1 Introduction

1.1 Notation and terminology

Suppose that \(X\) is a smooth manifold. For any vector space \(V\), we denote by \(\underline{V}_X\) the trivial bundle \(V\times X\rightarrow X\).

We denote by \(|\Lambda _X|\rightarrow X\) the line bundle of \(1\)-densities on \(X\), [10, 16], so that we have a well defined integration map

$$\begin{aligned} \int _X : C_0^\infty (|\Lambda _X|)\rightarrow {\mathbb R}, \quad C^\infty (|\Lambda _X|)\ni \rho \mapsto \int _X\rho (dx). \end{aligned}$$

Suppose that \(F\) is a smooth vector bundle over \(X\). We have two natural projections

$$\begin{aligned} \pi _x,\pi _y: X\times X\rightarrow X,\quad \pi _x(x,y)=x,\quad \pi _y(x,y)=y,\quad \forall x,y\in X. \end{aligned}$$

We set \(F\boxtimes F:= \pi _x^*F\otimes \pi _y^* F\), so that \(F\boxtimes F\) is vector bundle over \(X\times X\).

Following [10], Chap.VI,§1], we define a generalized section of \(F\) to be a continuous linear functional on the space \(C_0^\infty (F^*\otimes |\Lambda _X|)\) equipped with the natural locally convex topology. We denote by \(C^{-\infty }(F)\) the space of generalized sections of \(F\). We have a natural injection, [10], Chap.VI, §1]

$$\begin{aligned} i:C^\infty (F)\hookrightarrow C^{-\infty }(F). \end{aligned}$$

Recall that a Borel probability measure \(\mu \) on \({\mathbb R}\) is called (centered) Gaussian if has the form

$$\begin{aligned} \mu (dx)={\varvec{\gamma }}_v(dx):={\left\{ \begin{array}{ll} \frac{1}{\sqrt{2\pi v} }e^{-\frac{x^2}{2v}} dx, &{} v>0,\\ \delta _0, &{} v=0. \end{array}\right. } \end{aligned}$$

where \(\delta _0\) denotes the Dirac measure concentrated at the origin.

1.2 Gaussian ensembles of sections and correlators

The concept of Gaussian smooth random section of a vector bundle is very similar to the better known concept of Gaussian random function. Throughout this paper we fix a smooth compact connected manifold \(M\) of dimension \(m\) and a smooth real vector bundle \(E\rightarrow M\) of rank \(r\).

The notion of random section of a vector bundle is described in great detail in [3], Sec. 8]. This description relies on the concept of abstract Wiener space due to Gross, [9]. Since this concept may be less familiar to the readers with a more geometric bias, we decided to include an alternative approach, hopefully more palatable to geometers. From this point of view, a random smooth section of \(E\) is identified with a probability measure on the space of generalized sections \(C^{-\infty }(E)\) supported on the subspace \(C^\infty (E)\). The construction of such measures relies on the fundamental work of Minlos [14], Gelfand-Vilenkin [8], Fernique [7] and Schwartz [20]. We describe below the results relevant to our main investigation.

The space \(C^\infty (E^*\otimes |\Lambda _M|)\) is a nuclear countable Hilbert space in the sense of [8] and, as such, its dual \(C^{-\infty }(E)\) satisfies several useful measure theoretic properties. The next result follows from [7].

Proposition 1.1

  1. (i)

    The \({\sigma }\)-algebra of weakly Borel subsets of \(C^{-\infty }(E)\) is equal to the \({\sigma }\)-algebra of strongly Borel subsets. We will refer to this \({\sigma }\)-algebra as the Borel \({\sigma }\) -algebra of \(C^{-\infty }(E)\).

  2. (ii)

    Every Borel probability measure on \(C^{-\infty }(E)\) is Radon.

  3. (iii)

    Any Borel subset of \(C^\infty (E)\) (with its natural topology) belongs to the Borel \({\sigma }\)-algebra of \(C^{-\infty }(E)\).

Any section \({\varphi }\in C^\infty (E^*\otimes |\Lambda _M|)\) defines a continuous linear map \(L_{\varphi }: C^{-\infty }(E)\rightarrow {\mathbb R}\). Following [4, 8] we define a centered Gaussian measure Footnote 1 on \(C^{-\infty }(E)\) to be a Borel probability measure \({\varvec{\Gamma }}\) such that, for any section \({\varphi }\in C^\infty (E^*\otimes |\Lambda _M|)\) the pushforward \((L_{\varphi })_\#({\varvec{\Gamma }})\) is a centered Gaussian \({\varvec{\gamma }}_{\varphi }\) measure on \({\mathbb R}\).

The measure \({\varvec{\Gamma }}\) is completely determined by its covariance form which is the symmetric, nonnegative definite bilinear map

$$\begin{aligned} \mathcal {K}_{{\varvec{\Gamma }}}: C^\infty (E^*\otimes |\Lambda _M|)\times C^\infty (E^*\otimes |\Lambda _M|)\rightarrow {\mathbb R}\end{aligned}$$

given by

$$\begin{aligned} \mathcal {K}_{\varvec{\Gamma }}({\varphi },\psi )={\varvec{E}}_{\varvec{\Gamma }}\bigl (\,L_{\varphi }\cdot L_\psi \,\bigr ),\quad \forall {\varphi },\psi \in C^\infty (E^*\otimes |\Lambda _M|). \end{aligned}$$

Above, \({\varvec{E}}_{\varvec{\Gamma }}\) denotes the expectation with respect to the probability measure \({\varvec{\Gamma }}\) and we interpreted \(L_{\varphi }, L_\psi \) as random variables on the probability space \((C^{-\infty }(E),\Gamma )\),

Results of Fernique [7], Thm.II.2.3 + Thm.II.3.2] imply that \(\mathcal {K}_\Gamma \) is separately continuous. According to Schwartz’ kernel theorem [8], Chap.I, §3.5] the covariance form can be identified with a linear functional \(C_{\varvec{\Gamma }}\) on the topological vector space

$$\begin{aligned} C^\infty \bigl (\, (E^*\otimes |\Lambda _M|)\boxtimes (E^*\otimes |\Lambda _M|)\,\bigr )=C^\infty \bigl (\, (E^*\boxtimes E^*)\otimes |\Lambda _{M\times M}|\,\bigr ), \end{aligned}$$

i.e., \(C_{\varvec{\Gamma }}\in C^{-\infty }(E\boxtimes E)\). We will refer to \(C_{\varvec{\Gamma }}\) as the covariance kernel of \({\varvec{\Gamma }}\).

Theorem 1.2

(Minlos, [14]) Given a generalized section \(C\in C^{-\infty }(E\boxtimes E)\) such that the associated bilinear form

$$\begin{aligned} \mathcal {K}: C^\infty (E^*\otimes |\Lambda _M|)\times C^\infty (E^*\otimes |\Lambda _M|)\rightarrow {\mathbb R}\end{aligned}$$

is symmetric and nonnegative definite, there exists a unique Gaussian measure on \(C^{-\infty }(E)\) with covariance kernel \(C\).

Definition 1.3

A Gaussian measure \({\varvec{\Gamma }}\) on \(C^{-\infty } (E)\) is called smooth if \(C_{\varvec{\Gamma }}\) is given by a smooth section of \(E\boxtimes E\). We will refer to it as the covariance density. We will refer to the smooth Gaussian measures on \(C^{-\infty }(E)\) as a Gaussian ensemble of smooth sections of \(E\).

A smooth section \(C\) of \(E\boxtimes E\) can be viewed as a smooth family of bilinear maps

$$\begin{aligned} \tilde{C}_{{\varvec{x}},{\varvec{y}}}: E^*_{\varvec{x}}\times E^*_{\varvec{y}}\rightarrow {\mathbb R},\quad {\varvec{x}},{\varvec{y}}\in M, \end{aligned}$$

given by

$$\begin{aligned} \tilde{C}_{{\varvec{x}},{\varvec{y}}}({\varvec{u}}^*,{\varvec{v}}^*):=\bigl \langle \, {\varvec{u}}^*\otimes {\varvec{v}}^*, C_{{\varvec{x}},{\varvec{y}}}\,\bigr \rangle ,\quad \forall {\varvec{u}}^*\in E_{\varvec{x}}^*, \quad {\varvec{v}}^*\in E_{\varvec{y}}^*, \end{aligned}$$

where \(\langle -,-\rangle \) denotes the natural pairing between a vector space and its dual. In the sequel we will identify \(C_{{\varvec{x}},{\varvec{y}}}\) with the associated bilinear map \(\tilde{C}_{{\varvec{x}},{\varvec{y}}}\).

The next result, proved in Appendix A, explains the role of the smoothness condition.

Proposition 1.4

If the Gaussian measure \({\varvec{\Gamma }}\) on \(C^{-\infty }(E)\) is smooth, then \({\varvec{\Gamma }}\bigl (\, C^\infty (E)\,\bigr )\!=1\). In other words, a random generalized section in the Gaussian ensemble determined by \({\varvec{\Gamma }}\) is a.s. smooth.

Using Propositions 1.1 and 1.4 we deduce that a smooth Gaussian measure on \(C^{-\infty }(E)\) induces a Borel probability measure on \(C^\infty (E)\). Observe also that, for any \({\varvec{x}}\in M\), the induced map

$$\begin{aligned} C^\infty (E)\rightarrow E_{\varvec{x}},\quad C^\infty (E)\ni {\varphi }\mapsto {\varphi }({\varvec{x}})\in E_{\varvec{x}}\end{aligned}$$

is Borel measurable. The next result, proved in Appendix A, shows that the collection of random variables \((\,{\varphi }({\varvec{x}})\,)_{{\varvec{x}}\in M}\) is Gaussian.

Proposition 1.5

Suppose that \({\varvec{\Gamma }}\) is a smooth Gaussian measure on \(E\) with covariance density \(C\). Let \(n\) be a positive integer. Then for any points \({\varvec{x}}_1,\cdots , {\varvec{x}}_n\in M\) and any \({\varvec{u}}_i^*\in E_{{\varvec{x}}_i}^*\), \(i=1,\cdots , n\) the random vector

$$\begin{aligned} C^\infty (E)\!\ni \! {\varphi }\mapsto \bigl (\,X_1({\varphi }),\cdots , X_n({\varphi })\,\bigr )\in {\mathbb R}^n,\quad X_i({\varphi }) :=\langle \, {\varvec{u}}_i^*,{\varphi }({\varvec{x}}_i)\,\rangle ,\quad i=1,\cdots ,n, \end{aligned}$$

is Gaussian. Moreover

$$\begin{aligned} {\varvec{E}}(X_iX_j) =C_{{\varvec{x}}_i,{\varvec{x}}_j}({\varvec{u}}_i^*,{\varvec{u}}_j^*),\quad \forall i,j. \end{aligned}$$
(1.1)

A section \(C\in C^\infty (E\boxtimes E)\) is called symmetric if

$$\begin{aligned} C_{{\varvec{x}},{\varvec{y}}}({\varvec{u}}^*,{\varvec{v}}^*)=C_{{\varvec{y}},{\varvec{x}}}({\varvec{v}}^*,{\varvec{u}}^*),\quad \forall {\varvec{x}},{\varvec{y}}\in M,\quad \forall {\varvec{u}}^*\in E_{\varvec{x}}^*,\quad {\varvec{v}}^*\in E_{\varvec{y}}^*. \end{aligned}$$

If \(C\) is the covariance density of a smooth Gaussian measure \(\Gamma \) on \(C^{-\infty }(E)\), then Proposition 1.5 shows that \(C\) is symmetric.

