Abstract
We prove that a Gaussian ensemble of smooth random sections of a real vector bundle \(E\) over compact manifold \(M\) canonically defines a metric on \(E\) together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet theorem stating that if the bundle \(E\) and the manifold \(M\) are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble.
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
Suppose that \(F\) is a smooth vector bundle over \(X\). We have two natural projections
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]
Recall that a Borel probability measure \(\mu \) on \({\mathbb R}\) is called (centered) Gaussian if has the form
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
-
(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 \(C^{-\infty }(E)\).
-
(ii)
Every Borel probability measure on \(C^{-\infty }(E)\) is Radon.
-
(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
given by
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
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
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
given by
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
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
is Gaussian. Moreover
A section \(C\in C^\infty (E\boxtimes E)\) is called symmetric if
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
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
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
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

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.,

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.,
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
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
(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
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.
-
(i)
We will use the Latin letters \(i,j,k\) to denote indices in the range \(1,\cdots , m=\dim M\).
-
(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
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
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
Then for any \(i=1,\cdots , m\) the operator
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
We deduce that
\(\square \)
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
We obtain a \(1\)-form with matrix coefficients \(\Gamma (\underline{{\varvec{e}}}):=\sum _i \Gamma _i(\underline{{\varvec{e}}}) dy^i\). The operator
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
Then
We denote by \(d_x\) the differential with respect to the \(x\) variable. We deduce
Thus
This shows that for any local orthonormal frames \(\underline{\varvec{e}}\), \(\underline{\varvec{f}}\) of \(E|_\mathcal {O}\) we have
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
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
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
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 \({\varvec{p}}_0\),
Denote by \(F\) the curvature of \(\nabla ^C\),
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
where \(u_\alpha (x)\) is the random function
Proof
The random section \({\varvec{u}}\) has the local description
Then \(T(x,y)\) is a linear map \(E_y\rightarrow E_x\) given by the \(r\times r\) matrix
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
More precisely, we have
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
The coefficients \(F_{ij}(x)\) are \(r\times r\) matrices with entries \(F_{\alpha \beta |ij}(x)\), \(1\le \alpha ,\beta \le r\). Moreover
\(\square \)
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
is a Gaussian random vector. The section \({\varvec{u}}\) has the local description
Then
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
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
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
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
we deduce that \(Z_{\varvec{u}}\) is equipped with a natural orientation uniquely determined by the equalities
Thus, the zero set \(Z_{\varvec{u}}\) with this induced orientation defines an integration current \([Z_{\varvec{u}}]\in \Omega _{m-r}(M)\)
To a metric \((-,-)\) on \(E\) and a connection \(\nabla \) compatible with the metric we can associate a closed form
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
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
If we denote by \(\mathcal {S}_r\) the group of permutations of \(\{1,\cdots , r=2h\}\), then
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
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.,
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
In other words, we have to prove the equality
For any subset
we write \(dx^I:= dx^{i_1}\wedge \cdots \wedge dx^{i_k}\). We set
We can rewrite (3.3) in the more compact form
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
Each of the \(2\)-forms \(F_{\alpha \beta }\) admits a unique decomposition
For each subset \(I\subset \{1,\cdots m\}\) we write
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
The equality (3.4) is then equivalent to the equality
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 \(\mathcal {O}\)
where \(\rho (x)\) is a certain smooth function on \(\mathcal {O}\).
Step 2. Use the Gaussian computations in Appendix B to show that
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
where again \({\mathbb R}^r\) is equipped with the canonical Euclidean metric and orientation given by the volume form
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
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
We define the Jacobian at \(x\in \mathcal {O}\) of a smooth map \(F:\mathcal {O}\rightarrow E_0\) to be the scalar
where \(dF(x):{\mathbb R}^m\rightarrow E_0\) is the differential \(dF(x)\) of \(F\) at \(x\). We set
so that have a Gaussian random map
This random map is a.s. smooth. The random map
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
is integrable and
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.
-
(i)
The sequence \(g_n\) converges a.e. to \(g\).
-
(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
The random variable
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
A similar argument shows that
\(\square \)
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
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
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
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
Lemma 3.6 shows that if \(0\) is a regular value of \({\varvec{u}}\), then
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
This proves that
Now define
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
We have thus proved the equality
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
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.,
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
and thus we have to prove that
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
Its statistics are determined by the covariances
As in Appendix B, we consider the \((2,2)\)-double-form
where
Then
Now observe that (2.3) implies that
We deduce that
Using (5.6) and (5.9) we deduce
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
As in Appendix B, we have a natural associative multiplication
We have the mixed double-forms
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
Indeed, the only modification in the proof appears when we consider the random matrix (3.11),
In this case \(S\) is no longer a centered Gaussian random matrix. Its expectation is
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 \(\mu =\nabla _{I_0}{\varvec{u}}_0(0)\).
