# Crofton formulas in pseudo-Riemannian space forms

Andreas Bernig  ,  Dmitry Faifman  and  Gil Solanes<EMAIL_ADDRESS>frankfurt.de<EMAIL_ADDRESS><EMAIL_ADDRESS>Institut für
Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60629
Frankfurt, Germany  School of Mathematical Sciences, Tel Aviv University, Tel
Aviv 6997801, Israel  Departament de Matemàtiques, Universitat Autònoma de
Barcelona, 08193 Bellaterra, Spain, and
Centre de Recerca Matemàtica, Campus de Bellaterra, 08193 Bellaterra, Spain

###### Abstract.

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.

MSC classification: 53C65, 53C50, 52A22, 52A39
A.B. was supported by DFG grant BE 2484/5-2
D.F. was partially supported by ISF Grant 1750/20.
G.S. was supported by FEDER/MICINN grant PGC2018-095998-B-I00 and the Serra
Húnter Programme

###### Contents

  1. 1 Introduction
    1. 1.1 Crofton formulas
    2. 1.2 Results
    3. 1.3 Plan of the paper
  2. 2 Preliminaries
    1. 2.1 Notations
    2. 2.2 Smooth valuations
    3. 2.3 Generalized valuations
    4. 2.4 Intrinsic volumes on pseudo-Riemannian manifolds
  3. 3 Distributional Crofton formulas
    1. 3.1 The general setting
    2. 3.2 Distributional Crofton measures
    3. 3.3 Functorial properties of Crofton measures.
    4. 3.4 Applying generalized Crofton formulas to subsets
  4. 4 The Crofton wave front of LC-regular hypersurfaces
  5. 5 Construction of an invariant measure on the Grassmannian
    1. 5.1 Construction of a holomorphic family of Crofton measures
    2. 5.2 Construction of an $\operatorname{O}(p,q)$-invariant Crofton measure
    3. 5.3 The flat case
  6. 6 Crofton formulas for generalized pseudospheres

##  1\. Introduction

###  1.1. Crofton formulas

The classical Crofton formula computes the length of a rectifiable curve
$\gamma$ in $\mathbb{R}^{2}$ as

$\mathrm{Length}(\gamma)=\frac{\pi}{2}\int_{\overline{\operatorname{Gr}}_{1}(\mathbb{R}^{2})}\\#(\gamma\cap\overline{L})d\overline{L},$
(1)

where $\overline{\operatorname{Gr}}_{1}(\mathbb{R}^{2})$ is the space of lines
in $\mathbb{R}^{2}$ with a rigid motion invariant measure (which is normalized
in a suitable way).

A higher dimensional version states that for $M\subset\mathbb{R}^{n}$ a
compact submanifold with boundary, we have

$\mu_{k}(M)=c_{n,k}\int_{\overline{\operatorname{Gr}}_{n-k}(\mathbb{R}^{n})}\chi(M\cap\overline{E})d\overline{E},$

where $\overline{\operatorname{Gr}}_{n-k}(\mathbb{R}^{n})$ is the Grassmann
manifold of affine $(n-k)$-planes equipped with a rigid motion invariant
measure, $\chi$ is the Euler characteristic, and $\mu_{k}(M)$ is the $k$-th
intrinsic volume of $M$, which can be defined via Weyl’s tube formula [39].
The same formula also holds with the submanifold $M$ replaced by a compact
convex body $K$, in which case the $k$-th intrinsic volume can be defined via
Steiner’s tube formula [35]. We refer to [28, 33] for more information about
intrinsic volumes of convex bodies.

More generally, we can take an arbitrary translation-invariant measure $m$ on
$\overline{\operatorname{Gr}_{n-k}}(\mathbb{R}^{n})$ and consider the integral

$\mu(K):=\int_{\overline{\operatorname{Gr}_{n-k}}(\mathbb{R}^{n})}\chi(K\cap\overline{E})dm(\overline{E}).$

By the additivity of the Euler characteristic, we have

$\mu(K\cup L)+\mu(K\cap L)=\mu(K)+\mu(L)$

whenever $K,L,K\cup L$ are compact convex bodies, hence $\mu$ is a valuation.
Clearly $\mu$ belongs to the space $\operatorname{Val}$ of translation-
invariant valuations which are continuous with respect to the Hausdorff
metric. Additionally, $\mu$ is $k$-homogeneous and even, that is invariant
under $-\operatorname{Id}$. We thus get a map

$\operatorname{Cr}:\mathcal{M}(\overline{\operatorname{Gr}_{n-k}}(\mathbb{R}^{n}))^{tr}\to\operatorname{Val}_{k}^{+},$

where $\mathcal{M}^{tr}$ denotes the space of translation-invariant measures.

Alesker [2] has shown that the image of this map is dense with respect to the
natural Banach space topology on $\operatorname{Val}_{k}^{+}$. Therefore
Crofton formulas are a central tool in the study of valuations and in integral
geometry.

When restricted to smooth measures and valuations (see Section 2 for the
notion of smoothness of valuations), the map $\operatorname{Cr}$ is in fact a
surjection

$\operatorname{Cr}:\mathcal{M}^{\infty}(\overline{\operatorname{Gr}_{n-k}}(\mathbb{R}^{n}))^{tr}\twoheadrightarrow\operatorname{Val}_{k}^{\infty,+},$

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
$\mathbb{C}^{n}$ 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 [31], systolic geometry [37], algebraic geometry [1]
and more. Crofton formulas are employed also outside of pure mathematics, in
domains such as microscopy and stereology, see [27].

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 [23], 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
$\mathbb{R}^{3}$ is bigger than $4\pi$ if the knot is non-trivial.

In higher dimensions, the formula becomes slightly more complicated. On the
$n$-dimensional unit sphere we have

$\int_{\operatorname{Geod}_{n-k}(S^{n})}\chi(M\cap
E)dE=\sum_{j}\frac{1}{\pi\omega_{k+2j-1}}\binom{-\frac{k}{2}}{j}\mu_{k+2j}(M).$
(2)

Here $\operatorname{Geod}_{n-k}(S^{n})$ denotes the totally geodesic
submanifolds of dimension $(n-k)$, $\mu_{j}(M)$ is the $j$-th intrinsic volume
of $M$ (which can be defined as the restriction of the $j$-th intrinsic volume
on $\mathbb{R}^{n+1}$ under the isometric embedding
$S^{n}\hookrightarrow\mathbb{R}^{n+1}$), and $\omega_{n}$ denotes the volume
of the $n$-dimensional unit ball. A similar formula holds on hyperbolic space.
See [24, 25] 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, 29, 34, 41].

###  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 [32, 40] for more information.

######  Definition 1.1.

  1. i)

The _pseudo-Euclidean space of signature $(p,q)$_ is
$\mathbb{R}^{p,q}=\mathbb{R}^{p+q}$ with the quadratic form
$Q=\sum_{i=1}^{p}dx_{i}^{2}-\sum_{i=p+1}^{p+q}dx_{i}^{2}$.

  2. ii)

The _pseudosphere of signature $(p,q)$ and radius $r>0$_ is

$S_{r}^{p,q}=\\{v\in\mathbb{R}^{p+1,q}:Q(v)=r^{2}\\},$

equipped with the induced pseudo-Riemannian metric. Its sectional curvature
equals $\sigma=\frac{1}{r^{2}}$. The pseudosphere
$S_{1}^{n,1}\subset\mathbb{R}^{n+1,1}$ is called _de Sitter space_.

  3. iii)

The _pseudohyperbolic space of signature $(p,q)$ and radius $r>0$_ is

$H_{r}^{p,q}=\\{v\in\mathbb{R}^{p,q+1}:Q(v)=-r^{2}\\}.$

Its sectional curvature equals $\sigma=-\frac{1}{r^{2}}$. The pseudohyperbolic
space $H_{1}^{n,1}$ is called the _anti-de Sitter space_.

We will colloquially call these spaces _generalized pseudospheres_. The
isometry groups of generalized pseudospheres are given by

$\displaystyle\operatorname{Isom}(\mathbb{R}^{p,q})$
$\displaystyle\cong\overline{\operatorname{O}}(p,q)=\operatorname{O}(p,q)\ltimes\mathbb{R}^{p,q},$
$\displaystyle\operatorname{Isom}(S_{r}^{p,q})$
$\displaystyle\cong\operatorname{O}(p+1,q),$
$\displaystyle\operatorname{Isom}(H_{r}^{p,q})$
$\displaystyle\cong\operatorname{O}(p,q+1).$

In each case the action is transitive, and the isotropy group is conjugate to
$\operatorname{O}(p,q)$. These spaces are _isotropic_ in the sense that the
isotropy group acts transitively on the level sets of the metric in the
tangent bundle.

######  Definition 1.2.

A complete connected pseudo-Riemannian manifold of constant sectional
curvature is called a _space form_.

Up to taking connected components and universal coverings, any space form is a
generalized pseudosphere (cf. [32, Chapter 8, Corollary 24]).

On a generalized pseudosphere $M$, we will formulate Crofton formulas using
the space $\operatorname{Geod}_{n-k}(M)$ of totally geodesic subspaces.
However, there is no isometry-invariant 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

$\operatorname{Cr}:\mathcal{M}^{-\infty}(\operatorname{Geod}_{n-k}(M))\to\mathcal{V}^{-\infty}(M).$

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 [22] 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 [30] on analytic families of homogeneous generalized
functions on the space of symmetric matrices, one can construct such an
invariant generalized measure on $\operatorname{Geod}_{n-k}(M)$. 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

$\operatorname{Cr}_{k}^{M}:=\operatorname{Cr}_{M}(\widehat{m}_{k}),$

where $\sigma\neq 0$ is the curvature of $M$ and
$\widehat{m}_{k}:=\pi\omega_{k-1}\sqrt{\sigma^{-1}}^{k}m_{k}$. The flat case
$\sigma=0$ 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 $\sigma$. Then

$\operatorname{Cr}_{k}^{M}=\sum_{j\geq
0}\frac{\omega_{k-1}}{\omega_{k+2j-1}}{-\frac{k}{2}\choose
j}\sigma^{j}\mu_{k+2j}.$

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 $\mathbb{R}^{m+1}$, with
$M\subset\mathbb{R}^{m+1}$ embedded as in Definition 3.6. For the Riemannian
round sphere and hyperbolic space, this coincides with the standard definition
of strict convexity.

######  Corollary 1.3.

Let $A\subset M$ be a smooth and strictly convex domain in $M$. Then the
generalized measure $\widehat{m}_{k}$ can be applied to the function
$E\mapsto\chi(A\cap E),E\in\operatorname{Geod}_{n-k}(M)$, and

$\int_{\operatorname{Geod}_{n-k}(M)}\chi(A\cap
E)d\widehat{m}_{k}(E)=\sum_{j\geq
0}\frac{\omega_{k-1}}{\omega_{k+2j-1}}{-\frac{k}{2}\choose
j}\sigma^{j}\mu_{k+2j}(A).$

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
5.11 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.

### Acknowledgments

We wish to thank Gautier Berck for several inspiring talks and discussions.

##  2\. Preliminaries

###  2.1. Notations

By

$\omega_{n}=\frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)}$

we denote the volume of the $n$-dimensional unit ball. The space of smooth
$k$-forms on a manifold is denoted by $\Omega^{k}(M)$. The space of smooth
measures on $M$ is $\mathcal{M}^{\infty}(M)$. The space of generalized
measures, also called distributions, is denoted by

$\mathcal{M}^{-\infty}(M):=(C^{\infty}_{c}(M))^{*},$

where here and in the following the subscript $c$ denotes compactly supported
objects. Similarly, for $m=\dim M$, we denote the space of $k$-dimensional
currents on $M$ by

$\Omega_{-\infty}^{m-k}(M):=(\Omega^{k}_{c}(M))^{*}.$

The elements of this space can also be thought of as generalized
$(m-k)$-forms.

For an oriented $k$-dimensional submanifold $X\subset M$, we let $\llbracket
X\rrbracket$ be the $k$-current which is integration over $X$.

By $\mathbb{P}_{M}:=\mathbb{P}_{+}(T^{*}M)$ we denote the cosphere bundle of
$M$, which consists of all pairs $(p,[\xi]),p\in M,\xi\in T_{p}^{*}M\setminus
0$, where $[\xi]=[\xi^{\prime}]$ if there is some $\lambda>0$ with
$\xi=\lambda\xi^{\prime}$. When no confusion can arise, we use the same
notation for subsets of $\mathbb{P}_{M}$ and their lifts to $T^{*}M$. The
natural involution on $\mathbb{P}_{M}$ is the fiberwise antipodal map
$s(p,[\xi]):=(p,[-\xi])$.

The wave front set of a generalized form $\omega\in\Omega_{-\infty}(M)$ is a
closed subset of $\mathbb{P}_{M}$, denoted by $\operatorname{WF}(\omega)$, and
we refer to [26] or [21] for details.

For a generalized pseudo-sphere $M$, we denote by $\operatorname{Geod}_{k}(M)$
the space of totally geodesic $k$-dimensional submanifolds of $M$.

###  2.2. Smooth valuations

Let $M$ be a smooth manifold of dimension $m$, which we assume oriented for
simplicity.

Let $\mathcal{P}(M)$ be the set of compact differentiable polyhedra on $M$. To
$A\in\mathcal{P}(M)$ we associate two subsets of $\mathbb{P}_{M}$. The
conormal cycle, denoted $\operatorname{nc}(A)$, 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 $\operatorname{nc}(A)\subset N^{*}A$.
By definition, two stratified spaces intersect transversally if all pairs of
smooth strata are transversal.

A smooth valuation is a functional $\mu:\mathcal{P}(M)\to\mathbb{R}$ of the
form

$\mu(A)=\int_{A}\phi+\int_{\operatorname{nc}(A)}\omega,\quad\phi\in\Omega^{m}(M),\omega\in\Omega^{m-1}(\mathbb{P}_{M}).$

We will write $\mu=[[\phi,\omega]]$ in this case.

The space of smooth valuations is denoted by $\mathcal{V}^{\infty}(M)$. It
admits a natural filtration

$\mathcal{V}^{\infty}(M)=\mathcal{W}_{0}^{\infty}(M)\supset\mathcal{W}_{1}^{\infty}(M)\supset\ldots\supset\mathcal{W}_{m}^{\infty}(M)=\mathcal{M}^{\infty}(M).$

It is compatible with the Alesker product of valuations.

###  2.3. Generalized valuations

The space of generalized valuations is

$\mathcal{V}^{-\infty}(M):=(\mathcal{V}^{\infty}_{c}(M))^{*}.$

By Alesker-Poincaré duality we have a natural embedding
$\mathcal{V}^{\infty}(M)\hookrightarrow\mathcal{V}^{-\infty}(M)$.

There is a natural filtration

$\mathcal{V}^{-\infty}(M)=\mathcal{W}_{0}^{-\infty}(M)\supset\mathcal{W}_{1}^{-\infty}(M)\supset\ldots\supset\mathcal{W}_{m}^{-\infty}(M)=\mathcal{M}^{-\infty}(M).$

In particular, we may consider a generalized measure as a generalized
valuation.

A compact differentiable polyhedron $A$ defines a generalized valuation
$\chi_{A}$ by

$\langle\chi_{A},\mu\rangle=\mu(A),\quad\mu\in\mathcal{V}_{c}^{\infty}(M).$

A generalized valuation $\psi$ can be represented by two generalized forms
$\zeta\in C^{-\infty}(M),\tau\in\Omega_{-\infty}^{m}(\mathbb{P}_{M})$ such
that

$\langle\psi,[[\phi,\omega]]\rangle=\langle\zeta,\phi\rangle+\langle\tau,\omega\rangle.$

We refer to $\zeta$ and $\tau$ as the _defining currents_. For instance, the
defining currents of $\chi_{A},A\in\mathcal{P}(M)$ are
$\zeta=\mathbbm{1}_{A},\tau=\llbracket nc(A)\rrbracket$. The wave front set of
$\psi$ is defined as the pair
$\Lambda\subset\mathbb{P}_{M},\Gamma\subset\mathbb{P}_{\mathbb{P}_{M}}$ of the
wave front sets of $\zeta$ and $\tau$. The space of all generalized valuations
with wave front sets contained in $\Lambda,\Gamma$ is denoted by
$\mathcal{V}^{-\infty}_{\Lambda,\Gamma}(M)$.

Consider $\psi\in\mathcal{V}^{-\infty}(M)$. We say that $A\in\mathcal{P}(M)$
is WF-transversal to $\psi$, denoted $A\pitchfork\psi$, if the conditions of
[8, Theorem 8.3] hold for

$\displaystyle(\Lambda_{1},\Gamma_{1})$
$\displaystyle:=\operatorname{WF}(\chi_{A}),$
$\displaystyle(\Lambda_{2},\Gamma_{2})$
$\displaystyle:=\operatorname{WF}(\psi).$

These conditions imply that Alesker’s product of smooth valuations can be
extended to a jointly sequentially continuous product

$\mathcal{V}_{\Lambda_{1},\Gamma_{1}}^{-\infty}(M)\times\mathcal{V}_{\Lambda_{2},\Gamma_{2}}^{-\infty}(M)\to\mathcal{V}^{-\infty}(M),$

and in particular the pairing
$\psi(A):=\langle\psi,\chi_{A}\rangle=\int_{M}\psi\cdot\chi_{A}$ is well-
defined.

Let us write a sufficient set of conditions in a particular case.

######  Proposition 2.1.

Assume $\operatorname{WF}(\psi)\subset(N^{*}D,N^{*}L)$ for some submanifolds
with corners $D\subset M$, $L\subset\mathbb{P}_{M}$, and take
$A\in\mathcal{P}(M)$. Assume further

  1. i)

$A\pitchfork D$.

  2. ii)

$\operatorname{nc}(A)\pitchfork\pi^{-1}D$, where $\pi:\mathbb{P}_{M}\to M$ is
the natural projection.

  3. iii)

$\pi^{-1}A\pitchfork L$.

  4. iv)

$\operatorname{nc}(A)\pitchfork s(L)$, where
$s:\mathbb{P}_{M}\to\mathbb{P}_{M}$ is the antipodal map.

Then $A\pitchfork\psi$.

###### Proof.

These conditions imply the conditions in [8, Theorem 8.3]. ∎

###  2.4. Intrinsic volumes on pseudo-Riemannian manifolds

In [14] we constructed a sequence of complex-valued generalized valuations
$\mu_{0}^{M},\ldots,\mu_{m}^{M}$ 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 $\mu_{0}$ equals the
Euler characteristic, while $\mu_{m}$ is the volume measure of $M$, multiplied
by $\mathbf{i}^{q}$ where $q$ is the negative index of the signature. For
other values of $k$, $\mu_{k}$ is typically neither real nor purely imaginary.

The wave front set of $\mu_{k}$ is contained in
$(\emptyset,N^{*}(\operatorname{LC}^{*}_{M}))$, where
$\operatorname{LC}^{*}_{M}\subset\mathbb{P}_{M}$ is the dual light cone of the
metric, i.e. the set of all pairs $(p,[\xi])\in\mathbb{P}_{M}$ such that
$g|_{p}(\xi,\xi)=0$. Here we use the metric to identify $TM$ and $T^{*}M$.

A subset $A\in\mathcal{P}(M)$ is _LC-transversal_ if
$\operatorname{nc}(A)\pitchfork\operatorname{LC}^{*}_{M}$.

######  Lemma 2.2.

Assume $D=\emptyset,L=\operatorname{LC}^{*}_{M}\subset\mathbb{P}_{M}$, and
$A\in\mathcal{P}(M)$. Then the conditions of Proposition 2.1 are equivalent to
the LC-transversality of $A$. In particular, the intrinsic volume $\mu_{k}$
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 $\pi^{-1}A$ contains all vertical
directions, while the tangent space of $\operatorname{LC}^{*}_{M}$ 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 $TX$. We say that $(X,g)$ is LC-regular if $0$ is a regular value
of $g\in C^{\infty}(TX\setminus\underline{0})$.

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 $M\looparrowright\widetilde{M}$ of
pseudo-Riemannian manifolds we have

$\mu_{k}^{\widetilde{M}}|_{M}=\mu_{k}^{M},$

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.