A symmetric section \(C\in C^\infty (E\boxtimes E)\) is called nonnegative/positive definite if all the symmetric bilinear forms \(C_{{\varvec{x}},{\varvec{x}}}\) are such. Clearly the covariance density of a smooth Gaussian measure \(\Gamma \) on \(C^{-\infty }(E)\) is symmetric and nonnegative definite.

Definition 1.6

(a) A correlator on \(E\) is a section \(C\in C^\infty (E\boxtimes E)\) which is symmetric and nonnegative definite. The correlator is called nondegenerate if it is positive definite.

(b) A correlator \(C\in C^\infty (E\boxtimes E)\) is called stochastic if it is the covariance density of a Gaussian ensemble smooth sections of \(E\).

(c) A Gaussian ensemble of smooth sections of \(E\) is called nondegenerate if its covariance density is a nondegenerate correlator.

Remark 1.7

Minlos’ Theorem 1.2 shows that not all correlators are stochastic. The results in [3] show that a correlator \(C\) is stochastic if and only if it is a reproducing kernel, i.e., for any natural number \(n\), any points \({\varvec{x}}_1,\cdots , {\varvec{x}}_n\in M\) and any \({\varvec{u}}_i^*\in E_{{\varvec{x}}_i}^*\), \(i=1,\cdots , n\), the symmetric matrix

$$\begin{aligned} \bigl (\, C_{{\varvec{x}}_i,{\varvec{x}}_j}({\varvec{u}}_i^*,{\varvec{u}}_j^*),\bigr )_{1\le i,j\le n} \end{aligned}$$

is nonnegative definite.

Lemma 1.8

There exist nondegenerate Gaussian ensembles of smooth sections of \(E\).

Proof

Fix a finite dimensional subspace \({\varvec{U}}\in C^\infty (E)\) which is ample, i.e., for any \({\varvec{x}}\in M\) the evaluation map \({{\mathrm{\mathbf {ev}}}}_{\varvec{x}}:{\varvec{U}}\rightarrow E_{\varvec{x}}\), \({\varvec{u}}\mapsto {\varvec{u}}({\varvec{x}})\) is onto. (The existence of such spaces is a classical fact, proved e.g. in [5], Lemma 23.8].) By duality we obtain injections \({{\mathrm{\mathbf {ev}}}}_{\varvec{x}}^*:E_{\varvec{x}}^*\rightarrow {\varvec{U}}^*\).

Fix an Euclidean inner product \((-,-)_{\varvec{U}}\) on \({\varvec{U}}\) and denote by \({\varvec{\gamma }}\) the Gaussian measure on \({\varvec{U}}\) canonically determined by this product. Its covariance pairing \({\varvec{U}}^*\times {\varvec{U}}^*\rightarrow {\mathbb R}\) coincides with \((-,-)_{{\varvec{U}}^*}\), the inner product on \({\varvec{U}}^*\) induced by \((-,-)_{\varvec{U}}\). More precisely, this means that for any \(\xi ,\eta \in {\varvec{U}}^*\) we have

$$\begin{aligned} (\xi ,\eta )_{{\varvec{U}}^*}=\int _{\varvec{U}}\langle \xi ,s\rangle \langle \eta ,s\rangle {\varvec{\gamma }}(ds). \end{aligned}$$
(1.2)

The measure \({\varvec{\gamma }}\) defines a smooth Gaussian measure \(\hat{{\varvec{\gamma }}}\) on \(C^{-\infty }(E)\) such that \(\hat{{\varvec{\gamma }}}({\varvec{U}})=1\). Concretely, \(\hat{{\varvec{\gamma }}}\) is the pushforward of \({\varvec{\gamma }}\) via the natural inclusion \({\varvec{U}}\hookrightarrow C^\infty (E)\). This is a smooth measure. Its covariance density \(C\) is computed as follows: if \({\varvec{x}},{\varvec{y}}\in M\), \({\varvec{u}}^*\in E^*_{\varvec{x}}\), \({\varvec{v}}^*\in E^*_{\varvec{y}}\), then

$$\begin{aligned}&C_{{\varvec{x}},{\varvec{y}}}({\varvec{u}}^*,{\varvec{v}}^*)= \int _{\varvec{U}}\langle {\varvec{u}}^*,s({\varvec{x}})\rangle \langle {\varvec{v}}^*,s({\varvec{y}})\rangle \gamma (ds)= \int _{\varvec{U}}\langle {{\mathrm{\mathbf {ev}}}}_{\varvec{x}}^*{\varvec{u}}^*,s\rangle \langle {{\mathrm{\mathbf {ev}}}}^*_{\varvec{y}}{\varvec{v}}^*,s\rangle {\varvec{\gamma }}(ds)\\&\mathop {=}\limits ^{(1.2)} ({{\mathrm{\mathbf {ev}}}}^*_{\varvec{x}}{\varvec{u}}^*,{{\mathrm{\mathbf {ev}}}}^*_{\varvec{y}}{\varvec{v}}^*). \end{aligned}$$

In particular, when \({\varvec{x}}={\varvec{y}}\) we observe that \(C_{{\varvec{x}},{\varvec{x}}}\) coincides with the restriction to \(E_{\varvec{x}}^*\) of the inner product \((-,-)_{{\varvec{U}}^*}\) so the form \(C_{{\varvec{x}},{\varvec{x}}}\) is positive definite.\(\square \)

Definition 1.9

A Gaussian ensemble of smooth sections of \(E\) with associated Gaussian measure \({\varvec{\Gamma }}\) on \(C^{-\infty }(E)\) is said to have finite-type if there exists a finite dimensional subspace \({\varvec{U}}\subset C^\infty (E)\) such that \({\varvec{\Gamma }}({\varvec{U}})=1\).

Remark 1.10

The Gaussian ensemble constructed in Lemma 1.8 has finite type. All the nondegenerate finite type Gaussian ensembles of smooth sections can be obtained in this fashion and, as explained in [6, 19], the results of Narasimhan and Ramanan [15] show that any pair (metric, compatible connection) on \(E\) is determined by the correlator of a finite type ensemble of smooth sections on \(E\).

However, there exist nondegenerate gaussian ensembles which are not of finite type. They can be constructed using an approach conceptually similar to the one we used in Lemma 1.8. The only difference is that instead of a finite dimensional ample space of sections we use an ample Banach space of \(C^k\) sections equipped with a Gaussian measure. For details we refer to [3].

Definition 1.11

A Gaussian ensemble of smooth sections of \(E\) is called transversal, if a random section of this ensemble is a.s. transversal to the zero section of \(E\).

For the proof of the next result we refer to Appendix A.

Proposition 1.12

Any nondegenerate ensemble of smooth sections of \(E\) is transversal.

1.3 Statements of the main results

The main goal of this paper is to investigate some of the rich geometry of a nondegenerate Gaussian ensemble of smooth sections of \(E\). By definition, the correlator \(C\) of such an ensemble defines a metric on the dual bundle \(E^*\), and thus on \(E\) as well. In Example 2.1 we illustrate these abstract constructions on some familiar situations.

Less obvious is the fact that, in general a nondegenerate correlator \(C\), not necessarily stochastic, induces a connection \(\nabla ^C\) on \(E\) compatible with the canonical metric defined by \(C\). We will refer to this metric/connection as the correlator metric/connection. We prove this fact in Proposition 2.3.

This connection depends only on the first order jet of \(C\) along the diagonal of \(M\times M\). Using the correlator metric we can identify the bilinear form \(C_{{\varvec{x}},{\varvec{y}}}\) with a linear map \(T_{{\varvec{x}},{\varvec{y}}}: E_{\varvec{y}}\rightarrow E_{\varvec{x}}\). The definition of the connection shows that its infinitesimal parallel transport is given by the first order jet of \(T_{{\varvec{x}},{\varvec{y}}}\) along the diagonal \({\varvec{x}}={\varvec{y}}\).

If the correlator \(C\) is stochastic, then the connection \(\nabla ^C\) and its curvature can be given a probabilistic interpretation. Proposition 2.5 gives a purely probabilistic description of its curvature. This result contains as a special case Gauss’ Theorema Egregium.

Remark 1.13

The construction of \(\nabla ^C\) in Proposition 2.3 feels very classical, but we were not able to trace any reference. In the special case when \(C\) is a stochastic correlator, this connection is the \(L\)-\(W\) connection in [6], Prop. 1.1.1]. It can be given a probabilistic description, [6], Prop. 1.1.3], or a more geometric description obtained by using the stochastic correlator to canonically embed \(E\) in a trivial Hilbert bundle. These constructions use in an essential way the reproducing kernel property of a stochastic correlator. Proposition 2.3 shows that we need a lot less to produce a metric and a compatible connection.

Section 3 contains the main result of this paper, Theorem 3.2. In this section we need to assume that both \(M\) and \(E\) are oriented, and the rank of \(E\) is even and not greater than the dimension of \(M\). Let us digress to recall the classical Gauss–Bonnet–Chern theorem.

The Chern-Weil construction associates to a metric on \(E\) and connection \(\nabla \) compatible with this metric an Euler form \({\varvec{e}}(E,\nabla )\in \Omega ^r(M)\); see [16], Chap.8]. This form is closed, and the cohomology class it determines called is the geometric Euler class of \(E\). This cohomology class is independent of the choice of metric and compatible connection.

If \({\varvec{v}}\) is a smooth section of \(E\) transversal to the zero section, then its zero locus \(Z_{{\varvec{v}}}\) is a compact, codimension \(r\)-submanifold of \(M\) equipped with a canonical orientation. As such, it defines a closed integration current \([Z_{{\varvec{v}}}]\) of dimension \((m-r)\) whose homology class is independent of the choice of transversal section \({\varvec{v}}\). This means that, if \({\varvec{v}}_0,{\varvec{v}}_1\) are two sections of \(E\) transversal to the zero section, then

(1.3)

Indeed, using Sard’s theorem as in [13], Chap.5, Lemma 2], we can find an oriented \(C^1\)-submanifold with boundary \(\hat{Z}\subset [0,1]\times M\) such that, \(\partial \hat{Z}\cap (\{i\} \times M)=\{i\}\times Z_{{\varvec{v}}_i}\), \(i=0,1\), and we have an equality of oriented manifolds, \(\partial \hat{Z}= Z_{{\varvec{v}}_1}\sqcup -Z_{{\varvec{v}}_0}\). The equality (1.3) now follows from Stokes’ theorem.

The topological Euler class of the real, oriented vector bundle \(E\) is the cohomology class of \(M\) defined as the pullback of the Thom class of \(E\) via a(ny) section of this bundle. Equivalently, the topological Euler class of \(E\) is equal to Poincaré dual of the homology class of \(M\) determined by the zero-locus current \([Z_{{\varvec{v}}_0}]\) defined by a transversal section \({\varvec{v}}_0\); see [16], Exercise 8.3.21].