Notes
In the sequel, for simplicity, we will drop the attribute centered when referring to the various Gaussian measures since we will be working exclusively with such objects.
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Acknowledgments
I want to thank the anonymous referee for the very constructive and informative comments, suggestions and questions that helped improve the quality of this paper.
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Appendices
Appendix A: Proofs of various technical results
Proof of Proposition 1.4
Fix a metric \(g\) on \(M\), a metric and a compatible connection on \(E\). For each nonnegative integer \(k\) we can define the Sobolev spaces \(\mathcal {H}_k\) consisting of \(L^2\)-sections of \(E\) whose generalized derivatives up to order \(k\) are \(L^2\)-sections. We have a decreasing sequence of Hilbert spaces \(\mathcal {H}_0\supset \mathcal {H}_1\supset \cdots \) whose intersection is \(C^\infty (E)\). For \(k\ge 0\) we denote by \(\mathcal {H}_{-k}\) the topological dual of \(\mathcal {H}_k\) so that we have a decreasing family of Hilbert spaces \(\cdots \subset \mathcal {H}_1\subset \mathcal {H}_0\subset \mathcal {H}_{-1}\subset \cdots \!.\)
The results in [7] show that each of the subsets \(\mathcal {H}_k\subset C^{-\infty }(E)\), \(k\in {\mathbb Z}\), is a Borel subset. Using Minlos’s theorem [14], Sec.4, Thm.2] we deduce that if the covariance kernel \(C_{\varvec{\Gamma }}\) is smooth, then \({\varvec{\Gamma }}(\mathcal {H}_k)=1\), \(\forall k\in {\mathbb Z}\).\(\square \)
Proof of Proposition 1.5
Fix a Riemann metric \(g\) on \(M\). For each \(i=1,\cdots , n\) choose a sequence \((\delta _{\nu ,i})_{\nu \ge 0}\) of smooth functions on \(M\) supported in a coordinate neighborhood of \({\varvec{x}}_i\) such that
Fix trivializations of \(E\) near each \({\varvec{x}}_i\). Let \(t_1,\cdots , t_n\). Now define
and form the random variable \(C^{-\infty } (E)\ni {\varphi }\mapsto Y_\nu =Y_\nu ({\varphi })=L_{\Phi _\nu }({\varphi })\). This is a Gaussian random variable with variance
Now observe that
We deduce that \(Y_\nu \) converges in law to \(\sum _{i=1}^nt_iX_i\). In particular, this random variable is Gaussian and its variance is
This completes the proof of Proposition 1.5.\(\square \)
Proof of Proposition 1.12
Let us observe that when \(m=\dim M=\mathrm{rank}\,E\), then [2], Prop. 6.5] shows that any nondegenerate Gaussian ensemble of smooth sections of \(E\) is transversal. We will reduce the general case to this special situation.
The result is certainly local so it suffices to consider the case of nondegenerate Gaussian random maps
where \(\Omega =(\Omega , \mathcal {A}, \varvec{P})\) is a probability space, \(B\) is the unit open ball in \({\mathbb R}^m\) and \(r<m\). Throughout we assume that \(F\) is a.s. \(C^2\).
Fix a finite dimensional space \({\varvec{V}}\) of smooth functions \(B\rightarrow {\mathbb R}^{m-r}\) satisfying the ampleness condition
Equip \({\varvec{V}}\) with a nondegenerate Gaussian measure \({\varvec{\Gamma }}\). Form a new probability space
where \(\mathcal {B}\) denotes the \({\sigma }\)-algebra of Borel subsets of \({\varvec{V}}\). Denote by \(\Pi _\Omega \) the natural projection \(\widehat{\Omega }\rightarrow \Omega \). We consider a new Gaussian random map
Clearly the random map \(\widehat{F}\) is a.s. \(C^2\) and nondegenerate. We denote by \(\widehat{\Omega }_*\) the set of \(\hat{\omega }\in \widehat{\Omega }\) such that the map \(B\ni {\varvec{x}}\mapsto \hat{F}(\hat{\omega },{\varvec{x}})\in {\mathbb R}^m\) is \(C^2\) and, for any \({\varvec{x}}\in B\), its differential
is bijective. The random field \(\hat{F}\) satisfies the assumptions in [2], Prop. 6.5] and thus \(\widehat{\varvec{P}}(\widehat{\Omega }_*)=1\).
