and are called Lipschitz-Killing curvature measures. For instance, if (M, g) is a compact m-dimensional Riemannian submanifold without boundary, then Λ M m−2 (M, U ) = 1 4π U sc · dvol, where U ⊂ M is a Borel subset and sc is the scalar curvature of (M, g).
The structural similarity of the results by Steiner and Weyl is not a coincidence and can be explained with Alesker's much more recent theory of valuations on manifolds. In this language, the intrinsic volumes are valuations which are defined on arbitrary Riemannian manifolds and which behave naturally with respect to isometric embeddings; and the Lipschitz-Killing curvature measures are curvature measures naturally associated to Riemannian manifolds. Conversely, Fu and Wannerer [24] have recently shown in the spirit of Hadwiger's characterization that the intrinsic volumes/Lipschitz-Killing curvature measures are characterized by the Weyl principle, i.e. any valuation/curvature measure on Riemannian manifolds that satisfies the Weyl principle is a linear combination of intrinsic volumes/Lipschitz-Killing measures.
It is a very natural question to look for analogous results in pseudo-Riemannian geometry. In this case, the tubes are in general not compact, but nevertheless one can try to associate valuations and curvature measures to pseudo-Riemannian manifolds. In the flat case, this was achieved in [7] and [12]. It turns out that the continuity assumption is too restrictive and should be replaced by the notion of generalized valuations or curvature measures. The very rough idea is that a generalized valuation can not be evaluated on every compact differentiable polyhedron, but only on smooth enough sets which are transversal (in some precise sense) to the light cone of the metric. These sets are called LC-transversal. One obtains a sequence µ k , k ≥ 0 of complex-valued generalized valuations called intrinsic volumes, and a sequence Λ k , k ≥ 0 of complex-valued generalized curvature measures called Lipschitz-Killing curvature measures.
The extension to the curved case was carried out in [13]. To every pseudo-Riemannian manifold M p,q we associate a space LK(M ) of intrinsic volumes, which are generalized valuations on M ; and a space LK(M ) of Lipschitz-Killing curvature measures, which are generalized curvature measures on M . They can be evaluated on smooth enough LC-transversal sets. The most important property of these objects is the Weyl principle, which states that for every isometric immersion M N of pseudo-Riemannian manifolds, the restriction of the intrinsic volume µ N k to M equals µ M k and the restriction of the Lipschitz-Killing curvature measure Λ N k to M equals Λ M k . A characterization of the intrinsic volumes as the only generalized valuations on pseudo-Riemannian manifolds satisfying a Weyl principle appeared in [13,Theorem D], based on the results from [12].

1.2.
Results. To complete the analogy with the Euclidean/Riemannian case, we still need a characterization theorem for generalized curvature measures satisfying a Weyl principle. As noted above, each Lipschitz-Killing curvature measure satisfies the Weyl principle, i.e. it associates to each pseudo-Riemannian manifold (M, Q) a generalized curvature measure Λ M k such that whenever M N is an isometric immersion, then Λ N k | M = Λ M k . The same holds true for linear combinations ∞ k=0 a k Λ k + b kΛk . Our first main theorem states that, conversely, every assignment of a generalized curvature measure Λ M to each pseudo-Riemannian manifold M that satisfies Λ N | M = Λ M is of this form. In other words, the space LK(M ) of Lipschitz-Killing curvature measures is characterized by the Weyl principle.
To state the theorem more precisely, we need some terminology. Let ΨMet denote the category of pseudo-Riemannian manifolds with isometric immersions. Let GCrv be the category where the objects are pairs (M, Φ), with M a smooth manifold, and Φ ∈ C −∞ (M, C) (the space of generalized curvature measures). The morphisms e : (M, Φ M ) → (N, Φ N ) are immersions e : M N such that e * Φ N is well-defined, and Φ M = e * Φ N . The category GVal of manifolds with generalized valuations is defined similarly.
A Weyl functor on pseudo-Riemannian manifolds with values in generalized curvature measures is any covariant functor Λ : ΨMet → GCrv intertwining the forgetful functor to the category of smooth manifolds. More generally, we may similarly define Weyl functors between any two categories of manifolds equipped with a geometric structure, when natural restriction operations are available for both structures. Important examples of Weyl functors are the intrinsic volumes of Riemannian manifolds, taking values in smooth valuations, and the Lipschitz-Killing curvature measures [21,24]. For a different example, a family of Weyl functors on contact manifolds with values in generalized valuations was described in [20].