##  3\. Distributional Crofton formulas

Let $M^{m}$ be a manifold. A Crofton formula for a smooth valuation
$\phi\in\mathcal{V}^{\infty}(M)$ has the form $\phi(A)=\int_{S}\chi(X(s)\cap
A)d\mu(s)$, where $S$ is a smooth manifold parametrizing a smooth family of
submanifolds of $M$, and $\mu$ a smooth measure on $S$. Similarly, a
distributional Crofton formula has $\phi\in\mathcal{V}^{-\infty}(M)$, and
$\mu$ is a distribution.

In this section we study some general properties of such formulas, when
$M\subset V\setminus\\{0\\}$ is a submanifold without boundary in a
$d$-dimensional linear space $V$, and
$S=\operatorname{Gr}_{d-k}(V),k<d,X(s)=s\cap M,s\in S$.

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 $X\subset M$, define $Z_{X}\subset
X\times\operatorname{Gr}_{d-k}(V)$ by $Z_{X}=\\{(x,E):x\in X\cap E\\}$. Then
$Z_{X}$ is a manifold with corners, more precisely it is the total space of
the fiber bundle over $X$ with fiber
$\operatorname{Gr}_{d-k-1}(V/\mathbb{R}x)$ at $x\in X$. Write

$\textstyle{Z_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{X}}$$\scriptstyle{\tau_{X}}$$\textstyle{X}$$\textstyle{\operatorname{Gr}_{d-k}(V)}$

for the natural projections.

Denote by $W_{X}\subset\operatorname{Gr}_{d-k}(V)$ 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.

######  Lemma 3.1.

Let $V$ be a vector space,
$L_{0}\in\operatorname{Gr}_{l}(V),E_{0}\in\operatorname{Gr}_{k}(V)$ with
$L_{0}\subset E_{0}$. Denote by $i:L_{0}\hookrightarrow E_{0}$ the inclusion,
and $\pi:V/L_{0}\to V/E_{0}$ the projection.

  1. i)

Let $E(t)\in\operatorname{Gr}_{k}(V)$ be a smooth path with $E(0)=E_{0}$ and
$A:L_{0}\to V/L_{0}$ a linear map. Then there is a smooth path
$L(t)\in\operatorname{Gr}_{l}(E(t))$ with $L(0)=L_{0}$ and $L^{\prime}(0)=A$
if and only if the following diagram commutes:

$\textstyle{L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\scriptstyle{i}$$\textstyle{V/L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E^{\prime}(0)}$$\textstyle{V/E_{0}}$

  2. ii)

Let $L(t)\in\operatorname{Gr}_{l}(V)$ be a smooth path with $L(0)=L_{0}$. Let
$B:E_{0}\to V/E_{0}$ be a linear map. Then there is a smooth path
$E(t)\in\operatorname{Gr}_{k}(V)$ with $L(t)\subset E(t)$, $E(0)=E_{0}$ and
$E^{\prime}(0)=B$ if and only if the following diagram commutes:

$\textstyle{L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\prime}(0)}$$\scriptstyle{i}$$\textstyle{V/L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B}$$\textstyle{V/E_{0}}$

######  Remark 3.2.

The ’only if’ statement obviously remains true if instead of $L(t)\subset
E(t)$, we have $\measuredangle(L(t),E(t))=o(t)$ with respect to any Euclidean
structure.

###### Proof.

Consider the partial flag manifold $Z=\\{L\subset
E\\}\subset\operatorname{Gr}_{l}(V)\times\operatorname{Gr}_{k}(V)$. The group
$\operatorname{GL}(V)$ acts transitively on $Z$, and any smooth path
$F(t)=(L(t)\subset E(t))\in Z$ can be lifted to a smooth curve
$g(t)\in\operatorname{GL}(V)$ with $g(0)=\operatorname{Id}$ and
$F(t)=g(t)F(0)$. Thus $E^{\prime}(0):E_{0}\to V/E_{0}$ and
$L^{\prime}(0):L_{0}\to V/L_{0}$ are both projections of $g^{\prime}(0):V\to
V$, and the diagram commutes.

In the other direction, write $\pi_{W}:V\to V/W$ for the natural projection.
it follows by the above that the set of velocity vectors $L^{\prime}(0)$ for
all curves $L(t)\subset E(t)$ is the affine space $\\{\pi_{L_{0}}\circ
T|_{L_{0}}\in\mathrm{Hom}(L_{0},V/L_{0}):T\in\mathfrak{gl}(V),\pi_{E_{0}}\circ
T|_{E_{0}}=E^{\prime}(0)\\}$, which is of dimension
$\binom{k}{2}-\binom{l}{2}-\binom{k-l}{2}=l(k-l)$. 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].

######  Lemma 3.3.

The natural projection $\pi:N^{*}Z_{M}\setminus 0\to T^{*}M\setminus 0$ is a
submersion.

###### Proof.

Let $(p_{t},\xi_{t})$ be a smooth path in $T^{*}M\setminus 0$. We will lift it
to a smooth path $(p_{t},E_{t},\xi_{t},\eta_{t})\in
T^{*}(M\times\operatorname{Gr}_{d-k}(V))$ such that $p_{t}\in E_{t}$, and
$(\xi_{t},\eta_{t})\in N^{*}_{p_{t},E_{t}}Z_{M}$. Now for $v\in T_{p}M$, $B\in
T_{E}\operatorname{Gr}_{d-k}(V)=\mathrm{Hom}(E,V/E)$, we have by Lemma 3.1
(applied with $l=1,L_{0}=\mathbb{R}p$) that $(v,B)\in T_{p,E}Z_{M}$ if and
only if $v+E=B(p)$.

Hence

$N^{*}_{p,E}Z_{M}=\\{(\xi,\eta)\in T_{p}^{*}M\times
T_{E}^{*}\operatorname{Gr}_{d-k}(V):\langle\xi,v\rangle+\langle\eta,B\rangle=0\text{
whenever }v+E=B(p)\\}.$

Fix a Euclidean structure on $V$, inducing Euclidean structures on the spaces
$\mathrm{Hom}(E_{t},V/E_{t})$. Let us choose some $E_{t}$ such that $p_{t}\in
E_{t}$, and $T_{p_{t}}(M\cap E_{t})\subset\operatorname{Ker}(\xi_{t})$, which
evidently can be done. Consider the linear subspace

$W_{t}=\\{B\in T_{E_{t}}\operatorname{Gr}_{d-k}(V):B(p_{t})\in
T_{p_{t}}M+E_{t}\\},$

and for each $B\in W_{t}$ define
$\langle\eta_{t},B\rangle=-\langle\xi_{t},B(p_{t})\rangle$, which is well-
defined as $T_{p_{t}}M\cap E_{t}\subset\ker\xi_{t}$. Extend $\eta_{t}$ by zero
to $W_{t}^{\perp}$. It follows that $(\xi_{t},\eta_{t})\in
N^{*}_{p_{t},E_{t}}Z_{M}$, completing the proof. ∎

It follows by [7, Corollary 4.1.7] that the Radon transforms with respect to
the Euler characteristic,
$\mathcal{R}_{M}=(\tau_{M})_{*}\pi_{M}^{*}:\mathcal{V}_{c}^{-\infty}(M)\to\mathcal{V}^{-\infty}(\operatorname{Gr}_{d-k}(V))$
and
$\mathcal{R}_{M}^{T}=(\pi_{M})_{*}\tau_{M}^{*}:\mathcal{V}^{\infty}(\operatorname{Gr}_{d-k}(V))\to\mathcal{V}^{\infty}(M)$,
are well-defined and continuous.

######  Definition 3.4.

For any $\phi\in\mathcal{V}_{c}^{-\infty}(M)$, let $\widehat{\phi}\in
C^{-\infty}(\operatorname{Gr}_{d-k}(V))$ be the defining current of
$\mathcal{R}_{M}\phi$ (on the base manifold). Equivalently, using [6,
Proposition 7.3.6] we have

$\widehat{\phi}=[\mathcal{R}_{M}\phi]\in\mathcal{W}_{0}^{-\infty}(\operatorname{Gr}_{d-k}(V))/\mathcal{W}_{1}^{-\infty}(\operatorname{Gr}_{d-k}(V))=C^{-\infty}(\operatorname{Gr}_{d-k}(V)).$

######  Remark 3.5.

It is false in general that $\widehat{\phi}$ is a smooth function when $\phi$
is a smooth valuation, see Remark 3.10.

######  Definition 3.6.

The Crofton map
$\operatorname{Cr}_{M}:\mathcal{M}^{\infty}(\operatorname{Gr}_{d-k}(V))\to\mathcal{W}_{k}^{\infty}(M)$
is the restriction of $\mathcal{R}_{M}^{T}$ to
$\mathcal{M}^{\infty}(\operatorname{Gr}_{d-k}(V))$. More explicitly,

$\operatorname{Cr}_{M}(\mu)(X)=\int_{\operatorname{Gr}_{d-k}(V)}\widehat{\chi}_{X}(E)d\mu(E),\quad
X\in\mathcal{P}(M).$

We will see in Proposition 3.12 below that $\widehat{\chi}_{X}(E)=\chi(X\cap
E)$.

###  3.2. Distributional Crofton measures

To allow distributional Crofton measures, it seems essential to require that
all intersections $E\cap M$ are transversal, for
$E\in\operatorname{Gr}_{d-k}(V)$. This is easily seen to be equivalent, for
any $k>0$, to having $\mathbb{R}x\oplus T_{x}M=V$ for all $x\in M$. In
particular $\dim V=\dim M+1=m+1$. By further restricting attention to
connected manifolds, we deduce that $M$ is a hypersurface that is locally
diffeomorphic to an open subset of $\mathbb{P}_{+}(V)$ through the radial
projection. In other words, $M$ is locally a strictly star-shaped hypersurface
around the origin.

######  Proposition 3.7.

  1. i)

For all $0\leq k\leq m$ and $\psi\in\mathcal{V}^{\infty}_{c}(M)$, it holds
that $E\mapsto\psi(E\cap M)$ is a smooth function on
$\operatorname{Gr}_{m+1-k}(V)$.

  2. ii)

The image in $C^{-\infty}(\operatorname{Gr}_{m+1-k}(V))$ of this function
equals $\widehat{\psi}$.

###### Proof.

  1. i)

Let us first show $E\mapsto\psi(E\cap M)$ 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 $\mathbb{P}_{+}(V)$, which we henceforth identify with $M$.

By Boman’s theorem [18], it suffices to prove that $\psi(E_{t}\cap M)$ is a
smooth function of $t\in(-\epsilon,\epsilon)$ for all smooth curves
$E_{\bullet}:(-\epsilon,\epsilon)\to\operatorname{Gr}_{m+1-k}(V)$. 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 $g_{t}\in\mathrm{GL}(V)$ with
$g_{0}=\operatorname{Id}$ and $E_{t}=g_{t}E_{0}$. Then $\psi(E_{t}\cap
M)=g_{t}^{*}\psi(E_{0}\cap M)$ for sufficiently small $t$ such that
$g_{t}(\operatorname{Supp}(\psi))\subset M$, establishing the first part.

  2. ii)

Let us check $\psi(\bullet\cap M)=\widehat{\psi}$ in
$C^{-\infty}(\operatorname{Gr}_{m+1-k}(V))$. Take
$\mu\in\mathcal{M}^{\infty}(\operatorname{Gr}_{m+1-k}(V))$, and write

$\mu=\int_{\operatorname{Gr}_{m+1-k}(V)}\delta_{E}d\mu(E)=\int_{\operatorname{Gr}_{m+1-k}(V)}\chi_{\\{E\\}}d\mu(E)\in\mathcal{V}^{\infty}(\operatorname{Gr}_{m+1-k}(V)).$

Claim. $\tau_{M}^{-1}E\pitchfork\pi_{M}^{*}\psi$.

To see this, write $Z=Z_{M}$ and identify $W:=Z\times_{M}\mathbb{P}_{M}$ with
its image in $\mathbb{P}_{Z}$ under $d\pi_{M}^{*}$. Explicitly,
$W|_{(x,E)}=\mathbb{P}_{+}(\mathrm{Ker}(d_{(x,E)}\pi_{M})^{\perp})$, so $W$ is
the union of the conormal bundles to all fibers of $\pi_{M}$. It follows from
[7, Proposition 3.3.3] that
$\operatorname{WF}(\pi_{M}^{*}\psi)\subset(\emptyset,N^{*}W)$. By Proposition
2.1, it suffices to check that two intersections in $\mathbb{P}_{Z}$ are
transversal: $\pi^{-1}(\tau_{M}^{-1}E)\pitchfork W$ and
$N^{*}(\tau_{M}^{-1}E)\pitchfork W$.

Denote $z=(x,E)$, let $(z,\zeta)$ be an intersection point. The first
intersection is easy to analyze: $T_{z,\zeta}\pi^{-1}(\tau_{M}^{-1}E)$
contains all vertical directions of $\mathbb{P}_{Z}$, while $T_{z,\zeta}W$
contains all horizontal directions.

To analyze the second intersection, we lift all manifolds from
$\mathbb{P}_{Z}$ to $T^{*}Z$, and retain all notation for the corresponding
objects. As in the previous case, the image of $T_{z,\zeta}W$ under the
natural projection $\pi:T_{z,\zeta}T^{*}Z\to T_{z}Z$ is all of $T_{z}Z$, and
so it suffices to show $T_{z,\zeta}(T_{z}^{*}Z)\subset
T_{z,\zeta}W+T_{z,\zeta}N^{*}(\tau_{M}^{-1}E)$.

Since $N_{z}^{*}(\pi_{M}^{-1}z)\subset W$, it suffices to show that

$T_{z,\zeta}(T_{z}^{*}Z)\subset
T_{z,\zeta}N_{z}^{*}(\pi_{M}^{-1}x)+T_{z,\zeta}N_{z}^{*}(\tau_{M}^{-1}E),$

which is the same as

$T_{z}^{*}Z\subset
N_{z}^{*}(\pi_{M}^{-1}x)+N_{z}^{*}(\tau_{M}^{-1}E)=N_{z}^{*}(T_{z}\pi_{M}^{-1}x\cap
T_{z}\tau_{M}^{-1}E).$

The proof of the claim is completed by noting that the intersection
$T_{z}\pi_{M}^{-1}x\cap T_{z}\tau_{M}^{-1}E$ is trivial.

Consider the set

$X:=\left\\{(E,[\xi]):E\in\operatorname{Gr}_{m+1-k}(V),[\xi]\in\operatorname{WF}(\chi_{\tau_{M}^{-1}E})\right\\}\subset\operatorname{Gr}_{m+1-k}(V)\times\mathbb{P}_{+}\mathbb{P}_{Z}.$

We claim that it is compact. If $X\subset\bigcup_{i\in I}U_{i}$ is an open
cover, then for each $E\in\operatorname{Gr}_{m+1-k}(V)$ we find a finite
subcover $X_{E}\subset\bigcup_{i\in I_{E}}U_{i}$ of the compact set

$X_{E}:=X\cap(\\{E\\}\times\mathbb{P}_{+}\mathbb{P}_{Z})=\\{E\\}\times\operatorname{WF}(\chi_{\tau_{M}^{-1}E}).$

The map $g\mapsto X_{gE}$ is $\operatorname{GL}(V)$-equivariant, hence there
exists some open neighborhood $V_{E}\subset\operatorname{Gr}_{m+1-k}(V)$ of
$E$ such that $X_{E^{\prime}}\subset\bigcup_{i\in I_{E}}U_{i}$ for all
$E^{\prime}\subset V_{E}$. Now $\operatorname{Gr}_{m+1-k}(V)$ is compact,
hence finitely many $V_{E_{j}}$ cover $\operatorname{Gr}_{m+1-k}(V)$. Then
$X\subset\bigcup_{j}\bigcup_{i\in E_{j}}U_{i}$ is a finite subcover, proving
the claim. The image of $X$ in $\mathbb{P}_{+}\mathbb{P}_{Z}$ is then a
compact set disjoint from $\operatorname{WF}(\pi_{M}^{*}\psi)$.

Thus we can find a closed cone $\Gamma\subset
T^{*}\mathbb{P}_{Z}\setminus\underline{0}$ such that for all
$E\in\operatorname{Gr}_{m+1-k}(V)$,
$\chi_{\tau_{M}^{-1}E}\in\mathcal{V}^{-\infty}_{(\emptyset,\Gamma)}(Z)$, and
$\pi_{M}^{*}\psi$ acts as a continuous functional on the latter space. Thus we
can write

$\displaystyle\langle\widehat{\psi},\mu\rangle=\langle\pi_{M}^{*}\psi,\tau_{M}^{*}\int_{\operatorname{Gr}_{m+1-k}(V)}\chi_{\\{E\\}}d\mu(E)\rangle=\int_{\operatorname{Gr}_{m+1-k}(V)}\langle\pi_{M}^{*}\psi,\chi_{\tau_{M}^{-1}(E)}\rangle
d\mu(E).$

It remains to check that

$\langle\pi_{M}^{*}\psi,\chi_{\tau_{M}^{-1}(E)}\rangle=\psi(E\cap M).$ (3)

For a compact submanifold with boundary $A\subset M$ that is transversal to
$E\cap M$, we have by [8, Theorem 5],

$\langle\pi_{M}^{*}\chi_{A},\chi_{\tau^{-1}(E)}\rangle=\chi(\pi_{M}^{-1}A\cap\tau_{M}^{-1}(E))=\chi(A\cap
E)=\chi_{A}(E\cap M).$

It follows by linearity that any smooth valuation of the form
$\psi=\int_{\mathcal{A}}\chi_{A}d\nu(A)$, where $\mathcal{A}$ is a family of
submanifolds $A$ as above and $\nu$ a smooth measure, satisfies (3). This
family $\mathbf{Cr}_{E}$ of valuations spans a dense subset in
$\mathcal{V}^{\infty}(M)$. Indeed, we may approximate $\chi_{A}$ in
$\mathcal{V}^{-\infty}(M)$ by a sequence in $\mathbf{Cr}_{E}$ for any $A$
transversal to $E\cap M$. Were $\mathbf{Cr}_{E}$ not dense, by Alesker-
Poincaré duality one could find a non-zero smooth valuation $\phi$
annihilating $\mathbf{Cr}_{E}$, and thus also vanishing on all submanifolds
with boundary that are transversal to $E\cap M$. By the genericity of
transversality and continuity, $\phi$ would vanish on all submanifolds with
boundary. But this is impossible by [12]. It follows that equality in (3)
holds for all $\psi$.

∎

######  Corollary 3.8.

The map $\operatorname{Gr}_{m+1-k}(V)\to\mathcal{V}^{-\infty}(M)$,
$E\mapsto\chi_{E\cap M}$ is smooth, and for
$\mu\in\mathcal{M}^{\infty}(\operatorname{Gr}_{m+1-k}(V))$ it holds that
$\operatorname{Cr}(\mu)=\int_{\operatorname{Gr}_{m+1-k}(V)}\chi_{E\cap
M}d\mu(E)$.

######  Proposition 3.9.

The map
$\operatorname{Cr}:\mathcal{M}^{\infty}(\operatorname{Gr}_{m+1-k}(V))\to\mathcal{W}_{k}^{\infty}(M)$
extends to a continuous map
$\operatorname{Cr}:\mathcal{M}^{-\infty}(\operatorname{Gr}_{m+1-k}(V))\to\mathcal{W}_{k}^{-\infty}(M)$,
by setting, for all $\psi\in\mathcal{V}^{\infty}_{c}(M)$,

$\left\langle\operatorname{Cr}(\mu),\psi\right\rangle:=\int_{\operatorname{Gr}_{m+1-k}(V)}\psi(E\cap
M)d\mu(E).$

###### Proof.

The right hand side is well-defined for a generalized measure $\mu$, since the
function $E\mapsto\psi(E\cap M)$ is smooth by Proposition 3.7. Take
$\mu\in\mathcal{M}^{\infty}(\operatorname{Gr}_{m+1-k}(V))$,
$\psi\in\mathcal{V}_{c}^{\infty}(M)$. To verify this new definition extends
the smooth one, we ought to check that

$\langle(\pi_{M})_{*}\tau_{M}^{*}\mu,\psi\rangle=\int_{\operatorname{Gr}_{m+1-k}(V)}\psi(E\cap
M)d\mu(E),$

which is the content of Corollary 3.8. Continuity is equally evident. ∎

######  Remark 3.10.

It is tempting to define $\operatorname{Cr}(\mu)$ as a Radon transform:
$\operatorname{Cr}(\mu)=\mathcal{R}_{M}^{T}\mu$, 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.