The classical Gauss–Bonnet–Chern theorem states that the topological Euler class of \(E\) is equal to the geometric Euler class; see [11], Chap.IV, Thm.1.51] or [16], Thm. 8.3.17]. This means that, for any metric on \(E\), and any connection \(\nabla ^0\) compatible with the metric, the closed current \([Z_{{\varvec{v}}_0}]\) is homologous to the closed current defined by the Euler form \({\varvec{e}}(E,\nabla ^0)\), i.e.,

(1.4)

The main goal of this paper is to provide a probabilistic refinement of the above equality.

Fix a nondegenerate Gaussian ensemble of smooth sections of \(E\). This determines a metric and a compatible connection \(\nabla ^\mathrm{stoch}\) on \(E\). It thus determines an Euler form \({\varvec{e}}(E,\nabla ^\mathrm{stoch})\) on \(M\).

Since our ensemble of sections is nondegenerate it is also transversal according to Proposition 1.12 and thus the current \(Z_{\varvec{u}}\) will be well defined for almost all \({\varvec{u}}\) in the ensemble. In Theorem 3.2 we prove a stochastic Gauss-Bonnet formula stating that the expectation of the random current \([Z_{\varvec{u}}]\) is equal to the current defined by the Euler form \({\varvec{e}}(E,\nabla ^\mathrm{stoch})\), i.e.,

$$\begin{aligned} {\varvec{E}}\left( \int _{Z_{\varvec{u}}}\eta \right) =\int _M \eta \wedge {\varvec{e}}(E,\nabla ^\mathrm{stoch}),\quad \forall \eta \in \Omega ^{m-r}(M). \end{aligned}$$
(1.5)

Remark 1.14

(a) Let us point out that the cohomological formula (1.4) is a consequence of (1.5). To see this, fix an arbitrary metric \(h\) on \(E\) and a connection \(\nabla ^0\) compatible with \(h\). Denote by \({\varvec{e}}(E,\nabla ^0)\) the associate Euler form. Next, fix a smooth section \({\varvec{v}}_0\) of \(E\) that is transversal to \(0\).

As indicated in Remark 1.13, there exists a finite-type nondegenerate Gaussian ensemble of smooth sections of \(E\) whose associated metric is \(h\) and associated connection is \(\nabla ^0\), i.e., \(\nabla ^\mathrm{stoch}=\nabla ^0\). Then, a.s., a section \({\varvec{u}}\) in this ensemble is smooth and transversal to \(0\). From (1.3) we deduce that for any closed form \(\eta \in \Omega ^{m-r}(M)\) we have the a.s. equality

$$\begin{aligned} \int _{Z_{{\varvec{v}}_0}} \eta =\int _{Z_{{\varvec{u}}}}\eta . \end{aligned}$$

By taking the expectations of both sides and then invoking (1.5) we deduce (1.4).

(b) The stochastic formula (1.5) is stronger than the cohomological one because the Euler class of \(E\) could be zero (in cohomology), yet there exist metric connections on \(E\) whose associated Euler forms are nonzero.

(c) The stochastic Gauss–Bonnet–Chern formula (1.5) has a local character. It suffices to prove it for forms \(\eta \) supported on coordinate neighborhoods over which \(E\) is trivializable. The general case follows form these special ones by using partitions of unity and the obvious linearity in \(\eta \) of both sides of (1.5). This is in fact the strategy we adopt in our proof.

(d) In Remark 3.8(b) we explain what happens in the case when the Gaussian ensemble of random sections is no longer centered, say \({\varvec{E}}({\varvec{u}})={\varvec{u}}_0\in C^\infty (E)\). Formula (1.5) gets replaced by (3.15), where in the right hand side we get a different term that explicitly depends on the geometry of \(E\) and the bias \({\varvec{u}}_0\).

We prove the stochastic formula (1.5) by reducing it to the Kac-Rice formula [2], Thm. 6.4, 6.10] using a bit of differential geometry and certain Gaussian computations we borrowed from [1]. For the reader’s convenience we have included in Appendix B a brief survey of these facts.

1.4 Related results

In our earlier work [19] we proved a special case of this stochastic Gauss-Bonnet formula for nondegenerate Gaussian ensembles of finite type. The proof in [19] is differential geometric in nature and does not extend to the general situation discussed in the present paper.

In [18] we used related probabilistic techniques to prove a cohomological Gauss–Bonnet–Chern formula of the type (1.4) in the special case when \(E=TM\), and the connection \(\nabla \) is the Levi-Civita connection of a metric on \(TM\). Still in the case \(E=TM\), one can use rather different probabilistic ideas (Malliavin calculus) to prove the cohomological Gauss-Bonnet; the case when \(\nabla \) is the Levi-Civita connection of a metric on \(M\) was investigated by Hsu [12], while the case of a general metric connection on \(TM\) was recently investigated by H. Zhao [21].

2 The differential geometry of correlators

A correlator on a real vector bundle \(E\rightarrow M\) naturally induces additional geometric structures on \(E\). More precisely, we will show that it induces a metric on \(E\) together with a connection compatible with this metric. Here are a few circumstances that lead to correlators.

Example 2.1

(a) Suppose that \(M\) is a properly embedded submanifold of the Euclidean space \({\varvec{U}}\). Then the inner product \((-,-)_{\varvec{U}}\) on \({\varvec{U}}\) induces a correlator \(C\in C^\infty (T^*M\boxtimes T^*M)\) defined by the equalities

$$\begin{aligned} C_{{\varvec{x}},{\varvec{y}}}(X,Y)= (X,Y)_{\varvec{U}},\quad \forall {\varvec{x}},{\varvec{y}}\in M,\quad X\in T_{\varvec{x}}M\subset {\varvec{U}},\quad Y\in T_{\varvec{y}}M\subset {\varvec{U}}. \end{aligned}$$

(b) For any real vector space \({\varvec{U}}\) and any smooth manifold \(M\) we denote by \(\underline{{\varvec{U}}}_M:={\varvec{U}}\times M\rightarrow M\) the trivial vector bundle over \(M\) with fiber \({\varvec{U}}\).

Suppose that \({\varvec{U}}\) is a real, finite dimensional Euclidean space with inner product \((-,-)\). This induces an inner product \((-,-)_*\) on \({\varvec{U}}^*\). Suppose that \(E\rightarrow M\) is a smooth real vector bundle over \(M\) and \(P:\underline{{\varvec{U}}}_M\rightarrow E\) is a fiberwise surjective bundle morphism. In other words, \(E\) is a quotient bundle of a trivial real metric vector bundle. The dual \( P^*: E^*\rightarrow \underline{{\varvec{U}}}^*_M\) is an injective bundle morphism. Hence \(E^*\) is a subbundle of a trivial metric real vector bundle.

For any \({\varvec{x}}\in M\) and any \(u^*\in E^*_{\varvec{x}}\) we obtain a vector \(P_{\varvec{x}}^* u^*\in {\varvec{U}}^*_{\varvec{x}}=\) the fiber of \(\underline{{\varvec{U}}}^*_M\) at \({\varvec{x}}\in M\). This allows us to define a correlator \(C\in C^\infty (E\boxtimes E)\) given by

$$\begin{aligned} C_{{\varvec{x}}_1,{\varvec{x}}_2}(u_1^*,u_2^*)=\bigl (\, P_{{\varvec{x}}_1}^*u_1^*,\quad P^*_{{\varvec{x}}_2} u_2^*\,\bigr )_*,\quad \forall {\varvec{x}}_i\in M,\quad u_i^*\in E^*_{{\varvec{x}}_i}\quad i=1,2. \end{aligned}$$

Observe that, by definition, a correlator \(C\in C^\infty (E\boxtimes E)\) induces a metric on \(E^*\) and thus, by duality, a metric on \(E\). We will denote both these metric by \((-,-)_{E^*,C}\) and respectively \((-, -)_{E,C}\). When no confusion is possible will drop the subscript \(E\) or \(E^*\) from the notation. To simplify the presentation we adhere to the following conventions.

  1. (i)

    We will use the Latin letters \(i,j,k\) to denote indices in the range \(1,\cdots , m=\dim M\).

  2. (ii)

    We will use Greek letters \(\alpha ,\beta ,\gamma \) to denote indices in the range \(1,\cdots , r= \mathrm{rank}\,(E)\).

Using the metric \((-,-)_C\) we can identify \(C_{{\varvec{x}},{\varvec{y}}}\in E_{\varvec{x}}\otimes E_{\varvec{y}}\) with an element of

$$\begin{aligned} T_{{\varvec{x}},{\varvec{y}}}\in E_{\varvec{x}}\otimes E_{\varvec{y}}^*\cong {{\mathrm{Hom}}}(E_{\varvec{y}}, E_{\varvec{x}}). \end{aligned}$$

We will refer to \(T_{x,y}\) as the tunneling map associated to the correlator \(C\). Note that \(T_{{\varvec{x}},{\varvec{x}}}=\mathbb {1}_{E_{\varvec{x}}}\). If we denote by \(T_{{\varvec{x}},{\varvec{y}}}^*\in {{\mathrm{Hom}}}(E_{\varvec{y}},E_{\varvec{x}})\) the adjoint of \(T_{{\varvec{x}},{\varvec{y}}}\) with respect to the metric \((-,-)_{E, C}\), then the symmetry of \(C\) implies that

$$\begin{aligned} T_{{\varvec{y}},{\varvec{x}}}=T_{{\varvec{x}},{\varvec{y}}}^*. \end{aligned}$$

Lemma 2.2

Fix a point \({\varvec{p}}_0\in M\) and local coordinates \((x^i)_{1\le i\le m}\) in a neighborhood \(\mathcal {O}\) of \({\varvec{p}}_0\) in \(M\). Suppose that \( \underline{{\varvec{e}}}(x)=({\varvec{e}}_\alpha (x))_{1\le \alpha \le r}\) is a local \((-,-)_C\)-orthononomal frame of \(E|_{\mathcal {O}}\). We regard it as an isomorphism of metric bundles \( \underline{\mathbb R}^r_\mathcal {O}\rightarrow E|_\mathcal {O}\). We obtain a smooth map

$$\begin{aligned} T(\underline{{\varvec{e}}}): \mathcal {O}\times \mathcal {O}\rightarrow {{\mathrm{Hom}}}({\mathbb R}^r),\quad (x,y)\mapsto T(\underline{\varvec{e}})_{x,y}=\underline{{\varvec{e}}}(x)^{-1}T_{x,y}\underline{{\varvec{e}}}(y). \end{aligned}$$

Then for any \(i=1,\cdots , m\) the operator

$$\begin{aligned} \partial _{x^i} T(\underline{{\varvec{e}}})_{x,y}|_{x=y}:\underline{\mathbb R}^r_{y}\rightarrow \underline{\mathbb R}^r_{y}, \end{aligned}$$

is skew-symmetric.

