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Crofton formulas in pseudo-Riemannian space forms

Crofton formulas on simply-connected Riemannian space forms allow to compute the volumes, or more generally the Lipschitz-Killing curvature integrals of a submanifold with corners, by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

where Gr 1 (R 2 ) is the space of lines in R 2 with a rigid motion invariant measure (which is normalized in a suitable way). A higher dimensional version states that for M ⊂ R n a compact submanifold with boundary, we have the kernel of which coincides with the kernel of the cosine transform [9].
Among the many applications in integral geometry of such formulas, we mention the construction of a basis of the space of unitarily invariant valuations on C n by Alesker [3], the interpretation of Alesker's product of smooth and even valuations in terms of Crofton measures [11], the Holmes-Thompson intrinsic volumes of projective metrics [10]. Applications outside integral geometry include isoperimetric inequalities in Riemannian geometry [20], symplectic geometry [34], systolic geometry [40], algebraic geometry [1] and more. Crofton formulas are employed also outside of pure mathematics, in domains such as microscopy and stereology, see [30].
Crofton formulas do not only exist on flat spaces, but also on manifolds. In this case, we need a family of sufficiently nice subsets, endowed with a measure. Then the Crofton integral is given by the integral of the Euler characteristic of the intersection with respect to the measure. Under certain conditions which are given in [24], it yields a smooth valuation on the manifold in the sense of Alesker [5].
On spheres and hyperbolic spaces, a natural class of subsets are the totally geodesic submanifolds of a fixed dimension, endowed with the invariant measure. On the 2-dimensional unit sphere, we have a formula similar to (1), with affine lines replaced by equators. This formula is the main ingredient in the proof of the Fáry-Milnor theorem that the total curvature of a knot in R 3 is bigger than 4π if the knot is non-trivial.
In higher dimensions, the formula becomes slightly more complicated. On the n-dimensional unit sphere we have Here Geod n−k (S n ) denotes the totally geodesic submanifolds of dimension (n − k), µ j (M ) is the j-th intrinsic volume of M (which can be defined as the restriction of the j-th intrinsic volume on R n+1 under the isometric embedding S n ֒→ R n+1 ), and ω n denotes the volume of the n-dimensional unit ball. A similar formula holds on hyperbolic space. See [25,26] for more on the integral geometry of real space forms.
Moving on to Lorentzian signature, few results are available. The main challenge to overcome is the non-compactness of the isotropy group, which in general renders the Crofton integral divergent. Some special Crofton-type formulas in Lorentzian spaces of constant curvature, applicable under certain rather restrictive geometric conditions, appeared in [17,32,37,44].
1.2. Results. We are going to prove Crofton formulas on flat spaces, spheres and hyperbolic spaces of arbitrary signatures. Let us recall the definition of these manifolds, referring to [35,43] for more information. i) The pseudo-Euclidean space of signature (p, q) is R p,q = R p+q with the quadratic form Q = ii) The pseudosphere of signature (p, q) and radius r > 0 is S p,q r = {v ∈ R p+1,q : Q(v) = r 2 }, equipped with the induced pseudo-Riemannian metric. Its sectional curvature equals σ = 1 r 2 . The pseudosphere S n,1 1 ⊂ R n+1,1 is called de Sitter space.
Its sectional curvature equals σ = − 1 r 2 . The pseudohyperbolic space H n,1 1 is called the anti-de Sitter space.
We will colloquially call these spaces generalized pseudospheres. The isometry groups of generalized pseudospheres are given by Isom(R p,q ) ∼ = O(p, q) = O(p, q) ⋉ R p,q , Isom(S p,q r ) ∼ = O(p + 1, q), Isom(H p,q r ) ∼ = O(p, q + 1). In each case the action is transitive, and the isotropy group is conjugate to O(p, q). These spaces are isotropic in the sense that the isotropy group acts transitively on the level sets of the metric in the tangent bundle. Up to taking connected components and universal coverings, any space form is a generalized pseudosphere (cf. [35,Chapter 8,Corollary 24]).
On a generalized pseudosphere M , we will formulate Crofton formulas using the space Geod n−k (M ) of totally geodesic subspaces. However, there is no isometryinvariant Radon measure on this space. Therefore we will use an isometry-invariant generalized measure (also called distribution). This causes some technical problems, as the function that we want to integrate is not smooth. Nevertheless, in many cases the integral can still be evaluated. The result is not a valuation anymore, but a generalized valuation in the sense of Alesker [6]. The Crofton map is then a map Cr : M −∞ (Geod n−k (M )) → V −∞ (M ).
In the Riemannian case, any isometry invariant valuation admits an invariant Crofton measure. The corresponding statement in other signatures is also true, but much harder to prove. The second named author proved in [23] the statement first for certain signatures by an explicit, but difficult, computation and then used the behavior of Crofton formulas under restrictions and projections to handle the general case.
Furthermore, with the exception of Riemannian and Lorentzian signatures, the space of isometry-invariant generalized measures is of greater dimension than the space of isometry-invariant valuations. Thus we are forced to choose a distribution, and must take care to avoid the kernel of the Crofton map.
Using results by Muro [33] on analytic families of homogeneous generalized functions on the space of symmetric matrices, one can construct such an invariant generalized measure on Geod n−k (M ). We construct a particular generalized measure m k with the distinguishing property that it behaves well under restrictions of Crofton measures (see subsection 3.3), and is independent (in a precise sense) of signature and dimension.
There is some freedom in the normalization of a Crofton measure. We choose the normalization in such a way that the first coefficient in the Crofton formulas will always be 1.
Our Crofton formula will evaluate where σ = 0 is the curvature of M and m k := πω k−1 √ σ −1 k m k . The flat case σ = 0 appears through a careful limiting procedure.
The right hand side of the Crofton formula will be expressed in terms of the recently introduced intrinsic volumes on pseudo-Riemannian manifolds [14], which are complex-valued generalized valuations on M . They satisfy a Hadwiger-type classification [15], which allows us to use the template method to compute the coefficients in this formula. However, the resulting computations lead to distributional integrals on the space of symmetric matrices, that can be evaluated directly essentially only for the Lorentzian signature. To conclude the general case, we use techniques of meromorphic continuation and distributional boundary values of analytic functions.
Due to the functorial properties of the constructed Crofton distribution mirroring those of the intrinsic volumes, namely their adherence to the Weyl principle, the resulting Crofton formulas are signature independent. Remarkably, they are also holomorphic; i.e. they involve the intrinsic volumes only and not their complex conjugates.
Main theorem. Let M be a generalized pseudosphere of sectional curvature σ. Then The Crofton formulas should be understood formally, namely as the correspondence between distributions on the Grassmannian and the intrinsic volumes through an abstractly defined Crofton map. However they can also be interpreted as explicit Crofton-type formulas applicable to sufficiently nice subsets of the generalized pseudospheres.
By a strictly convex subset of a non-flat generalized pseudosphere M m we mean its intersection with a strictly convex cone in R m+1 , with M ⊂ R m+1 embedded as in Definition 1.1. For the Riemannian round sphere and hyperbolic space, this coincides with the standard definition of strict convexity.
Note that the spherical Crofton formula (2) is a special case of our theorem. We also note that we will prove a slightly more general statement in Corollary 5.14 using the notion of LC-regularity from [14].
1.3. Plan of the paper. After covering the preliminaries, we turn in section 3 to study general Crofton formulas with a distributional Crofton measure, utilizing the Alesker-Radon transform on valuations. In particular, we study under which conditions such formulas can be applied directly to a given subset.
In Section 4 we consider LC-regular domains and hypersurfaces of space forms, and deduce that they would be in good position for the evaluation of intrinsic volumes through Crofton integrals, once the corresponding distributions are constructed. The latter construction is carried out in Section 5. Moreover, these distributions are embedded in a meromorphic family of measures on a complex domain as a distributional boundary value, and some delicate -though central to our analysis -convergence questions are investigated and settled. Finally in Section 6, the Hadwiger-type description of intrinsic volumes combined with the template method are applied to yield the explicit Crofton formulas in all cases. For an oriented k-dimensional submanifold X ⊂ M , we let X be the k-current which is integration over X.
By P M := P + (T * M ) we denote the cosphere bundle of M , which consists of all pairs (p, if there is some λ > 0 with ξ = λξ ′ . When no confusion can arise, we use the same notation for subsets of P M and their lifts to T * M . The natural involution on P M is the fiberwise antipodal map s(p, The wave front set of a generalized form ω ∈ Ω −∞ (M ) is a closed subset of P M , denoted by WF(ω), and we refer to [29] or [22] for details.
For a generalized pseudo-sphere M , we denote by Geod k (M ) the space of totally geodesic k-dimensional submanifolds of M .