In this language, the intrinsic volumes and Lipschitz-Killing curvature measures on Riemannian manifolds were extended in [13] to Weyl functors µ k : ΨMet → GVal and Λ k : ΨMet → GCrv, respectively. It was moreover shown that the Weyl functors ΨMet → GVal are spanned over R by µ 0 = χ and {µ k , µ k } k≥1 . Our first result is a similar classification for the curvature measures.
Theorem A. Any Weyl functor Λ : ΨMet → GCrv is given by a unique infinite linear combination Λ = ∞ k=0 a k Λ k + b kΛk . Theorem A may be used to prove geometric formulas by the template method, where the templates are special pseudo-Riemannian manifolds. As a first application of this method, we prove the following formula, which is well-known in the Riemannian case [17,Equation 3.34]. .
•) denotes the exterior product of the generalized measures Λ M1 k1 (A 1 , •) and Λ M2 k2 (A 2 , •). By the Weyl principle, we have for every pseudo-Riemannian manifold where Isom(M ) is the isometry group.
A (pseudo-Riemannian) space form is a complete connected pseudo-Riemannian manifold of constant sectional curvature. We refer to Section 2.1 for the classification of space forms. A connected space form is isotropic if the isometry group acts transitively on the level sets of the metric. Examples include all pseudospheres, pseudohyperbolic spaces, and flat pseudo-Euclidean spaces. Our third main theorem states that the displayed inclusions become equalities if M is an isotropic space form. It thus gives a complete description of isometry invariant generalized valuations and curvature measures for these space forms, and can be considered as a Hadwiger-type theorem. The following special case of the theorem completes the classification of isometry invariant generalized valuations from [12] and will be the main ingredient in the proof of Theorem C.
In each case, a basis is given by the real and/or imaginary parts of the Lipschitz-Killing curvature measure Λ k .
1.3. Acknowledgements. Part of this work was carried out during the second named author's stay at CRM -Université de Montréal, which is gratefully acknowledged.

Preliminaries
2.1. Pseudo-Riemannian space forms. We refer to [32,39] for the material in this subsection.
The pseudosphere of signature (p, q) and radius r > 0 is The pseudosphere S n,1 1 ⊂ R n+1,1 is called de Sitter space and denoted by dS n,1 .
iii) The pseudohyperbolic space of signature (p, q) and radius r > 0 is The pseudohyperbolic space H n,1 1 is called the anti-de Sitter space.
The isometry groups of these spaces are given by In each case, the action is transitive and the stabilizer at any point is conjugate to O(p, q).

Definition 2.2.
A complete connected pseudo-Riemannian manifold of constant sectional curvature is called space form. A connected space form whose isometry group acts transitively on the level sets of the metric is called isotropic.
By a theorem of Wolf [39], the stabilizer of a point in the isometry group of an isotropic space form acts on the tangent space by the full orthogonal group.
The next theorem gives a classification of simply connected space forms. They are all isotropic. Theorem 2.3. Let (M, Q) be a pseudo-Riemannian space form of signature (p, q) and curvature K. Then the universal cover of M is isometric to

2.2.
Valuations and curvature measures on manifolds. In this subsection, we recall the definitions of the basic objects of this paper: smooth and generalized valuations and smooth and generalized curvature measures on manifolds. We refer to [13, Section 2] for more details.
Let M be a smooth manifold, assumed oriented for simplicity. By π : P M → M we denote the cosphere bundle. We let P(M ) denote the set of compact differentiable polyhedra. A smooth valuation on a manifold M is a functional µ : Here nc(A) is the normal cycle of A, which is an integral current in P M . The space of smooth valuations is a Fréchet space which is denoted by V ∞ (M ). Examples are the Euler characteristic χ, the volume, and more generally the intrinsic volumes of a Riemannian manifold (M, g).