Proof

We identify \(\mathcal {O}\,\times \,\mathcal {O}\) with an open neighborhood of \((0,0)\in {\mathbb R}\times {\mathbb R}\) with coordinates \((x^i,y^j)\). Introduce new coordinates \(z^i:=x^i-y^i\), \(s^j:=x^j+y^j\), so that \(\partial _{x^i}=\partial _{z^i}+\partial _{s^i}\). We view the map \(T(\underline{\varvec{e}})\) as depending on the variables \(z,s\). Note that

$$\begin{aligned} T(\underline{\varvec{e}})_{0,s}=\mathbb {1},\quad T(\underline{\varvec{e}})_{-z,s}= T(\underline{\varvec{e}})_{z,s}^*,\quad \forall z,s. \end{aligned}$$

We deduce that

$$\begin{aligned}&\displaystyle \partial _{s^i}T(\underline{\varvec{e}})|_{0,s}=\partial _{s^i}T(\underline{\varvec{e}})|_{0,s}^*=0,\\&\displaystyle \partial _{x^i} T(\underline{\varvec{e}})|_{0,s} =\partial _{z^i} T(\underline{\varvec{e}})|_{0,s}+\partial _{s^i}T({\varvec{e}})|_{0,s}=\partial _{z^i} T(\underline{\varvec{e}})|_{0,s},\\&\displaystyle \bigl (\,\partial _{x^i} T(\underline{\varvec{e}})|_{0,s}\,\bigr )^*=\partial _{x^i} T(\underline{\varvec{e}})^*|_{0,s}= -\partial _{z_i} T(\underline{\varvec{e}})|_{0,s} +\partial _{s^i} T(\underline{\varvec{e}})|_{0,s}=-\partial _{x^i} T(\underline{\varvec{e}})|_{0,s}. \end{aligned}$$

\(\square \)

Given a coordinate neighborhood with coordinates \((x^i)\) and a local isomorphism of metric vector bundles (local orthonormal frame) \(\underline{{\varvec{e}}}: \underline{\mathbb R}^r_\mathcal {O}\rightarrow E|_\mathcal {O}\) as above, we define the skew-symmetric endomorphisms

$$\begin{aligned} \Gamma _i(\underline{{\varvec{e}}}) :\underline{\mathbb R}^r_\mathcal {O}\rightarrow \underline{\mathbb R}^r_\mathcal {O},\quad i=1,\cdots , m=\dim M,\quad \Gamma _i(\underline{{\varvec{e}}})_y= -\partial _{x^i}T_{x,y}|_{x=y}. \end{aligned}$$
(2.1)

We obtain a \(1\)-form with matrix coefficients \(\Gamma (\underline{{\varvec{e}}}):=\sum _i \Gamma _i(\underline{{\varvec{e}}}) dy^i\). The operator

$$\begin{aligned} \nabla ^{\underline{{\varvec{e}}}}= d+\Gamma (\underline{{\varvec{e}}}) \end{aligned}$$
(2.2)

is then a connection on \(\underline{\mathbb R}^r_\mathcal {O}\) compatible with the metric natural metric on this trivial bundle. The isomorphism \(\underline{\varvec{e}}\) induces a metric connection \(\underline{\varvec{e}}_*\nabla ^{\underline{\varvec{e}}}\) on \(E|_\mathcal {O}\).

Suppose that \(\underline{\varvec{f}}:\underline{\mathbb R}^r_\mathcal {O}\rightarrow E|_\mathcal {O}\) is another orthonormal frame of \(E_\mathcal {O}\) related to \(\underline{\varvec{e}}\) via a transition map

$$\begin{aligned} g: \mathcal {O}\rightarrow O(r),\quad \underline{\varvec{f}}=\underline{\varvec{e}}\cdot g. \end{aligned}$$

Then

$$\begin{aligned} T(\underline{\varvec{f}})_{x,y} =g^{-1}(x)T(\underline{\varvec{e}})_{x,y} g(y). \end{aligned}$$

We denote by \(d_x\) the differential with respect to the \(x\) variable. We deduce

$$\begin{aligned} \Gamma (\underline{\varvec{f}})_y= & {} -d_x T(\underline{\varvec{f}})_{x,y}|_{x=y}\\= & {} -\bigl (\,d_xg^{-1}(x)\,\bigr )_{x=y}\cdot \underbrace{T(\underline{\varvec{e}})_{y,y}}_{=\mathbb {1}}\cdot g(y)- g^{-1}(y)\bigl (\,d_xT(\underline{\varvec{e}})_{x,y} \bigr )|_{x=y} g(y)\\= & {} g^{-1}(y) dg(y) g^{-1}(y)\cdot g(y) + g^{-1}(y) \Gamma (\underline{\varvec{e}})_y g(y)\\= & {} g(y)^{-1} dg(y) + g^{-1}(y) \Gamma (\underline{\varvec{e}})_y g(y). \end{aligned}$$

Thus

$$\begin{aligned} \Gamma (\underline{\varvec{e}}\cdot g)= g^{-1}dg +g^{-1}\Gamma (\underline{\varvec{e}}) g. \end{aligned}$$

This shows that for any local orthonormal frames \(\underline{\varvec{e}}\), \(\underline{\varvec{f}}\) of \(E|_\mathcal {O}\) we have

$$\begin{aligned} \underline{\varvec{e}}_*\nabla ^{\underline{\varvec{e}}}=\underline{\varvec{f}}_*\nabla ^{\underline{\varvec{f}}}. \end{aligned}$$

We have thus proved the following result.

Proposition 2.3

If \(E\rightarrow M\) is a smooth real vector bundle, then any correlator \(C\) on \(M\) induces a canonical metric \((-,-)_{C}\) on \(E\) and a connection \(\nabla ^C\) compatible with this metric. More explicitly, if \(\mathcal {O}\subset M\) is an coordinate neighborhood on \(M\) and \(\underline{\varvec{e}}:\underline{\mathbb R}^r_\mathcal {O}\rightarrow E|_\mathcal {O}\) is an orthogonal trivialization, then \(\nabla ^C\) is described by

$$\begin{aligned} \nabla ^C =d +\sum _i \Gamma _i(\underline{\varvec{e}}) dx^i, \end{aligned}$$

where the skew-symmetric \(r\times r\)-matrix \(\Gamma _i(\underline{\varvec{e}})\) is given by (2.1).

Remark 2.4

(a) In the special case described in Example 2.1, the connection associated to the corresponding correlator coincides with the Levi-Civita connection of the metric induced by the correlator. As mentioned earlier, for a stochastic correlator there is an alternative, probabilistic description of the associated connection; see [6], §1.1] for details.

(b) Suppose that we fix local coordinates \((x^i)\) near a point \({\varvec{p}}_0\) such that \(x^i({\varvec{p}}_0)=0\). We denote by \(P_{x,0}\) the parallel transport of \(\nabla ^C\) from \(0\) to \(x\) along the line segment from \(0\) to \(x\). Then

$$\begin{aligned} P_{0,0}=\mathbb {1}_{E_0}=T_{0,0},\quad \partial _{x^i}P_{x,0}|_{x=0}=-\Gamma _i(0)=\partial _{x^i,0} T_{x,0}|_{x=0}. \end{aligned}$$

We see that the tunneling map \(T_{x,0}\) is a first order approximation at \(0\) of the parallel transport map \(P_{x,0}\) of the connection \(\nabla ^C\).

(c) Note that we have proved a slightly stronger result. Suppose that \(E\rightarrow M\) is a real vector bundle equipped with a metric. An integral kernel on \(E\) is a section \(T\in C^\infty (E^*\boxtimes E)\) and defines a smooth family of linear operators \(T_{{\varvec{x}},{\varvec{y}}}\in {{\mathrm{Hom}}}(E_{\varvec{y}}, E_{\varvec{x}})\), \({\varvec{x}},{\varvec{y}}\in M\). We say that an integral kernel \(T\) is a symmetric tunneling if

$$\begin{aligned} T_{x,x}= \mathbb {1}_{E_x}, \quad T^*_{{\varvec{x}},{\varvec{y}}}=T_{{\varvec{y}},{\varvec{x}}},\quad \forall {\varvec{x}},{\varvec{y}}\in M. \end{aligned}$$

The proof of Proposition 2.3 shows that any symmetric tunneling on a metric vector bundle naturally determines a connection compatible with the metric.

For later use, we want to give a more explicit description of the curvature of the connection \(\nabla ^C\) in the special case when the correlator \(C\) stochastic and thus it is the covariance density of a nondegenerate Gaussian ensemble of smooth sections of \(E\).

Proposition 2.5

Suppose that \(C\) is a stochastic correlator on \(E\) defined by the nondegenerate Gaussian ensemble smooth random sections of \(E\). Denote by \({\varvec{u}}\) a random section in this ensemble. Fix a point \({\varvec{p}}_0\), local coordinates \((x^i)\) on \(M\) near \({\varvec{p}}_0\) such that \(x^i({\varvec{p}}_0)=0\) \(\forall i\), and a local \((-,-)_C\)-orthonormal frame \(\bigl (\,{\varvec{e}}_\alpha (x) \,\bigr )_{1\le \alpha \le r}\) of \(E\) in a neighborhood of \({\varvec{p}}_0\) which is synchronous at \({\varvec{p}}_0\),

$$\begin{aligned} \nabla ^C{\varvec{e}}_\alpha |_{{\varvec{p}}_0}=0,\quad \forall \alpha . \end{aligned}$$

Denote by \(F\) the curvature of \(\nabla ^C\),

$$\begin{aligned} F=\sum _{ij} F_{ij}(x) dx^i\wedge dx^j,\quad F_{ij}(x)\in {{\mathrm{End}}}(E_{{\varvec{p}}_0}). \end{aligned}$$

Then \(F_{ij}(0)\) is the endomorphism of \(E_{{\varvec{p}}_0}\) which in the frame \({\varvec{e}}_\alpha ({\varvec{p}}_0)\) is described by the \(r\times r\) matrix with entries

$$\begin{aligned} F_{\alpha \beta |ij}(0)\!:=\! {\varvec{E}}\bigl (\,\partial _{x^i}u_\alpha (x)\partial _{x^j}u_\beta (x)\,\bigr )|_{x=0}\!-\!{\varvec{E}}\bigl (\,\partial _{x^j}u_\alpha (x)\partial _{x^i}u_\beta (x)\,\bigr )|_{x=0},\quad 1\!\le \!\alpha ,\beta \!\le \! r, \end{aligned}$$
(2.3)

where \(u_\alpha (x)\) is the random function

$$\begin{aligned} u_\alpha (x):=\bigl (\,{\varvec{u}}(x),{\varvec{e}}_\alpha (x)\,\bigr )_C. \end{aligned}$$

Proof

The random section \({\varvec{u}}\) has the local description

$$\begin{aligned} {\varvec{u}}=\sum _\alpha u_\alpha (x){\varvec{e}}_\alpha (x). \end{aligned}$$

Then \(T(x,y)\) is a linear map \(E_y\rightarrow E_x\) given by the \(r\times r\) matrix

$$\begin{aligned} T(x,y)= \bigl (\, T_{\alpha \beta }(x,y)\,\bigr )_{1\le \alpha ,\beta \le r},\quad T_{\alpha \beta }(x, y) = {\varvec{E}}(\, u_\alpha (x)u_\beta (y)\,\bigr ). \end{aligned}$$