Smooth valuations.
Let M be a smooth manifold of dimension m, which we assume oriented for simplicity.
Let P(M ) be the set of compact differentiable polyhedra on M . To A ∈ P(M ) we associate two subsets of P M . The conormal cycle, denoted nc(A), is the union of all conormal cones to A. It is an oriented closed Lipschitz submanifold of dimension (m − 1), and naturally stratified by locally closed smooth submanifolds corresponding to the strata of A.
The conormal bundle, denoted N * A, is the union of the conormal bundles to all smooth strata of A. It holds that nc(A) ⊂ N * A. By definition, two stratified spaces intersect transversally if all pairs of smooth strata are transversal.
A smooth valuation is a functional µ : P(M ) → R of the form We will write µ = [[φ, ω]] in this case. The space of smooth valuations is denoted by V ∞ (M ). It admits a natural filtration . It is compatible with the Alesker product of valuations.

Generalized valuations. The space of generalized valuations is
There is a natural filtration . In particular, we may consider a generalized measure as a generalized valuation.
A compact differentiable polyhedron A defines a generalized valuation χ A by We refer to ζ and τ as the defining currents. For instance, the defining currents of χ A , A ∈ P(M ) are ζ = ½ A , τ = nc(A) . The wave front set of ψ is defined as the pair Λ ⊂ P M , Γ ⊂ P PM of the wave front sets of ζ and τ . The space of all generalized valuations with wave front sets contained in Λ, Γ is denoted by V −∞ Λ,Γ (M ). Consider ψ ∈ V −∞ (M ). We say that A ∈ P(M ) is WF-transversal to ψ, denoted A ⋔ ψ, if the conditions of [8,Theorem 8.3] hold for These conditions imply that Alesker's product of smooth valuations can be extended to a jointly sequentially continuous product Let us write a sufficient set of conditions in a particular case. Then A ⋔ ψ.
2.4. Intrinsic volumes on pseudo-Riemannian manifolds. In [14] we constructed a sequence of complex-valued generalized valuations µ M 0 , . . . , µ M m naturally associated to a pseudo-Riemannian manifold M of dimension m. They are invariant under isometries and called the intrinsic volumes of M . The intrinsic volume µ 0 equals the Euler characteristic, while µ m is the volume measure of M , multiplied by i q where q is the negative index of the signature. For other values of k, µ k is typically neither real nor purely imaginary.
The wave front set of µ k is contained in (∅, N * (LC * M )), where LC * M ⊂ P M is the dual light cone of the metric, i.e. the set of all pairs (p, [ξ]) ∈ P M such that g| p (ξ, ξ) = 0. Here we use the metric to identify T M and T * M .
Then the conditions of Proposition 2.1 are equivalent to the LC-transversality of A. In particular, the intrinsic volume µ k may be evaluated at LC-transversal A.
Proof. The first two conditions are empty. The third condition is satisfied for arbitrary A, since the tangent space to π −1 A contains all vertical directions, while the tangent space of LC * M contains all horizontal directions. The fourth condition is precisely LC-transversality.
We will also need the notion of LC-regularity, which was introduced in [14]. Definition 2.3. Let X be a smooth manifold equipped with a smooth field g of quadratic forms over T X. We say that (X, g) is LC-regular if 0 is a regular value of g ∈ C ∞ (T X \ 0).
It was shown in [14,Proposition 4.9] that the extrinsic notion of LC-transversality and the intrinsic notion of LC-regularity coincide: a submanifold of a pseudo-Riemannian manifold, equipped with the field of quadratic forms induced from the metric, is LC-regular if and only if it is LC-transversal.
The most important property of the intrinsic volumes is that they satisfy a Weyl principle: for any isometric immersion M M of pseudo-Riemannian manifolds we have µ M k | M = µ M k , in particular the restriction on the left hand side is well-defined. Conversely, we have shown in [15] that any family of valuations associated to pseudo-Riemannian manifolds that satisfies the Weyl principle must be a linear combination of intrinsic volumes.

Distributional Crofton formulas
Let M m be a manifold. A Crofton formula for a smooth valuation φ ∈ V ∞ (M ) has the form φ(A) = S χ(X(s)∩A)dµ(s), where S is a smooth manifold parametrizing a smooth family of submanifolds of M , and µ a smooth measure on S. Similarly, a distributional Crofton formula has φ ∈ V −∞ (M ), and µ is a distribution.
In this section we study some general properties of such formulas, when M ⊂ V \ {0} is a submanifold without boundary in a d-dimensional linear space V , and S = Gr d−k (V ), k < d, X(s) = s ∩ M, s ∈ S.
We utilize the Radon transform on valuations, introduced in [7]. Loosely speaking, the Crofton map is but the Radon transform of a measure with respect to the Euler characteristic. However, there are technical difficulties in applying this formalism directly to distributions, and a large part of this section is concerned with resolving those difficulties. The main results to this end are Propositions 3.7 and 3.12. In the last part, we describe the Crofton wave front of sufficiently nice sets in Proposition 3.17, which controls the applicability of an explicit Crofton integral to the given set.
3.1. The general setting. For a submanifold with corners Then Z X is a manifold with corners, more precisely it is the total space of the fiber bundle over X with fiber for the natural projections. Denote by W X ⊂ Gr d−k (V ) the set of subspaces intersecting X transversally in V .
We will need a simple fact from linear algebra, which we state in a rather general form that will be useful for us in several places.
Denote by i : L 0 ֒→ E 0 the inclusion, and π : V /L 0 → V /E 0 the projection.  L 0 In the other direction, write π W : V → V /W for the natural projection. it follows by the above that the set of velocity vectors L ′ (0) for all curves . This is also the dimension of the affine space of all A such that the diagram commutes, which finishes the proof of the first part. The second part follows from the first one by taking orthogonal complements.
We need the following technical statement appearing in [7, Proposition 5.1.3]. Proof. Let (p t , ξ t ) be a smooth path in T * M \ 0. We will lift it to a smooth path Hence Fix a Euclidean structure on V , inducing Euclidean structures on the spaces Hom(E t , V /E t ). Let us choose some E t such that p t ∈ E t , and T pt M ∩E t ⊂ Ker(ξ t ), which evidently can be done. Consider the linear subspace and recall the natural isomorphism q t : We now may define η t ∈ W * t by η t , B = − ξ t , q t (B(p t )) for each B ∈ W t , as T pt M ∩ E t ⊂ ker ξ t . Extend η t by zero to W ⊥ t . It follows that (ξ t , η t ) ∈ N * pt,Et Z M , completing the proof.
It follows by [7,Corollary 4.1.7] that the Radon transforms with respect to the Euler characteristic, , are well-defined and continuous.
) be the defining current of R M φ (on the base manifold). Equivalently, using [6, Proposition 7.3.6] we have Remark 3.5. It is false in general that φ is a smooth function when φ is a smooth valuation, see Remark 3.10.