###  3.3. Functorial properties of Crofton measures.

The following is a partial summary of the results of [22, Appendix B] (adapted
from the affine to the linear Grassmannian), whereto we refer the reader for
further details.

Let $j:U^{r}\hookrightarrow V^{d}$ be an inclusion of a linear subspace. There
is then a well-defined operation of restriction

$j^{*}:\mathcal{M}^{\infty}(\operatorname{Gr}_{k}(V))\to\mathcal{M}^{\infty}(\operatorname{Gr}_{k-(d-r)}(U)),$

which is the pushforward under the (almost everywhere defined) map
$J_{U}:E\mapsto j^{-1}(E)=E\cap U$.

Let $S_{U}\subset\operatorname{Gr}_{k}(V)$ be the collection of subspaces
intersecting $U$ non-generically, and fix a closed cone $\Gamma\subset
T^{*}\operatorname{Gr}_{k}(V)\setminus 0$ such that $\Gamma\cap
N^{*}S_{U}=\emptyset$. Given $k\geq d-r$, let
$\mathcal{M}^{-\infty}_{\Gamma}(\operatorname{Gr}_{k}(V))$ denote the set of
generalized measures (distributions) $\mu$ whose wave front sets lie in
$\Gamma$, equipped with the Hörmander topology.

The map $j^{*}$ extends as a sequentially continuous map

$j^{*}:\mathcal{M}_{\Gamma}^{-\infty}(\operatorname{Gr}_{k}(V))\to\mathcal{M}^{-\infty}(\operatorname{Gr}_{k-(d-r)}(U)).$

Similarly, if $\pi:V\to W$ is a quotient map, there is a natural pushforward
operation

$\pi_{*}:\mathcal{M}^{\infty}(\operatorname{Gr}_{k}(V))\to\mathcal{M}^{\infty}(\operatorname{Gr}_{k}(W)),$

which is the pushforward under the (almost everywhere defined) map
$\Pi_{W}:E\mapsto\pi(E)$. It extends to distributions whose wave front sets
are disjoint from the conormal cycle of the collection of subspaces
intersecting $\operatorname{Ker}\pi$ 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.

######  Proposition 3.11.

Take a submanifold $M^{r}\subset V^{d}$, a subspace $j:U\hookrightarrow V$
such that $Z:=M\cap j(U)$ is a submanifold, and a distribution
$\mu\in\mathcal{M}_{\Gamma}^{-\infty}(\operatorname{Gr}_{d-k}(V))$. Assume
$\operatorname{Cr}_{M}(\mu)$ is transversal to $Z$ in the sense of [7,
Definition 3.5.2]. Then
$\operatorname{Cr}_{M}(\mu)|_{Z}=\operatorname{Cr}_{Z}(j^{*}\mu)$.

###### Proof.

Choose an approximate identity
$\rho_{i}\in\mathcal{M}^{\infty}(\textrm{GL}(V))$ as $i\to\infty$, and set
$\mu_{i}=\mu\ast\rho_{i}\in\mathcal{M}^{\infty}(\operatorname{Gr}_{d-k}(V))$.
For all $A\in\mathcal{P}(Z)$ we have

$\operatorname{Cr}_{M}(\mu_{i})(A)=\int_{\operatorname{Gr}_{d-k}(V)}\chi(A\cap
E)d\mu_{i}(E)=\int_{\operatorname{Gr}_{r-k}(U)}\chi(A\cap
E)d((J_{U})_{*}\mu_{i})(E),$

and therefore
$\operatorname{Cr}_{M}(\mu_{i})|_{Z}=\operatorname{Cr}_{Z}(j^{*}\mu_{i})$. The
restriction of valuations to a submanifold is continuous in the Hörmander
topology on the space of valuations with wave front set contained in
$\textrm{WF}(\operatorname{Cr}_{M}(\mu))$, see [7, Claim 3.5.4]. Thus the left
hand side weakly converges to $\operatorname{Cr}_{M}(\mu)|_{Z}$. The right
hand side weakly converges to $\operatorname{Cr}_{Z}(\mu)$. ∎

###  3.4. Applying generalized Crofton formulas to subsets

Let $M^{m}\subset V=\mathbb{R}^{m+1}$ be a strictly star-shaped hypersurface
around the origin. Given $A\in\mathcal{P}(M)$ and a Crofton distribution
$\mu\in\mathcal{M}^{-\infty}(\operatorname{Gr}_{m+1-k}(V))$, we would like to
evaluate $\operatorname{Cr}(\mu)$ on $A$ using an explicit Crofton integral,
whenever $A\pitchfork\operatorname{Cr}(\mu)$.

The following proposition provides some a-priori regularity for
$\widehat{\chi}_{A}$.

######  Proposition 3.12.

For $A\in\mathcal{P}(M)$, it holds that $\chi(A\cap\bullet)\in
L^{1}(\operatorname{Gr}_{m+1-k}(V))$, is finite and locally constant on
$W_{A}:=\\{E:E\pitchfork A\\}$. Furthermore,
$\widehat{\chi}_{A}=\chi(A\cap\bullet)$.

###### Proof.

Let us first check that $\chi(A\cap\bullet)\in
L^{1}(\operatorname{Gr}_{m+1-k}(V))$. Fix a Euclidean structure on $V$ and
identify $M$ with the unit sphere. By [16, Lemma A.2], for a fixed
$E_{0}\in\operatorname{Gr}_{m+1-k}(V)$ we have $[g\mapsto\chi(A\cap
gE_{0})]\in L^{1}(\operatorname{SO}(V))$. Let $dg,dE$ be the Haar measures on
$\operatorname{SO}(V)$ and $\operatorname{Gr}_{m+1-k}(V)$ respectively, and
$p:\operatorname{SO}(V)\to\operatorname{Gr}_{m+1-k}(V)$ given by $g\mapsto
gE_{0}$. Then $p_{*}(\chi(A\cap gE_{0})dg)=\chi(A\cap E)dE$, and so
$\chi(A\cap E)$ is integrable. It is evidently finite and locally constant on
$W_{A}$

It remains to check that $\widehat{\chi}_{A}=\chi(A\cap\bullet)$. Take an
approximate identity
${\rho}_{j}\in\mathcal{M}^{\infty}(\operatorname{SO}(V))$, which for
convenience we assume invariant under inversion.

Consider the convolution
$\phi_{j}:=\chi_{A}\ast{\rho_{j}}\in\mathcal{V}^{-\infty}(M)$. As
$\operatorname{SO}(V)$ is transitive on $M$ and $\mathbb{P}_{M}$, it follows
that the defining currents of $\phi_{j}$ are smooth, and therefore
$\phi_{j}\in\mathcal{V}^{\infty}(M)$.

By [16, Theorem A.1],
$\tilde{\phi}_{j}:=\int_{\operatorname{SO}(V)}\chi(gA\cap\bullet)d{\rho}_{j}(g)$
is a well-defined smooth valuation. Let us show that
$\tilde{\phi}_{j}=\phi_{j}$. Take $\psi\in\mathcal{V}_{c}^{\infty}(M)$ and
compute:

$\langle\tilde{\phi}_{j},\psi\rangle=\int_{\operatorname{SO}(V)}\psi(gA\cap
M)d{\rho}_{j}(g)=\int_{\operatorname{SO}(V)}\psi(gA)d{\rho}_{j}(g)$

by [16], while

$\langle\phi_{j},\psi\rangle=\langle\chi_{A},\psi\ast{\rho}_{j}\rangle=(\psi\ast\nu_{j})(A)=\int_{\operatorname{SO}(V)}\psi(gA)d{\rho}_{j}(g).$

Equality now follows by Alesker-Poincaré duality. We thus have the following
equalities of functions on $\operatorname{Gr}_{m+1-k}(V)$:

$\phi_{j}({\bullet}\cap M)=\tilde{\phi}_{j}({\bullet}\cap
M)=\int_{\operatorname{SO}(V)}\chi(gA\cap{\bullet})d{\rho}_{j}(g)=\chi(A\cap\bullet)\ast{\rho}_{j},$

where the right hand side is the convolution of $\chi(A\cap\bullet)\in
L^{1}(\operatorname{Gr}_{m+1-k}(V))$ with ${\rho}_{j}$. It follows that
$\phi_{j}(E\cap M)\to\chi(A\cap E)$ in $L^{1}(\operatorname{Gr}_{m+1-k}(V))$.

Fix $\mu\in\mathcal{M}^{\infty}(\operatorname{Gr}_{m+1-k}(V))$. By Proposition
3.7 and $\operatorname{GL}(V)$-equivariance,

$\displaystyle\langle\widehat{\chi}_{A},\mu\rangle=\lim_{j\to\infty}\langle\widehat{\chi}_{A}\ast{\rho}_{j},\mu\rangle=\lim_{j\to\infty}\langle\widehat{\chi_{A}\ast{\rho}_{j}},\mu\rangle$
$\displaystyle=\lim_{j\to\infty}\int_{\operatorname{Gr}_{m+1-k}(V)}\phi_{j}(E\cap
M)d\mu(E)$ $\displaystyle=\int_{\operatorname{Gr}_{m+1-k}(V)}\chi(A\cap
E)d\mu(E),$

and so $\widehat{\chi}_{A}=\chi(A\cap\bullet)$. ∎

######  Definition 3.13.

The _$k$ -Crofton wave front_ of $A\in\mathcal{P}(M)$ is
$\operatorname{Cr}\operatorname{WF}^{k}(A):=\operatorname{WF}(\widehat{\chi}_{A})\subset
T^{*}\operatorname{Gr}_{m+1-k}(V)$.

######  Proposition 3.14.

Assume $\operatorname{Cr}\operatorname{WF}^{k}(A)\cap
s^{*}\operatorname{WF}(\mu)=\emptyset$ where $s$ is the fiberwise antipodal
map, while $A\pitchfork\operatorname{Cr}(\mu)$. Then

$\operatorname{Cr}(\mu)(A)=\int_{\operatorname{Gr}_{m+1-k}(V)}\chi(A\cap
E)d\mu(E).$ (4)

###### Proof.

We identify $M$ with $\mathbb{P}_{+}(V)$. Then $\mathcal{V}^{-\infty}(M)\to
C^{-\infty}(\operatorname{Gr}_{m+1-k}(V))$, $\phi\mapsto\widehat{\phi}$ is
$\operatorname{GL}(V)$-equivariant. Consider the sequence of smooth valuations
$\psi_{j}$ given by $\psi_{j}=\int_{\operatorname{GL}(V)}g^{*}\chi_{A}\cdot
d\rho_{j}(g)$, where $\rho_{j}$ is a compactly supported approximate identity
on $\operatorname{GL}(V)$. Clearly $\psi_{j}\to\chi_{A}$ in the Hörmander
topology of $\mathcal{V}^{-\infty}_{\operatorname{WF}(\chi_{A})}(M)$. By
$\operatorname{GL}(V)$-equivariance we have that

$\widehat{\psi}_{j}=\int_{\operatorname{GL}(V)}g^{*}\widehat{\chi}_{A}\cdot
d\rho_{j}(g)\to\widehat{\chi}_{A}$

in
$C^{-\infty}_{\operatorname{WF}(\widehat{\chi}_{A})}(\operatorname{Gr}_{m+1-k}(V))$.

It holds by Propositions 3.9 and 3.7 that

$\langle\operatorname{Cr}(\mu),\psi_{j}\rangle=\int_{\operatorname{Gr}_{m+1-k}(V)}\psi_{j}(E\cap
M)d\mu(E)=\langle\mu,\widehat{\psi}_{j}\rangle.$

As $j\to\infty$, the left hand side converges to
$\langle\operatorname{Cr}(\mu),\chi_{A}\rangle=\operatorname{Cr}(\mu)(A)$, as
$A\pitchfork\operatorname{Cr}(\mu)$. The right hand side converges to
$\langle\mu,\widehat{\chi}_{A}\rangle$ (since
$\operatorname{WF}(\mu)\cap\operatorname{WF}(\widehat{\chi}_{A})=\emptyset$),
which is the same as $\int_{\operatorname{Gr}_{m+1-k}(V)}\chi(A\cap E)d\mu(E)$
by Proposition 3.12. ∎

Determining $\operatorname{Cr}\operatorname{WF}^{k}(A)$ precisely appears to
be difficult in general. Let us focus on a subset $A\in\mathcal{P}(M)$ which
is either a compact domain with smooth boundary, or a compact hypersurface
without boundary.

For the following, we write $H=H(A)$ for $\partial A$ if $A$ is of full
dimension, and for $A$ when it is a hypersurface. Write $\widehat{E}:=E\cap
M$, and note that $E$ intersects $H$ transversally in $V$ if and only if
$\widehat{E}$ intersects $H$ transversally in $M$. Denote

$\displaystyle\widetilde{B}_{H}$ $\displaystyle:=\\{(x,E)\in
Z_{H}:T_{x}\widehat{E}\subset T_{x}H\\},$ $\displaystyle B_{H}$
$\displaystyle:=\tau_{H}(\widetilde{B}_{H})\subset\operatorname{Gr}_{m+1-k}(V).$

It is not hard to see that $\widetilde{B}_{H}$ is an embedded submanifold of
$Z_{H}$ of dimension

$\dim\widetilde{B}_{H}=\dim H+(m-k)(\dim
H-(m-k))=k(m+1-k)-1=\dim\operatorname{Gr}_{m+1-k}(V)-1.$ (5)

If $(x,E)\in\widetilde{B}_{H}$, we say that $x\in H$ is a tangent point for
$E$. Observe also that $W_{A}=B_{H}^{c}$.

Write $\tilde{\tau}_{H}$ for the restriction
$\tau_{H}|_{\widetilde{B}_{H}}:\widetilde{B}_{H}\to\operatorname{Gr}_{m+1-k}(V)$.
We sometimes write $B^{m+1-k}_{H}$, etc. to specify the dimension.

######  Definition 3.15.

We say that $E\in\operatorname{Gr}_{m+1-k}(V)$ is a _regular tangent_ to $A$
if $\tilde{\tau}_{H}$ is immersive on $\tilde{\tau}_{H}^{-1}(E)$.

Note that if $E\notin B_{H}$ then it is automatically regular.

For a subset $A\subset V$ we denote by $\mathbb{P}(A)$ its image in the
projective space $\mathbb{P}(V)$. The regularity of the tangent is equivalent
to the non-vanishing of the Gauss curvature of the corresponding section, as
follows.

######  Lemma 3.16.

Fix $(p,E)\in\widetilde{B}_{H}$. Choose any line $N\subset T_{p}M\setminus
T_{p}H$, and set $F=E\oplus N$. Then
$\tilde{\tau}_{H}:\widetilde{B}_{H}\to\operatorname{Gr}_{m+1-k}(V)$ is an
immersion at $(p,E)$ if and only if $\mathbb{P}(H\cap F)\subset\mathbb{P}(F)$
has non-degenerate second fundamental form at $p$.

In particular, all tangents to $A$ are regular if and only if
$\mathbb{P}(H)\subset\mathbb{P}(V)$ is a strictly convex hypersurface.

###### Proof.

Let us sketch the argument, see [36, Lemma 1(ii)] for details. Clearly
$d\tilde{\tau}_{H}$ is injective on the subspace of directions where $p$ moves
transversally to $E$. Namely, fixing any subspace $\overline{E}\subset T_{p}H$
such that $E\oplus\overline{E}=T_{p}H\oplus\mathbb{R}p$, $d\tilde{\tau}_{H}$
is injective on $\\{(v,A)\in T_{p}H\times
T_{E}\operatorname{Gr}_{m+1-k}(V):v\in\overline{E}\\}\cap T\widetilde{B}_{H}$.
That injectivity is retained as the remaining directions are added,
corresponds to the non-degeneracy of the Gauss map of the section $H\cap F$. ∎

We now describe the Crofton wave front near regular tangents. For an immersed
manifold $X\subset Y$, by $N^{*}X$ we understand the union of all conormal
spaces to individual embedded parts of $X$.

######  Proposition 3.17.

Assume $E_{0}\in B^{m+1-k}_{H}$ is a regular tangent. Then
$\operatorname{Cr}\operatorname{WF}^{k}_{E_{0}}(A)\subset N^{*}_{E_{0}}B_{H}$.

That $\operatorname{Cr}\operatorname{WF}^{k}_{E_{0}}(A)$ is contained in the
sum of the conormal spaces of the embedded parts of $B_{H}$ follows from the
fact that $\widehat{\chi}_{A}$ 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 $\widehat{\chi}_{A}$.

###### Proof.

In the following, by a ball (centered at a point) we mean a compact
contractible neighborhood (of the point) with smooth boundary. Since
$\widetilde{B}_{H}$ is a submanifold of $Z_{H}$, by assumption
$B_{H}\subset\operatorname{Gr}_{m+1-k}(V)$ is an immersed submanifold in a
neighborhood around $E_{0}$, which is a hypersurface by (5).

The preimage $\tilde{\tau}_{H}^{-1}(E_{0})$ must be finite, or else we could
find a sequence of distinct points $(q_{j},E_{0})\in\widetilde{B}_{H}$, which
then has a limit point $(q_{0},E_{0})$, and $\tilde{\tau}_{H}$ would fail to
be injective in a neighborhood of $(q_{0},E_{0})$, contradicting the assumed
immersivity of $\tilde{\tau}_{H}$ there. Denote
$\tilde{\tau}_{H}^{-1}(E_{0})=\\{(q_{j},E_{0}),1\leq j\leq N\\}$.

We can now find a ball $W\subset\operatorname{Gr}_{m+1-k}(V)$ centered at
$E_{0}$, such that $B_{H}\cap W$ is the finite union of embedded hypersurfaces
$F_{j}$, each diffeomorphic to a Euclidean ball, with $E_{0}\in F_{j}$ and
$\partial F_{j}\subset\partial W$ for all $j$. Note that we have no control on
how these hypersurfaces intersect each other. Denote by $C_{j}^{\pm}$ the
connected components of $W\setminus F_{j}$. The indices are matched by
requiring that a neighborhood of $(q_{j},E_{0})\in\widetilde{B}_{H}$ is mapped
to $F_{j}$ by $\tilde{\tau}_{H}$.

Fix small balls $K_{j}\subset M$ around $q_{j}$ such that

$\tilde{\tau}_{H}:\pi_{H}^{-1}(K_{j})\cap\tilde{\tau}_{H}^{-1}(W)\to F_{j}$

is an embedding, $\partial K_{j}\pitchfork H$ and $\partial
K_{j}\pitchfork\widehat{E}_{0}$ in $M$, and $(\widehat{E}_{0}\cap
H)\pitchfork(\partial K_{j}\cap H)$ in $H$. We may further assume all $K_{j}$
are pairwise disjoint, and $H\cap K_{j}$ is diffeomorphic to a Euclidean ball.

Denote by $\frac{1}{2}K_{j}$ a smaller ball centered at $q_{j}$. Taking $W$
sufficiently small, we may assume that for any $E\in W$,
$\widehat{E}\pitchfork(H\setminus\cup_{j}\frac{1}{2}K_{j})$, and for all $j$
we have $\widehat{E}\pitchfork\partial K_{j}$ in $M$ and $(\widehat{E}\cap
H)\pitchfork(\partial K_{j}\cap H)$ in $H$.

For $\epsilon\in\\{\pm\\}^{N}$, denote
$C_{\epsilon}=\cap_{j=1}^{N}C_{j}^{\epsilon_{j}}$. Recall that
$\widehat{\chi}_{A}$ is locally constant on $W_{A}=B_{H}^{c}$, and so is
constant on any connected component of a non-empty set $C_{\epsilon}$. Let us
show there are integers $e_{j}=e_{j}(E_{0})$ such that for any
$\epsilon,\epsilon^{\prime}\in\\{\pm\\}^{N}$ and any connected components
$C\subset C_{\epsilon}$, $C^{\prime}\subset C_{\epsilon^{\prime}}$ one has

$\widehat{\chi}_{A}|_{C^{\prime}}-\widehat{\chi}_{A}|_{C}=\sum_{j:\epsilon_{j}<\epsilon_{j}^{\prime}}e_{j}-\sum_{j:\epsilon_{j}^{\prime}<\epsilon_{j}}e_{j}.$

For $E\in W$, denote $\Sigma_{j}(E):=\widehat{E}\cap\partial K_{j}\cap H$. As
it is the transversal intersection of $\widehat{E}\cap H$ and $\partial
K_{j}\cap H$ in $H$, it is a closed manifold of dimension $(m-k-2)$, and
$\chi(\Sigma_{j}(E))$ is independent of $E\in W$.