The coefficients of the connection \(1\)-form \(\Gamma =\sum _i\Gamma _i dx^i\) are endomorphisms of \(E_x\) given by \(r\times r\) matrices

$$\begin{aligned} \Gamma _i(x)=\bigl ( \, \Gamma _{\alpha \beta |i}(x)\,\bigr )_{1\le \alpha ,\beta \le r}. \end{aligned}$$

More precisely, we have

$$\begin{aligned} \Gamma _{\alpha \beta |i}(x) = -{\varvec{E}}\bigl (\,\partial _{x^i}u_\alpha (x)u_\beta (x)\,\bigr ). \end{aligned}$$
(2.4)

Because the frame \(\bigl (\,{\varvec{e}}_\alpha (x)\,\bigr )\) is synchronous at \(x=0\) we deduce that, at \({\varvec{p}}_0\), we have \(\Gamma _i(0)=0\) and

$$\begin{aligned} F({\varvec{p}}_0)= & {} \sum _{i<j} F_{ij} (x)dx^i\wedge dx^j\in {{\mathrm{End}}}(E_{{\varvec{p}}_0})\otimes \Lambda ^2 T^*_{{\varvec{p}}_0} M,\\ F_{ij}= & {} \partial _{x^i}\Gamma _j({\varvec{p}}_0)-\partial _{x^j}\Gamma _i({\varvec{p}}_0). \end{aligned}$$

The coefficients \(F_{ij}(x)\) are \(r\times r\) matrices with entries \(F_{\alpha \beta |ij}(x)\), \(1\le \alpha ,\beta \le r\). Moreover

$$\begin{aligned}&F_{\alpha \beta |ij}(0)=\partial _{x^j}\Gamma _{\alpha \beta |j}(0)-\partial _{x^j}\Gamma _{\alpha \beta |i}(0)\\&\mathop {=}\limits ^{(2.4)}\partial _{x^j}{\varvec{E}}\bigl (\,\partial _{x^i}u_\alpha (x)u_\beta (x)\,\bigr )|_{x=0}-\partial _{x^i} {\varvec{E}}\bigl (\,\partial _{x^j}u_\alpha (x)u_\beta (x)\,\bigr )|_{x=0}\\&\quad ={\varvec{E}}\bigl (\,\partial ^2_{x^jx^i}u_\alpha (x)u_\beta (x)\,\bigr )|_{x=0}+{\varvec{E}}\bigl (\,\partial _{x^i}u_\alpha (x)\partial _{x^j}u_\beta (x)\,\bigr )|_{x=0}\\&\quad \quad - {\varvec{E}}\bigl (\,\partial ^2_{x^ix^j}u_\alpha (x)u_\beta (x)\,\bigr )|_{x=0}-{\varvec{E}}\bigl (\,\partial _{x^j}u_\alpha (x)\partial _{x^i}u_\beta (x)\,\bigr )|_{x=0}\\&\quad ={\varvec{E}}\bigl (\,\partial _{x^i}u_\alpha (x)\partial _{x^j}u_\beta (x)\,\ \bigr )|_{x=0}-{\varvec{E}}\bigl (\,\partial _{x^j}u_\alpha (x)\partial _{x^i}u_\beta (x)\,\bigr )|_{x=0}. \end{aligned}$$

\(\square \)

Remark 2.6

When \(C\) is the stochastic correlator defined in Example 2.1(a), Proposition 2.5 specializes to Gauss’ Theorema Egregium.

Corollary 2.7

Suppose that \({\varvec{u}}\) is a nondegenerate, Gaussian smooth random section of \(E\) with covariance density \(C\in C^\infty (E\boxtimes E)\). Denote by \((-,-)_C\) and respectively \(\nabla ^C\) the metric and respectively the connection on \(E\) defined by \(C\). Then for any \({\varvec{p}}_0\in M\) the random variables \({\varvec{u}}({\varvec{p}}_0)\) and \(\nabla ^C{\varvec{u}}({\varvec{p}}_0)\) are independent.

Proof

We continue to use the same notations as in the proof of Proposition 2.5. Observe first that

$$\begin{aligned} \bigl (\,{\varvec{u}}({\varvec{p}}_0),\nabla {\varvec{u}}({\varvec{p}}_0)\,\bigr )\in E_{{\varvec{p}}_0}\oplus E_{{\varvec{p}}_0}\otimes T_{{\varvec{p}}_0}^* M, \end{aligned}$$

is a Gaussian random vector. The section \({\varvec{u}}\) has the local description

$$\begin{aligned} {\varvec{u}}(x)=\sum _\beta u_\alpha (x){\varvec{e}}_\beta (x). \end{aligned}$$

Then

$$\begin{aligned} \nabla ^C_{x^i}{\varvec{u}}({\varvec{p}}_0)=\sum _\alpha \partial _{x^i} u_\alpha (0){\varvec{e}}_\alpha (0),\quad 0=\Gamma _{\alpha \beta |i}(x) \mathop {=}\limits ^{(2.4)} -{\varvec{E}}\bigl (\,\partial _{x^i}u_\alpha (0)u_\beta (0)\,\bigr ). \end{aligned}$$

Since \((u_\beta (0),\partial _{x^i} u_\alpha (0)\,)\) is a Gaussian vector, we deduce that the random variables \(u_\beta (0),\partial _{x^i} u_\alpha (0)\) are independent.\(\square \)

Remark 2.8

The local definition of the connection coefficients \(\Gamma _i\) shows that the above independence result is a special case of a well known fact in the theory of Gaussian random vectors: if \(X,Y\) are finite dimensional Gaussian vectors such that the direct sum \(X\oplus Y\) is also Gaussian, then for a certain deterministic linear operator \(A\) the random vector \(X-AY\) is independent of \(Y\); see e.g. [2], Prop. 1.2]. More precisely, this happens when

$$\begin{aligned} A={{\mathrm{\varvec{cov}}}}(X,Y)\cdot {{\mathrm{\varvec{cov}}}}(Y)^{-1}. \end{aligned}$$

Corollary 2.7 follows from this fact applied in the special case \(X=d{\varvec{u}}(0)\) and \(Y={\varvec{u}}(0)\).

If we use local coordinates \((x^i)\) and a local orthonormal frame \(({\varvec{e}}_\alpha )\) in a neighborhood \(\mathcal {O}\), then we can view \(\mathcal {O}\) as an open subset \({\mathbb R}^m\) and \({\varvec{u}}\) as a map \({\varvec{u}}:\mathcal {O}\rightarrow {\mathbb R}^r\). As such, it has a differential \(d{\varvec{u}}(x)\) at any \(x\in \mathcal {O}\). The formula (2.4) defining the coefficients of the correlator connection \(\nabla ^C\) and the classical regression formula [2], Prop. 1.2] yield the following a.s. equality: for any point \(x\in \mathcal {O}\) we have

$$\begin{aligned} \nabla ^C{\varvec{u}}(x) =d{\varvec{u}}(x)-{\varvec{E}}\bigl ( d{\varvec{u}}(x)\,\vert \; {\varvec{u}}(x)\,\bigr ). \end{aligned}$$
(2.5)

Above, the notation \({\varvec{E}}(\,\mathbf {var}\;|\; \mathbf {cond}\;)\) stands for the conditional expectation of the variable \(\mathbf {var}\) given the conditions \(\mathbf {cond}\). The above equality implies immediately that the random vectors \(d{\varvec{u}}(x)\) and \({\varvec{u}}(x)\) are independent.

We want to mention that (2.5) is a special case of the probabilistic description in [6], Prop. 1.13].

3 Kac-Rice implies Gauss–Bonnet–Chern

In this section we will prove a refined Gauss–Bonnet–Chern equality involving a nondegenerate Gaussian ensembles of smooth sections of \(E\). We will make the following additional assumption.

  • The manifold \(M\) is oriented.

  • The bundle \(E\) is oriented and its rank is even, \(r=2h\).

  • \(r\le m=\dim M\).

3.1 The setup

We denote by \(\Omega _k(M)\) the space of \(k\)-dimensional currents, i.e., the space of linear maps \(\Omega ^k(M)\rightarrow {\mathbb R}\) that are continuous with respect to the natural locally convex topology on the space of smooth \(k\)-forms on \(M\). If \(C\) is a \(k\)-current and \(\eta \) is a smooth \(k\)-form, then we denote by \(\langle \eta , C\rangle \) the value of \(C\) at \(\eta \).

Suppose that we are given a metric on \(E\) and a connection \(\nabla \) compatible with the metric. Observe that if \({\varvec{u}}:M\rightarrow E\) is a smooth section of \(E\) transversal to the zero section, then its zero set \(Z_{{\varvec{u}}}\) is a smooth codimension \(r\) submanifold of \(M\) and there is a canonical adjunction isomorphism

$$\begin{aligned} \mathfrak {a}_{\varvec{u}}: T_{Z_{\varvec{u}}}M\rightarrow E|_{Z_{\varvec{u}}}, \end{aligned}$$

where \(T_{Z_{\varvec{u}}}M= TM|_{Z_{\varvec{u}}}/TZ_{\varvec{u}}\) is the normal bundle of \(Z_{\varvec{u}}\hookrightarrow M\). For more details about this map we refer to [16], Exercise 8.3.21] or [19], Sec.2]. From the orientability of \(M\) and \(E\), and from the adjunction induced isomorphism

$$\begin{aligned} TM|_{Z_{\varvec{u}}}\cong E|_{Z_{\varvec{u}}}\oplus TZ_{\varvec{u}}\end{aligned}$$

we deduce that \(Z_{\varvec{u}}\) is equipped with a natural orientation uniquely determined by the equalities

$$\begin{aligned} \varvec{or}(TM|_{Z_{\varvec{u}}})= \varvec{or}(E|_{Z_{\varvec{u}}})\wedge \varvec{or}( TZ_{\varvec{u}}) \mathop {=}\limits ^{(r\in 2{\mathbb Z})} \varvec{or}( TZ_{\varvec{u}})\wedge \varvec{or}(E|_{Z_{\varvec{u}}}). \end{aligned}$$

Thus, the zero set \(Z_{\varvec{u}}\) with this induced orientation defines an integration current \([Z_{\varvec{u}}]\in \Omega _{m-r}(M)\)

$$\begin{aligned} \Omega ^{m-r}(M)\ni \eta \mapsto \langle \eta ,[Z_{\varvec{u}}]\rangle :=\int _{Z_{\varvec{u}}}\eta . \end{aligned}$$

To a metric \((-,-)\) on \(E\) and a connection \(\nabla \) compatible with the metric we can associate a closed form

$$\begin{aligned} {\varvec{e}}(E,\nabla )\in \Omega ^r(M). \end{aligned}$$

Its construction involves the concept of Pfaffian discussed in great detail in Appendix B and it goes as follows.