There is a natural notion of the emphsupport of a valuation, and the space of compactly supported smooth valuations is denoted by V ∞ c (M ) and equipped with a natural LF-topology. A generalized valuation is an element of the dual In this case, (2) still makes sense for particularly nice A. Every A ∈ P(M ) defines a generalized valuation χ A by setting χ A , µ = µ(A), and every smooth valuation can also be considered as a generalized valuation by Alesker-Poincaré duality. The wave front of a generalized valuation describes its singularities, we refer to [4,Section 8] for the definition. Given closed subsets Λ ⊂ P + (T * M ), Γ ⊂ P + (T * P M ), the space of generalized valuations with wave front included in (Λ, Γ) is denoted by V −∞ Λ,Γ (M ). A smooth curvature measure is a functional of the form Here A ∈ P(M ) and U ⊂ M is a Borel subset. The Fréchet space of smooth curvature measures is denoted by C ∞ (M ). Sometimes we also write [φ, ω] for the curvature measure defined by (3). The pairs of forms (φ, ω) such that the valuation µ from (2) is trivial were described in [11] in terms of the contact structure on P M . We need a (simpler) version of this description for curvature measures.
Note that a local contact form α is unique up to multiplication by a non-zero function. In the following, we will do some constructions using α, and will leave it to the reader to check that each construction is independent of the choice of α.
The Lefschetz decomposition of a form ω ∈ Ω k (P M ) is given by where for these notions.
The following conditions are equivalent.
ii) φ = 0 and ω belongs to the ideal generated by α and dα.
Proof. ii) =⇒ i) This follows from the fact that normal cycles are Legendrian.
Taking f be supported in intA, we find that φ = 0. Letting the support of f shrink to a point on the boundary, it follows that ω vanishes on all tangent spaces to nc(A). Since these tangent spaces are dense in the set of all Legendrian planes, it follows that ω vanishes on all Legendrian planes. By [11,Lemma 1.4] be the Lefschetz decompositions. Then ω i ∧ τ 0 ≡ 0 and τ i ∧ ω 0 ≡ 0 for all i > 0. The assumption is thus equivalent to which implies by Poincaré duality that α ∧ ω 0 = 0. Hence ω 0 is a multiple of α. Since each ω i , i > 0 is a multiple of dα, the statement follows.
A generalized curvature measure is given by We write Φ(µ, f ) for this evaluation. The space of generalized curvature measures is denoted by C −∞ (M ). As for generalized valuations, the singularities of a generalized curvature measure can be described by its wave front set [13, Section 2.3]. The set of generalized curvature measures with wave front set contained in (Λ, Γ), where Λ ⊂ P + (T * M ), Γ ⊂ P + (T * P M ) are closed subsets, is denoted by C −∞ Λ,Γ (M ). If A ∈ P(M ) satisfies certain transversality conditions (which are given in terms of wave front sets), then is well-defined.
The following conditions are equivalent.
For the implication i) =⇒ ii), we work locally and with coordinates and assume that M = R n . Convolve with an approximate identity ρ j ∈ C ∞ c (GL(n)). Then ρ j * Φ is the smooth curvature measure represented by the smooth forms (ρ j * φ, ρ j * ω), but obviously it is the trivial curvature measure. By Proposition 2.4, is closed in the weak topology (which follows from the implication ii) ⇐⇒ iii)), it follows that ω belongs to this space.
Sometimes we also need C-valued valuations and curvature measures, which are defined in an analogous way. In all the following, the range which is either R or C is often omitted from notation, and should be determined from context. A k-homogeneous element in one of these spaces can be represented by a pair (0, ω) with ω translation-invariant and of bidegree (k, n − k − 1) if k < n; and by a pair (φ, 0) with φ translation-invariant if k = n.
Proposition 2.6. The (smooth or generalized) curvature measure induced by a translation-invariant (smooth or generalized) form ω of bidegree (k, n − k − 1) with k < n vanishes if and only if ω belongs to the ideal in Ω * (P V ) tr or Ω * −∞ (P V ) tr generated by α and dα.