Let us distinguish the two cases under consideration. Assume first $A=H$ is a
hypersurface. Since $K_{i}\cap K_{j}=\emptyset$, we have

$\mathbbm{1}_{M}=\prod_{j=1}^{N}(\mathbbm{1}_{K_{j}}+\mathbbm{1}_{\overline{K_{j}^{c}}}-\mathbbm{1}_{\partial
K_{j}})=\sum_{j=1}^{N}(\mathbbm{1}_{K_{j}}-\mathbbm{1}_{\partial
K_{j}})+\mathbbm{1}_{\overline{K^{c}}},$

with $K:=\bigcup_{j=1}^{N}K_{j}$. Hence for $E\in W\setminus B_{H}$ we have

$\chi(E\cap A)=\sum_{j=1}^{N}\chi(E\cap K_{j}\cap
A)-\sum_{j=1}^{N}\chi(\Sigma_{j}(E))+\chi(E\cap\overline{K^{c}}\cap A).$

The last summand is constant on $W$ by construction. Consequently, for $E\in
C$, $E^{\prime}\in C^{\prime}$ we have

$\widehat{\chi}_{A}(E^{\prime})-\widehat{\chi}_{A}(E)=\sum_{j=1}^{N}\left(\chi(E^{\prime}\cap
A\cap K_{j})-\chi(E\cap A\cap K_{j})\right).$

The function $\chi(\bullet\cap A\cap K_{j})$ is locally constant on
$W\setminus F_{j}$, and it remains to define

$e_{j}:=\chi(\bullet\cap A\cap K_{j})|_{C_{j}^{+}}-\chi(\bullet\cap A\cap
K_{j})|_{C_{j}^{-}}.$

The case of full-dimensional $A$ is only slightly more involved. If $(m-k)$ is
odd, we have $\widehat{\chi_{A}}=\frac{1}{2}\widehat{\chi_{\partial A}}$,
reducing to the previous case. Thus assume $(m-k)$ is even.

Write as before, whenever $E\in W\setminus B_{H}$,

$\chi(E\cap A)=\sum_{j=1}^{N}\chi(E\cap K_{j}\cap
A)-\sum_{j=1}^{N}\chi(E\cap\partial K_{j}\cap
A)+\chi(E\cap\overline{K^{c}}\cap A).$

Note that for $E\in W\setminus B_{H}$, all intersections are manifolds with
corners. We have $\chi(E\cap\partial K_{j}\cap
A)=\frac{1}{2}\chi(E\cap\partial K_{j}\cap\partial
A)=\frac{1}{2}\chi(\Sigma_{j}(E))$, thus it is constant in $W$.

Set

$S_{j}:=\widehat{E}\cap\partial K_{j},\quad S:=\widehat{E}\cap\partial A.$

$S_{j}$ is a transversal intersection in $M$ for all $E\in W$ and hence a
smooth hypersurface, while $S$ is given by a transversal intersection in $M$
and hence smooth for $E\in W\setminus B_{H}$. Moreover, $S$ is a smooth
hypersurface outside of $\frac{1}{2}K_{j}$ for all $E\in W$.

We claim that the intersection $S_{j}\cap S=\Sigma_{j}(E)$ is transversal in
$\widehat{E}$ for all $E\in W$. For if the intersection is not transversal at
$x$, then $T_{x}(\widehat{E}\cap\partial K_{j})=T_{x}(\widehat{E}\cap\partial
A)$. But by assumption $\widehat{E}\cap\partial A$ and $\partial
K_{j}\cap\partial A$ intersect transversally in $\partial A$, in particular

$T_{x}(\widehat{E}\cap\partial A)+T_{x}(\partial K_{j}\cap\partial
A)=T_{x}\partial A.$

In conjunction with the previous equality, we get $T_{x}\partial A\subset
T_{x}\partial K_{j}$, which is false.

Let $X,Y\subset P$ be smooth domains in a manifold $P$, and assume $\partial
X\pitchfork\partial Y$ and $X$ is compact. Let $Z\subset X\cap Y$ be the
closure of a connected component of $X\cap Y$. Then $\chi(Z)$ is constant as
$X,Y$ are perturbed while maintaining transversality.

Taking $P=\widehat{E}$, $X=E\cap A$ with $\partial X=S$ (which is a manifold
for $E\in W\setminus B_{H}$), and $Y=E\cap K_{j}$ with $\partial Y=S_{j}$ we
get that $\chi(E\cap A\cap K_{j})$ is locally constant in $W\setminus B_{H}$.
Taking $X=E\cap A$ with $\partial X=S$ (which is a manifold outside
$\bigcup_{j}\frac{1}{2}K_{j}$ for all $E\in W$), $Y=E\cap\overline{K^{c}}$
with $\partial Y=\bigcup_{j}S_{j}$, it follows that $\chi(E\cap
A\cap\overline{K^{c}})$ is constant in $W$.

It follows in both cases that for $E\in W$, $\widehat{\chi}_{A}(E)$ is a
linear combination of the indicator functions of the complements of the
hypersurfaces $F_{j}$. Therefore
$\operatorname{WF}_{E_{0}}(\widehat{\chi}_{A})\subset\bigcup_{j}N^{*}F_{j}$,
concluding the proof.

∎

##  4\. The Crofton wave front of LC-regular hypersurfaces

Let $(W,Q)$ be a vector space equipped with a quadratic form. We denote by
$\Lambda_{k}^{\nu}(W)\subset\operatorname{Gr}_{k}(W)$ the collection of
subspaces $E\subset W$ where $Q|_{E}$ has nullity $\nu$. We will need to
describe those sets in several cases.

######  Proposition 4.1.

Assume $(V,Q)$ has dimension $d$.

  1. i)

If $Q$ is non-degenerate, then
$\Lambda_{k}^{\nu}(V)\subset\operatorname{Gr}_{k}(V)$ is a submanifold of
dimension

$\dim\Lambda_{k}^{\nu}(V)=k(d-k)-\binom{\nu+1}{2}.$ (6)

Writing $E_{0}:=E\cap E^{Q}$, we have

$T_{E}\Lambda_{k}^{\nu}(V)=\\{A\in\operatorname{Hom}(E,V/E):Q(Au,u)=0,\forall
u\in E_{0}\\}.$ (7)

  2. ii)

If $Q$ has nullity $1$ and $E\in\Lambda_{k}^{\nu}(V)$ is such that
$\operatorname{Ker}Q\cap E=\\{0\\}$, then $\Lambda_{k}^{\nu}(V)$ is a manifold
near $E$ whose dimension is given by Eq. (6).

###### Proof.

  1. i)

See [13, Proposition 4.2].

  2. ii)

Write $L_{0}:=\operatorname{Ker}(Q)$. Consider $W:=V\oplus\mathbb{R}$, and
extend $Q$ as a non-degenerate quadratic form $\widetilde{Q}$ on $W$. Let us
verify that the submanifolds $\Lambda_{k}^{\nu}(W)$ and
$\operatorname{Gr}_{k}(V)$ intersect transversally in
$\operatorname{Gr}_{k}(W)$ at $E$.

As $L_{0}\not\subset E$, also $E^{\widetilde{Q}}\not\subset
L_{0}^{\widetilde{Q}}=V$. Thus we can find a line $L\subset
E^{\widetilde{Q}}\setminus V$. Now any linear map $A:E\to W/E$ decomposes as a
sum $A=A_{1}+A_{2}$ with
$A_{1}\in\operatorname{Hom}(E,V/E)=T_{E}\operatorname{Gr}_{k}(V),A_{2}\in\operatorname{Hom}(E,(E+L)/E)\subset\operatorname{Hom}(E,W/E)=T_{E}\operatorname{Gr}_{k}(W)$.
Since $\widetilde{Q}(A_{2}u,v)=0$ for all $u,v\in E_{0}$, we have $A_{2}\in
T_{E}\Lambda^{\nu}_{k}(W)$ by (7).

This proves the claim. As
$\Lambda_{k}^{\nu}(V)=\Lambda_{k}^{\nu}(W)\cap\operatorname{Gr}_{k}(V)$, it is
a manifold near $E$. The formula for the dimension then follows from the
previous case.

######  Corollary 4.2.

Let $B$ be a smooth manifold, and $W$ a real vector bundle of rank $d$ over
$B$. Let $Q\in\Gamma(B,\operatorname{Sym}^{2}(W^{*}))$ be a smooth field of
quadratic forms, of nullity at most $1$ for all $x$. Let
$\operatorname{Gr}_{k}(W)$ be the corresponding bundle of $k$-subspaces over
$B$, and consider
$\Lambda_{k}^{\nu}(W)=\\{(x,E)\in\operatorname{Gr}_{k}(W):E\in\Lambda^{\nu}_{k}(W_{x},Q_{x})\\}$.
If $(p,E)\in\Lambda_{k}^{\nu}(W)$ and $\operatorname{Ker}(Q_{x})\cap
E=\\{0\\}$, then $\Lambda_{k}^{\nu}(W)$ is a manifold near $(p,E)$, of
dimension

$\dim\Lambda_{k}^{\nu}(W)=\dim B+k(d-k)-{\nu+1\choose 2}.$

###### Proof.

Using a local trivialization, this reduces to Proposition 4.1. ∎

######  Lemma 4.3.

Let $W$ be a $d$-dimensional vector space equipped with a quadratic form $Q$
of nullity $1$ with kernel $L_{0}$. Assume $L_{0}\subset
E_{0}\in\Lambda_{k}^{\nu}(W)$, and define the set $C\subset
T_{E_{0}}\operatorname{Gr}_{k}(W)$ of all velocity vectors $E^{\prime}(0)$ of
smooth curves $E(t)\in\Lambda_{k}^{\nu}(W)$ with $E(0)=E_{0}$. Then $C$ is a
cone of dimension $k(d-k)-{\nu+1\choose 2}$.

###### Proof.

Assume $L_{0}=\operatorname{Span}(v_{0})$, and fix a $(d-1)$-complement
$W_{0}$ such that $W=W_{0}\oplus L_{0}$. Denote
$\Lambda_{k}^{\nu}(W,W_{0})=\\{E\in\Lambda_{k}^{\nu}(W):\operatorname{Ker}(Q|_{E})\subset
W_{0}\\}$. Consider
$I:\operatorname{Gr}_{k}(W)\setminus\operatorname{Gr}_{k}(W_{0})\to\operatorname{Gr}_{k-1}(W_{0})$
given by $I(E)=E\cap W_{0}$.

We claim that the restriction

$\tilde{I}:=I:\Lambda_{k}^{\nu}(W)\setminus\Lambda_{k}^{\nu}(W,W_{0})\to\Lambda_{k-1}^{\nu-1}(W_{0})$

is well-defined. Indeed, for $E$ of nullity $\nu$, the nullity of $I(E)$ is at
least $(\nu-1)$. Now if also $E\notin\Lambda_{k}^{\nu}(W,W_{0})$, take
$w\in\operatorname{Ker}(Q|_{E})\setminus W_{0}$ so that
$E=I(E)\oplus\operatorname{Span}(w)$. If $I(E)$ contains a $\nu$-dimensional
subspace $U$ that is $Q$-orthogonal to $I(E)$, then $U$ is also $Q$-orthogonal
to $E$ as $Q(w,I(E))=0$. Hence
$U\oplus\operatorname{Span}(w)\subset\operatorname{Ker}(Q|_{E})$, and
consequently the nullity of $E$ is at least $(\nu+1)$, a contradiction.

Let us describe the fiber $\tilde{I}^{-1}(F)$ of
$F\in\Lambda_{k-1}^{\nu-1}(W_{0})$. Denote
$\Lambda_{k}^{\nu}(W,L_{0})=\\{E\in\Lambda_{k}^{\nu}(W):L_{0}\subset E\\}$,
and $\pi_{0}:W\to W_{0}$ is the projection along $L_{0}$. If
$E\in\tilde{I}^{-1}(F)\cap\Lambda^{\nu}_{k}(W,L_{0})$, then $E=F\oplus L_{0}$.

If $E\in\tilde{I}^{-1}(F)\setminus\Lambda^{\nu}_{k}(W,L_{0})$, we take $w$ as
above and decompose $w=w_{1}+w_{2},w_{1}\in W_{0},w_{2}\in L_{0}$. Then
$w_{1}\neq 0$ and $w_{1}=w-w_{2}\in F^{Q}$. It follows that there is a unique
line $L\in\Lambda_{1}^{1}(F^{Q}\cap W_{0}/F^{Q}\cap F)$ such that
$\pi_{0}(E)=F+\tilde{L}$ for any lift $\tilde{L}\in\Lambda_{1}^{1}(F^{Q}\cap
W_{0})$ of $L$. Thus for any fixed $L$, the corresponding part of
$\tilde{I}^{-1}(F)\setminus\Lambda^{\nu}_{k}(W,L_{0})$ can be identified with
$\mathbb{P}_{+}(\tilde{L}\oplus L_{0})\simeq\mathbb{P}_{+}(L\oplus L_{0})$
with $\pm L,\pm L_{0}$ excluded.

Now let us see how the various parts of the fiber fit together. Consider

$\widetilde{M}_{F}=\widetilde{\Lambda}_{1}^{1}(F^{Q}\cap W_{0}/F^{Q}\cap
F)\subset\mathbb{P}_{+}(F^{Q}\cap W_{0}/F^{Q}\cap F),$

where we write $\widetilde{\Lambda}^{1}_{1}$ for oriented null lines.

Let $\widetilde{\Sigma}\widetilde{M}_{F}$ be its suspension with poles at two
copies of $L_{0}$ with opposite orientations. Finally, set $\Sigma
M_{F}=\widetilde{\Sigma}\widetilde{M}_{F}/\mathbb{Z}_{2}$, the action by the
antipodal map, and let $L_{F}\in\Sigma M_{F}$ be the image of $L_{0}$. Put

$U_{\Sigma}(F)=\Sigma M_{F}\setminus\mathbb{P}(F^{Q}\cap W_{0}/F^{Q}\cap F),$

which is a neighborhood of $L_{F}$. There is then a natural diffeomorphism
$U_{\Sigma}(F)\simeq\tilde{I}^{-1}(F)\subset\Lambda_{k}^{\nu}(W)\setminus\Lambda_{k}^{\nu}(W,W_{0})$.

We get the fibration

$\textstyle{\tilde{I}^{-1}(F)\simeq
U_{\Sigma}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{k}^{\nu}(W)\setminus\Lambda_{k}^{\nu}(W,W_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I}$$\textstyle{\Lambda_{k-1}^{\nu-1}(W_{0})}$

and the section $F\mapsto L_{F}$, or equivalently $F\mapsto F\oplus L_{0}$,
coincides with $\Lambda_{k}^{\nu}(W,L_{0})$. In particular, the cone $C\subset
T_{E_{0}}\operatorname{Gr}_{k}(W)$ has a linear factor that can be identified
with $T_{E_{0}\cap W_{0}}\Lambda_{k-1}^{\nu-1}(W_{0})$. Denoting
$F_{0}=E_{0}\cap W_{0}$, the cone $C/T_{F_{0}}\Lambda_{k-1}^{\nu-1}(W_{0})$ is
then naturally identified with the abstract cone over
$\widetilde{\Lambda}_{1}^{1}(F_{0}^{Q}\cap W_{0}/F_{0}^{Q}\cap F_{0})$.

The dimension of $C$ can now be readily computed. We have
$\dim\Lambda_{k-1}^{\nu-1}(W_{0})=(k-1)(d-k)-\binom{\nu}{2}$ by Proposition
4.1. Since the restriction of $Q$ to $F_{0}^{Q}\cap W_{0}/F_{0}^{Q}\cap F_{0}$
is non-degenerate, we have $\dim\widetilde{\Lambda}_{1}^{1}(F_{0}^{Q}\cap
W_{0}/F^{Q}_{0}\cap F_{0})=\dim F^{Q}_{0}\cap W_{0}/F^{Q}_{0}\cap
F_{0}-2=d-k-(\nu-1)-2$. Hence

$\dim C=\Lambda_{k-1}^{\nu-1}(W_{0})+\dim\tilde{\Lambda}_{1}^{1}(F^{Q}_{0}\cap
W_{0}/F^{Q}_{0}\cap F_{0})+1=k(d-k)-\binom{\nu+1}{2}.$

∎

We now turn to LC-regular submanifolds. First, we will need a simple fact on
LC-regular metrics.

######  Lemma 4.4.

Let $(M,g)$ be LC-regular, and assume $g$ is degenerate on $T_{p}M$. Let
$v_{1}(x),\dots,v_{m}(x)$ be any local frame near $p$, with Gram matrix
$A(x)=(g(v_{i},v_{j}))_{i,j=1}^{m}\in\operatorname{Sym}_{m}(\mathbb{R})$. Then
the condition $d_{p}(\det A)\neq 0$ is independent of the choice of the frame
$(v_{j})$. Moreover, if the nullity of $g_{p}$ is $\nu=1$, then $d_{p}(\det
A)\neq 0$, and the degenerate subset of the metric near $p$ is a smooth
hypersurface.

###### Proof.

Let $\tilde{v}_{1}(x),\dots,\tilde{v}_{m}(x)$ be a different local frame with
corresponding Gram matrix $\widetilde{A}(x)$. Then the change of basis matrix
$U(x)\in\operatorname{GL}(m)$ satisfies $\widetilde{A}(x)=U(x)^{T}A(x)U(x)$.
By assumption, $\det A(p)=\det\widetilde{A}(p)=0$. Thus
$d_{p}(\det\widetilde{A})=\det U(p)^{2}d_{p}(\det A)$, which implies the first
statement.

For the second statement, choose coordinates $x_{1},\dots,x_{m}$ on $M$ near
$p$, and take $v_{j}=\frac{\partial}{\partial x_{j}}$. We may assume that
$\operatorname{Ker}(g_{p})=\operatorname{Ker}A(p)=\operatorname{Span}(v_{m})$,
and by assumption $A(p)$ has non degenerate principal $(m-1)$-minor. By LC-
regularity, we can choose a curve $p(t)\in M$ with $p(0)=p$ and a smooth
vector field $v(t)$ along it with $v(0)=v_{m}$ such that
$\left.\frac{d}{dt}\right|_{t=0}g(v(t),v(t))=\left.\frac{d}{dt}\right|_{t=0}\langle
A(p(t))e_{m},e_{m}\rangle\neq 0$. It follows that

$\left.\frac{d}{dt}\right|_{t=0}\det
A(p(t))=\left.\frac{d}{dt}\right|_{t=0}\langle
A(p(t))e_{m},e_{m}\rangle\cdot\det(A(p))_{i,j=1}^{m-1}\neq 0.$

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.

######  Proposition 4.5.

Let $(V,Q)$ be a pseudo-Euclidean vector space of dimension $(n+1)$, and
$M=Q^{-1}(r)$ with $r\in\\{\pm 1\\}$ a pseudo-Riemannian space form. Let
$H\subset M$ be an LC-regular hypersurface, and
$E\in\operatorname{Gr}_{n-k+1}(V)$. Assume $E\in B_{H}$ is a regular tangent
to $H$ of nullity $\nu$. Then each embedded part of $B_{H}$ through $E$
intersects $\Lambda^{\nu}_{n-k+1}(V)$ transversally at $E$.

###### Proof.

Denote the signature of $M$ by $(p_{M},q_{M})$. Write $g=Q|_{H}$,
$\widehat{E}=E\cap M$. Since $H$ is a hypersurface, the nullity of $g_{x}$ is
at most one for all $x\in H$. Define
$\widetilde{\Lambda}^{\nu}_{n+1-k}:=\tau_{M}^{-1}\Lambda^{\nu}_{n+1-k}(V)$,
which is a submanifold of $Z_{M}$ as
$\tau_{M}:Z_{M}\to\operatorname{Gr}_{n+1-k}(V)$ is a submersion.

As $B_{H}$ is a hypersurface, one should show for every embedded part $F$ of
$B_{H}$ through $E$ that $T_{E}\Lambda^{\nu}_{n+1-k}(V)\not\subset T_{E}F$.
Assuming the contrary, there is $p\in H\cap E$ such that
$T_{E}\Lambda^{\nu}_{n+1-k}(V)\subset d\tau_{H}(T_{p,E}\widetilde{B}_{H})$.
Observe that $\operatorname{Ker}(Q|_{E})\subset T_{p}\widehat{E}$.