Denote by \(F\) the curvature of \(\nabla \) and set

$$\begin{aligned} {\varvec{e}}(E,\nabla ):=\frac{1}{(2\pi )^h} {{\mathrm{\mathbf {Pf}}}}(-F)\in \Omega ^r(M), \end{aligned}$$

where the Pfaffian \({{\mathrm{\mathbf {Pf}}}}(-F)\) has the following local description. Fix a positively oriented, local orthonormal frame \({\varvec{e}}_1(x),\cdots ,{\varvec{e}}_r(x)\) of \(E\) defined on some open coordinate neighborhood \(\mathcal {O}\) of \(M\). Then \(F|_{\mathcal {O}}\) is described by a skew-symmetric \(r\times r\) matrix \((F_{\alpha \beta })_{1\le \alpha ,\beta \le r}\), where

$$\begin{aligned} F_{\alpha \beta }\in \Omega ^2(\mathcal {O}),\quad \forall \alpha ,\beta . \end{aligned}$$

If we denote by \(\mathcal {S}_r\) the group of permutations of \(\{1,\cdots , r=2h\}\), then

$$\begin{aligned} {{\mathrm{\mathbf {Pf}}}}\bigl (-F\bigr )=\frac{1}{2^h h!}\sum _{{\sigma }\in \mathcal {S}_r}{\epsilon }({\sigma }) F_{{\sigma }_1{\sigma }_2}\wedge \cdots \wedge F_{{\sigma }_{2h-1}{\sigma }_{2h}}\in \Omega ^{2h}(\mathcal {O}), \end{aligned}$$
(3.1)

where \({\epsilon }({\sigma })\) denotes the signature of the permutation \({\sigma }\in \mathcal {S}_r\).

Remark 3.1

The \(r\times r\)-matrix \((F_{\alpha \beta })\) depends on the choice of positively oriented local orthonormal frame \(({\varvec{e}}_\alpha (x))\). However, the Pfaffian \({{\mathrm{\mathbf {Pf}}}}(-F)\) is a degree \(r\)-form on \(\mathcal {O}\) that is independent of the choice of positively oriented local orthonormal frame.

As explained in [16], Chap.8], the degree \(2h\)-form \({\varvec{e}}(E,\nabla )\) is closed and it is called the Euler form of the connection \(\nabla \). Moreover, its DeRham cohomology class is independent of the choice of the metric connection \(\nabla \). The Euler form defines an \((m-r)\)-dimensional current

$$\begin{aligned} {\varvec{e}}(E,\nabla )^\dag \in \Omega _{m-r}(M),\quad \Omega ^{m-r}(M)\ni \eta \mapsto \langle \eta ,{\varvec{e}}(E,\nabla )^\dag \rangle :=\int _M\eta \wedge {\varvec{e}}(E,\nabla ). \end{aligned}$$

3.2 A stochastic Gauss–Bonnet–Chern theorem

We can now state the main theorem of this paper.

Theorem 3.2

(Stochastic Gauss–Bonnet–Chern) Assume that the manifold \(M\) is oriented, the bundle \(E\) is oriented and has even rank \(r=2h\le m=\dim M\). Fix a nondegenerate Gaussian ensemble of smooth sections of \(E\). Denote by \({\varvec{u}}\) a random section of this ensemble, by \(C\) the correlator of this Gaussian ensemble, by \((-,-)_C\) the metric on \(E\) induced by \(C\) and by \(\nabla \) the connection on \(E\) determined by this correlator. Then the expectation of the random \((m-r)\)-dimensional current \([Z_{\varvec{u}}]\) is equal to the current \({\varvec{e}}(E,\nabla )^\dag \), i.e.,

$$\begin{aligned} {\varvec{E}}\bigl (\, \langle \eta , [Z_{\varvec{u}}]\rangle \,\bigr )=\int _M\eta \wedge {\varvec{e}}(E,\nabla ),\quad \forall \eta \in \Omega ^{m-r}(M). \end{aligned}$$
(3.2)

Proof

The linearity in \(\eta \) of (3.2) shows that it suffices to prove this equality in the special case when \(\eta \) is compactly supported on a coordinate neighborhood \(\mathcal {O}\) of a point \({\varvec{p}}_0\in M\). Fix coordinates \(x^1,\cdots , x^m\) on \(\mathcal {O}\) with the following properties.

  • \(x^i({\varvec{p}}_0)=0\), \(\forall i=1,\cdots , m\).

  • The orientation of \(M\) along \(\mathcal {O}\) is given by the top degree form \(\omega _\mathcal {O}:=dx^1\wedge \cdots \wedge dx^m\).

Invoking again the linearity in \(\eta \) of (3.2) we deduce that it suffices to prove it in the special case when \(\eta \) has the form

$$\begin{aligned} \eta =\eta _0 dx^{r+1}\wedge \cdots \wedge dx^m,\quad \eta _0\in C^\infty _0(\mathcal {O}). \end{aligned}$$

In other words, we have to prove the equality

$$\begin{aligned} {\varvec{E}}\bigl (\,\bigr \langle \, \eta _0dx^{r+1}\wedge \cdots \wedge dx^m, [Z_{\varvec{u}}]\,\bigr \rangle \,\bigr )\!=\!\int _\mathcal {O}\eta _0 dx^{r+1}\wedge \cdots \wedge dx^m\wedge {\varvec{e}}(E,\nabla ),\, \forall \eta _0\in C_0^\infty (\mathcal {O}). \end{aligned}$$
(3.3)

For any subset

$$\begin{aligned} I=\{i_1<\cdots <i_k\}\subset \{1,\cdots , m\} \end{aligned}$$

we write \(dx^I:= dx^{i_1}\wedge \cdots \wedge dx^{i_k}\). We set

$$\begin{aligned} I_0:=\{1,\cdots , r\},\quad J_0:=\{r+1,\cdots , m\}. \end{aligned}$$

We can rewrite (3.3) in the more compact form

$$\begin{aligned} {\varvec{E}}\bigl (\,\bigr \langle \, \eta _0dx^{J_0}, [Z_{\varvec{u}}]\,\bigr \rangle \,\bigr )=\int _\mathcal {O}\eta _0 dx^{J_0}\wedge {\varvec{e}}(E,\nabla ),\quad \forall \eta _0\in C_0^\infty (\mathcal {O}). \end{aligned}$$
(3.4)

Fix a local, positively oriented, \(-(,-)_C\)-orthonormal frame \(({\varvec{e}}_\alpha (x))_{1\le \alpha \le r}\) of \(E|_\mathcal {O}\). The restriction to \(\mathcal {O}\) of the curvature \(F\) of \(\nabla \) is then a skew-symmetric \(r\times r\)-matrix

$$\begin{aligned} F=(F_{\alpha \beta })_{1\le \alpha ,\beta \le r},\quad F_{\alpha \beta }\in \Omega ^2(\mathcal {O}),\quad \forall \alpha ,\beta . \end{aligned}$$

Each of the \(2\)-forms \(F_{\alpha \beta }\) admits a unique decomposition

$$\begin{aligned} F_{\alpha \beta }=\sum _{1\le i<j\le m} F_{\alpha \beta |ij} dx^i\wedge dx^j. \end{aligned}$$

For each subset \(I\subset \{1,\cdots m\}\) we write

$$\begin{aligned} F^I_{\alpha \beta }:= \sum _{\begin{array}{c} i<j\\ i,j\in I \end{array}} F_{\alpha \beta |ij} dx^i\wedge dx^j\in \Omega ^2(\mathcal {O}). \end{aligned}$$

We denote by \(F^I\) the skew-symmetric \(r\times r\) matrix with entries \(F^I_{\alpha \beta }\).

The degree \(r\) form \({{\mathrm{\mathbf {Pf}}}}(-F)\) admits a canonical decomposition

$$\begin{aligned} {{\mathrm{\mathbf {Pf}}}}(-F)=\sum _{|I|=r}{{\mathrm{\mathbf {Pf}}}}\bigl (\,-F^I\,\bigr )=\sum _{|I|=r}{{\mathrm{\mathbf {pf}}}}(-F)_I \, dx^I,\quad {{\mathrm{\mathbf {pf}}}}(-F)_I\in C^\infty (\mathcal {O}). \end{aligned}$$

The equality (3.4) is then equivalent to the equality

$$\begin{aligned} {\varvec{E}}\bigl (\,\bigr \langle \, \eta _0dx^{J_0}, [Z_{\varvec{u}}]\,\bigr \rangle \,\bigr )=\frac{1}{(2\pi )^h}\int _\mathcal {O}\eta _0 {{\mathrm{\mathbf {pf}}}}(-F)_{I_0}\omega _\mathcal {O},\quad \forall \eta _0\in C_0^\infty (\mathcal {O}), \end{aligned}$$
(3.5)

where we recall that \(\omega _\mathcal {O}= dx^1\wedge \cdots \wedge dx^m\). To prove the above equality we will use the following two-step strategy.

Step 1. Invoke the Kac-Rice formula to express the left-hand side of (3.5) as an integral over \(\mathcal {O}\)

$$\begin{aligned} {\varvec{E}}\bigl (\,\bigr \langle \, \eta _0dx^{J_0}, [Z_{\varvec{u}}]\,\bigr \rangle \,\bigr )=\int _\mathcal {O}\eta _0(x) \rho (x) \omega _\mathcal {O}, \end{aligned}$$

where \(\rho (x)\) is a certain smooth function on \(\mathcal {O}\).

Step 2. Use the Gaussian computations in Appendix B to show that

$$\begin{aligned} \rho (x)={{\mathrm{\mathbf {pf}}}}(-F)_{I_0}(x),\quad \forall x\in \mathcal {O}. \end{aligned}$$

Let us now implement this strategy. We view \(\mathcal {O}\) as an open neighborhood of the origin in \({\mathbb R}^m\) equipped with the canonical Euclidean metric and the orientation given by \(\omega _\mathcal {O}\). Denote by \(E_0\) the fiber of \(E\) over the origin. Using the oriented, orthonormal local frame \(({\varvec{e}}_\alpha )\) we can view the restriction to \(\mathcal {O}\) of the random section \({\varvec{u}}\) as smooth Gaussian random map

$$\begin{aligned} {\varvec{u}}: \mathcal {O}\rightarrow E_0\cong {\mathbb R}^r, \quad x\mapsto (u_\alpha (x))_{1\le \alpha \le r}, \end{aligned}$$

where again \({\mathbb R}^r\) is equipped with the canonical Euclidean metric and orientation given by the volume form

$$\begin{aligned} \omega _E= du_1\wedge \cdots \wedge du_r. \end{aligned}$$

The fact that the frame \((\,{\varvec{e}}_\alpha (x)\,)\) is orthonormal with respect to the metric \((-,-)_C\) implies that for any \(x\in \mathcal {O}\) the probability distribution of the random vector \({\varvec{u}}(x)\) is the standard Gaussian measure on the Euclidean space \({\mathbb R}^r\). We denote by \(p_{{\varvec{u}}(x)}\) the probability density of this vector so that

$$\begin{aligned} p_{{\varvec{u}}(x)}(y) =\frac{1}{(2\pi )^h} e^{-\frac{1}{2}|y|^2},\quad y\in E_0\cong {\mathbb R}^r, \end{aligned}$$
(3.6)

where \(|-|\) denotes the canonical Euclidean norm on \({\mathbb R}^r\), \(h=r/2\).

The zero set \(Z_{\varvec{u}}\) is a.s. a submanifold of \(\mathcal {O}\) and, as such, it is equipped with an induced Riemann metric with associated volume density \(|dV_{Z_{{\varvec{u}}}}|\).

