Kac-Rice formula for transverse intersections

Abstract

We prove a generalized Kac-Rice formula that, in a well defined regular setting, computes the expected cardinality of the preimage of a submanifold via a random map by expressing it as the integral of a density. Our proof starts from scratch and although it follows the guidelines of the standard proofs of Kac-Rice formula, it contains some new ideas coming from the point of view of measure theory. Generalizing further, we extend this formula to any other type of counting measure, such as the intersection degree. We discuss in depth the specialization to smooth Gaussian random sections of a vector bundle. Here, the formula computes the expected number of points where the section meets a given submanifold of the total space, it holds under natural non-degeneracy conditions and can be simplified by using appropriate connections. Moreover, we point out a class of submanifolds, that we call sub-Gaussian, for which the formula is locally finite and depends continuously with respect to the covariance of the first jet. In particular, this applies to any notion of singularity of sections that can be defined as the set of points where the jet prolongation meets a given semialgebraic submanifold of the jet space. Various examples of applications and special cases are discussed. In particular, we report a new proof of the Poincaré kinematic formula for homogeneous spaces and we observe how the formula simplifies for isotropic Gaussian fields on the sphere.

Appendices

Densities

Let M be a smooth manifold. The density bundle or, as we call it, the bundle of density elements $\varDelta M$ is the vector bundle:

$$\begin{aligned} \varDelta M=\wedge ^m(T^*M)\otimes L, \end{aligned}$$

where L is the orientation bundle (see [6, Sect. 7]); $\varDelta M$ is a smooth real line bundle and the fiber can be identified canonically with

$$\begin{aligned} \begin{aligned} \varDelta _pM =\{\delta :(T_pM)^m\rightarrow {\mathbb {R}}:\delta =\pm |{\omega }|, \hbox { for some}\ {\omega }\in \wedge ^mT^*_pM\}. \end{aligned} \end{aligned}$$

We call an element $\delta \in \varDelta _pM$, a density element at p.

Given a set of coordinates $x^1,\dots ,x^m$ on U, we denote

$$\begin{aligned} dx=dx^1\dots dx^m=|dx^1\wedge \dots \wedge dx^m|. \end{aligned}$$

Using the language of [6], dx is the section $(dx_1\wedge \cdots \wedge dx_m)\otimes e_U$, where $e_U$ is the section of $L|_{U}$ which in the trivialization induced by the chart $(x_1, \ldots , x_m)$ corresponds to the constant function 1.

We have, directly from the definition, that

$$\begin{aligned} dx=\left| \det \left( \frac{\partial x}{\partial y}\right) \right| dy. \end{aligned}$$

for any other set of coordinates $y^1,\dots ,y^m$. It follows that an atlas for M with transition functions $g_{a,b}$ defines a trivializing atlas for the vector bundle $\varDelta M$, with transition functions for the fibers given by $|\det (Dg_{a,b})|$.

Each density element $\delta$, is either positive or negative, respectively if $\delta =|{\omega }|$ or $\delta =-|{\omega }|$, for some skew-symmetric multilinear form $w\in \wedge ^mT^*_pM$. In other words, the bundle $\varDelta M$ is canonically oriented and thus trivial (but not canonically trivial) and we denote the subbundle of positive elements by $\varDelta ^+ M\cong M\times [0,+\infty )$ (again, not canonically).

The modulus of a density element is defined in coordinates, by the identity $|({\omega }(x)dx)|=|{\omega }(x)|dx$. This defines a continuous map $|\cdot |:\varDelta M\rightarrow \varDelta ^+ M$.

The sections of $\varDelta M$ are called densities and we will usually denote them as maps $p\mapsto \delta (p)$. We define ${\mathscr {D}}^r(M)$ to be the space of ${\mathscr {C}}^r$ densities and by ${\mathscr {D}}^r_c(M)$ the subset of the compactly supported ones. For smooth densities, we just write ${\mathscr {D}}={\mathscr {D}}^\infty$. From the formula for transition functions it is clear that there is a canonical linear function

$$\begin{aligned} \int _M :{\mathscr {D}}_c(M)\rightarrow {\mathbb {R}}\qquad \int _M \delta =:\int _{M}\delta (p)dp. \end{aligned}$$

If N is Riemannian and $f:M\rightarrow N$ is a ${\mathscr {C}}^1$ map with $\dim M\le \dim N$, we define the jacobian density $\delta f\in {\mathscr {D}}(M)$ by the following formula, in coordinates:

$$\begin{aligned} \delta _u f= \sqrt{\det \left( \frac{\partial f}{\partial u}^Tg(f(u))\frac{\partial f}{\partial u}\right) }du\in \varDelta _p^+M, \end{aligned}$$

(A.1)

In particular $\delta (\text {id}_M)$ is the Riemannian volume density of M and we denote it by dM. Similarly, if f is a Riemannian inclusion $M\subset N$, then $\delta f=dM$.