Before proceeding with the more complicated general case, we consider the case
$\nu=1$. Since $\dim\Lambda^{1}_{n+1-k}(V)=\dim B_{H}=k(n+1-k)-1$ by
Proposition 4.1, we have
$T_{E}\Lambda^{1}_{n+1-k}(V)=d\tau_{H}(T_{p,E}\widetilde{B}_{H})$. Let
$v_{0}\in E\cap T_{p}H$ be in the kernel of ${g|_{T_{p}\widehat{E}}}$. For any
smooth curve $v_{t}\in TH$ through $v_{0}$, we may find a smooth curve
$E_{t}\subset B_{H}$ through $E$ such that $v_{t}\in E_{t}$. Then
$A:=\left.\frac{d}{dt}\right|_{t=0}E_{t}\in\operatorname{Hom}(E,V/E)$
satisfies $Q(Av_{0},v_{0})=0$, and by (7) we have $A\in
T_{E}\Lambda^{1}_{n+1-k}(V)$.

It follows that $\left.\frac{d}{dt}\right|_{t=0}g(v_{t})=2Q(v_{0},Av_{0})=0$,
contradicting the LC-regularity of $H$.

Let us now consider the general case. Fix an auxiliary Riemannian metric $h$
on $M$, and let $\rho_{H}$ be the least distance projection to $H$, defined
and smooth in a neighborhood of $p$.

By our assumption $\tilde{\tau}_{H}$ is an immersion at $(p,E)$. Using
Proposition 4.1 we see that

$\dim(d\tilde{\tau}_{H})^{-1}T_{E}\Lambda^{\nu}_{n+1-k}(V)=\dim\Lambda^{\nu}_{n+1-k}(V)=k(n-k+1)-{\nu+1\choose
2}.$ (8)

By definition, $d_{p,E}\tilde{\tau}_{H}^{-1}T_{E}\Lambda^{\nu}_{n+1-k}(V)$
consists of all velocity vectors $(p^{\prime}(0),E^{\prime}(0))$ to curves
$(p(t),E(t))\in\widetilde{B}_{H}$ through $(p,E)$ such that $E^{\prime}(0)\in
T_{E}\Lambda^{\nu}_{n-k+1}$. Let us check that equivalently one may take all
velocity vectors to curves
$(\tilde{p}(t),\widetilde{E}(t))\in\widetilde{\Lambda}^{\nu}_{n-k+1}$ with
$(\tilde{p}^{\prime}(0),\widetilde{E}^{\prime}(0))\in
T_{p,E}\widetilde{B}_{H}$. Indeed, given $(p(t),E(t))$ as above, one can find
$\widetilde{E}(t)\in\Lambda^{\nu}_{n-k+1}(V)$ with
$\widetilde{E}^{\prime}(0)=E^{\prime}(0)$, and then choose
$\tilde{p}(t)\in\widetilde{E}(t)\cap M$ such that
$\tilde{p}^{\prime}(0)=p^{\prime}(0)$. In the other direction, given
$(\tilde{p}(t),\widetilde{E}(t))$, define $p(t)=\rho_{H}(\tilde{p}(t))$ and
$E(t)=d_{\tilde{p}(t)}\rho_{H}(T_{\tilde{p}(t)}\widehat{\widetilde{E}}(t))\oplus\mathbb{R}p(t)$.

Case 1: $\operatorname{Ker}(g_{p})\cap E=\\{0\\}$. Define

$\widetilde{B}^{\nu}_{H}:=\tilde{\tau}_{H}^{-1}\Lambda^{\nu}_{n-k+1}(V)=\\{(q,F)\in\widetilde{B}_{H}:T_{q}\widehat{F}\in\Lambda^{\nu}_{n-k}(T_{q}H)\\}.$

By Corollary 4.2 we see that $\widetilde{B}^{\nu}_{H}$ is a smooth manifold
near $(p,E)$, of dimension

$\displaystyle\dim T_{p,E}\widetilde{B}_{H}^{\nu}$ $\displaystyle=\dim
H+(n-k)(n-1-(n-k))-{\nu+1\choose 2}$ (9)
$\displaystyle=k(n-k+1)-1-{\nu+1\choose 2}.$

Claim 1. $(d_{p,E}\tilde{\tau}_{H})^{-1}T_{E}\Lambda^{\nu}_{n+1-k}(V)\subset
T_{p,E}\widetilde{B}_{H}^{\nu}$.

We postpone the proof. Combined with eqs. (8) and (9) we get a contradiction.

Case 2: $T_{p}H$ has nullity one, and $\operatorname{Ker}(g_{p})\subset E$.
Let $S\subset H$ be the degenerate subset of $g$. It follows from Lemma 4.4
that $S$ is a smooth hypersurface. Define
$\widetilde{B}^{\nu}_{H}(S)=\\{(q,F)\in\widetilde{B}_{H}:q\in
S,F\in\Lambda^{\nu}_{n-k+1}(V)\\}$. It is a fiber bundle over $S$ with fiber
$\Lambda^{\nu}_{n-k}(\mathbb{R}^{p_{M}-1,q_{M}-1,1})$.

Claim 2. For any $(w,\xi)\in
d_{p,E}\tilde{\tau}_{H}^{-1}T_{E}\Lambda^{\nu}_{n-k+1}(V)$ with $w\in T_{p}S$
there is a curve $(q(t),F(t))\in\widetilde{B}^{\nu}_{H}(S)$ with
$(q^{\prime}(0),F^{\prime}(0))=(w,\xi)$.

Again we postpone the proof of the claim. The set of all vectors
$(q^{\prime}(0),F^{\prime}(0))$ as in the claim defines, by Lemma 4.3, a cone
in $T_{p,E}\widetilde{B}_{H}$ of dimension

$\displaystyle N$ $\displaystyle=\dim S+(n-k)((n-2)-(n-k-1))-{\nu+1\choose 2}$
$\displaystyle=(n-k)(k-1)-{\nu+1\choose 2}+n-2<\dim
d_{p,E}\tilde{\tau}_{H}^{-1}T_{E}\Lambda^{\nu}_{n-k+1}(V).$

It follows by the claim that we can find a curve
$(p(t),E(t))\in\widetilde{B}_{H}$ through $(p,E)$ with $E^{\prime}(0)\in
T_{E}\Lambda_{n-k+1}^{\nu}(V)$ and $p^{\prime}(0)\notin T_{p}S$.

Let $v_{n-1}\in T_{p}H$ span $\operatorname{Ker}(g_{p})$, and recall that
$v_{n-1}\in T_{p}\widehat{E}$, in particular
$v_{n-1}\in\operatorname{Ker}(Q|_{E})$. Choosing any smooth vector field
$v_{n-1}(t)\in T_{p(t)}\widehat{E}(t)$, we find
$\left.\frac{d}{dt}\right|_{t=0}Q(v_{n-1}(t))=2Q(v_{n-1}^{\prime}(0),v_{n-1}(0))=0$.

Choose a frame $(v_{j}(t))_{j=1}^{n-1}$ for $H$ along $p(t)$ with
$v_{n-1}(0)=v_{n-1}$, such that
$T_{p(t)}\widehat{E}(t)=\operatorname{Span}(v_{k}(t),\dots,v_{n-1}(t))$. Then

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}\det(Q(v_{i}(t),v_{j}(t)))_{i,j=1}^{n-1}$
$\displaystyle=\det(Q(v_{i}(t),v_{j}(t)))_{i,j=1}^{n-2}\cdot\left.\frac{d}{dt}\right|_{t=0}Q(v_{n-1}(t))=0,$

which means that $d_{p}(\det g)(p^{\prime}(0))=0$ by Lemma 4.4. Since
$\operatorname{Ker}d_{p}(\det g)=T_{p}S$ and $p^{\prime}(0)\notin T_{p}S$, we
get a contradiction. This completes the proof of the proposition, modulo the
two claims we now proceed to prove.∎

###### Proof of Claim 1.

Consider a curve $(p(t),E(t))\in\widetilde{\Lambda}^{\nu}_{n-k+1}$ with
$(p^{\prime}(0),E^{\prime}(0))\in T_{p,E}\widetilde{B}_{H}$. We ought to find
a curve $(q(t),F(t))\in\widetilde{B}^{\nu}_{H}$ through $(p,E)$ with
$(q^{\prime}(0),F^{\prime}(0))=(p^{\prime}(0),E^{\prime}(0))$.

Set $q(t)=\rho_{H}(p(t))$, evidently $q^{\prime}(0)=p^{\prime}(0)$. Fix a
subspace $W_{0}\subset T_{p}H$ which is non-degenerate and contains
$T_{p}\widehat{E}$. 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
$W_{0}\subset T_{p}H$ which contains $T_{p}\widehat{E}$ and satisfies
$W_{0}\cap\ker g_{p}=\\{0\\}$. Now fix any linear map $A:W_{0}\to V/W_{0}$
which makes the following diagram commutative.

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E^{\prime}(0)}$$\scriptstyle{i}$$\textstyle{V/E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\scriptstyle{i}$$\textstyle{V/W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{T_{p}H\oplus\mathbb{R}p\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left.\frac{d}{dt}\right|_{t=0}(T_{q(t)}H\oplus\mathbb{R}q(t))}$$\textstyle{V/(T_{p}H\oplus\mathbb{R}p)}$

and use Lemma 3.1 to find smooth paths $W(t)\supset E(t)$,
$\widetilde{W}(t)\subset T_{q(t)}H\oplus\mathbb{R}q(t)$ with
$W(0)=\widetilde{W}(0)=W_{0}$ and $W^{\prime}(0)=\widetilde{W}^{\prime}(0)=A$.
For small $t$, $W(t),\widetilde{W}(t)$ are non-degenerate of fixed signature
$(\alpha,\beta)$.

Consider the manifold $Z=\\{(x,W)\in
M\times\operatorname{Gr}_{\alpha+\beta}(V),x\in
W,\mathrm{sign}(Q|_{W})=(\alpha,\beta)\\}$. Clearly $Z$ is a homogeneous space
for $\operatorname{O}(V,Q)$, with the equivariant projection
$\pi_{Z}:\operatorname{O}(V,Q)\to Z$ normalized by
$\pi_{Z}(\operatorname{Id})=(p,W_{0})$. We can fix a smooth section
$X_{Z}:Z\to\operatorname{O}(V,Q)$ near $(p,W_{0})$ with
$X_{Z}(p,W_{0})=\operatorname{Id}$ such that $\pi_{Z}\circ
X_{Z}=\operatorname{Id}$. Now define the smooth path
$R_{t}\in\operatorname{O}(V,Q)$ by

$R_{t}=X_{Z}(q(t),\widetilde{W}(t))\circ X_{Z}(p(t),W(t))^{-1}.$

Then $R_{t}p(t)=q(t)$, and $\left.\frac{d}{dt}\right|_{t=0}R_{t}=0$ since
$\left.\frac{d}{dt}\right|_{t=0}W(t)=\left.\frac{d}{dt}\right|_{t=0}\widetilde{W}(t)$.

Setting $F(t)=R_{t}E(t)$, we have
$(q^{\prime}(0),F^{\prime}(0))=(p^{\prime}(0),E^{\prime}(0))$, and
$(q(t),F(t))\in\widetilde{B}^{\nu}_{H}$. This proves the claim. ∎

###### Proof of Claim 2.

Consider a curve $(p(t),E(t))\in\widetilde{\Lambda}^{\nu}_{n-k+1}(V)$ through
$(p,E)$, with $p^{\prime}(0)=w\in T_{p}S$ and
$(p^{\prime}(0),E^{\prime}(0))=(w,\xi)\in T_{p,E}\widetilde{B}_{H}$. Let
$\rho_{S}:M\to S$ be the least distance projection with respect to $h$, well-
defined and smooth in some neighborhood of $p$. Set $q(t)=\rho_{S}(p(t))$,
clearly $q^{\prime}(0)=p^{\prime}(0)$. Denote
$L_{0}=\operatorname{Ker}(g_{p})\subset T_{p}H\cap E$, and extend to a smooth
path of lines $L_{t}\subset E(t)\cap
E(t)^{Q}\in\Lambda_{\nu}^{\nu}(T_{p(t)}M)$. Consider the manifold of pairs

$Z=\\{(x,L):x\in M,L\in\Lambda^{1}_{1}(T_{x}M)\\}.$

Clearly $Z$ is a homogeneous space for $\operatorname{O}(V,Q)$, with the
equivariant projection $\pi_{Z}:\operatorname{O}(V,Q)\to Z$ normalized by
$\pi_{Z}(\operatorname{Id})=(p,L_{0})$. We can fix a smooth section
$X_{Z}:Z\to\operatorname{O}(V,Q)$ near $(p,L_{0})$ with
$X_{Z}(p,L_{0})=\operatorname{Id}$ such that $\pi_{Z}\circ
X_{Z}=\operatorname{Id}$. Now define the smooth path
$R_{t}\in\operatorname{O}(V,Q)$ by

$R_{t}=X_{Z}(q(t),\operatorname{Ker}(g_{q(t)}))\circ X_{Z}(p(t),L_{t})^{-1}.$

Then $R_{t}p(t)=q(t)$, and $\left.\frac{d}{dt}\right|_{t=0}R_{t}=0$, provided
that
$\left.\frac{d}{dt}\right|_{t=0}L_{t}=\left.\frac{d}{dt}\right|_{t=0}\operatorname{Ker}(g_{q(t)})$.

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 $x,y\in X$. We will also write, for two
subspaces $E,F\subset V$, $\measuredangle(E,F)$ 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.

As $(p^{\prime}(0),E^{\prime}(0))\in T_{p,E}\widetilde{B}_{H}$, we may find a
curve $(\tilde{p}(t),\widetilde{E}(t))\in\widetilde{B}_{H}$ through $(p,E)$
with
$(p^{\prime}(0),E^{\prime}(0))=(\tilde{p}^{\prime}(0),\widetilde{E}^{\prime}(0))$.
Define $\widetilde{H}(t):=T_{\tilde{p}(t)}H\oplus\mathbb{R}\tilde{p}(t)$. It
follows that $\measuredangle(E(t),\widetilde{H}(t))=O(t^{2})$, and by Lemma
3.1 we have the commutative diagram

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E^{\prime}(0)}$$\scriptstyle{i}$$\textstyle{V/E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\widetilde{H}(0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{H}^{\prime}(0)}$$\textstyle{V/\widetilde{H}(0)}$

Taking the dual diagram and identifying $V=V^{*}$ using $Q$, we get

$\textstyle{L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{i}$$\textstyle{V/L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{E^{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(E^{Q})^{\prime}(0)}$$\textstyle{V/E^{Q}}$
(10)

where $f_{0}=(\widetilde{H}^{Q})^{\prime}(0)$. As
$\tilde{p}^{\prime}(0)=p^{\prime}(0)=q^{\prime}(0)$, it is clear that

$f_{0}=\left.\frac{d}{dt}\right|_{t=0}(T_{\tilde{p}(t)}H\oplus\mathbb{R}\tilde{p}(t))^{Q}=\left.\frac{d}{dt}\right|_{t=0}(T_{q(t)}H\oplus\mathbb{R}q(t))^{Q}=\left.\frac{d}{dt}\right|_{t=0}\operatorname{Ker}g_{q(t)}.$

By Lemma 3.1, we can find $L_{t}\subset E(t)^{Q}$ with
$\left.\frac{d}{dt}\right|_{t=0}L_{t}=f_{0}$. 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 $q\in S$, $\widetilde{E}\in\operatorname{Gr}_{n+1-k}(V)$ and
$\operatorname{Ker}(g_{q})\subset\widetilde{E}$, then
$\widetilde{E}^{Q}\subset T_{q}H\oplus\mathbb{R}q$. We have

$\displaystyle|\tilde{p}(t)-q(t)|_{M}$ $\displaystyle=O(t^{2}),$
$\displaystyle|T_{\tilde{p}(t)}H-T_{q(t)}H|_{\operatorname{Gr}_{n-1}(V)}$
$\displaystyle=O(t^{2}),$
$\displaystyle|L_{t}-\operatorname{Ker}(g_{q(t)})|_{\mathbb{P}(V)}$
$\displaystyle=O(t^{2}).$

It follows that $\measuredangle(E(t)^{Q},\widetilde{H}(t))=O(t^{2})$, and so
we may apply Lemma 3.1 to get the commutative square

$\textstyle{E^{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(E^{Q})^{\prime}(0)}$$\scriptstyle{i}$$\textstyle{V/E^{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\widetilde{H}(0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{H}^{\prime}(0)}$$\textstyle{V/\widetilde{H}(0),}$

and by duality also

$\textstyle{L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{i}$$\textstyle{V/L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E^{\prime}(0)}$$\textstyle{V/E.}$
(11)

Denote $K(t)=E(t)\cap E(t)^{Q}$, $K_{0}=K(0)$. Observe there is a natural
inclusion $\alpha_{K}:V/K_{0}\hookrightarrow V/E\oplus V/E^{Q}$. It follows
from Lemma 3.1 applied to the inclusions $K(t)\subset E(t)$, $K(t)\subset
E(t)^{Q}$ that $\alpha_{K}\circ K^{\prime}(0):K_{0}\to V/E\oplus V/E^{Q}$
coincides with $E^{\prime}(0)\oplus(E^{Q})^{\prime}(0):K_{0}\to V/E\oplus
V/E^{Q}$.

Combining diagrams (10) and (11) then yields the commutative diagram

$\textstyle{L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{i}$$\textstyle{V/L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{K_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{K}\circ
K^{\prime}(0)}$$\textstyle{V/E\oplus V/E^{Q},}$

and so also

$\textstyle{L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{i}$$\textstyle{V/L_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{K_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K^{\prime}(0)}$$\textstyle{V/K_{0}.}$

By Lemma 3.1, we may redefine $L_{t}$ such that
$\left.\frac{d}{dt}\right|_{t=0}L_{t}=f_{0}$ and $L_{t}\subset K(t)=E(t)\cap
E(t)^{Q}$. In particular, $L_{t}$ is a null line of $Q$.

Setting $F(t):=R_{t}E(t)$ we have $q(t)\in F(t)$,
$T_{q(t)}\widehat{F}(t)\subset T_{q(t)}H$ since $E(t)\subset L_{t}^{Q}$, and
$F^{\prime}(0)=E^{\prime}(0)$ since $\left.\frac{d}{dt}\right|_{t=0}R_{t}=0$.
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 $H$.

######  Corollary 4.7.

Let $V,M,H,E$ be as in Proposition 4.5 with $H$ compact without boundary, and
$A\subset M$ is either $H$ itself or a domain with $\partial A=H$. Then
$\operatorname{Cr}\operatorname{WF}_{E}(A)\cap
N_{E}^{*}\Lambda^{\nu}_{n-k+1}(V)=\emptyset$.

###### Proof.

Follows from Propositions 4.5 and 3.17. ∎

##  5\. Construction of an invariant measure on the Grassmannian

For $X\in\operatorname{Sym}_{r}(\mathbb{R})$ and $\lambda\in\mathbb{C}$ we
set, as in [30],

$|\det X|_{p}^{\lambda}:=\begin{cases}|\det X|^{\lambda}&\text{if
}\mathrm{sign}(X)=(p,r-p)\\\ 0&\text{otherwise.}\end{cases}$

It is well-known, essentially due to Cayley and Gärding, that $|\det
X|_{q}^{\lambda}$ extends as a meromorphic in $\lambda$ family of generalized
functions, which are in fact tempered distributions.

We will use the set

$U_{\mathbb{C}}:=\\{\mathrm{Re}\zeta>\frac{1}{2}\\}\cup\left\\{\mathrm{Im}\zeta>0\right\\}\subset\mathbb{C},$

and write $\sqrt{z}$ for the unique branch of the square root function on
$U_{\mathbb{C}}$ such that $\sqrt{z}>0$ for $z>\frac{1}{2}$.

###  5.1. Construction of a holomorphic family of Crofton measures

For the following, let
$\operatorname{Sym}_{r}^{+}(\mathbb{R})\subset\operatorname{Sym}_{r}(\mathbb{R})$
be the cone of positive-definite matrices, and
$\mathfrak{h}_{r}=\operatorname{Sym}_{r}(\mathbb{R})\oplus\mathbf{i}\operatorname{Sym}_{r}^{+}(\mathbb{R})\subset\operatorname{Sym}_{r}(\mathbb{C})$
the Siegel upper half space. The following is well-known.