Distributional Crofton measures.
To allow distributional Crofton measures, it seems essential to require that all intersections E ∩ M are transversal, for E ∈ Gr d−k (V ). This is easily seen to be equivalent, for any k > 0, to having Rx ⊕ T x M = V for all x ∈ M . In particular dim V = dim M + 1 = m + 1. We deduce that M is a hypersurface that is locally diffeomorphic to an open subset of P + (V ) through the radial projection. In other words, M is locally a strictly star-shaped hypersurface around the origin.

Proof.
i) Let us first show E → ψ(E ∩ M ) is smooth. By choosing an open cover of M by star-shaped charts, and using the partition of unity property of smooth valuations [6], we may assume M projects diffeomorphically to an open subset of P + (V ), which we henceforth identify with M . By Boman's theorem [18], it suffices to prove that ψ(E t ∩M ) is a smooth function of t ∈ (−ǫ, ǫ) for all smooth curves E • : (−ǫ, ǫ) → Gr m+1−k (V ). It suffices in fact to show smoothness in some open interval around 0 for any such given curve.
Let us lift E t to a smooth curve g t ∈ GL(V ) with g 0 = Id and Claim. τ −1 M E ⋔ π * M ψ. To see this, write Z = Z M and identify W := Z × M P M with its image in P Z under dπ * M . Explicitly, W | (x,E) = P + (Ker(d (x,E) π M ) ⊥ ), so W is the union of the conormal bundles to all fibers of π M . It follows from [7, Proposition 3.3.3] that WF(π * M ψ) ⊂ (∅, N * W ). By Proposition 2.1, it suffices to check that two intersections in P Z are transversal: π −1 (τ −1 M E) ⋔ W and N * (τ −1 M E) ⋔ W . Denote z = (x, E), let (z, ζ) be an intersection point. The first intersection is easy to analyze: T z,ζ π −1 (τ −1 M E) contains all vertical directions of P Z , while T z,ζ W contains all horizontal directions.
To analyze the second intersection, we lift all manifolds from P Z to T * Z, and retain all notation for the corresponding objects. As in the previous case, the image of T z,ζ W under the natural projection π : . It follows by linearity that any smooth valuation of the form ψ = A χ A dν(A), where A is a family of submanifolds A as above and ν a smooth measure, satisfies (3). This family Cr E of valuations spans a dense subset in V ∞ (M ). Indeed, we may approximate χ A in V −∞ (M ) by a sequence in Cr E for any A transversal to E ∩ M . Were Cr E not dense, by Alesker-Poincaré duality one could find a non-zero smooth valuation φ annihilating Cr E , and thus also vanishing on all submanifolds with boundary that are transversal to E ∩ M . By the genericity of transversality and continuity, φ would vanish on all submanifolds with boundary. But this is impossible by [12]. It follows that equality in (3) holds for all ψ.
Proof. The right hand side is well-defined for a generalized measure µ, since the To verify this new definition extends the smooth one, we ought to check that which is the content of Corollary 3.8. Continuity is equally evident.
Remark 3.10. It is tempting to define Cr(µ) as a Radon transform: Cr(µ) = R T M µ, as defined in [7]. Unfortunately the conditions of [7, Corollary 4.1.7], which guarantee that the transform is well-defined on generalized valuations, do not hold for general k, as can be seen by a simple dimension count.

Functorial properties of Crofton measures.
The following is a partial summary of the results of [23, Appendix B] (adapted from the affine to the linear Grassmannian), whereto we refer the reader for further details.
Let j : U r ֒→ V d be an inclusion of a linear subspace. There is then a well-defined operation of restriction ) denote the set of generalized measures (distributions) µ whose wave front sets lie in Γ, equipped with the Hörmander topology.
The map j * extends as a sequentially continuous map which is the pushforward under the (almost everywhere defined) map Π W : E → π(E). It extends to distributions whose wave front sets are disjoint from the conormal cycle of the collection of subspaces intersecting Ker π non-generically.
The following proposition captures the intuitively obvious fact that the pullback of distributions/valuations under embeddings commutes with the Crofton map. We prove a weak version which suffices for our purposes.
Recall that M is a locally star-shaped hypersurface around the origin. and therefore Cr M (µ i )| Z = Cr Z (j * µ i ). The restriction of valuations to a submanifold is continuous in the Hörmander topology on the space of valuations with wave front set contained in WF(Cr M (µ)), see [7,Claim 3.5.4]. Thus the left hand side weakly converges to Cr M (µ)| Z . The right hand side weakly converges to Cr Z (µ).

3.4.
Applying generalized Crofton formulas to subsets. Let M m ⊂ V = R m+1 be a strictly star-shaped hypersurface around the origin. Given A ∈ P(M ) and a Crofton distribution µ ∈ M −∞ (Gr m+1−k (V )), we would like to evaluate Cr(µ) on A using an explicit Crofton integral, whenever A ⋔ Cr(µ).
The following proposition provides some a-priori regularity for χ A .
Proof. Let us first check that . Fix a Euclidean structure on V and identify M with the unit sphere. By [16,Lemma A.2], for a fixed . Let dg, dE be the Haar measures on SO(V ) and Gr m+1−k (V ) respectively, and p : It is evidently finite and locally constant on W A It remains to check that χ A = χ(A ∩ •). Take an approximate identity ρ j ∈ M ∞ (SO(V )), which for convenience we assume invariant under inversion. Consider the convolution φ j := χ A * ρ j ∈ V −∞ (M ). As SO(V ) is transitive on M and P M , it follows that the defining currents of φ j are smooth, and therefore valuation. Let us show thatφ j = φ j . Take ψ ∈ V ∞ c (M ) and compute: by [16], while Equality now follows by Alesker-Poincaré duality. We thus have the following equalities of functions on Gr m+1−k (V ): Fix µ ∈ M ∞ (Gr m+1−k (V )). By Proposition 3.7 and GL(V )-equivariance, and so χ A = χ(A ∩ •).
As χ A is real-valued, Cr WF k (A) must be symmetric under the fiberwise antipodal map.
Determining Cr WF k (A) precisely appears to be difficult in general. Let us focus on a subset A ∈ P(M ) which is either a compact domain with smooth boundary, or a compact hypersurface without boundary.
For the following, we write H = H(A) for ∂A if A is of full dimension, and for A when it is a hypersurface. Write E := E ∩ M , and note that E intersects H transversally in V if and only if E intersects H transversally in M . Denote If (x, E) ∈ B H , we say that x ∈ H is a tangent point for E. Observe also that , etc. to specify the dimension.
Note that if E / ∈ B H then it is automatically regular. For a subset A ⊂ V we denote by P(A) its image in the projective space P(V ). The regularity of the tangent is equivalent to the non-vanishing of the Gauss curvature of the corresponding section, as follows.
In particular, all tangents to A are regular if and only if P(H) ⊂ P(V ) is a strictly convex hypersurface.
Proof. Let us sketch the argument, see [39, Lemma 1(ii)] for details. Clearly dτ H is injective on the subspace of directions where p moves transversally to E. Namely, That injectivity is retained as the remaining directions are added, corresponds to the non-degeneracy of the Gauss map of the section H ∩ F .
We now describe the Crofton wave front near regular tangents. For an immersed manifold i : X Y and y ∈ i(X), we denote That Cr WF k E0 (A) is contained in the sum of the conormal spaces of the embedded parts of B H follows from the fact that χ A is locally constant on the complement of B H . However, to show that it is actually contained in the union of those conormal spaces, in the following proof we will need a more precise description of χ A .
Proof. In the following, by a ball (centered at a point) we mean a compact contractible neighborhood (of the point) with smooth boundary. Since B H is a submanifold of Z H , by assumption B H ⊂ Gr m+1−k (V ) is an immersed submanifold in a neighborhood around E 0 , which is a hypersurface by (5).
The preimageτ −1 H (E 0 ) must be finite, or else we could find a sequence of distinct points (q j , E 0 ) ∈ B H , which then has a limit point (q 0 , E 0 ), andτ H would fail to be injective in a neighborhood of (q 0 , E 0 ), contradicting the assumed immersivity We can now find a ball W ⊂ Gr m+1−k (V ) centered at E 0 , such that B H ∩W is the finite union of embedded hypersurfaces F j , each diffeomorphic to a Euclidean ball, with E 0 ∈ F j and ∂F j ⊂ ∂W for all j. Note that we have no control on how these hypersurfaces intersect each other. Denote by C ± j the connected components of W \ F j . The indices are matched by requiring that a neighborhood of (q j , is a locally closed submanifold in H. We assume the K j small enough so that they are pairwise disjoint, and in particular ∂K j ∩ Z 0 = ∅. We may moreover assume that H ∩ K j is diffeomorphic to a Euclidean ball. By the transversality theorem, we may perturb K j if necessary to Denote by 1 2 K j a smaller ball centered at q j . Taking W sufficiently small, we may assume that This follows by the stability of transversal intersections, because E is a smooth perturbation of E 0 , which intersects H transversally in an open neighborhood of and H , and so is constant on any connected component of a non-empty set C ǫ . Let us show there are integers e j = e j (E 0 ) such that for any ǫ, ǫ ′ ∈ {±} N and any Let us distinguish the two cases under consideration.
The last summand is constant on W by properties (6) and (8). Consequently, The function χ(• ∩ A ∩ K j ) is locally constant on W \ F j , and it remains to define The case of full-dimensional A is only slightly more involved. If (m − k) is odd, we have χ A = 1 2 χ ∂A , reducing to the previous case. Thus assume (m − k) is even.
Write as before, whenever E ∈ W \ B H , Note that for E ∈ W \ B H , all intersections are manifolds with corners. We have S j is a transversal intersection in M for all E ∈ W and hence a smooth hypersurface, while S is given by a transversal intersection in M and hence smooth for In conjunction with the previous equality, we get T x ∂A ⊂ T x ∂K j , which is false.
Let X, Y ⊂ P be smooth domains in a manifold P , and assume ∂X ⋔ ∂Y and Thus we may define e j as in the previous case by eq. (10).
It follows from eq. (9) that for E ∈ W , χ A (E) is a linear combination of the indicator functions of the connected components of the complements of the hypersurfaces F j in W . Therefore WF E0 ( χ A ) ⊂ j N * F j , concluding the proof.