The proof is similar to the proofs of Propositions 2.4 and 2.5 and we omit the details. In this case, instead of taking the usual Poincaré pairing on manifolds, we have to use the Poincaré pairing on translation-invariant forms as follows. Take the wedge product of two translation-invariant forms of complementary degrees, and push-forward to V . Then we obtain a translation-invariant n-form on V , hence a multiple of the volume form. The corresponding factor is then the pairing of the two forms. This pairing is non-degenerate.
2.4. LC-transversality. Let (M, Q) be a pseudo-Riemannian manifold. We denote by LC * M ⊂ P M = P + (T * M ) the set of null-directions in the cosphere bundle and by LC M ⊂ P + (T M ) the set of null-directions in the sphere bundle. For a submanifold X ⊂ M , we let N * X ⊂ T * M be the conormal bundle, which we consider often as a subset of P + (T * M ). Using the metric Q, we may identify this set with a subset in T M (or P + (T M )), denoted by N Q X.
We will need the following generalization of LC-transversality.

Uniqueness of the Lipschitz-Killing functors
In this section we will prove Proposition 1.1, which is the technical heart of the proof of Theorems A and C. We will need two technical lemmas. The first one is from [12], see also [7,Section 4.4] for some of the notation. To state it, we need some preparation.
Let X be a smooth manifold, and E a smooth vector bundle over X. For any ν ≥ 0 and a locally closed submanifold Y ⊂ X, define the vector bundle F ν Y over Y with fiber be the space of all generalized sections supported on Y with differential order not greater that ν ≥ 0 in directions normal to Y . One then has a natural isomorphism . Now let a Lie group G act on X in such a way that there are finitely many orbits, all of which are locally closed submanifolds. We will assume that E is a G-vector .
For a G-module X and a character χ on G, we write X G,χ = {ω ∈ X : gω = χ(g)ω, g ∈ G} and call its elements (G, χ)-invariant. By tensorizing all representations with the one-dimensional representation C on which G acts by χ(g), a similar upper bound holds for (G, χ) resp. (Stab(y j ), χ)-invariant subspaces. We will use the character det : O(p, q) → R.
Differentiating the functional equation at α = 0, we find Write F (σ) = f (x) in an interval |x| < ǫ 2 ≤ ǫ 1 where σ is invertible. Then f ′ (x) = F ′ (σ)(s(x) + xs ′ (x)), and the equation becomes Proof of Proposition 1.1. A translation-invariant generalized curvature measure of degree n is induced by a translation-invariant generalized n-form on V . Since translations act transitively on V , such forms are actually smooth and hence multiples of the volume form. Hence Curv −∞,O(p,q) n is spanned by the volume. Let us assume in the following that 0 ≤ k ≤ n − 1. Consider first the case min(p, q) = 0, i.e. the case of a Euclidean (or anti-Euclidean) vector space. The action of the Euclidean motion group O(p, q) on the cosphere bundle is transitive, which implies that all invariant generalized forms are smooth. However, from the classification of the invariant smooth forms in [22,23] it follows immediately that Curv , and the latter space is onedimensional by Hadwiger's theorem.
In the remaining case min(p, q) > 0 we proceed as in [12, Section 5.3]. Let P + (V * ) := V * \ {0}/R + and let P V := V × P + (V * ) be the cosphere bundle over V . For ξ ∈ P + (V * ) we denote by ξ Q the Q-orthogonal complement, which is a hyperplane in V . Denote by D k,l the vector bundle over P + (V * ) whose fiber over ξ is given by Then the space of translation-invariant forms is given by The space of horizontal translation-invariant generalized forms (i.e. the quotient of all generalized translation-invariant forms by the vertical translation-invariant generalized forms) is We claim that the space Γ −∞ M 0 (P + (V * ), D k,n−k−1 p ) G,det of (G, det)-invariant and primitive generalized forms supported on the light cone is trivial if (n − k) is odd, and at most one-dimensional if (n − k) is even.
To prove the claim, we fix ξ ∈ M 0 and denote by H ⊂ G the stabilizer of ξ. By Lemma 3.1 and [12, Lemma 5.7], Set Since we have an exact sequence we get an exact sequence Note that (ξ Q /ξ) * ∼ = ξ Q /ξ, since the restriction of Q to ξ Q /ξ is non-degenerate. The action of H on this space is that of O(p − 1, q − 1). Hence By [12, Lemma 2.1] we have As in [12] we define the subspaces and the quotients If A is an H-representation and B a subrepresentation, then dim A H,det ≤ dim B H,det + dim(A/B) H,det . We thus obtain that Hence W u can only contain an (H, det)-invariant if β = 1. Let us show by a direct argument as in [12,Prop. 5.10] that also in the case β = 1, there is no such invariant.