######  Lemma 5.1.

For $Z\in\mathfrak{h}_{r}$, $\det Z\neq 0$. In particular, we can define for
every $\lambda\in\mathbb{C}$ the holomorphic function $Z\mapsto(\det
Z)^{\lambda}$, normalized by $\lim_{\epsilon\to
0^{+}}\det(I_{r}+\mathbf{i}\epsilon I_{r})^{\lambda}=1$. Moreover, all
eigenvalues of $Z\in\mathfrak{h}_{r}$ lie in the upper half plane of
$\mathbb{C}$.

###### Proof.

Write $Z=X+\mathbf{i}Y$, $Y>0$. Let $Q_{X}(v)=\langle
Xv,v\rangle,Q_{Y}=\langle Yv,v\rangle$ 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: $D=\mathrm{diag}(d_{j})$. Since $D+\mathbf{i}I_{r}=U^{T}ZU$ with
$U$ invertible, and $\det(D+\mathbf{i}I_{r})=\prod(d_{j}+\mathbf{i})\neq 0$,
it follows that $\det Z\neq 0$. Since $\mathfrak{h}_{r}$ 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, $\det(X+\mathbf{i}Y-\lambda I_{r})=\det((X-\lambda
I_{r})+\mathbf{i}Y)\neq 0$ for $\lambda\in\mathbb{R}$. Next we argue as before
and select a diagonalizing basis, given by $U\in\mathrm{GL}(r)$. We
furthermore may assume that $\det U>0$, by interchanging two basis elements.
Choose a smooth path $U_{t}\in\mathrm{GL}(r)$ with $U_{0}=\mathrm{Id}$ and
$U_{1}=U$. Then $U_{t}^{T}ZU_{t}\in\mathfrak{h}_{r}$ is a smooth path. For
$t=1$, the endpoint is $D+\mathbf{i}I_{r}$, 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 $V=V_{+}\oplus V_{-}$ which is both $P$\- and
$Q$-orthogonal, and $Q|_{V_{\pm}}=\pm P|_{V_{\pm}}$.

From here on, let $V=\mathbb{R}^{p}\oplus\mathbb{R}^{q}=\mathbb{R}^{n+1}$ 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 $Q_{\zeta}$ on $V$ with $\zeta\in\mathbb{C}$, by

$Q_{\zeta}:=Q+2\zeta P_{0}.$

We then have

$Q_{\zeta}(x,y):=\begin{cases}(2\zeta+1)P_{0}(x,y)&x,y\in\mathbb{R}^{p}\\\
(2\zeta-1)P_{0}(x,y)&x,y\in\mathbb{R}^{q}\\\
0&x\in\mathbb{R}^{p},y\in\mathbb{R}^{q}.\end{cases}$

Observe that $Q_{\zeta}$ is real and positive-definite for
$\zeta>\frac{1}{2}$, and $Q_{0}=Q$. Furthermore by Lemma 5.1, $\det
Q_{\zeta}\neq 0$ for $\zeta\in U_{\mathbb{C}}$, as either $Q_{\zeta}$ or
$\mathbf{i}Q_{\zeta}$ lies in $\mathfrak{h}_{n+1}$. Note that a complex-valued
non-degenerate quadratic form $Q$ on a real vector space $E$ defines an
element
$\operatorname{vol}_{Q}^{2}\in\operatorname{Dens}_{\mathbb{C}}(E)^{2}$, and
given a branch of square root we also get a complex-valued density
$\operatorname{vol}_{Q}\in\operatorname{Dens}_{\mathbb{C}}(E)$.

For a subspace $E\in\operatorname{Gr}_{n+1-k}(V)$ and $\zeta\in
U_{\mathbb{C}}$, let $X^{P}_{\zeta}(E)$ be the Gram matrix of $Q_{\zeta}|_{E}$
with respect to a Euclidean structure $P$. It depends on a local choice of a
section of $P$-orthonormal bases, and locally defines a map
$X^{P}_{\zeta}:\operatorname{Gr}_{n+1-k}(V)\to\operatorname{Sym}_{n+1-k}(\mathbb{C})$.
Observe that $\det(X^{P}_{\zeta})$ is independent of the choice of
$P$-orthonormal bases of $E$. Moreover, either the real or imaginary part of
$Q_{\zeta}|_{E}$ is positive-definite, and consequently by Lemma 5.1, $\det
X^{P}_{\zeta}(E)\neq 0$.

The function $\det(X^{P}_{\zeta})^{\lambda}\in
C^{\infty}(\operatorname{Gr}_{n+1-k}(V),\mathbb{C})$ is thus well-defined for
all $P$ and $\lambda\in\mathbb{C}$, and analytic in $\zeta\in U_{\mathbb{C}}$,
once the normalization $\det(X^{P}_{1})^{\lambda}>0$ is fixed, as
$U_{\mathbb{C}}$ is simply-connected.

Define the smooth measure $\widetilde{m}^{P}_{\zeta}$ on the Grassmannian
$\operatorname{Gr}_{n+1-k}(V)$ by

$d\widetilde{m}^{P}_{\zeta}(E):=\det(X^{P}_{\zeta})^{-\frac{n+1}{2}}(E)d\sigma_{P}(E),$

where $d\sigma_{P}(E)$ is the $\operatorname{O}(P)$-invariant probability
measure on the Grassmannian.

######  Proposition 5.2.

The complex-valued smooth measure

$m^{\zeta}_{k}:=(2\zeta+1)^{\frac{p(n+1-k)}{2}}(2\zeta-1)^{\frac{q(n+1-k)}{2}}\widetilde{m}_{\zeta}^{P_{0}},\quad\zeta\in
U_{\mathbb{C}}.$

depends analytically on $\zeta$ and is normalized, i.e.

$\int_{\operatorname{Gr}_{n+1-k}(V)}dm^{\zeta}_{k}=1.$

###### Proof.

The first statement is clear. For the second, we first see how
$\widetilde{m}^{P}_{\zeta}$ depends on $P$. Let $P_{1},P_{2}$ be two Euclidean
structures on $V$. From the natural identification
$T_{E}\operatorname{Gr}_{n+1-k}(V)=E^{*}\otimes V/E$ we obtain that

$\operatorname{Dens}(T_{E}\operatorname{Gr}_{n+1-k}(V))=\operatorname{Dens}^{*}(E)^{n+1}\otimes\operatorname{Dens}(V)^{n+1-k}.$

Spelling this out gives

$\frac{d\sigma_{P_{1}}(E)}{d\sigma_{P_{2}}(E)}=\left(\frac{\operatorname{vol}_{P_{1}|_{E}}}{\operatorname{vol}_{P_{2}|_{E}}}\right)^{-(n+1)}\left(\frac{\operatorname{vol}_{P_{1}}}{\operatorname{vol}_{P_{2}}}\right)^{n+1-k},$

Since

$\det
X_{\zeta}^{P_{i}}(E)=\frac{\operatorname{vol}^{2}_{Q_{\zeta}|_{E}}}{\operatorname{vol}^{2}_{P_{i}|_{E}}},$

we find that

$\widetilde{m}^{P_{1}}_{\zeta}=\left(\frac{\operatorname{vol}_{P_{1}}}{\operatorname{vol}_{P_{2}}}\right)^{n+1-k}\widetilde{m}^{P_{2}}_{\zeta}.$

For $\zeta>\frac{1}{2}$, $Q_{\zeta}$ is a Euclidean structure. Then

$\displaystyle 1$
$\displaystyle=\int\widetilde{m}_{\zeta}^{Q_{\zeta}}=\left(\frac{\operatorname{vol}_{Q_{\zeta}}}{\operatorname{vol}_{P_{0}}}\right)^{n+1-k}\int\widetilde{m}_{\zeta}^{P_{0}}$
$\displaystyle=\sqrt{2\zeta+1}^{p(n+1-k)}\sqrt{2\zeta-1}^{q(n+1-k)}\int\widetilde{m}_{\zeta}^{P_{0}}=\int
m^{\zeta}_{k}.$

By analytic continuation, this formula also holds for general $\zeta\in
U_{\mathbb{C}}$. ∎

###  5.2. Construction of an $\operatorname{O}(p,q)$-invariant Crofton measure

######  Lemma 5.3.

The meromorphic family of generalized functions

$f_{\lambda}(X):=\sum_{h=0}^{r}e^{\mathbf{i}\pi h\lambda}|\det
X|^{\lambda}_{r-h}\in C^{-\infty}(\operatorname{Sym}_{r}(\mathbb{R}))$

is analytic in $\lambda\in\mathbb{C}$ and satisfies

$f_{\lambda}(-X)=e^{\mathbf{i}\pi r\lambda}\overline{f_{\lambda}(X)}.$ (12)

###### Proof.

We recall some results from [30]. Consider a linear combination

$g_{\lambda}(X):=\sum_{h=0}^{r}a_{h}|\det X|^{\lambda}_{r-h}$

with constant coefficients $a_{h}\in\mathbb{C}$ and set
$\vec{a}:=(a_{0},\ldots,a_{r})\in\mathbb{C}^{r+1}$. Then $g_{\lambda}\in
C^{-\infty}(\operatorname{Sym}_{r}(\mathbb{R}))$ is meromorphic with possible
poles in the set $\left\\{-m,-\frac{2m+1}{2}:m\geq 1\right\\}$.

The order of the pole at $s$ in this set can be obtained as follows. Set
$\epsilon=-1$ if $s$ is an even integer and $\epsilon=1$ otherwise. Define
inductively linear maps
$d^{(m)}=(d^{(m)}_{0},\ldots,d^{(m)}_{r+1-m}):\mathbb{C}^{r+1}\to\mathbb{C}^{r+1-m}$
by setting

$\displaystyle d^{(0)}_{h}(\vec{a})$ $\displaystyle:=a_{h}$ $\displaystyle
d^{(1)}_{h}(\vec{a})$ $\displaystyle:=a_{h}+\epsilon a_{h+1}$ $\displaystyle
d_{h}^{(2l+1)}(\vec{a})$ $\displaystyle:=d^{(2l-1)}_{h}-d^{(2l-1)}_{h+2},\quad
l=1,2,\ldots$ $\displaystyle d_{h}^{(2l)}(\vec{a})$
$\displaystyle:=d^{(2l-2)}_{h}+d^{(2l-2)}_{h+2},\quad l=1,2,\ldots$

Then $g_{\lambda}$ has a pole of order $p$ at $s=-\frac{2m+1}{2}$ if and only
if $d^{2p}(\vec{a})\neq 0,d^{2p+2}(\vec{a})=0$. Similarly, $g_{\lambda}$ has a
pole of order $p$ at $s=-m$ if and only if $d^{2p-1}(\vec{a})\neq
0,d^{2p+1}(\vec{a})=0$. Here we use the convention that $d^{(m)}=0$ if $m>r+1$
and that a pole of order $0$ is a point of analyticity.

In our situation, the coefficients $a_{h}=a_{h}(\lambda)=e^{\mathbf{i}\pi
h\lambda}$ depend on $\lambda$ and we cannot apply Muro’s result directly.
However, writing

$f_{\lambda}(X)=\sum_{h=0}^{r}a_{h}(\lambda)|\det
X|^{\lambda}_{r-h}=\sum_{j=0}^{\infty}\frac{(\lambda-s)^{j}}{j!}\sum_{h=0}^{r}a_{h}^{(j)}(s)|\det
X|^{\lambda}_{r-h},$

we see that it is enough to prove that the order of the pole of
$\sum_{h=0}^{r}a_{h}^{(j)}(s)|\det X|^{\lambda}_{r-h}$ at $\lambda=s$ is at
most $j$ for all $j$.

By induction we find that for all $l=0,1,\ldots$

$\displaystyle d^{2l}_{h}(\vec{a}(\lambda))$ $\displaystyle=e^{\mathbf{i}\pi
h\lambda}(1+e^{2\pi\mathbf{i}\lambda})^{l},$ $\displaystyle
d^{2l+1}_{h}(\vec{a}(\lambda))$ $\displaystyle=e^{\mathbf{i}\pi
h\lambda}(1+\epsilon
e^{\mathbf{i}\pi\lambda})(1-e^{2\pi\mathbf{i}\lambda})^{l},$

and hence

$\displaystyle
d^{2j+2}_{h}\left(\vec{a}^{(j)}\left(-\frac{2m+1}{2}\right)\right)$
$\displaystyle=\left.\frac{d^{j}}{d\lambda^{j}}\right|_{\lambda=-\frac{2m+1}{2}}e^{\mathbf{i}\pi
h\lambda}(1+e^{2\pi\mathbf{i}\lambda})^{j+1}=0,$ $\displaystyle
d^{2j+1}_{h}\left(\vec{a}^{(j)}(-m)\right)$
$\displaystyle=\left.\frac{d^{j}}{d\lambda^{j}}\right|_{\lambda=-m}e^{\mathbf{i}\pi
h\lambda}(1+\epsilon
e^{\mathbf{i}\pi\lambda})(1-e^{2\pi\mathbf{i}\lambda})^{j}=0,$

which finishes the proof.

∎

######  Proposition 5.4.

Let $C^{\prime}\subset\operatorname{Sym}_{r}^{+}(\mathbb{R})$ be a closed
convex cone. Then

$\lim_{Y\to 0,Y\in C^{\prime}}\det(X+\mathbf{i}Y)^{\lambda}=f_{\lambda}(X)$

in the strong topology on tempered distributions on
$\operatorname{Sym}_{r}(\mathbb{R})$.

###### Proof.

For $\mathrm{Re}\lambda\geq 0$ the statement is easy, so in the following we
assume $\mathrm{Re}\lambda<0$.

First we claim that the limit exists. We will show that there are constants
$\alpha=\alpha(\lambda)\geq 0$ and $b^{\prime}=b(\lambda,C^{\prime})$ such
that

$|\det(X+\mathbf{i}Y)^{\lambda}|\leq b^{\prime}\|Y\|^{-\alpha},\qquad\forall
X\in\operatorname{Sym}_{r}(\mathbb{R}),\forall Y\in
C^{\prime}\setminus\\{0\\}$ (13)

First take $Y=I_{r}$. Letting $(\mu_{j})_{j=1}^{r}\subset\mathbb{R}$ be the
eigenvalues of $X$, we get

$\displaystyle|\det(X+\mathbf{i}I_{r})^{\lambda}|$
$\displaystyle=\prod|\mu_{j}+\mathbf{i}|^{{\mathrm{Re}}\lambda}e^{-\mathrm{Im}\lambda\cdot\mathrm{Arg}(\mu_{j}+i)}$
$\displaystyle\leq e^{\pi
r|\mathrm{Im}\lambda|}\prod(\mu_{j}^{2}+1)^{\frac{\mathrm{Re}\lambda}{2}}$
$\displaystyle\leq e^{\pi r|\mathrm{Im}\lambda|}.$

Now for general $Y$, we have

$|\det(X+\mathbf{i}Y)^{\lambda}|=|\det
Y|^{\mathrm{Re}\lambda}|\det(\sqrt{Y}^{-1}X\sqrt{Y}^{-1}+\mathbf{i}I)^{\lambda}|\leq
e^{\pi r{|\mathrm{Im}\lambda|}}|\det Y|^{\mathrm{Re}\lambda},$

and letting $c:=\sup\left\\{\frac{\|Y\|^{r}}{|\det Y|}:Y\in
C^{\prime}\right\\}$, we conclude that (13) holds with
$b^{\prime}=c^{-\mathrm{Re}\lambda}e^{\pi r|\mathrm{Im}\lambda|}$, and
$\alpha=-r\mathrm{Re}\lambda\geq 0$.

It now follows from [38, Section 26.3] that the limit

$\det(X+\mathbf{i}0)^{\lambda}:=\lim_{Y\to 0,Y\in
C^{\prime}}\det(X+\mathbf{i}Y)^{\lambda}\in\mathcal{S}^{\prime}$

exists in the strong topology on the space of tempered distributions of order
$\lceil r|\mathrm{Re}\lambda|\rceil+{r+1\choose 2}+3$.

It remains to verify that $\det(X+\mathbf{i}0)^{\lambda}=f_{\lambda}(X)$ for
$\mathrm{Re}\lambda<0$. Denote

$H_{\epsilon}=\\{X+\mathbf{i}\epsilon
I_{r}:X\in\operatorname{Sym}_{r}(\mathbb{R})\\}\subset\operatorname{Sym}_{r}(\mathbb{C}).$

Let $\psi(X)$ be a Schwartz function on $\operatorname{Sym}_{r}(\mathbb{R})$,
which is the Fourier transform of a compactly supported smooth function, in
particular it has an analytic extension to
$\operatorname{Sym}_{r}(\mathbb{C})$. Writing
$dZ=\mbox{\Large$\wedge$}_{i=1}^{r}\mbox{\Large$\wedge$}_{j=i}^{r}dz_{ij}$,
the integral $\int_{H_{\epsilon}}\psi(Z)\det(Z)^{\lambda}dZ$ is convergent,
since $\psi$ is rapidly decaying at infinity and $\det(Z)^{\lambda}$ of
polynomial growth. It is clearly analytic in $\lambda\in\mathbb{C}$.
Furthermore, its value is independent of $\epsilon$ as the integrand is a
closed form, rapidly decaying at infinity. For $\lambda>0$, we have

$\det(X+\mathbf{i}\epsilon
I_{r})^{\lambda}=\prod_{j=1}^{r}(\mu_{j}+\mathbf{i}\epsilon)^{\lambda}\to\prod|\mu_{j}|^{\lambda}e^{\mathbf{i}\pi\\#\\{\mu_{j}<0\\}\lambda}=f_{\lambda}(X),$

and so

$\int_{H_{\epsilon}}\psi(Z)\det(Z)^{\lambda}dZ\to\int_{\operatorname{Sym}_{r}(\mathbb{R})}\psi(X)f_{\lambda}(X)dX$

for $\lambda>0$. By analytic extension we conclude that for all
$\lambda\in\mathbb{C}$ and $\epsilon>0$,

$\int_{H_{\epsilon}}\psi(Z)\det(Z)^{\lambda}dZ=\int_{\operatorname{Sym}_{r}(\mathbb{R})}\psi(X)f_{\lambda}(X)dX,$

that is

$\int_{\operatorname{Sym}_{r}(\mathbb{R})}\psi(X+\mathbf{i}\epsilon
I_{r})\det(X+\mathbf{i}\epsilon
I_{r})^{\lambda}dX=\int_{\operatorname{Sym}_{r}(\mathbb{R})}\psi(X)f_{\lambda}(X)dX.$

As $\epsilon\to 0$, we have $\psi(X+\mathbf{i}\epsilon I_{r})\to\psi(X)$ in
$\mathcal{S}$, while $\det(X+\mathbf{i}\epsilon
I_{r})^{\lambda}\to\det(X+\mathbf{i}0)^{\lambda}$ in $\mathcal{S}^{\prime}$.
It follows by continuity that

$\int_{\operatorname{Sym}_{r}(\mathbb{R})}\psi(X)\det(X+\mathbf{i}0)^{\lambda}dX=\int_{\operatorname{Sym}_{r}(\mathbb{R})}\psi(X)f_{\lambda}(X)dX.$

Finally, noting that the set of Schwartz functions such as $\psi$ is dense, we
conclude that $\det(X+\mathbf{i}0)^{\lambda}=f_{\lambda}$ for all
$\lambda\in\mathbb{C}$, as claimed. ∎

Henceforth we use $f_{\lambda}$ and $\det(X+\mathbf{i}0)^{\lambda}$
interchangeably. The following statement shows that the convergence along
$Y\in\mathbb{R}_{+}I_{r}$ 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.

######  Proposition 5.5.

Denote $N^{*}\Gamma^{r}=\cup_{\nu=0}^{r}N^{*}\Gamma^{r}_{\nu}\subset
T^{*}\operatorname{Sym}_{r}(\mathbb{R})$, where $\Gamma^{r}_{\nu}$ consists of
all matrices of nullity $\nu$. It then holds for all $\lambda\in\mathbb{C}$
that $\det(X+\mathbf{i}\epsilon I_{r})^{\lambda}\to f_{\lambda}(X)$ in
$C^{-\infty}_{N^{*}\Gamma^{r}}(\operatorname{Sym}_{r}(\mathbb{R}))$ in the
normal Hörmander topology .