The Crofton wave front of LC-regular hypersurfaces
Let (W, Q) be a vector space equipped with a quadratic form. We denote by Λ ν k (W ) ⊂ Gr k (W ) the collection of subspaces E ⊂ W where Q| E has nullity ν. We will need to describe those sets in several cases.
ii) If Q has nullity 1 and is a manifold near E whose dimension is given by Eq. (11).
ii) Write L 0 := Ker(Q). Consider W := V ⊕ R, and extend Q as a nondegenerate quadratic form Q on W . Let us verify that the submanifolds Λ ν k (W ) and Gr k (V ) intersect transversally in Gr k (W ) at E. (12). This proves the claim.
The formula for the dimension then follows from the previous case.
Proof. Using a local trivialization, this reduces to Proposition 4.1.
Then C is a cone over a closed manifold, and has dimension at most , whence the statement is trivial. We henceforth assume that is not the case, that is Q has both positive and negative directions.
We claim that the restrictioñ If I(E) contains a ν-dimensional subspace U that is Q-orthogonal to I(E), then U is also Q-orthogonal to E as Q(w E , I(E)) = 0. Hence U ⊕ Span(w E ) ⊂ Ker(Q| E ), and consequently the nullity of E is at least (ν + 1), a contradiction.
We will describe the fiberĨ inherits a non-degenerate quadratic form, also denoted Q. Let π K : F Q ∩ W 0 ։ V F be the projection, and observe that Q(π K x) = Q(x), so that the projection to the second summand, and observe that L 0 ⊕ V F is naturally equipped with a quadratic form Q with nullity 1 and kernel L 0 .
The manifold Λ 1 Define a smooth map Proof. For the first statement, we consider two cases. If N = L 0 then L 0 ⊕ F ∈ Λ ν k (W ), and I(L 0 ⊕ F ) = F . If N = L 0 , N / ∈ P(V F ) and E = N + F , then one easily verifies that Q| N = 0 implies Q| N = 0 and consequently Ker(Q| E ) = Ker(Q| F )⊕ N , so that again E ∈ Λ ν k (W ), and clearly I(E) = F . For injectivity, first note that Φ F (N ) = L 0 ⊕ F ⇐⇒ N = L 0 . All other points N ∈ U 0 (F ) lie inside a unique projective line P(L 0 ⊕ L) with L ∈ Λ 1 1 (V F ), and Since π 0 (E) = π 0 ( N ) + F , it follows that L E := π K π 0 ( N ) ∈ Λ 1 1 (V F ) is uniquely defined by E, and it holds that N ∈ P(L 0 ⊕ L E ). It remains to observe that Φ F is obviously injective when restricted to P(L 0 ⊕ L E ).
We verify that Φ F is onto in the non-trivial case it follows that w 1 ∈ π 0 (E) \ F and so we can write π 0 (E) = F +L,L = Span(w 1 ) ∈ Λ 1 1 (F Q ∩ W 0 ). Put L = π KL ∈ Λ 1 1 (V F ). Now take N ∈ P(L 0 ⊕ L) to be the projection of the line Span(w E ) ∈ P(L 0 ⊕L ⊕ F K ). Note also that P(L 0 ⊕ L) ⊂ Λ 1 1 (L 0 ⊕ V F ). Clearly Φ F (N ) = E, concluding the proof of the claim.
Denote by Ψ F :Ĩ −1 (F ) → U 0 (F ) the inverse map of Φ F . We may fix an auxiliary Euclidean structure on W , and choose w E ∈ Ker(Q| E ) as the unit vector forming the least angle with L. Now the map E → Ker(Q| E ), Λ ν k (W ) → Gr ν (W ) is smooth, in the sense that it restricts to a smooth map on every smooth curve in Λ ν k (W ) ⊂ Gr ν (W ). This is because Ker(Q| E ) is the eigenspace of 0, which has fixed dimension along the smooth curve. It follows that Define the cone C 0 ⊂ T L0 P(L 0 ⊕ V F ) of tangent vectors to all curves L t through L 0 belonging to Λ 1 1 (L 0 ⊕ V F ), as well as the cone C F ⊂ T L0⊕F Gr k (W ) that consists of all tangent vectors to smooth curves through L 0 ⊕ F inside Λ ν k (W ). We conclude that the differential D L0 Φ F : Considering all subspaces F ∈ Λ ν−1 k−1 (W 0 ) simultaneously, we have the fibratioñ The image of the section F → L 0 ⊕ F coincides with Λ ν k (W, L 0 ). Therefore, the cone C ⊂ T E0 Gr k (W ) has a linear factor that can be identified with The dimension of C 0 can be readily computed. It can be identified with the abstract cone with base Λ 1 1,+ (V F ), the manifold of oriented null lines in V F . Since the form Q on V F is non-degenerate and indefinite, we have dim Λ 1 1, We have dim Λ ν−1 k−1 (W 0 ) = (k − 1)(d − k) − ν 2 by Proposition 4.1, and so We now turn to LC-regular submanifolds. First, we will need a simple fact on LC-regular metrics. For the second statement, choose coordinates x 1 , . . . , x m on M near p, and take v j = ∂ ∂xj . We may assume that Ker(g p ) = Ker A(p) = Span(v m ), and by assumption A(p) has non degenerate principal (m − 1)-minor. By LC-regularity, we can choose a curve p(t) ∈ M with p(0) = p and a smooth vector field v(t) along it with As the degenerate subset of g near p is {x : det A(x) = 0}, the last assertion follows.
The following is the main result of the section. We use the notation and terminology of Sections 3.1 and 3.4.
is a submersion. The relevant maps are given by the following diagram.
Before proceeding with the more complicated general case, we consider the case Let v 0 ∈ E ∩ T p H be in the kernel of g| Tp E . For any smooth curve v t ∈ T H through v 0 , we may find a smooth curve dt t=0 E t ∈ Hom(E, V /E) satisfies Q(Av 0 , v 0 ) = 0, and by (12) we have A ∈ T E Λ 1 n+1−k (V ). It follows that d dt t=0 g(v t ) = 2Q(v 0 , Av 0 ) = 0, contradicting the LC-regularity of H.
Let us now consider the general case. Fix an auxiliary Riemannian metric h on M , and let ρ H be the least distance projection to H, defined and smooth in a neighborhood of p.
By our assumptionτ H is an immersion at (p, E). Using Proposition 4.1 we see that Claim 1. Let X, Y be manifolds, let F : X → Y be a submersion at p ∈ Z. Let Z ⊂ X and W ⊂ Y be embedded submanifolds, and p ∈ Z. Denote f :