Let ρ ∈ W H,det u . Let Y ⊂ ξ Q be a complement of ξ and H Y ⊂ H := Stab(ξ) be the stabilizer of Y . The decomposition is compatible with the action of H Y . The projection of ρ to the second summand is H Y -invariant. Since there are elements in H Y acting by the identity on Y (and hence on ξ Q /ξ) and by rescaling ξ, the second summand can not contain non-zero (H Y , det)-invariant elements. Hence ρ must belong to the first summand. Since Y was an arbitrary complement of ξ, our invariant must belong to the intersection Fix some Y that gives a minimal-length representation By the minimality of the representation, the η i are linearly independent.
The subgroup of H acting by Id on ξ Q /ξ is transitive on all hyperplanes Y complementing ξ. Acting by such an element g on ρ, we get the equality It follows that y i = y ′ i for all i, hence y i ∈ Y ∧ n−k−1 Y . It is elementary to prove that this intersection is trivial, and hence y i = 0 for all i and therefore ρ = 0.
On the other hand, we have We distinguish two cases. If (n − k) is odd, then β = n − k − 2ν − 3 = −1 for all ν. By (4), the space of (G, det)-invariant generalized primitive forms of degree (k, n − k − 1) supported on the light cone is trivial. Let φ ∈ Ω k,n−k−1 p,−∞ (P V ) tr be (G, det)-invariant. The restriction of φ to each open orbit of P V must be a multiple of the form φ ± k,r constructed in [13,Section 5.1]. Hence there are constants c ± such that φ = c + φ + k,r on S + V and φ = c − φ − k,r on S − V . It follows that the global form φ − c + φ 0 k,r − c − φ 1 k,r is supported on the light cone. Since this form is (G, det)invariant, it vanishes.
Let (n − k) be even. Let φ ∈ Ω k,n−k−1 p,−∞ (P V ) tr be (G, det)-invariant. In the following, we identify V = V * using the quadratic form Q. By the above, we obtain that φ = c + φ + k,r on S + V and φ = c − φ − k,r on S − V for some constants c ± . We will show that c + = c − . Fix a compatible Euclidean structure P on V (see [12,Definition 2.7]). From [13,Equation (60)], we have where ρ k,r is a globally defined smooth form, and σ + , σ − are the positive and negative parts of σ = σ + − σ − = Q P . Write V = R p,q = R 1,1 ⊕ R p−1,q−1 and L := P + (R 1,1 ) ⊂ P + (V ), ξ 0 , ξ ′ 0 ∈ L the degenerate lines. Take g α ∈ SO + (Q) fixing R p−1,q−1 and acting by an α-boost We will use the standard Euclidean structure on R 1,1 and introduce the polar angle θ for which θ(ξ 0 ) = π 4 . Consider the interval L ′ with 0 ≤ θ ≤ π 2 . Then σ = cos 2θ is a coordinate in the interior of L ′ . With respect to this coordinate, we Since φ k,r = σ − n−k 2 ρ k,r is SO(p, q)-invariant, we find that By O(p, q)-invariance it follows that the wave front set of φ is disjoint from N * L. Letting j L : L ֒→ P + (V ) be the inclusion and denoting D = j * L D k,n−1−k , we may therefore define φ L = j * L φ ∈ Γ −∞ (L, D).