###### Proof.

First note that $g^{*}f_{\lambda}=\det(g)^{2\lambda}f_{\lambda}$ for all
$g\in\operatorname{GL}(r)$. Thus we have the differential equations
$(\underline{A}-2\lambda\operatorname{tr}(A))f_{\lambda}=0$, where
$\underline{A}$ is the vector field defined by the infinitesimal action of
$A\in\mathfrak{gl}(r)$. It follows from [26, Theorem 8.3.1] that
$\operatorname{WF}(f_{\lambda})\subset N^{*}\Gamma^{r}$.

We proceed by induction on $r$, the case $r=1$ being trivial. Since
$\det(X+\mathbf{i}\epsilon I_{r})^{\lambda}\to\det(X+\mathbf{i}0)^{\lambda}$
in the strong topology by Proposition 5.4 and
$N^{*}_{0}\Gamma^{r}_{r}=T_{0}^{*}\operatorname{Sym}_{r}(\mathbb{R})$, it
remains to consider convergence in
$\operatorname{Sym}_{r}(\mathbb{R})\setminus\\{0\\}$. Consider a matrix
$Y\in\operatorname{Sym}_{r}(\mathbb{R})$ of nullity $\nu<r$. Let $E_{0}$ be
its kernel, and $F_{0}=E_{0}^{\perp}$. There is then a unique map
$E:U\to\operatorname{Gr}_{\nu}(\mathbb{R}^{r})$ 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.

We claim $E=E(X)$ is smooth. Indeed, consider
$Z=\\{(X,F):X(F)=F\\}\subset\operatorname{Sym}_{r}(\mathbb{R})\times\operatorname{Gr}_{r-\nu}(\mathbb{R}^{r})$.
Clearly $Z$ is the graph of a unique function $F=F(X)$ near $(Y,F_{0})$. Let
us check that $Z$ is a manifold near $(Y,F_{0})$. Define
$\alpha:U\times\operatorname{Gr}_{r-\nu}(\mathbb{R}^{r})\to\operatorname{Gr}_{r-\nu}(\mathbb{R}^{r})\times\operatorname{Gr}_{r-\nu}(\mathbb{R}^{r})$
by $\alpha(X,F)=(F,X(F))$. Then $Z=\alpha^{-1}(\Delta)$, where $\Delta$ is the
diagonal. Let us verify that $\alpha$ is a submersion at $(Y,F_{0})$.

For $M\in\operatorname{Sym}_{r}(\mathbb{R})$ and $H\in
T_{F_{0}}\operatorname{Gr}_{r-\nu}(\mathbb{R}^{r})=\mathrm{Hom}(F_{0},\mathbb{R}^{r}/F_{0})$,
one computes $d_{Y,F_{0}}\alpha(M,H)=(H,Y\circ
H+M|_{F_{0}\to\mathbb{R}^{r}/F_{0}})=(H,M|_{F_{0}\to\mathbb{R}^{r}/F_{0}})$,
since by construction $Y:\mathbb{R}^{r}/F_{0}\to\mathbb{R}^{r}/F_{0}$ is the
zero map. Noting that any linear map $F_{0}\to\mathbb{R}^{r}/F_{0}$ is induced
by a symmetric matrix mapping $M:\mathbb{R}^{r}\to\mathbb{R}^{r}$, it follows
that $\alpha$ is submersive and $Z$ is a manifold. Further,

$T_{Y,F_{0}}Z=\\{(M,H):M\in\operatorname{Sym}_{r}(\mathbb{R}),H=M|_{F_{0}\to\mathbb{R}^{r}/F_{0}}\\}.$

In particular if $(0,H)\in T_{Y,F_{0}}Z$, then we must have $H=0$. It follows
that $F(X)$ is smooth in $U$, and therefore so is $E(X)=F(X)^{\perp}$.

Choose arbitrary orthonormal frames $e_{i}(X)$ for $E(X)$ and $f_{i}(X)$ for
$F(X)=E(X)^{\perp}$ depending smoothly on $X$. Define

$A:U\to\operatorname{Sym}_{\nu}(\mathbb{R}),\quad
B:U\to\operatorname{Sym}_{r-\nu}(\mathbb{R})$

by

$A(X)=(\langle Xe_{i}(X),e_{j}(X)\rangle),\quad B(X)=(\langle
Xf_{i}(X),f_{j}(X)\rangle).$

Then $A$ is a submersion in $U$. Indeed one has

$d_{Y}A(M)_{i,j}=\langle Me_{i}(Y),e_{j}(Y)\rangle+\langle
Ye_{i}(Y),d_{Y}e_{j}(M)\rangle+\langle Ye_{j}(Y),d_{Y}e_{i}(M)\rangle,$

and the last two summands vanish as $e_{i}(Y),e_{j}(Y)\in E_{0}$. It follows
that
$d_{Y}A:\operatorname{Sym}_{r}(\mathbb{R})\to\operatorname{Sym}_{\nu}(\mathbb{R})$
is surjective, and so $A$ is submersive near $Y$.

It holds that

$\det(X+\mathbf{i}\epsilon I_{r})^{\lambda}=A^{*}\det(X_{1}+\mathbf{i}\epsilon
I_{\nu})^{\lambda}\det(B(X)+\mathbf{i}\epsilon I_{r-\nu})^{\lambda},\quad
X_{1}\in\operatorname{Sym}_{\nu}(\mathbb{R}).$

As $B(X)$ is non-degenerate, the second factor is a smooth function in
$(X,\epsilon)\in U\times\mathbb{R}$.

For the first factor, we have by the induction assumption that
$\det(X_{1}+\mathbf{i}\epsilon
I_{\nu})^{\lambda}\to\det(X_{1}+\mathbf{i}0)^{\lambda}$ in the normal topology
on $C^{-\infty}_{N^{*}\Gamma^{\nu}}(\operatorname{Sym}_{\nu}(\mathbb{R}))$. It
then holds that

$\operatorname{WF}(A^{*}\det(X_{1}+\mathbf{i}0)^{\lambda})\subset
A^{*}(N^{*}\Gamma^{\nu})=N^{*}(A^{-1}\Gamma^{\nu})=N^{*}(\Gamma^{r}\cap U),$

and by [19], $A^{*}\det(X_{1}+\mathbf{i}\epsilon I_{\nu})^{\lambda}\to
A^{*}\det(X_{1}+\mathbf{i}0)^{\lambda}$ in the normal Hörmander topology on
$C^{-\infty}_{N^{*}\Gamma^{r}}(\operatorname{Sym}_{r}(\mathbb{R}))$. We
conclude that

$\det(X+\mathbf{i}\epsilon I_{r})^{\lambda}\to\det(X+\mathbf{i}0)^{\lambda}$

in the normal Hörmander topology on
$C^{-\infty}_{N^{*}\Gamma^{\nu}}(\operatorname{Sym}_{r}(\mathbb{R}))$. ∎

######  Remark 5.6.

Using the Hilbert-Schmidt inner product to identify
$T_{0}^{*}\operatorname{Sym}_{r}(\mathbb{R})=\operatorname{Sym}_{r}(\mathbb{R})$,
the statement of the proposition in fact holds with all conormal cones
intersected with $\operatorname{Sym}_{r}^{+}(\mathbb{R})$, which follows from
[26, Theorem 8.1.6].

In [22, Proposition 4.9], an $\operatorname{O}(p,q)$-invariant distribution
was constructed on $\operatorname{Gr}_{n+1-k}(V)$. Let us briefly recall the
construction.

Write

$\kappa=\min(p,q,k,n+1-k),\quad p^{\prime}=\max(0,p-k),\quad
q^{\prime}=\max(0,q-k).$

Then

$\kappa+p^{\prime}+q^{\prime}=n+1-k.$ (14)

Let $P$ be a $Q$-compatible Euclidean structure, and let $\sigma_{P}$ be the
corresponding $\operatorname{O}(P)$-invariant probability measure on
$\operatorname{Gr}_{n+1-k}(V)$. Decompose $V=V_{P}^{+}\oplus V_{P}^{-}$ such
that $Q|_{V_{P}^{\pm}}=\pm P|_{V_{P}^{\pm}}$. For a subspace
$E\in\operatorname{Gr}_{n+1-k}(V)$, choose a $P$-orthonormal basis
$u_{1},\dots,u_{\kappa},v^{+}_{1},\dots,v^{+}_{p^{\prime}},v^{-}_{1},\dots
v^{-}_{q^{\prime}}$ such that $v^{\pm}_{\nu}\in V^{\pm}_{P}$. Let
$\widehat{X}^{P}_{\zeta}(E)$ be the Gram matrix of $Q_{\zeta}$ with respect to
$P$, restricted to $\operatorname{Span}(u_{\nu})$. It holds that $\det
X_{\zeta}^{P}(E)=(2\zeta+1)^{p^{\prime}}(2\zeta-1)^{q^{\prime}}\det\widehat{X}^{P}_{\zeta}(E)$.
Then
$\widetilde{m}_{k}^{0}|_{U_{P}}:=(\widehat{X}^{P}_{0})^{*}f_{-\frac{n+1}{2}}\cdot
d\sigma_{P}(E)$ is a well-defined distribution in the open and dense set
$U_{P}$ where
$\widehat{X}_{0}^{P}:\operatorname{Gr}_{n+1-k}(V)\to\operatorname{Sym}_{\kappa}(\mathbb{R})$
is a submersion, explicitly $U_{P}=\\{E:\pm
1\notin\mathrm{Spec}(X^{P}_{0}(E))\\}$. We now choose several Euclidean
structures $P_{j}$ as above, $1\leq j\leq N$, such that the sets $U_{P_{j}}$
cover the Grassmannian. Then $\widetilde{m}_{k}^{0}|_{U_{P_{j}}}$ patch to a
globally defined distribution $\widetilde{m}_{k}^{0}$, which is invariant
under $\operatorname{O}(Q)$.

We will be approximating $\widetilde{m}_{k}^{0}$ by smooth measures. For this
purpose we will need to simplify the construction above, namely we will only
use a single Euclidean structure. This is made possible by monitoring the wave
front set.

Write $P=P_{0}$, $X_{0}=X^{P}_{0}$, $\widehat{X}_{0}=\widehat{X}^{P}_{0}$,
$dE=d\sigma_{P}(E)$.

######  Proposition 5.7.

The distribution

$\widetilde{m}^{0}_{k}:=\widehat{X}_{0}^{*}f_{-\frac{n+1}{2}}(E)\cdot
dE\in\mathcal{M}^{-\infty}(\operatorname{Gr}_{n+1-k}(V))$

is well-defined and $\operatorname{O}(Q)$-invariant.

###### Proof.

Denote by $\mathrm{mult}(\mu,X)$ the multiplicity of $\mu$ in the spectrum of
$X$, and by $E_{\mu}(X)$ the eigenspace of $X$ with eigenvalue $\mu$. Consider

$B^{a}_{\mu}=\\{X:\mathrm{mult}(\mu,X)=a\\}\subset\operatorname{Sym}_{\kappa}(\mathbb{R}).$

This is a locally closed submanifold: $B^{a}_{0}$ locally coincides with an
orbit of the action of $\operatorname{GL}(\kappa)$ on
$\operatorname{Sym}_{\kappa}(\mathbb{R})$ by $(g,X)\mapsto g^{T}Xg$, and
$B^{a}_{\mu}=\mu I+B^{a}_{0}$.

Using the Hilbert-Schmidt Euclidean structure $\langle
X,Y\rangle=\operatorname{tr}(XY)$ to identify
$T_{X}\operatorname{Sym}_{\kappa}(\mathbb{R})=T_{X}^{*}\operatorname{Sym}_{\kappa}(\mathbb{R})=\operatorname{Sym}_{\kappa}(\mathbb{R})$,
let us describe the set $N^{*}_{X}B_{\mu}^{a}$. As
$T_{X}B_{0}^{a}=\\{A^{T}X+XA:A\in\mathfrak{gl}_{\kappa}(\mathbb{R})\\}$, we
have $\Xi\in N^{*}_{X}B_{0}^{a}\iff\operatorname{tr}(\Xi A^{T}X+\Xi XA)=0$ for
all $A$, or equivalently
$N^{*}_{X}B_{0}^{a}=\\{\Xi\in\operatorname{Sym}_{\kappa}(\mathbb{R}):\Xi
X=0\\}$. It follows that $N^{*}_{X}B_{\mu}^{a}=N^{*}_{X-\mu
I}B_{0}^{a}=\\{\Xi:\Xi X=\mu\Xi\\}$, and one easily computes that
$\mathrm{codim}B_{\mu}^{a}=\binom{a+1}{2}$.

Let us check that $\widetilde{m}_{k}^{0}$ is well-defined. For this purpose,
writing

$L_{E,Y}=\mathrm{Ker}(d\widehat{X}_{0}^{*}:T^{*}_{Y}\operatorname{Sym}_{\kappa}(\mathbb{R})\to
T_{E}^{*}\operatorname{Gr}_{n+1-k}(V)),$

we should check [21] that $\operatorname{WF}(f_{\lambda})\cap
L_{E,Y}=\emptyset$ whenever $\widehat{X}_{0}(E)=Y$, where
$\lambda=-\frac{n+1}{2}$. We may assume $Y$ lies in the singular support of
$f_{\lambda}$, and $\widehat{X}_{0}$ is not submersive at $E$. This implies
that $Y\in B^{r}_{0}\cap B^{s}_{1}\cap B^{t}_{-1}$ with $r>0$ and $s+t>0$.

The intersections $B^{s,t}_{1,-1}:=B^{s}_{1}\cap B^{t}_{-1}$ and
$B^{r}_{0}\cap B^{s,t}_{1,-1}$ are transversal. Indeed,
$N_{Y}^{*}B_{1}^{s}\cap N_{Y}^{*}B_{-1}^{t}=\\{\Xi:\Xi Y=\Xi=-\Xi\\}=\\{0\\}$,
so $B^{s}_{1}\pitchfork B^{t}_{-1}$,
$N^{*}_{Y}B^{s,t}_{1,-1}=\\{\Xi_{1}+\Xi_{2}:\Xi_{1}Y=\Xi_{1},\Xi_{2}Y=-\Xi_{2}\\}$
and
$\mathrm{codim}T_{Y}B^{s,t}_{1,-1}=\mathrm{codim}T_{Y}B_{1}^{s}+\mathrm{codim}T_{Y}B_{-1}^{t}={s+1\choose
2}+{t+1\choose 2}$. Next if $\Xi=\Xi_{1}+\Xi_{2}\in
N^{*}_{Y}B^{s,t}_{1,-1}\cap N_{Y}^{*}B_{0}^{r}$, then
$\Xi_{1}-\Xi_{2}=\Xi_{1}Y+\Xi_{2}Y=\Xi Y=0$, so that $\Xi_{1}=\Xi_{2}$, which
can only happen if $\Xi_{1}=\Xi_{2}=0$, thus $\Xi=0$. Therefore
$B_{0}^{r}\pitchfork B_{1,-1}^{s,t}$.

By Proposition 5.5, $\operatorname{WF}_{Y}(f_{\lambda})\subset
N_{Y}^{*}B_{0}^{r}$. Set $E_{Y}:=E_{1}(Y)\oplus E_{-1}(Y)$, and define
$W\subset B^{s,t}_{1,-1}$ as the set of matrices $X$ of the same spectrum as
$Y$ that satisfy $E_{\mu}(X)=E_{\mu}(Y)$ for all $\mu\neq\pm 1$. It is clearly
a manifold that can be identified with
$\operatorname{Gr}_{s}(E_{Y})=\operatorname{Gr}_{s}(\mathbb{R}^{s+t})$, in
particular $\dim W=st$. Evidently $W\subset B^{r}_{0}$.

For $1\leq j\leq\kappa$, choose a curve $\gamma_{j}$ through $E$ given by

$\gamma_{j}(t)=\operatorname{Span}(u_{1},u_{2},\dots,u_{j}(t),\ldots,u_{\kappa},v^{+}_{1},...,v^{+}_{p^{\prime}},v_{1}^{-},\dots,v_{q^{\prime}}^{-})$

with all vectors orthonormal and fixed except for $u_{j}$, such that
$u_{\kappa-s-t+1},\dots,u_{\kappa-t}\in V_{P}^{+}$,
$u_{\kappa-t+1},\dots,u_{\kappa}\in V_{P}^{-}$, and $\xi=u_{j}^{\prime}(0)\in
E^{P}$ arbitrary. By [22, Lemma 4.1], we have

$d_{E}\widehat{X}_{0}(\gamma_{j}^{\prime}(0))=\begin{pmatrix}0&\cdots&0&Q(\xi,u_{1})&0&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&\cdots&0&Q(\xi,u_{j-1})&0&\cdots&0\\\
Q(\xi,u_{1})&\cdots&Q(\xi,u_{j-1})&Q(\xi,2u_{j})&Q(\xi,u_{j+1})&\cdots&Q(\xi,u_{\kappa})\\\
0&\cdots&0&Q(\xi,u_{j+1})&0&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&\cdots&0&Q(\xi,u_{\kappa})&0&\cdots&0\end{pmatrix}$

Note that $E\cap(E^{P})^{Q}=\operatorname{Span}(u_{\kappa-
s-t+1},\dots,u_{\kappa},v^{+}_{1},...,v^{+}_{p^{\prime}},v^{-}_{1},...,v_{q^{\prime}}^{-})$
by assumption. Hence $Q(\xi,u_{1}),Q(\xi,u_{2}),\ldots,Q(\xi,u_{\kappa-s-t})$
are linearly independent functionals in $\xi\in E^{P}$. Thus the bottom right
$(s+t)\times(s+t)$ submatrix of $d_{E}\widehat{X}_{0}(\gamma_{j}^{\prime}(0))$
vanishes, while all entries on row $j$ in the first $(\kappa-s-t)$ columns are
arbitrary. It follows that
$\mathrm{codim}(\mathrm{Image}(d_{E}\widehat{X}_{0}))={s+t+1\choose 2}$, and
$\mathrm{Image}(d_{E}\widehat{X}_{0})\cap T_{Y}W=\\{0\\}$.

As $-P\leq Q\leq P$, it holds that
$\mathrm{Image}(d_{E}\widehat{X}_{0})\subset T_{Y}B^{s,t}_{1,-1}$. Furthermore
${s+1\choose 2}+{t+1\choose 2}+st={s+t+1\choose 2}$, so that
$\dim\mathrm{Image}(d_{E}\widehat{X}_{0})+\dim T_{Y}W=\dim
T_{Y}B^{s,t}_{1,-1}$, and so

$\mathrm{Image}(d_{E}\widehat{X}_{0})\oplus T_{Y}W=T_{Y}B^{s,t}_{1,-1}.$

Since $T_{Y}W\subset T_{Y}B^{r}_{0}$ and
$T_{Y}B^{r}_{0}+T_{Y}B^{s,t}_{1,-1}=T_{Y}\operatorname{Sym}_{\kappa}(\mathbb{R})$,
we conclude that
$\mathrm{Image}(d_{E}\widehat{X}_{0})+T_{Y}B^{r}_{0}=T_{Y}\operatorname{Sym}_{\kappa}(\mathbb{R})$
and so $N^{*}B_{0}^{r}\cap L_{E,Y}=\\{0\\}$. We conclude that
$\operatorname{WF}(f_{\lambda})\cap L_{E,Y}=\emptyset$, and so
$\widetilde{m}_{k}^{0}$ is well-defined.