By Corollary 4.2 we see that
We postpone the proof. Combined with eqs. (13) and (14) we get a contradiction.
Case 2: T p H has nullity one, and Ker(g p ) ⊂ E. Let S ⊂ H be the degenerate subset of g. It follows from Lemma 4.4 that S is a smooth hypersurface.
It is a fiber bundle over S with fiber Λ ν n−k (R pM −1,qM −1,1 ). Claim 3. For any (w, ξ) ∈ d p,Eτ . Again we postpone the proof of the claim. The set of all vectors (q ′ (0), F ′ (0)) as in the claim defines, by Lemma 4.3, a cone in T p,E B H of dimension It follows by the claim that we can find a curve (p(t), E(t)) ∈ B H through (p, E) with E ′ (0) ∈ T E Λ ν n−k+1 (V ) and p ′ (0) / ∈ T p S.
Let v n−1 ∈ T p H span Ker(g p ), and recall that v n−1 ∈ T p E, in particular v n−1 ∈ Ker(Q| E ). Choosing any smooth vector field v n−1 (t) ∈ T p(t) E(t), we find which means that d p (det g)(p ′ (0)) = 0 by Lemma 4.4. Since Ker d p (det g) = T p S and p ′ (0) / ∈ T p S, we get a contradiction. This completes the proof of the proposition, modulo the two claims we now proceed to prove.
Proof of Claim 2. Consider a curve (p(t), E(t)) ∈ Λ ν n−k+1 with (p ′ (0), E ′ (0)) ∈ T p,E B H . We ought to find a curve (q(t), Set q(t) = ρ H (p(t)), evidently q ′ (0) = p ′ (0). Fix a subspace W 0 ⊂ T p H which is non-degenerate and contains T p E. If T p H is non-degenerate, we can just take W 0 = T p H. Otherwise, dim ker g p = 1 and we may take any hyperplane W 0 ⊂ T p H which contains T p E and satisfies W 0 ∩ ker g p = {0}. Now fix any linear map A : W 0 → V /W 0 which makes the following diagram commutative.
Proof of Claim 3. Consider a curve (p(t), E(t)) ∈ Λ ν n−k+1 (V ) through (p, E), with p ′ (0) = w ∈ T p S and (p ′ (0), E ′ (0)) = (w, ξ) ∈ T p,E B H . Let ρ S : M → S be the least distance projection with respect to h, well-defined and smooth in some neighborhood of p. Set q(t) = ρ S (p(t)), clearly q ′ (0) = p ′ (0). Denote L 0 = Ker(g p ) ⊂ T p H ∩ E, and extend to a smooth path of lines L t ⊂ E(t) ∩ E(t) Q ∈ Λ ν ν (T p(t) M ). Consider the manifold of pairs Clearly Z is a homogeneous space for O(V, Q), with the equivariant projection π Z : O(V, Q) → Z normalized by π Z (Id) = (p, L 0 ). We can fix a smooth section Then R t p(t) = q(t), and d dt t=0 R t = 0, provided that d dt t=0 L t = d dt t=0 Ker(g q(t) ). Let us verify that L t can be chosen in this fashion. In the following, we fix some Riemannian metric on various manifolds, and write |x − y| X for the corresponding distance between x, y ∈ X. We will also write, for two subspaces E, F ⊂ V , ∡(E, F ) for the angle between them with respect to some Euclidean metric. This should not create ambiguity, as we will be concerned only with rough small scale asymptotics.

t). It follows that ∡(E(t), H(t)) = O(t 2 ), and by Lemma 3.1 we have the commutative diagram
Taking the dual diagram and identifying V = V * using Q, we get where Ker g q(t) .
By Lemma 3.1, we can find L t ⊂ E(t) Q with d dt t=0 L t = f 0 . Note that L t is not in general a null line of Q. We now proceed to modify the definition of L t to force it to be a null line.
Observe that if q ∈ S, E ∈ Gr n+1−k (V ) and Ker(g q ) ⊂ E, then E Q ⊂ T q H ⊕ Rq. We have It follows that ∡(E(t) Q , H(t)) = O(t 2 ), and so we may apply Lemma 3.1 to get the commutative square , and by duality also Combining diagrams (15) and (16) then yields the commutative diagram By Lemma 3.1, we may redefine L t such that d dt t=0 L t = f 0 and L t ⊂ K(t) = E(t) ∩ E(t) Q . In particular, L t is a null line of Q.
Setting F (t) := R t E(t) we have q(t) ∈ F (t), T q(t) F (t) ⊂ T q(t) H since E(t) ⊂ L Q t , and F ′ (0) = E ′ (0) since d dt t=0 R t = 0. This proves the claim. Remark 4.6. It is easy to see that the conclusion of the proposition with k = n − 1 is equivalent to the LC-regularity of H.