Under the action of SO + (1, 1), we can decompose into equivariant summands whereξ ⊂ R 1,1 . The first summand can be further written as a sum of lines on which SO + (1, 1) acts trivially. It follows that D can be SO + (1, 1)-equivariantly decomposed into the sum of line bundles, D = ⊕ N i=1 D j . Let π j : D → D j be the projection. Since ρ k,r is a smooth and non-vanishing form, there exists j 0 such that π j0 (ρ k,r ) is a smooth non-vanishing section of D j0 over a neighborhood of ξ 0 , denotedL 0 . It follows that π j0 (φ L ) = f π j0 (ρ k,r ) for some f ∈ C −∞ (L 0 ). By (5) and the invariance of φ L we obtain that and applying Proposition 3.2 shows that outside of ξ 0 , f must coincide with a multiple of σ − n−k 2 , that is c + = c − . On the other hand, since (n − k) is even, β = −1 if and only if ν = n−k 2 − 1. By (4) it follows that the space of (G, det)-invariant generalized primitive forms of degree (k, n − k − 1) supported on the light cone is at most 1-dimensional.
In both cases we find dim Curv Proof of Theorem A. Consider a Weyl functor Λ : ΨMet → GCrv. Fix p, q > 0. From Proposition 1.1 we obtain that for some constants a k , b k . Since the Λ k ,Λ k are linearly independent and satisfy a Weyl principle, the a k , b k are unique and independent of the choice of R p,q provided that k < p + q.
on each pseudo-Euclidean space R p ′ ,q ′ . By functoriality and the pseudo-Riemannian Nash embedding theorem [18], we then have on each pseudo-Riemann manifold M

4.
A Künneth-type formula for Lipschitz-Killing curvature measures 4.1. Disintegration of curvature measures. We start with a general proposition.
Proposition 4.1. Let X 1 , X 2 be smooth manifolds and X := . Then there exists a unique smooth curvature measureΨ ∈ C ∞ (X 1 ) such that Proof. Uniqueness is clear. To prove existence, we use the notations and maps from [5, Section 4.1]. The relevant diagram is . According to [3,5], the exterior product

The second equation implies that
The signs come from the fact that the degree of the antipodal map is given by (−1) n1 on P X1 × X 2 ; by (−1) n2 on X 1 × P X2 ; and (−1) n1+n2 on P X and onP X . Let Ψ ∈ C ∞ (X 1 × X 2 ) be a smooth curvature measure, given by forms ρ ∈ Ω n1+n2 (X 1 × X 2 ), η ∈ Ω n1+n2−1 (P X ). Then Using (7), we see that the first summand is given by According to (8), the second summand splits as the sum T 1 + T 2 + T 3 where Recall from [5] the set M = M 1 ∪ M 2 ⊂ P X with where Γ runs over all closed subsets in T * P X \ 0 that are disjoint from T * M P X . By [5, Proposition 4.1], a transversal curvature measure can be applied to a pair . Then there exists a unique generalized curvature measureΨ ∈ C −∞ (X 1 ) such that . As in the previous proof, we are going to define the formsφ ∈ Ω n1 −∞ (X),ω ∈ Ω n1−1 −∞ (P X ) by (9) and (10).
We have to check that this is possible. Forφ we note that the wave front set of i 2 * p * 2 (Dω 2 + π * X2 φ 2 ) is contained in T * M2 P X ⊂ T * M P X , hence the wedge product with η is defined, and thenφ is well-defined.
The first term in the definition ofω is well-defined since F is a submersion. The second term is well-defined since i 1 is transversal to Γ.
From the same arguments as in the previous proof, we see that the generalized curvature measureΨ := [φ,ω] satisfies (11).
is continuous.
We omit the proof of the general case of the statement. In this case, a careful analysis of the wave front set of the exterior product ψ 1 ⊠ ψ 2 is needed. By [5,Proposition 4.1] it consists of the union of three sets, and each of these sets can be shown to be disjoint from the wave front set of LC * M1×M2 by arguments which are similar to the given ones.
Proof of Theorem B. By [13, Proposition 7.1], we have is a generalized curvature measure on M 1 , which will be denoted by Λ M1×M2 If i : M 1 →M 1 is an isometric embedding, then i × id : M 1 × M 2 →M 1 × M 2 is also an isometric embedding. By Weyl's principle [13, Theorem D] we have for all M 1 . By induction on k 1 we will show that for each fixed M 2 , are Weyl functors. Suppose that this is true for all k 1 < n.

Generalized valuations and curvature measures on isotropic space forms
The aim of this section is to prove Theorem C. The following special case of valuations on pseudo-Euclidean spaces R p,q was considered in [7] for min(p, q) = 1 and in [12] in general.