Given any Riemannian metric on the manifold M, one can make the identification $\varDelta _p M= {\mathbb {R}}(dM(p))$ and treat densities as if they were functions. Moreover this identification preserves the sign, since dM is always a positive density: $dM(p)\in \varDelta _p^+M$, for every $p\in M$. We denote by B(M) the set of all Borel measurable functions $M\rightarrow [-\infty ,+\infty ]$ and by $L(M)=B(M)dM$ the set of all measurable, not necessarily finite, densities (the identification with B(M) depends on the choice of the metric). Let $B^+(M)$ be the set of positive measurable functions $M\rightarrow [0,+\infty ]$ and $L^+(M)$ be the set of densities of the form $\rho dM$, for some $\rho \in B^+(M)$. In other words

$$\begin{aligned} L^+(M)=\left\{ \text {measurable functions }M\ni p\mapsto \delta (p)\in \varDelta ^+_pM \cup \{+\infty \}\right\} \end{aligned}$$

is the set of all nonnegative, non necessarily finite measurable densities. The integral can be extended in the usual way (by monotone convergence) to a linear function $\int _M:L^+(M)\rightarrow {\mathbb {R}}\cup \{\infty \}$. Similarly, we define the spaces $L^1(M),L^\infty (M),L^1_{loc}(M),L^\infty _{loc}(M)$ and their respective topologies by analogy with the standard case $M={\mathbb {R}}^m$.

Definition A.1

We say that a real Radon measure (see [2]) $\mu$ on M is absolutely continuous if $\mu (A)=0$ on any zero measure subset $A\subset M$. In other words $\mu$ is absolutely continuous if and only if $\varphi _*\left( \mu |_{U}\right)$ is absolutely continuous with respect to the Lebesgue measure ${\mathscr {L}}^m$ for any chart $\varphi :U\subset M\rightarrow {\mathbb {R}}^m$.

In this language, the Radon-Nikodym Theorem takes the following form.

Theorem A.2

Let $\mu$ be an absolutely continuous real Radon measure on M. Then there is a density $\delta \in L^1_{loc}(M)$ such that, for all Borel subsets $A\subset M$,

$$\begin{aligned} \mu (A)=\int _A\delta . \end{aligned}$$

Angle between subspaces

Let $E, \langle \cdot ,\cdot \rangle$ be a euclidean vector space (i.e. a finite dimensional real Hilbert space).

Definition B.1

Let $f=(f_1,\dots ,f_k)$ be a tuple (in row) of vectors $f_i\in E$. We define its volume as

$$\begin{aligned} \mathrm {vol}(f)=\sqrt{\det \langle f^T, f\rangle }. \end{aligned}$$

If the vectors $f_1,\dots ,f_k$ are independent, then f is called a frame. The span $\text {span}(f)$ of the tuple f is the subspace of E spanned by the vectors $f_1,\dots ,f_k$. A tuple f is a basis of a subspace $V\subset E$ if and only if it is a frame and its span is V. Given two frames v, w, we can form the tuple (v, w).

Definition B.2

Let $V,W\subset E$ be subspaces. Let v and w be frames in E such that:

• $\text {span}(v)=V\cap (V\cap W)^\perp$;

• $\text {span}(w)=W\cap (V\cap W)^\perp$.

We define the angle between V and W as

$$\begin{aligned} \sigma (V,W)={\left\{ \begin{array}{ll} \frac{\mathrm {vol}(v w)}{\mathrm {vol}(v)\mathrm {vol}(w)} \quad &{}\text { if }V\not \subset W\text { and }W\not \subset V;\\ 1 &{}\text { otherwise.} \end{array}\right. } \end{aligned}$$

It is easy to see that the definition is well posed, independently from the choices of the frames. This definition corresponds to Howard’s [18] in the case when $V\cap W=\{0\}$. Observe that $\sigma$ is symmetric and that we have $\sigma (V,W)=1$ if and only if $V=A\oplus _\perp B$ and $W=A\oplus _\perp C$ with and $B\perp C$.

When V and W are one dimensional, $\sigma (V,W)=|\sin \theta |$, where $\theta$ is the angle between the two lines. In general $\sigma (V,W)\in [0,1]$ is equal to the product of the sines of the nontrivial principal angles between V and W (see [19, 28, 40]). In particular, notice that $\sigma (V,W)>0$ always.

It is important to notice that $\sigma :\text {Gr}_k(E)\times \text {Gr}_h(E)\rightarrow [0,1]$ is not a continuous function. However the restriction to the subset of the pairs of subspaces (V, W) such that $\dim (V+W)=n$ is continuous.

Proposition B.3

Assume that $W\not \subset V$. Let w be a basis for $W\cap (V\cap W)^\perp$, as in Definition B.2, then

$$\begin{aligned} \sigma (V,W)=\frac{\mathrm {vol}\left( \varPi _{V^\perp }(w)\right) }{\mathrm {vol}(w)} \end{aligned}$$

Proof

First, observe that the projected frame $\varPi _{V^\perp }(w)$ is a basis of the space $(V+W)\cap V^\perp$. Let $\nu$ be an orthonormal basis of the same space, let v be an orthonormal basis for $V\cap (V\cap W)^\perp$ and let $\tau$ be a basis for $V\cap W$. It follows that $(\tau , v,\nu )$ is a basis for $V+W$. Therefore, there is an invertible matrix B and a matrix A such that

$$\begin{aligned} w=\begin{pmatrix}\tau&v&\nu \end{pmatrix}\begin{pmatrix} 0\\ A\\ B\end{pmatrix}. \end{aligned}$$