Recall that if \(T: U\rightarrow V\) is a linear map between two Euclidean spaces such that \(\dim U\ge \dim V\), then its Jacobian is the scalar

$$\begin{aligned} {{\mathrm{Jac}}}_T:=\sqrt{\det (TT^*)}. \end{aligned}$$

We define the Jacobian at \(x\in \mathcal {O}\) of a smooth map \(F:\mathcal {O}\rightarrow E_0\) to be the scalar

$$\begin{aligned} J_F(x)={{\mathrm{Jac}}}_{dF(x)}=\sqrt{\det d F(x) dF(x)^*}, \end{aligned}$$

where \(dF(x):{\mathbb R}^m\rightarrow E_0\) is the differential \(dF(x)\) of \(F\) at \(x\). We set

$$\begin{aligned} \mathcal {T}:={{\mathrm{Hom}}}({\mathbb R}^m,E_0), \end{aligned}$$

so that have a Gaussian random map

$$\begin{aligned} d{\varvec{u}}: \mathcal {O}\rightarrow \mathcal {T}, \quad x\mapsto d{\varvec{u}}(x). \end{aligned}$$

This random map is a.s. smooth. The random map

$$\begin{aligned} \mathcal {O}\rightarrow E_0\times \mathcal {T},\quad x\mapsto \bigl ( {\varvec{u}}(x), d{\varvec{u}}(x)\,\bigr ), \end{aligned}$$

is also a Gaussian random map. We have the following Kac-Rice formula, [2], Thm. 6.4,6.10].

Theorem 3.3

(Kac-Rice) Let \(g: \mathcal {T}\rightarrow {\mathbb R}\) be a bounded continuous function. Then, for any \(\lambda _0\in C_0(\mathcal {O})\), the random variable

$$\begin{aligned} {\varvec{u}}\mapsto \int _{Z_{\varvec{u}}} \lambda _0(x) g\bigl (\,d{\varvec{u}}(x)\,\bigr ) |dV_{Z_{\varvec{u}}}(x)|\ \end{aligned}$$

is integrable and

$$\begin{aligned} {\varvec{E}}\left( \int _{Z_{\varvec{u}}} \lambda _0(x) g\bigl (\,d{\varvec{u}}(x)\,\bigr ) |dV_{Z_{\varvec{u}}}(x)|\,\right) =\int _\mathcal {O}\lambda _0(x){\varvec{w}}(x) \omega _\mathcal {O}(x),\quad \forall \lambda _0\in C_0(\mathcal {O}), \nonumber \\ \end{aligned}$$
(3.7a)
$$\begin{aligned} {\varvec{w}}(x)= & {} {\varvec{E}}\Bigl (\, J_{\varvec{u}}(x) g(\,d{\varvec{u}}(x)\,)\,\bigr |\; {\varvec{u}}(x)=0\,\Bigr )p_{{\varvec{u}}(x)}(0)\nonumber \\&\mathop {=}\limits ^{(3.6)}\frac{1}{(2\pi )^h}{\varvec{E}}\Bigl (\, J_{\varvec{u}}(x) g(\,d{\varvec{u}}(x)\,)\,\bigr |\; {\varvec{u}}(x)=0\,\Bigr ). \end{aligned}$$
(3.7b)

In particular, the function \(x\mapsto \lambda _0(x) \rho (x)\) is also integrable.

The above equality extends to more general functions \(g\).

Definition 3.4

We say that a bounded measurable function \(g:\mathcal {T}\rightarrow {\mathbb R}\) is admissible if there exists a sequence of bounded continuous functions \(g_n:\mathcal {T}\rightarrow {\mathbb R}\) with the following properties.

  1. (i)

    The sequence \(g_n\) converges a.e. to \(g\).

  2. (ii)

    \(\sup _n\Vert g_n\Vert _{L^\infty }<\infty \).

Lemma 3.5

Theorem 3.3 continues to hold if \(g\) is an admissible function \(\mathcal {T}\rightarrow {\mathbb R}\).

Proof

Fix an admissible function \(g:\mathcal {T}\rightarrow {\mathbb R}\) and a sequence of bounded continuous functions \(g_n:\mathcal {T}\rightarrow {\mathbb R}\) satisfying the conditions in Definition 3.4. Set \(K:= \sup _n\Vert g_n\Vert _{L^\infty }\). Then

$$\begin{aligned} \left| \int _{Z_{\varvec{u}}} \lambda _0(x) g_n\bigl (\,d{\varvec{u}}(x)\,\bigr ) |dV_{Z_{\varvec{u}}}(x)|\,\right| \le K\int _{Z_{\varvec{u}}} |\lambda _0(x)| |dV_{Z_{\varvec{u}}}(x)|. \end{aligned}$$

The random variable

$$\begin{aligned} {\varvec{u}}\mapsto K \int _{Z_{\varvec{u}}} |\lambda _0(x)| |dV_{Z_{\varvec{u}}}(x)| \end{aligned}$$

is integrable according to the Theorem 3.3 in the special case \(g\equiv K\) and \(\lambda _0=|\lambda _0|\). The dominated converge theorem implies that

$$\begin{aligned} \lim _{n\rightarrow \infty } {\varvec{E}}\left( \int _{Z_{\varvec{u}}} \lambda _0(x) g_n\bigl (\,d{\varvec{u}}(x)\,\bigr ) |dV_{Z_{\varvec{u}}}(x)|\,\right) ={\varvec{E}}\left( \int _{Z_{\varvec{u}}} \lambda _0(x) g\bigl (\,d{\varvec{u}}(x)\,\bigr ) |dV_{Z_{\varvec{u}}}(x)|\,\right) . \end{aligned}$$

A similar argument shows that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }{\varvec{E}}\Bigl (\, J_{\varvec{u}}(x) g(\,d{\varvec{u}}(x)\,)\,\bigr |\; {\varvec{u}}(x)=0\,\Bigr )={\varvec{E}}\Bigl (\, J_{\varvec{u}}(x) g(\,d{\varvec{u}}(x)\,)\,\bigr |\; {\varvec{u}}(x)=0\,\Bigr ). \end{aligned}$$

\(\square \)

To apply the above Kac-Rice formula we need to express the integral over \(Z_{\varvec{u}}\) of a form as an integral of a function with respect to the volume density. More precisely, we seek an equality of the type

$$\begin{aligned} \int _{Z_{\varvec{u}}} dx^{J_0} =\int _{Z_{\varvec{u}}} \eta _0(x) g\bigl (\, d{\varvec{u}}(x)\,\bigr ) |dV_{Z_{\varvec{u}}}(x)|, \end{aligned}$$

for some admissible function \(g\). This is achieved in the following technical result whose proof can be found in Appendix A.

Lemma 3.6

Suppose that \(0\) is a regular value of \({\varvec{u}}\). Set \(u_\alpha (x)= ({\varvec{u}},{\varvec{e}}_\alpha (x)\,)\). Then

$$\begin{aligned} dx^{J_0}|_{Z_{\varvec{u}}}=\frac{\Delta _{I_0}(d{\varvec{u}})}{J_{\varvec{u}}}d V_{Z_u}, \end{aligned}$$

where \(J_{\varvec{u}}:\mathcal {O}\rightarrow {\mathbb R}_{\ge 0}\) is the Jacobian of \({\varvec{u}}\) and \(\Delta _{I_0}(d{\varvec{u}})\) is the determinant of the \(r\times r\) matrix \(\frac{\partial {\varvec{u}}}{\partial x^{I_0}}\) with entries

$$\begin{aligned} \frac{\partial u_\alpha }{\partial x^j},\quad \alpha ,j\in I_0. \end{aligned}$$

Any linear map \(T\in \mathcal {T}={{\mathrm{Hom}}}({\mathbb R}^m,E_0)\) is represented by an \(r\times m\) matrix. For any subset \(J\) of \(\{ 1,\cdots , m\}\) of cardinality \(r\) we denote by \(\Delta _J(T)\) the determinant of the \(r\times r\) minor \(T_J\) determined by the columns indexed by \(J\).

Denote by \(\mathcal {T}_*\) the subset of \(\mathcal {T}\) consisting of surjective linear maps \({\mathbb R}^m\rightarrow E_0\). The complement \(\mathcal {T}{\setminus } \mathcal {T}_*\) is a negligible subset of \(\mathcal {T}\). Observe that \(T\in \mathcal {T}^*{\Longleftrightarrow }{{\mathrm{Jac}}}_T\ne 0\). Define

$$\begin{aligned} G: \mathcal {T}\rightarrow {\mathbb R},\quad g(T)={\left\{ \begin{array}{ll} \frac{\Delta _{I_0}(T)}{{{\mathrm{Jac}}}_T},&{} T\in \mathcal {T}_*,\\ 0, &{} T\in \mathcal {T}{\setminus } \mathcal {T}_*. \end{array}\right. } \end{aligned}$$

Lemma 3.6 shows that if \(0\) is a regular value of \({\varvec{u}}\), then

$$\begin{aligned} \int _{Z_{\varvec{u}}} \eta _0 dx^{J_0}= \int _{Z_{\varvec{u}}} \eta _0(x) G(d{\varvec{u}}(x)) |dV_{Z_{\varvec{u}}}(x)|,\quad \forall \eta _0\in C_0(\mathcal {O}). \end{aligned}$$

Lemma 3.7

The measurable function \(G: \mathcal {T}\rightarrow {\mathbb R}\) is admissible.

Proof

We first prove that \(G\) is bounded on \(\mathcal {T}_*\). This follows from the classical identity

$$\begin{aligned} {{\mathrm{Jac}}}_T^2=\sum _{|J|=r}\Delta _{J}(T)^2. \end{aligned}$$

This proves that

$$\begin{aligned} \left| \frac{\Delta _{I_0}(T)}{{{\mathrm{Jac}}}_T}\right| \le 1. \end{aligned}$$

Now define

$$\begin{aligned} G_n( T):=\frac{\Delta _{I_0}(T)}{ \sqrt{n^{-2}+{{\mathrm{Jac}}}_T^2}},\quad \forall T\in \mathcal {T}. \end{aligned}$$

Observe that \(G_n(T)\nearrow G(T)\) for \(T\in \mathcal {T}_*\) as \(n\rightarrow \infty \) and \(\sup _n\Vert G_n\Vert _{L^\infty }\le 1\).\(\square \)

We deduce that

$$\begin{aligned}&{\varvec{E}}\Bigl (\, \bigl \langle \, \eta _0(x) dx^{J^0}, [Z_{\varvec{u}}]\,\bigl )={\varvec{E}}\left( \int _{Z_{\varvec{u}}} \eta _0(x) G\bigl (\,d{\varvec{u}}(x)\,\bigr ) |dV_{Z_{\varvec{u}}}(x)|\,\right) \\&\mathop {=}\limits ^{(3.7a)}\frac{1}{(2\pi )^h}\int _\mathcal {O}\eta _0(x){\varvec{E}}\Bigl (\,J_{\varvec{u}}({\varvec{x}}) G(\,d{\varvec{u}}(x)\,)\; \Bigl |\;{\varvec{u}}(x)=0\,\Bigr ) \omega _\mathcal {O}\\&\quad =\frac{1}{(2\pi )^h}\int _\mathcal {O}\eta _0(x)\,\underbrace{{\varvec{E}}\Bigl (\,\Delta _{I_0}\bigl (\,d{\varvec{u}}(x)\,\bigr )\;\Bigl |\; {\varvec{u}}(x)=0\,\Bigr )}_{=:\rho (x)} \omega _\mathcal {O}. \end{aligned}$$