Construction of an invariant measure on the Grassmannian
For X ∈ Sym r (R) and λ ∈ C we set, as in [33], It is well-known, essentially due to Cayley and Gärding [27], that | det X| λ q extends as a meromorphic in λ family of generalized functions, which are in fact tempered distributions.
We will use the set and write √ z for the unique branch of the square root function on U C such that √ z > 0 for z > 1 2 . 5.1. A holomorphic family of Crofton measures. For the following, let Sym + r (R) ⊂ Sym r (R) be the cone of positive-definite matrices, and h r = Sym r (R)⊕i Sym + r (R) ⊂ Sym r (C) the Siegel upper half space. The following is well-known.
Lemma 5.1. For Z ∈ h r , det Z = 0. In particular, we can define for every λ ∈ C the holomorphic function Z → (det Z) λ , normalized by lim ǫ→0 + det(I r + iǫI r ) λ = 1. Moreover, all eigenvalues of Z ∈ h r lie in the upper half plane of C.
be the corresponding quadratic forms. Choose a basis u j such that the Gram matrix of Y is I r , and of X is diagonal: D = diag(d j ). Since D +iI r = U T ZU with U invertible, and det(D + iI r ) = (d j + i) = 0, it follows that det Z = 0. Since h r is simply connected, the second statement follows.
For the last statement, we first note there can be no real eigenvalues. Indeed by the first statement, det(X + iY − λI r ) = det((X − λI r ) + iY ) = 0 for λ ∈ R. Next we argue as before and select a diagonalizing basis, given by U ∈ GL(r). We furthermore may assume that det U > 0, by interchanging two basis elements. Choose a smooth path U t ∈ GL(r) with U 0 = Id and U 1 = U . Then U T t ZU t ∈ h r is a smooth path. For t = 1, the endpoint is D + iI r , which has all eigenvalues in the upper half plane. If Z has eigenvalues in the lower half-plane, then by continuity for some t there will be a real eigenvalue, a contradiction.
Recall for the following that given a non-degenerate quadratic form Q on V , a compatible Euclidean form is any positive-definite form P such that V admits a decomposition V = V + ⊕ V − which is both P -and Q-orthogonal, and Q| V± = ±P | V± .
From here on, let V = R p ⊕ R q = R n+1 with the standard quadratic form Q of signature (p, q) and the corresponding compatible Euclidean form P 0 . Define a family of complex-valued quadratic forms Q ζ on V with ζ ∈ C, by We then have Observe that Q ζ is real and positive-definite for ζ > 1 2 , and Q 0 = Q. Furthermore by Lemma 5.1, det Q ζ = 0 for ζ ∈ U C , as either Q ζ or iQ ζ lies in h n+1 . Note that a complex-valued non-degenerate quadratic form Q on a real vector space E defines an element vol 2 Q ∈ Dens C (E) 2 , and given a branch of square root we also get a complex-valued density vol Q ∈ Dens C (E).
By a (P -)frame on an open subset U ⊂ Gr n+1−k (V ) we understand a smooth section of the Stiefel manifold of P -orthonormal (n + 1 − k)-frames in V defined over U . For a subspace E ∈ Gr n+1−k (V ) and ζ ∈ U C , choose a frame u i (E) on U and define X P ζ : U → Sym n+1−k (C) to be the corresponding Gram matrix of Q ζ , namely Observe that by eq. (17), det(X P ζ ) is independent of the choice of P -orthonormal bases of E. Moreover, either the real or imaginary part of Q ζ | E is positive-definite, and consequently by Lemma 5.1, det X P ζ (E) = 0. The function det(X P ζ ) λ ∈ C ∞ (Gr n+1−k (V ), C) is thus well-defined for all P and λ ∈ C, and analytic in ζ ∈ U C , once the normalization det(X P 1 ) λ > 0 is fixed, as U C is simply-connected.
Define the smooth measure m ζ,P k on the Grassmannian Gr n+1−k (V ) by depends analytically on ζ and is normalized, i.e.
Proof. The first statement is clear. For the second, we first see how m ζ,P k depends on P . Let P 1 , P 2 be two Euclidean structures on V . From the natural identification T E Gr n+1−k (V ) = E * ⊗ V /E we obtain that Spelling this out gives For ζ > 1 2 , Q ζ is a Euclidean structure. Then By uniqueness of the analytic continuation, this formula also holds for general ζ ∈ U C .