As the proof above is independent of the value of $\lambda$, invariance under
$\operatorname{O}(Q)$ follows by analytic continuation as in the original
construction in [22], and we omit the details. ∎

######  Definition 5.8.

Set

$m_{k}:=e^{\frac{\mathbf{i}\pi}{2}\left((n+1)\min(k,q)-qk\right)}\widetilde{m}_{k}^{0}\in\mathcal{M}^{-\infty}(\operatorname{Gr}_{n+1-k}(V))^{\operatorname{O}(Q)}.$

######  Lemma 5.9.

Let $j\colon\mathbb{R}^{p,q}\to\mathbb{R}^{q,p}$ be given by $j(x,y)=(y,x)$
where $x\in\mathbb{R}^{p},y\in\mathbb{R}^{q}$. Let us also denote by $j$ the
induced map
$\operatorname{Gr}_{p+q-k}(\mathbb{R}^{p,q})\to\operatorname{Gr}_{p+q-k}(\mathbb{R}^{q,p})$.
Then

$j^{*}m_{k}=\overline{m_{k}}.$

###### Proof.

We have
$m_{k}=\mathbf{i}^{(n+1)\min(k,p)-pk}{\widehat{X}_{0}^{*}f_{-\frac{n+1}{2}}}d\sigma_{P_{0}}$
on $\mathbb{R}^{q,p}$. Since $\widehat{X}_{0}\circ j=-\widehat{X}_{0}$, (12)
implies that
$j^{*}\widehat{X}_{0}^{*}f_{-\frac{n+1}{2}}=\mathbf{i}^{-\kappa(n+1)}\overline{\widehat{X}_{0}^{*}f_{-\frac{n+1}{2}}}$.
Using (14) we get

$\displaystyle j^{*}m_{k}$
$\displaystyle=\mathbf{i}^{(n+1)\min(k,p)-pk-\kappa(n+1)}\overline{\widehat{X}_{0}^{*}f_{-\frac{n+1}{2}}}d\sigma_{P_{0}}$
$\displaystyle=\mathbf{i}^{-(n+1)\min(k,q)+qk}\overline{\widehat{X}_{0}^{*}f_{-\frac{n+1}{2}}}d\sigma_{P_{0}}$

which is the conjugate of $m_{k}$ in $\mathbb{R}^{p,q}$. ∎

######  Proposition 5.10.

Define

$N^{*}\Lambda:=\cup_{\nu\geq 1}N^{*}\Lambda^{\nu}_{n+1-k}(V).$

  1. i)

The wave front set of $m_{k}$ is contained in $N^{*}\Lambda$.

  2. ii)

$m^{\mathbf{i}\epsilon}_{k}\to m_{k}$ in
$\mathcal{M}_{N^{*}\Lambda}^{-\infty}(\operatorname{Gr}_{n+1-k}(V))$ as
$\epsilon\to 0^{+}$ in the normal Hörmander topology.

###### Proof.

Write $\lambda=-\frac{n+1}{2}$. For $\zeta\in U_{\mathbb{C}}$ we have

$(2\zeta+1)^{-\frac{n+1-k}{2}p}(2\zeta-1)^{-\frac{n+1-k}{2}q}m^{\zeta}_{k}(E)=\det(X^{P_{0}}_{\zeta})^{\lambda}dE.$

We compute using the special basis described above adapted to $P_{0}$,

$\displaystyle\det(X^{P_{0}}_{\mathbf{i}\epsilon}(E))^{\lambda}$
$\displaystyle=\det(X_{0}(E)+2\mathbf{i}\epsilon
I_{n+1-k})^{\lambda}=X_{0}^{*}\det(X+2\mathbf{i}\epsilon I_{n+1-k})^{\lambda}$
$\displaystyle=(1+2\mathbf{i}\epsilon)^{\lambda
p^{\prime}}(-1+2\mathbf{i}\epsilon)^{\lambda
q^{\prime}}\widehat{X}_{0}^{*}\det(X+2\mathbf{i}\epsilon
I_{\kappa})^{\lambda}.$

By Proposition 5.5, we have $\det(X+2\mathbf{i}\epsilon
I_{\kappa})^{\lambda}\to f_{\lambda}(X)$ in the normal Hörmander topology on
$C^{-\infty}_{\Gamma^{\kappa}}(\operatorname{Sym}_{\kappa}(\mathbb{R}))$. By
the proof of Proposition 5.7, we may use the continuity of the pull-back
$\widehat{X}_{0}^{*}$ in the normal Hörmander topology [19]. Noting that
$\widehat{X}_{0}^{-1}(\Gamma^{\kappa}_{\nu})\subset\Lambda_{n-k+1}^{\nu}(V)$
so that $\widehat{X}_{0}^{*}N^{*}\Gamma^{\kappa}\subset N^{*}\Lambda$, we find
that

$\widehat{X}_{0}^{*}\det(X+2\mathbf{i}\epsilon
I_{\kappa})^{\lambda}dE\to\widetilde{m}_{k}^{0}$

in the normal Hörmander topology as stated. ∎

######  Corollary 5.11.

Let $M\subset V^{n+1}$ be a pseudosphere or a pseudohyperbolic space, and
$A\subset M$ either a smooth domain with LC-regular boundary, or a smooth LC-
regular hypersurface without boundary. Assume all $Q$-degenerate tangents to
$A$ of codimension $k$ are regular. Then

$\operatorname{Cr}(m_{k})(A)=\int_{\operatorname{Gr}_{n+1-k}(V)}\chi(A\cap
E)dm_{k}(E).$ (15)

###### Proof.

First note that $\operatorname{Cr}(m_{k})\in\mathcal{V}^{-\infty}(M)$ is
isometry invariant and by [15, Theorem C] is given by a linear combination of
the intrinsic volumes. Thus $A$ is WF-transversal to
$\operatorname{Cr}(m_{k})$. The assertion now follows from Corollary 4.7, and
Propositions 5.10 part i) and 3.14. ∎

######  Corollary 5.12.

Let $M^{n}\subset V^{n+1}$ be a pseudosphere or a pseudohyperbolic space, and
$A\subset M$ either a smooth domain or a hypersurface without boundary. Denote
$H=H(A)$.

  1. i)

Assume that for each $x\in H$, $H$ is either pseudo-Riemannian at $x$ or
tangentially regular at $x$. Then (15) holds for $k=1$.

  2. ii)

If $\mathbb{P}(H)\subset\mathbb{P}(V)$ is strictly convex, then (15) holds for
all $k$.

###### Proof.

In both cases, it follows from [14, Lemma 4.7] that $H$ is LC-regular, and we
can apply Corollary 5.11. ∎

Example. The complex-valued distribution
$m_{n}\in\mathcal{M}^{-\infty}(\mathbb{P}(V))$ is invariant under the group of
projective transformations preserving the quadric $[Q]=\\{Q=0\\}$. Its
singular support is $[Q]$, and $\operatorname{WF}(m_{n})\subset N^{*}[Q]$. In
particular, $m_{n}(A)$ is well defined for any domain $A\subset\mathbb{P}(V)$
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.

###  5.3. The flat case

Next we construct a translation- and $\operatorname{O}(p,q)$-invariant
distribution on the affine Grassmannian
$\overline{\operatorname{Gr}}_{p+q-k}(\mathbb{R}^{p,q})$.

######  Proposition 5.13.

Let $P$ be a $Q$-compatible Euclidean structure in
$W=\mathbb{R}^{p+1,q}=W_{+}\oplus W_{-}$. Let $x\in W_{+}\cap S^{p,q}$,
$T=T_{x}S^{p,q}$, and define

$s\colon\overline{\operatorname{Gr}}_{p+q-k}(T)\longrightarrow\operatorname{Gr}_{p+q+1-k}(W),\qquad
s(v+F)=F\oplus\mathbb{R}(x+v),\quad{F\in\operatorname{Gr}_{p+q-k}(T)},$

which is a diffeomorphism onto its open image. Given $t>0$, consider the
homothety $v\mapsto tv$ on $T$ and the induced map $h_{t}$ on
$\overline{\operatorname{Gr}}_{p+q-k}(T)$.

  1. i)

Let $dE$ be an $\operatorname{O}(P)$-invariant measure on
$\operatorname{Gr}_{p+q+1-k}(V)$, thus given by a smooth density. Then

$d\overline{F}=\frac{1}{k!}\left.\frac{d^{k}}{dt^{k}}\right|_{t=0}h_{t}^{*}s^{*}dE$

is an $\overline{\operatorname{O}(P|_{T})}$-invariant measure on
$\overline{\operatorname{Gr}}_{p+q-k}(T)$.

  2. ii)

Let
$\widehat{X}_{0}\colon\operatorname{Gr}_{p+q+1-k}(W)\to\operatorname{Sym}_{\kappa}(\mathbb{R})$
where $\kappa=\min(p+1,q,k,p+q+1-k)$ be as in Proposition 5.7, and let
$\widehat{X}_{0}^{\prime}$ be the corresponding map on
$\operatorname{Gr}_{p+q-k}(T)$. Then

$\left.\frac{1}{k!}\frac{d^{k}}{dt^{k}}\right|_{t=0}(h_{1/t})_{*}(s^{-1})_{*}(\widehat{X}_{0}^{*}f_{-\frac{p+q+1}{2}}(E)dE)=\widehat{X}_{0}^{\prime*}f_{-\frac{p+q+1}{2}}(F)d\overline{F},$

and this generalized measure is
$\overline{\operatorname{O}(Q|_{T})}$-invariant. Here $(s^{-1})_{*}$ denotes
the push-forward by $s^{-1}$ of the restriction to the open set
$\mathrm{Image}(s)$.

###### Proof.

  1. i)

For
$g\in\operatorname{O}(P|_{T})=\operatorname{Stab}_{\operatorname{O}(P)}(x)\subset\operatorname{O}(P)$,
since $g\circ s\circ h_{t}=s\circ h_{t}\circ g$ and $g^{*}dE=dE$, we have
$g^{*}h_{t}^{*}s^{*}dE=h_{t}^{*}s^{*}g^{*}dE=h_{t}^{*}s^{*}dE$, which yields
$\operatorname{O}(P|_{T})$-invariance. As for translation invariance, let
$\rho_{U}\colon
U\times\mathbb{R}^{k}\to\overline{\operatorname{Gr}}_{p+q-k}(T)$ be a local
trivialization of the bundle
$\pi\colon\overline{\operatorname{Gr}}_{p+q-k}(T)\to{\operatorname{Gr}_{p+q-k}}(T)$,
and put $\eta=\rho_{U}^{*}s^{*}dE$. Then
$h_{t}\circ\rho_{U}(F,w)=\rho_{U}(F,tw)$ and thus

$\rho_{U}^{*}h_{t}^{*}s^{*}(dE)_{(F,w)}=t^{k}\eta_{(F,tw)}=t^{k}\eta_{(F,0)}+O(t^{k+1}).$

Since the induced action of a translation of $T$ on $U\times\mathbb{R}^{k}$
has the form $(F,w)\mapsto(F,w+\varphi(F))$, the translation invariance of
$\overline{dF}$ follows.

  2. ii)

On each $E\in\operatorname{Gr}_{p+q+1-k}(W)$ take a $P$-orthonormal basis
$u_{1},\dots,u_{\kappa}$, $v^{+}_{1},\dots,v^{+}_{p^{\prime}}$,
$v^{-}_{1},\dots v^{-}_{q^{\prime}}$ such that $v_{i}^{+}\in
W^{+},v_{i}^{-}\in W^{-}$. Let $X_{0}$ be the Gram matrix of $Q$ on this
basis, and let $\widehat{X}_{0}$ be the submatrix corresponding to the
$u_{i}$. Then
$X_{0}=\operatorname{Diag}(\widehat{X}_{0},1,\stackrel{{\scriptstyle
p^{\prime}}}{{\ldots}},1,-1\stackrel{{\scriptstyle
q^{\prime}}}{{\ldots}},-1)$. In particular $X_{0}$ and $\widehat{X}_{0}$ have
the same determinant up to sign, and their signatures differ by
$(p^{\prime},q^{\prime})$.

Let now $f_{1},\ldots,f_{p+q-k}$ be a $P$-orthonormal basis of
$F\in\operatorname{Gr}_{p+q-k}(T)$, and let $w\in T$ be $P$-orthogonal to $F$.
A $P$-orthonormal basis of $s(tw+F)$ is
$(1+t^{2}P(w))^{-\frac{1}{2}}(x+tw),f_{1},\ldots,f_{p+q-k}$. Hence, the Gram
matrices $X_{0}^{\prime},X_{0}$ of $Q$ restricted to $F,s(tw+F)$ have the same
signature and

$\det X_{0}=(1+t^{2}P(w))^{-1}Q(x+tw)\det X_{0}^{\prime}.$

Moreover, arguing as in the previous paragraph we have $\det
X_{0}^{\prime}=\pm\det\widehat{X}_{0}^{\prime}$ and
$\operatorname{sign}(X_{0}^{\prime})=\operatorname{sign}(\widehat{X}_{0}^{\prime})+(p^{\prime},q^{\prime})$
whenever $Q(x+tw)>0$.

Therefore, for $\lambda>0$, $F\in\operatorname{Gr}_{p+q-k}(T)$ and $w\in
F^{P}\cap T$ we have

$\displaystyle f_{\lambda}(\widehat{X}_{0}(s(tw+F)))$
$\displaystyle=e^{\mathbf{i}\pi q^{\prime}\lambda}f_{\lambda}(X_{0}(s(tw+F)))$
$\displaystyle=e^{\mathbf{i}\pi
q^{\prime}\lambda}(1+t^{2}P(w))^{-\lambda}Q({x+tw})^{\lambda}f_{\lambda}(X_{0}^{\prime}(F))$
$\displaystyle=(1+t^{2}P(w))^{-\lambda}Q({x+tw})^{\lambda}f_{\lambda}(\widehat{X}_{0}^{\prime}(F))$

whenever $Q(x+tw)>0$, where $f_{\lambda}$ is defined by Lemma 5.3 on
$\operatorname{Sym}_{\kappa}(\mathbb{R})$ or
$\operatorname{Sym}_{p+q-k}(\mathbb{R})$ depending on the argument.

By analytic continuation we get

$s^{*}\widehat{X}_{0}^{*}f_{\lambda}(tw+F)=(1+t^{2}P(w))^{-\lambda}Q({x+tw})^{\lambda}(\widehat{X}_{0}^{\prime})^{*}f_{\lambda}(tw+F)$

for all $\lambda$. Hence,

$\lim_{t\to
0}h_{t}^{*}s^{*}\widehat{X}_{0}^{*}f_{\lambda}=(\widehat{X}_{0}^{\prime})^{*}f_{\lambda}.$

In the proof of $i)$ we have seen $(h_{1/t})_{*}(s^{-1})_{*}dE=O(t^{k})$.
Hence, by continuity

$\displaystyle\left.\frac{d^{k}}{dt^{k}}\right|_{t=0}(h_{1/t})_{*}(s^{-1})_{*}(\widehat{X}_{0}^{*}f_{-\frac{p+q+1}{2}}(E)dE)$
$\displaystyle=\lim_{t\to
0}h_{t}^{*}s^{*}\widehat{X}_{0}^{*}f_{-\frac{p+q+1}{2}}\left.\frac{d^{k}}{dt^{k}}\right|_{t=0}(h_{1/t})_{*}(s^{-1})_{*}dE$
$\displaystyle=k!(\widehat{X}_{0}^{\prime})^{*}f_{-\frac{p+q+1}{2}}d\overline{F}.$

Translation invariance is clear. Further, if
$g\in\operatorname{O}(Q|_{T})\subset\operatorname{O}(Q)$, then $g\circ s\circ
h_{t}=s\circ h_{t}\circ g$. Since
$\widehat{X}_{0}^{*}f_{-\frac{p+q+1}{2}}(E)dE$ is
$\operatorname{O}(Q)$-invariant, this yields
$\operatorname{O}(Q|_{T})$-invariance.

∎

Translation-invariance and ${\operatorname{O}(P|_{T})}$-invariance
characterize $d\overline{F}$ uniquely up to normalization. The normalization
can be deduced from Theorem 6.5 in the case $q=0$. As for the translation-
invariant and $\operatorname{O}(Q|_{T})$-invariant generalized measure
obtained on $T\cong\mathbb{R}^{p,q}$, we take the normalization of Definition
5.8 as follows.

######  Definition 5.14.

On $\overline{\operatorname{Gr}}_{p+q-k}(\mathbb{R}^{p,q})$ we fix the
following translation-invariant and ${\operatorname{O}(p,q)}$-invariant
generalized measure

$\check{m}_{k}:=e^{\frac{\mathbf{i}\pi}{2}[(p+q+1)\min(k,q)-qk]}(\widehat{X}_{0}^{\prime}(F))d\overline{F}.$

The Crofton map in the flat case is

$\operatorname{Cr}:\mathcal{M}^{-\infty}(\overline{\operatorname{Gr}}_{p+q-k}(V))\to\mathcal{V}^{-\infty}(V),$

given by
$\langle\operatorname{Cr}(\mu),\psi\rangle=\int_{\overline{\operatorname{Gr}}_{p+q-k}(V)}\psi(\overline{E})d\mu(\overline{E})$
for all $\psi\in\mathcal{V}_{c}^{\infty}(V)$.

The results of the present section and sections 3, 4 can be easily adapted to
the flat pseudo-Euclidean setting. Let us state explicitly Corollary 5.12 in
the flat case.

######  Corollary 5.15.

Let $A\subset\mathbb{R}^{p,q}$ be either a smooth domain or a hypersurface
without boundary. Denote by $H=H(A)$ the corresponding closed hypersurface.

  1. i)

Assume that for each $x\in H$, $H$ is either pseudo-Riemannian near $x$ or has
non-zero Gauss curvature at $x$. Then

$\operatorname{Cr}(\check{m}_{k})(A)=\int_{\overline{\operatorname{Gr}}_{p+q-k}(V)}\chi(A\cap
E)d\check{m}_{k}(E).$ (16)

holds for $k=1$.

  2. ii)

If $H$ is strictly convex, then (16) holds for all $k$.

##  6\. 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 $\zeta>\frac{1}{2}$, we denote by
$S_{\zeta}={Q_{\zeta}^{-1}(1)}\subset\mathbb{R}^{p+q}=\mathbb{R}^{n+1}$ the
unit sphere in the Euclidean space $(\mathbb{R}^{n+1},Q_{\zeta})$. For
$\zeta=0$ we have $S^{p-1,q}=Q_{0}^{-1}(1)$ with the induced pseudo-Riemannian
metric $Q_{0}$. We will also denote by $S^{n}$ the unit sphere in
$\mathbb{R}^{n+1}$ with respect to some fixed Euclidean structure (which is
independent of $\zeta$).

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
$e\colon\mathbb{R}^{p,q}\hookrightarrow\mathbb{R}^{p+l,q+r}$, we have
$e^{*}m_{k}=m_{k}$.

###### Proof.

For $\zeta>\frac{1}{2}$, $Q_{\zeta}$ is positive definite, and so
$e^{*}m_{k}^{\zeta}=m_{k}^{\zeta}$ by the uniqueness of probability measure on
the Grassmannian invariant under the positive definite orthogonal group, as
$e^{*}:\mathcal{M}^{\infty}(\operatorname{Gr}_{p+q+l+r-k}(\mathbb{R}^{p+l,q+r}))\to\mathcal{M}^{\infty}(\operatorname{Gr}_{p+q-k}(\mathbb{R}^{p,q}))$
is essentially the pushforward operation under intersection with
$\mathbb{R}^{p,q}$. The statement then follows by analytic extension in
$\zeta$, combined with Proposition 5.10. ∎

######  Proposition 6.2.

Given $A\in\mathcal{P}(S^{p-1,q})$, let $\overline{A}\in\mathcal{P}(S^{n})$ 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 $\chi(A\cap E)$
is constant for a.e. plane $E$ of codimension $k$. Then

$\lim_{\epsilon\to
0^{+}}\operatorname{Cr}_{S^{n}}(m_{k}^{\mathbf{i}\epsilon})(\overline{A})=\operatorname{Cr}_{S^{p-1,q}}(m_{k})(A).$

###### Proof.

We have by Proposition 3.14

$\operatorname{Cr}_{S^{n}}(m_{k}^{\mathbf{i}\epsilon})(\overline{A})=\langle
m_{k}^{\mathbf{i}\epsilon},\chi(\overline{A}\cap\bullet)\rangle.$

By Corollary 5.11, resp. by Proposition 3.14, it holds that

$\displaystyle\operatorname{Cr}_{k}^{p-1,q}(A)=\operatorname{Cr}(m_{k})(A)=\langle
m_{k},\chi(A\cap\bullet)\rangle.$

By part ii) of Proposition 5.10, $m_{k}^{\mathbf{i}\epsilon}$ tends to $m_{k}$
as $\epsilon\to 0^{+}$ in the normal Hörmander topology on
$\mathcal{M}^{-\infty}_{N^{*}\Lambda}(\operatorname{Gr}_{n+1-k}(V))$.
Combining Corollary 4.7 and Proposition 5.10 part i), we see that evaluating