We have thus proved the equality

$$\begin{aligned} {\varvec{E}}\Bigl (\, \bigl \langle \, \eta _0(x) dx^{J^0}, \, [Z_{\varvec{u}}]\,\bigr \rangle \,\Bigl )=\frac{1}{(2\pi )^h}\int _\mathcal {O}\rho (x) \eta _0(x) \omega _\mathcal {O},\quad \forall \eta _0\in C_0^\infty (\mathcal {O}). \end{aligned}$$
(3.8)

The density \(\rho (x)\) in the right-hand-side of the above equality could a priori depend on the choice of the \((-,-)_C\)-orthonormal frame because it involved the frame dependent matrix \(\frac{\partial {\varvec{u}}}{\partial x^{I_0}}\). On the other hand, the left-hand-side of the equality (3.8) is plainly frame independent. This shows that the density \(\rho \) is also frame independent. To prove (3.5) and thus Theorem 3.2 it suffices to show that

$$\begin{aligned} {\varvec{E}}\Bigl (\,\Delta _{I_0}\bigl (\,d{\varvec{u}}(x)\,\bigr )\;\Bigl |\; {\varvec{u}}(x)=0\,\Bigr )= {{\mathrm{\mathbf {pf}}}}(-F)_{I_0}(x),\quad \forall x\in \mathcal {O}. \end{aligned}$$
(3.9)

We will prove the above equality for \(x=0\). Both sides are frame invariant and thus we are free to choose the frame \((\,{\varvec{e}}_\alpha (x)\,)\) as we please. We assume that it is synchronous at \(x=0\), i.e.,

$$\begin{aligned} \nabla {\varvec{e}}_\alpha (0)=0,\quad \forall \alpha . \end{aligned}$$

Then \(\nabla ^C{\varvec{u}}(0)=d{\varvec{u}}(0)\). Corollary 2.7 now implies that the Gaussian vectors \(d{\varvec{u}}(0)\) and \({\varvec{u}}(0)\) are independent. Hence

$$\begin{aligned} {\varvec{E}}\Bigl (\,\Delta _{I_0}\bigl (\,d{\varvec{u}}(0)\,\bigr )\;\Bigl |\; {\varvec{u}}(0)=0\,\Bigr )={\varvec{E}}\Bigl (\,\Delta _{I_0}\bigl (\,d{\varvec{u}}(0)\,\bigr )\,\Bigr ), \end{aligned}$$

and thus we have to prove that

$$\begin{aligned} {\varvec{E}}\Bigl (\,\Delta _{I_0}\bigl (\,d{\varvec{u}}(0)\,\bigr )\,\Bigr )= {{\mathrm{\mathbf {pf}}}}(-F)_{I_0}(0). \end{aligned}$$
(3.10)

The random variable \(\Delta _{I_0}\bigl (\,d{\varvec{u}}(0)\,\bigr )\) is the determinant of the \(r\times r\) Gaussian matrix \(S:=\frac{\partial {\varvec{u}}}{\partial x^{I_0}}\) with entries

$$\begin{aligned} S_{\alpha i}:=\partial _{x^i} u_\alpha (0),\quad 1\le \alpha ,i\le r. \end{aligned}$$
(3.11)

Its statistics are determined by the covariances

$$\begin{aligned} K_{\alpha i|\beta j}:= {\varvec{E}}\bigl (\, S_{\alpha i} S_{\beta j} \,\bigr )={\varvec{E}}\bigl (\, \partial _{x^i} u_\alpha (0)\partial _{x^j} u_\beta (0)\,\bigr ). \end{aligned}$$
(3.12)

As in Appendix B, we consider the \((2,2)\)-double-form

$$\begin{aligned} \varvec{\Xi }_K=\sum _{\alpha <\beta ,\;i<j} \varvec{\Xi }_{\alpha \beta |ij} {\varvec{v}}^\alpha \wedge {\varvec{v}}^\beta \otimes {\varvec{v}}^i\wedge {\varvec{v}}^j\in \Lambda ^{2,2}{\varvec{V}}^*, \end{aligned}$$

where

$$\begin{aligned} \varvec{\Xi }_{\alpha \beta |ij}:=\bigl ( K_{\alpha i|\beta j}- K_{\alpha j|\beta i}\,\bigr ),\quad \forall 1\le \alpha ,\beta \le r,\quad 1\le i,j\in I_0. \end{aligned}$$

Then

$$\begin{aligned} {\varvec{E}}\Bigl (\,\Delta _{I_0}\bigl (\,d{\varvec{u}}(0)\,\bigr )\,\Bigr )\mathop {=}\limits ^{(5.10)} \frac{1}{h!} {{\mathrm{\mathrm{tr}}}}\varvec{\Xi }_K^{\wedge h}. \end{aligned}$$
(3.13)

Now observe that (2.3) implies that

$$\begin{aligned} \varvec{\Xi }_{\alpha \beta |ij}=F_{\alpha \beta |ij}(0)=\forall 1\le \alpha ,\beta \le r,\quad 1\le i,j\in I_0. \end{aligned}$$

We deduce that

$$\begin{aligned} \varvec{\Xi }_K= \Omega _{-F^{I_0}(0)} \mathop {:=}\limits ^{(5.2)} \sum _{\begin{array}{c} \alpha <\beta ,\; i<j,\\ i,j\in I_0 \end{array}} F_{\alpha \beta |ij} du_\alpha \wedge du_\beta \otimes dx^i\wedge dx^j. \end{aligned}$$

Using (5.6) and (5.9) we deduce

$$\begin{aligned} {{\mathrm{\mathbf {Pf}}}}(-F^{I_0})_{x=0}= & {} {{\mathrm{\mathbf {pf}}}}(-F)_{I_0}(0) \,dx^{I_0}= \frac{1}{h!}\Bigl (\, {{\mathrm{\mathrm{tr}}}}\Omega _{-F^{I_0}(0)}^{\wedge h}\,\Bigr ) dx^{I_0}\nonumber \\= & {} \frac{1}{h!} \Bigl (\, {{\mathrm{\mathrm{tr}}}}\varvec{\Xi }_K^{\wedge h}\,\Bigr ) dx^{I_0}\mathop {=}\limits ^{(3.13)}{\varvec{E}}\Bigl (\,\Delta _{I_0}\bigl (\,d{\varvec{u}}(0)\,\bigr )\,\Bigr )dx^{I_0}. \end{aligned}$$
(3.14)

This proves (3.10) and thus completes the proof of Theorem 3.2.\(\square \)

Remark 3.8

(a) When the rank of \(E\) is odd, the topological Euler class with real coefficients is trivial, [16], Thm. 8.3.17]. In this case, if \({\varvec{u}}\) is a section of \(E\) transversal to the zero section, then we have the equality of currents \([Z_{-{\varvec{u}}}]=-[Z_{{\varvec{u}}}]\). If \({\varvec{u}}\) is a random section of a smooth, nondegenerate Gaussian ensemble, then the above equality implies \({\varvec{E}}([Z_{{\varvec{u}}}])=0\).

(b) Theorem 3.2 deals with centered Gaussian ensembles of smooth sections of \(E\). However its proof can be easily modified to produce results for noncentered ensembles as well.

Suppose that \({\varvec{u}}\) is a centered nondegenerate random Gaussian smooth section of \(E\) with associated induced metric \(h\) and connection \(\nabla \) as in Theorem 3.2. Fix a smooth section \({\varvec{u}}_0\) of \(E\) and form the noncentered Gaussian random section \({\varvec{v}}={\varvec{u}}_0+{\varvec{u}}\). Then \({\varvec{v}}\) is a.s. transversal to the zero section and we obtain a random current \([Z_{\varvec{v}}]\).

Fix \({\varvec{p}}\in M\) and define the spaces of mixed double-forms

$$\begin{aligned} \Lambda ^{j,k}(T^*_{\varvec{p}}M, E_{\varvec{p}}):=\Lambda ^j T^*_{\varvec{p}}M\otimes \Lambda ^k E_{\varvec{p}}. \end{aligned}$$

As in Appendix B, we have a natural associative multiplication

$$\begin{aligned} \wedge : \Lambda ^{j,k}(T^*_{\varvec{p}}M, E_{\varvec{p}})\otimes \Lambda ^{j',k'}(T^*_{\varvec{p}}M, E_{\varvec{p}})\rightarrow \Lambda ^{j+j',j+k'}(T^*_{\varvec{p}}M, E_{\varvec{p}}). \end{aligned}$$

We have the mixed double-forms

$$\begin{aligned} \nabla {\varvec{u}}_0({\varvec{p}})\in C^\infty \bigl (\,\Lambda ^{1,1}(T^* M, E)\,\Bigr ),\quad F(\nabla )\in C^\infty \bigl (\,\Lambda ^{2,2} (T^*M, E)\,\bigl ). \end{aligned}$$

Fix a point \({\varvec{p}}\in M\). Observe that the metric and orientation on \(E_p\) canonically determine a unit vector \(\omega _{E_{\varvec{p}}}\) of the top exterior power \(\Lambda ^r E^*_{\varvec{p}}\), \(r=2h\). The canonical map \(\beta :\Lambda ^r E_{\varvec{p}}\otimes \Lambda ^r E^*_{\varvec{p}}\rightarrow {\mathbb R}\) is an isomorphism, and we obtain natural contractions

Now define

Note that \({{\mathrm{\mathbf {Pf}}}}(-F(\nabla ))={{\mathrm{\mathbf {Pf}}}}(-F(\nabla ), {\varvec{u}}_0)_{{\varvec{u}}_0=0}\). The proof of Theorem 3.2 shows that

$$\begin{aligned} {\varvec{E}}\bigl (\, \langle \eta , [Z_{\varvec{v}}]\rangle \,\bigr )= \frac{1}{(2\pi )^h}\int _M {{\mathrm{\mathbf {Pf}}}}(-F(\nabla ),{\varvec{u}}_0)\wedge \eta ,\quad \forall \eta \in \Omega ^{m-r}(M). \end{aligned}$$
(3.15)

Indeed, the only modification in the proof appears when we consider the random matrix (3.11),

$$\begin{aligned} S=\Bigl (\,\frac{\partial v^\alpha }{\partial x^i}(0)\,\Bigr )_{\begin{array}{c} 1\le \alpha \le r\\ i\in I_0 \end{array}},\quad I_0=\{1,\cdots , r\}. \end{aligned}$$

In this case \(S\) is no longer a centered Gaussian random matrix. Its expectation is

$$\begin{aligned} {\varvec{E}}( S)=\nabla _{I_0} {\varvec{u}}_0(0):=\sum _{i\in I_0} dx^i\otimes \nabla _{x^i}{\varvec{u}}_0(0), \end{aligned}$$

while its covariances are still given by (3.12). The equality (3.15) now follows by using the same argument as in the last part of the proof of Theorem 3.2, with one notable difference: in the equality (3.13) we must invoke the (5.11) with \(\mu =\nabla _{I_0}{\varvec{u}}_0(0)\).