5.2.
Homogeneous distributions on the space of symmetric matrices.
Lemma 5.3. The meromorphic family of generalized functions is analytic in λ ∈ C and satisfies Proof. We recall some results from [33]. Consider a linear combination In our situation, the coefficients a h = a h (λ) = e iπhλ depend on λ and we cannot apply Muro's result directly. However, writing we see that it is enough to prove that the order of the pole of By induction we find that for all l = 0, 1, . . .
which finishes the proof.
in the strong topology on tempered distributions on Sym r (R).
Proof. For Reλ ≥ 0 the statement is easy, so in the following we assume Reλ < 0.
First we claim that the limit exists. We will show that there are constants First take Y = I r . Letting (µ j ) r j=1 ⊂ R be the eigenvalues of X, we get (19) holds with b ′ = c −Reλ e πr|Imλ| , and α = −rReλ ≥ 0.
As ǫ → 0, we have ψ(X +iǫI r ) → ψ(X) in S, while det(X +iǫI r ) λ → det(X +i0) λ in S ′ . It follows by continuity that Finally, noting that the set of Schwartz functions such as ψ is dense, we conclude that det(X + i0) λ = f λ for all λ ∈ C, as claimed.
Henceforth we use f λ and det(X + i0) λ interchangeably. The following statement shows that the convergence along Y ∈ R + I r holds in a finer topology, namely the normal Hörmander topology. We refer to [19] for its definition (where it is called normal topology). The main point for us is that the operation of pull-back of generalized sections is continuous in this topology, provided some condition on wave fronts is satisfied.
We do not know if an analogue of the following proposition holds for an arbitrary distributional boundary value; the proof below is tailored to our particular case, and in essence leverages strong convergence by induction on dimension.
Proof. First note that g * f λ = det(g) 2λ f λ for all g ∈ GL(r). Thus we have the differential equations (A − 2λ tr(A))f λ = 0, where A is the vector field defined by the infinitesimal action of A ∈ gl(r). It follows from [29,Theorem 8.3.1] that WF(f λ ) ⊂ N * Γ r . We proceed by induction on r, the case r = 1 being trivial. Since det(X + iǫI r ) λ → det(X + i0) λ in the strong topology by Proposition 5.4 and N * 0 Γ r r = T * 0 Sym r (R), it remains to consider convergence in Sym r (R) \ {0}. Consider a matrix Y ∈ Sym r (R) of nullity ν < r. Let E 0 be its kernel, and F 0 = E ⊥ 0 . There is then a unique map E : U → Gr ν (R r ) in a neighborhood U of Y such that E(Y ) = E 0 , and E(X) is an invariant subspace of X. Here and in the following, U is assumed sufficiently small for various purposes.
For M ∈ Sym r (R) and H ∈ T F0 Gr r−ν (R r ) = Hom(F 0 , R r /F 0 ), one computes d Y,F0 α(M, H) = (H, Y • H + M | F0→R r /F0 ) = (H, M | F0→R r /F0 ), since by construction Y : R r /F 0 → R r /F 0 is the zero map. Noting that any linear map F 0 → R r /F 0 is induced by a symmetric matrix mapping M : R r → R r , it follows that α is submersive and Z is a manifold. Further, In particular if (0, H) ∈ T Y,F0 Z, then we must have H = 0. It follows that F (X) is smooth in U , and therefore so is E(X) = F (X) ⊥ .
Choose arbitrary orthonormal frames e i (X) for E(X) and f i (X) for F (X) = E(X) ⊥ depending smoothly on X. Define A(X) = ( Xe i (X), e j (X) ), B(X) = ( Xf i (X), f j (X) ). Then A is a submersion in U . Indeed one has and the last two summands vanish as e i (Y ), e j (Y ) ∈ E 0 . It follows that d Y A : Sym r (R) → Sym ν (R) is surjective, and so A is submersive near Y .
It holds that As B(X) is non-degenerate, the second factor is a smooth function in (X, ǫ) ∈ U ×R. For the first factor, we have by the induction assumption that det( and by [19], A * det(X 1 + iǫI ν ) λ → A * det(X 1 + i0) λ in the normal Hörmander topology on C −∞ N * Γ r (Sym r (R)). We conclude that det(X + iǫI r ) λ → det(X + i0) λ in the normal Hörmander topology on C −∞ N * Γ ν (Sym r (R)). Remark 5.6. Using the Hilbert-Schmidt inner product to identify T * 0 Sym r (R) = Sym r (R), the statement of the proposition in fact holds with all conormal cones intersected with Sym + r (R), which follows from [29, Theorem 8.1.6]. 5.3. Construction of an O(p, q)-invariant Crofton distribution. In [23, Proposition 4.9], an O(p, q)-invariant distribution was constructed on Gr n+1−k (V ). To avoid singularities, it made use of several auxiliary Euclidean structures that gave rise to several locally defined distributions that were then patched together. For the present paper, we will need an alternative construction making use of a single Euclidean structure. To handle the singularities, we must carefully monitor the wave front set.
Write P = P 0 for the Euclidean structure on V . Let dE = dσ P denote the O(P )-invariant probability measure on Gr n+1−k (V ).
. We denote κ = n + 1 − k. An orthonormal basis u 1 , . . . , u κ spanning E ∈ Gr κ (V ) will be called adapted if, denoting s = dim E ∩ V + P , t = dim E ∩ V − P , the vectors u κ−s−t+1 , . . . , u κ−t form a basis of E ∩ V + P , while u κ−t+1 , . . . , u κ form a basis of E ∩V − P . A frame u i (E ′ ), i = 1, . . . , κ given near E and adapted at E is well-adapted in a neighborhood W if, whenever E ′ ∈ W is such that E ′ ∩ E is spanned by the subset (u i (E)) i =j for some 1 ≤ j ≤ κ, then u i (E ′ ) = u i (E) for all i = j.
It is easy to see that a well-adapted frame can always be chosen to extend a given adapted orthonormal basis u i (E) of E to a small neighborhood -define the frame u i (E ′ ) by orthogonally projecting u i (E) to E ′ , and then applying the Gram-Schmidt process.
Claim. Let M 0 ∈ Mat n×n (R) be a matrix. Then for any ǫ > 0 there is a neighborhood W ǫ of M 0 such that for any M ∈ W ǫ , Ker(M ) ⊥ contains a subspace that is ǫ-close to Ker(M 0 ) ⊥ .
Proof. Assume that rankM 0 = r, and the first r rows u 1 (M 0 ) T , . . . , u r (M 0 ) T are linearly independent. Therefore, Ker(M 0 ) ⊥ = Span(u 1 (M 0 ), . . . , u r (M 0 )). By choosing W ǫ small enough, we may ensure that Span(u 1 (M ), . . . , u r (M )) is rdimensional, and ǫ-close to Ker(M 0 ) ⊥ . Since Span(u 1 (M ), . . . , u r (M )) ⊂ Ker(M ) ⊥ , this concludes the proof. Proof. WriteṀ 0 = d dτ 0 M τ . By Lemma 5.7, we ought to show that for all unit vectors v ∈ E ǫ (M 0 ), Ṁ 0 , vv T = 0. Now for any such v, M 0 v, v = ǫ. We know by assumption | M τ v, v | ≤ 1 for all τ , and so Proposition 5.10. Let X 0 be locally defined by a well-adapted frame at E ∈ Gr n−k+1 (V ). Then there is a neighborhood W E of E where the distribution is well-defined. Furthermore, the corresponding distributions agree on non-empty intersections W E1 ∩ W E2 for all E 1 , E 2 , giving rise to a globally defined distribution . Let us prove the claim. We may assume that Y lies in the singular support of To this end simply observe that by Lemma 5.7 Similarly, the intersection . As X ∈ W Y is uniquely determined by its eigenspace E X (1), W Y is evidently a manifold that can be identified with Gr s (E Y ) = Gr s (R s+t ), in particular dim W Y = st. By definition, W Y ⊂ B r 0 . Now since −P ≤ Q ≤ P , the spectrum of X 0 (E) lies in [−1, 1]. By Lemma 5.9 we have Image(d E X 0 ) ⊂ T Y B s,t 1,−1 . For 1 ≤ j ≤ κ, choose a smooth curve γ j (τ ) through E given by γ j (τ ) = Span(u 1 (E), u 2 (E), . . . , cos τ u j (E) + sin τ ξ, . . . , u κ (E)), where ξ ∈ E P is arbitrary. Observe that T E Gr κ (V ) = Span{γ ′ j (0) : 1 ≤ j ≤ κ}. Since the frame is well-adapted to E, u i (γ j (τ )) = u i (E) for i = j, and so u j (γ j (τ )) = cos τ u j (E) + sin τ ξ. One computes we may choose ξ j ∈ E P to get arbitrary κ − s − t first entries in the j-th row and column. Thus the entries of a matrix in Image(d E X 0 ) can be made arbitrary outside of the bottom right (s + t) × (s + t) minor. Consequently, . This is because T Y W Y consists of all matrices that vanish outside of the bottom right (s + t) × (s + t)-minor M , which has zeros in its top left s × s minor and bottom right t × t minor.
One easily verifies that s+1 For E ′ in a sufficiently small neighborhood W E of E, Image d E ′ X 0 must contain a subspace that is ǫ-close to Image(d E X 0 ). We may moreover by Lemma 5.8 assume W E is such that for all E ′ ∈ W E , Y ′ = X 0 (E ′ ) has nullity r ′ ≤ r, and T Y ′ B r ′ 0 contains a subspace that is ǫ-close to T Y B r 0 . Thus for sufficiently small ǫ we find a neighborhood W E such that for all 0 defines a sequentially continuous linear operator on C −∞ N * Γ κ (Sym κ (R)), where f λ lies. Moreover, X * 0 f λ must itself be an analytic family: for a smooth compactly supported test measure ψ on W E we have X * 0 f λ , ψ = f λ , (X 0 ) * ψ , and by [28, Chapter VI, Proposition 3.9] we have WF((X 0 ) * ψ) ⊂ ∪ E ′ ∈WE L E ′ ,X0(E ′ ) . Now analyticity of a vector-valued function coincides with weak analyticity in quasi-complete locally convex vector spaces, and C −∞ N * Γ κ (Sym κ (R)) is quasi-complete [21,Proposition 29]. Furthermore (X 0 ) * ψ defines a continuous linear functional on C −∞ N * Γ κ (Sym κ (R)) by [21,Lemma 3], confirming that the family of generalized functions X * 0 f λ ∈ C −∞ (W E ) is analytic. Now observe that the continuous functions X * 0 f λ defined separately for W E1 and W E2 coincide on non-empty intersections W E1 ∩ W E2 for Reλ > 0 by eq. (17). If follows by uniqueness of the analytic extension that this holds for all λ ∈ C. In particular, m 0 k is a globally well-defined distribution. For invariance, we note that for g ∈ O(Q), g * X * 0 f λ = ψ g (E) λ X * 0 f λ for λ > 0, and consequently by uniqueness of analytic extension for all λ ∈ C. Here ψ g (E) = Jac(g : E → gE) −2 , which for g ∈ O(Q) satisfies ψ g (E) = det X0(gE) det X0(E) , see [13,Proposition 4.7] for details. Using the identification Henceforth whenever X 0 appears, a local well-adapted frame should be chosen arbitrarily unless an explicit choice is provided. 5.4. Some properties of the invariant distributions. We will use the following rescaling of the invariant distribution constructed above, which brings the total integral to 1 as will be later seen.
. Lemma 5.12. Let j : R p,q → R q,p be given by j(x, y) = (y, x) where x ∈ R p , y ∈ R q . Let us also denote by j the induced map Gr p+q−k (R p,q ) → Gr p+q−k (R q,p ). Then dσ P0 on R q,p . Since X 0 • j = −X 0 (where X 0 is defined using j-corresponding frames), (18) . It follows that which is the conjugate of m k in R p,q .
By Proposition 5.5, we have det(X + 2iǫI n+1−k ) λ → f λ (X) in the normal Hörmander topology on C −∞ Γ n+1−k (Sym n+1−k (R)). By the proof of Proposition 5.10, we may use the continuity of the pull-back X * 0 in the normal Hörmander topology [19]. Noting that in the normal Hörmander topology as stated.
Corollary 5.14. Let M ⊂ V n+1 be a pseudosphere or a pseudohyperbolic space, and A ⊂ M either a smooth domain with LC-regular boundary, or a smooth LCregular hypersurface without boundary. Assume all Q-degenerate tangents to A of codimension k are regular. Then Proof. First note that Cr(m k ) ∈ V −∞ (M ) is isometry invariant and by [15, Theorem C] is given by a linear combination of the intrinsic volumes. Thus A is WF-transversal to Cr(m k ). The assertion now follows from Corollary 4.7, and Propositions 5.13 part i) and 3.14. Proof. In both cases, it follows from [14,Lemma 4.7] that H is LC-regular, and we can apply Corollary 5.14.
Example. The complex-valued distribution m n ∈ M −∞ (P(V )) is invariant under the group of projective transformations preserving the quadric [Q] = {Q = 0}. Its singular support is [Q], and WF(m n ) ⊂ N * [Q]. In particular, m n (A) is well defined for any domain A ⊂ P(V ) that is smooth near [Q] and transversal to it. When Q is definite, the quadric [Q] has no real points and m n is the Haar measure on the round projective space. 5.5. The flat case. Next we construct a translation-and O(p, q)-invariant distribution on the affine Grassmannian Gr p+q−k (R p,q ).
which is a diffeomorphism onto its open image. Given t > 0, consider the homothety v → tv on T and the induced map h t on Gr p+q−k (T ).
i) Let dE be an O(P )-invariant measure on Gr p+q+1−k (V ), thus given by a smooth density. Then ii) Let X 0 : Gr p+q+1−k (W ) → Sym p+q+1−k (R) be as in Proposition 5.10, and let X ′ 0 be the corresponding map on Gr p+q−k (T ). Then and this generalized measure is O(Q| T )-invariant. Here (s −1 ) * denotes the push-forward by s −1 of the restriction to the open set Image(s).
By analytic continuation we get In the proof of i) we have seen (h 1/t ) * (s −1 ) * dE = O(t k ). Hence, by continuity Translation invariance is clear.
Translation-invariance and O(P | T )-invariance characterize dF uniquely up to normalization. The normalization can be deduced from Theorem 6.5 in the case q = 0. As for the translation-invariant and O(Q| T )-invariant generalized measure obtained on T ∼ = R p,q , we take the normalization of Definition 5.11 as follows.
Definition 5.17. On Gr p+q−k (R p,q ) we fix the following translation-invariant and O(p, q)-invariant generalized measurě The Crofton map in the flat case is given by Cr(µ), holds for k = 1. ii) If H is strictly convex, then (21) holds for all k.