Then, by Definition B.2, we have

$$\begin{aligned} \begin{aligned} \sigma (V,W)=\frac{\mathrm {vol}(v,w)}{\mathrm {vol}(v)\mathrm {vol}(w)} =\frac{\left| \det \begin{pmatrix}\mathbb {1}&{} A \\ 0 &{} B \end{pmatrix}\right| }{\mathrm {vol}(w)}=\frac{\mathrm {vol}\left( \varPi _{V^\perp }(w)\right) }{\mathrm {vol}(w)}. \end{aligned} \end{aligned}$$

$\square$

Proposition B.4

$\sigma (V^\perp ,W^\perp )=\sigma (V,W).$

Proof

The statement is trivially true if $V\subset W$ or $W \subset V$, so let us assume that this is not the case. Let $\nu$ be an orthonormal basis of the space $(V+W)\cap V^\perp$ and v be an orthonormal basis of $V\cap (V\cap W)^\perp$. Besides, let $(\tau ,w)$ be an orthonormal basis of W, such that $\tau$ is a basis for $V\cap W$. We have

$$\begin{aligned} \begin{aligned} \sigma (V^\perp ,W^\perp )&=\sigma (W^\perp ,V^\perp )\\&=\mathrm {vol}\left( \varPi _{W}(\nu )\right) \\&=\det \langle w^T,\nu \rangle \\&=\det \langle \nu ^T,w\rangle \\&= \mathrm {vol}\left( \varPi _{V^\perp }(w)\right) =\sigma (V,W). \end{aligned} \end{aligned}$$

$\square$

Area and Coarea formula

Definition C.1

Let $\varphi :M\rightarrow N$ be a ${\mathscr {C}}^1$ map between ${\mathscr {C}}^1$ Riemannian manifolds. The Jacobian (often called normal Jacobian when f is a submersion) of $\varphi$ at $p\in M$ is

$$\begin{aligned} J_p\varphi :={\left\{ \begin{array}{ll} 0 &{} \text {if rank}(d_p\varphi )\text { is not maximal;} \\ \frac{\mathrm {vol}_{N}\left( d_p\varphi (e)\right) }{\mathrm {vol}_{M}(e)} &{} \text {otherwise;} \end{array}\right. } \end{aligned}$$

where $e=(e_1,\dots , e_k)$ is any basis of $\ker (d_p\varphi )^\perp \subset T_pM$.

If $L:V_1\rightarrow V_2$ is a linear map between metric vector spaces, then we write $J L:=J_0L$ (Clearly $J_p\varphi =Jd_p\varphi$). If $V_1,V_2$ have the same dimension, then, to stress this fact, we may write $|\det L|:=JL$, although the sign of $\det L$ is not defined, unless we specify orientations.

In particular, let $\varphi :({\mathbb {R}}^m;g_1)\rightarrow ({\mathbb {R}}^n;g_2)$ with differential $\frac{\partial \varphi }{\partial u}(u)=A$ having maximal rank, then

$$\begin{aligned} J_u\varphi :={\left\{ \begin{array}{ll} \sqrt{\frac{\det (A^Tg_2A)}{\det (g_1)}} &{} \text {if }m\le n; \\ \sqrt{\frac{\det (Ag_1^{-1}A^T)}{\det (g_2)^{-1}}} &{} \text {if }m\ge n. \end{array}\right. } \end{aligned}$$

(C.1)

Remark C.1

In the case $m\le n$, the density induced on M by a map $\varphi$, defined in (A.1) corresponds to $\delta _p\varphi =(J_p\varphi ) dM$.

Theorem C.2

(Area formula) Let $f:M\rightarrow N$ be a Lipschitz map between ${\mathscr {C}}^1$ Riemannian manifolds, with $\dim M=\dim N$. Let $g:M\rightarrow [0,+\infty ]$ be a Borel function, then

$$\begin{aligned} \int _M g(p)(J_pf) dM(p)=\int _N\left[ \sum _{p\in f^{-1}(q)}g(p)\right] dN(q). \end{aligned}$$

Proof

See [11, Theorem 3.2.3]. $\square$

The Area formula is actually much more general than this, in that it holds for $\dim M\le \dim N$ and with the Hausdorff measure instead of dN. However, this simplified statement is all that we need in this paper. It also can be thought as a generalization of the following, in the case $\dim M=\dim N$.

Theorem C.3

(Coarea formula) Let $f:M\rightarrow N$ be a ${\mathscr {C}}^1$ submersion between smooth Riemannian manifolds, with $\dim M\ge \dim N$. Let $g:M\rightarrow [0,+\infty ]$ be a Borel function, then

$$\begin{aligned} \int _M g(p) (J_pf) dM(p)=\int _N\int _{f^{-1}(q)}g(p) d\left( f^{-1}(q)\right) (p) dN(q). \end{aligned}$$

Proof

See [8] or deduce it from [11, Theorem 3.2.12]. $\square$