Crofton formulas for generalized pseudospheres
For the de Sitter space embedded in Lorentz space, one can compute the Crofton formulas through a direct computation of the restriction of the measures to subspaces, combined with the Hadwiger theorem and the template method. However for general signatures, an explicit computation appears to be hard. Instead, we carry out an analytic extension argument, which recovers the Crofton formulas for all signatures in a unified fashion.
For ζ > 1 2 , we denote by S ζ = Q −1 ζ (1) ⊂ R p+q = R n+1 the unit sphere in the Euclidean space (R n+1 , Q ζ ). For ζ = 0 we have S p−1,q = Q −1 0 (1) with the induced pseudo-Riemannian metric Q 0 . We will also denote by S n the unit sphere in R n+1 with respect to some fixed Euclidean structure (which is independent of ζ).
In the following we make use of the operation of restriction of Crofton distributions, as described in Section 3.1.
Proposition 6.1. For the standard inclusion e : R p,q ֒→ R p+l,q+r , we have e * m k = m k .
Proof. Note first that the restriction e * m k is well-defined by [23,Remark 2.13]. For ζ > 1 2 , Q ζ is positive definite, and so e * m ζ k = m ζ k by the uniqueness of probability measure on the Grassmannian invariant under the positive definite orthogonal group, as e * : M ∞ (Gr p+q+l+r−k (R p+l,q+r )) → M ∞ (Gr p+q−k (R p,q )) is essentially the pushforward operation under intersection with R p,q . By analytic extension in ζ, we get e * m iǫ k = m iǫ k . The statement then follows from Proposition 5.13 ii). Proposition 6.2. Given A ∈ P(S p−1,q ), let A ∈ P(S n ) be its radial projection. Assume A is either an LC-regular hypersurface, or a smooth domain with LC-regular boundary. Assume further that either all Q 0 -degenerate tangents of codimension k are regular, or χ(A ∩ E) is constant for a.e. plane E of codimension k. Then . Proof. We have by Proposition 3.14 . By Corollary 5.14, it holds that By part ii) of Proposition 5.13, m iǫ k tends to m k as ǫ → 0 + in the normal Hörmander topology on M −∞ N * Λ (Gr n+1−k (V )). Combining Corollary 4.7 and Proposition 5.13 part i), we see that evaluating at χ(A ∩ •) = χ(A ∩ •) is continuous in this topology, and the statement follows.
In order to normalize the leading coefficient in the Crofton formulas we rescale the measures m k ,m k as follows.
Definition 6.4. Let M ⊂ R p,q be the pseudosphere of curvature σ > 0, or the pseudohyperbolic space of curvature σ < 0. We define In the flat pseudo-Euclidean space M = R p,q we take Cr M k = πω k−1 Cr M (m k ).
Theorem 6.5 (Crofton formula). Let M be a pseudosphere, a pseudohyperbolic space or a pseudo-Euclidean space. Then, independently of the signature of M , where σ is the sectional curvature of M and n its dimension.
Proof. Take first the pseudosphere M = S p−1,q of curvature σ = 1. We can assume q > 0 as the formula is known in S n (cf. e.g. [25]). We know that (a j,p,q µ k+2j + b j,p,q µ k+2j ) for certain coefficients a j,p,q , b j,p,q ∈ C. Indeed, by [15, Theorem C] we may express Cr M k as a linear combination of the intrinsic volumes and their complex conjugates. Since both µ r and Cr M r are the restrictions of elements in Val −∞,+ r and thus belong to the (−1) r -eigenspace of the Euler-Verdier involution, only the displayed terms appear.
By the linear independence of {µ i } i ∪ {µ i } i [14,Corollary 7.4], and taking l ≥ q − 1, this yields a j,p,q = a j,p+l,q = a j,p+l,1 , b j,p,q = b j,p+l,q = b j,p+l,1 for all j ≤ p−1+q−k 2 . It suffices then to determine a j := a j,p,1 , b j := b j,p,1 ; i.e. to prove the statement in the de Sitter sphere M = S p−1,1 . To this end we evaluate both sides on the templates R p−1,s ⊂ M with s = 0, 1. In order to compute Cr M k (R p−1,s ) we use the spherical Crofton formula: for ζ > 1 2 . Given p ≥ k and s = 0, 1, let S p be the unit sphere of an arbitrary Euclidean structure in R p,1 = R p+1 and let T p−1,s be the radial projection on S p of R p−1,s . By Definition 6.4, Proposition 6.2, and applying analytic continuation to (26)  (a j + (−1) s b j )µ k+2j (R k+2l−s,s ), since µ k+2j (R k+2l−s,s ) ∈ i s R by (22).
For l = 0 we have µ k+2j (R k−s,s ) = 0 for all j ≥ 1 and hence c 0 = a 0 + (−1) s b 0 for s = 0, 1, and thus a 0 = c 0 , b 0 = 0. Suppose that a j = c j , b j = 0 for all j < j 0 . Taking l = j 0 we deduce c j0 = a j0 + (−1) s b j0 for s = 0, 1. By induction we deduce a j = c j and b j = 0 for all j, which completes the proof for σ = 1.
Let us now turn to σ = −1, i.e. to H p,q−1 ⊂ R p,q . Note that the anti-isometry j : R q,p → R p,q of Lemma 5.12 maps S q−1,p to H p,q−1 . Therefore, by Lemma 5.12 and the homogeneity of the µ k , This proves the statement for σ = −1. The case σ < 0 follows as before from the homogeneity of the µ k .
Finally we consider the case σ = 0. Let us identify M = R p−1,q with the tangent space T x S p−1,q at some x ∈ S p−1,q . Let Λ On the other hand, denoting by g the metric on S p−1,q , since µ k ∈ W −∞ k , behaves naturally with respect to isometries and is k-homogeneous, we have Since (φ • h t ) * g/t 2 converges, C ∞ -uniformly on compact sets, to the flat metric g 0 , we conclude by [14, Proposition 1.2 ii)] that Λ x k µ g k = µ g0 k . Applying Λ x k to both sides of (25) the case σ = 0 follows.
Recall from [14] that the intrinsic volumes µ k were defined in terms of certain generalized curvature measures C 0 k,p , C 1 k,p . On a manifold of constant curvature σ, Remark 6.6. It is interesting to note that (27) yields Cr M 2 = i q glob(C 0 0,0 + iC 1 0,0 ), which can be seen as a generalization of the fact that the angular excess of a spherical triangle is proportional to its area.