If we denote by \(\pi _r\) the natural projection \({\mathbb R}^{m-r}\oplus {\mathbb R}^r\rightarrow {\mathbb R}^r\) we observe that
Hence, if \(\omega \in \Pi _\Omega (\widehat{\Omega })\), the differential \(D_{\varvec{x}}F(\omega , -): T_{\varvec{x}}B\rightarrow {\mathbb R}^r\) is onto for any \({\varvec{x}}\in B\). Clearly \(\varvec{P}\bigl (\, \Pi _\Omega (\widehat{\Omega })\,\bigr ) =1\). This proves that the random map \(F\) is transversal. \(\square \)
Proof of Lemma 3.6
We follow a strategy similar to the one used in the proof of [17], Cor. 2.11]. Fix a point \(p_0\in Z_{{\varvec{u}}}\). Now choose local coordinates \((t^1,\cdots , t^m)\) on \(\mathcal {O}\) near \(p_0\) and local coordinates \(y^1,\cdots , y^r\) on \(E_0\) near \(0\in E_0\) with the following properties.
-
In the \((t,y)\)-coordinates the map \({\varvec{u}}\) is given by the linear projection
$$\begin{aligned} y^j=t^j, \quad j=1,\cdots , r. \end{aligned}$$ -
The orientation of \(E_0\) is given by \(dy=dy^1\wedge \cdots \wedge dy^r\).
We set
The coordinates \(t^{J_0}\) can be used as local coordinates on \(Z_{\varvec{u}}\) near \(p_0\) and we assume that \(dt^{J_0}\) defines the induced orientation of \(Z_{{\varvec{u}}}\). We can then write
where \(\rho _\mathcal {O}\), \(\rho _E\) and \(\rho _{\varvec{u}}\) are positive smooth functions on their respective domains. In the \(t\)-coordinates we have
where \(\lambda \) is the determinant of the \((m-r)\times (m-r)\) matrix \( \frac{\partial x^{J_0}}{\partial t^{J_0}}\) with entries \(\frac{\partial x^i}{\partial t^j}\), \(i,j\in J_0\). Thus
We have
On the other hand,
Using this in (4.2) we deduce
Now observe that, along \(Z_{\varvec{u}}\), we have
On the other hand, [17], Lemma 1.2] shows that \(\frac{\rho _E\rho _{\varvec{u}}}{J_{\varvec{u}}\rho _\mathcal {O}}=1\) which proves that
\(\square \)
Appendix B: Pfaffians and Gaussian computations
We collect here a few facts about Pfaffians needed in the main body of the paper.
Fix a positive even integer \(r=2h>0\). Given a commutative \({\mathbb R}\)-algebra \(\mathcal {A}\) we denote by \({{\mathrm{Skew}}}_r(\mathcal {A})\) the space of skew-symmetric \(r\times r\)-matrices with entries in \(\mathcal {A}\). The Pfaffian of a matrix \(F\in {{\mathrm{Skew}}}_r(\mathcal {A})\) is a certain universal homogeneous polynomial of degree \(h=r/2\) in the entries of \(F\). More precisely, if we denote by \(\mathcal {S}_r\) the group of permutations of \(\{1,\cdots , r=2h\}\), then
where \({\epsilon }({\sigma })\) denotes the signature of the permutation \({\sigma }\in \mathcal {S}_r\). The Pfaffian can be given an equivalent alternative description.
Fix an oriented real, \(r\)-dimensional Euclidean space \(E\) and an oriented orthonormal basis \(e_1,\cdots , e_r\) of \(E\). Denote by \(e^1,\cdots , e^r\) the dual basis of \(E^*\) and consider the \(\mathcal {A}\)-valued \(2\)-form
then the Pfaffian of \(F\) is uniquely determined by the equality, [16], Sec. 2.2.4],
We are interested only in a certain special case when
where \({\varvec{V}}\) is a real Euclidean space of dimension \(m\ge r\) and
In this case \({{\mathrm{\mathbf {Pf}}}}(F)\in \Lambda ^r {\varvec{V}}^*\) and has the following alternative description.
Fix an orthonormal basis \(\{{\varvec{v}}_1,\cdots ,{\varvec{v}}_m)\) of \({\varvec{V}}\). For \(1\le \alpha _1,\alpha _2\le r\) and \(1\le j_1,j_2\le m\) we set
Denote by \(\mathcal {S}_r'\) the subset of \(\mathcal {S}_r\) consisting of permutations \(({\sigma }_1,\cdots ,{\sigma }_{2h})\) such that
Then
For every subset \(I=\{i_1<\cdots <i_r\} \subset \{1,\cdots , m\}\) we write
where \(\{{\varvec{v}}^1,\cdots ,{\varvec{v}}^m\}\) is the orthonormal basis of \({\varvec{V}}^*\) dual to \(\{{\varvec{v}}_1,\cdots , {\varvec{v}}_m\}\).
For an ordered multiindex \(I\) we denote by \({\varvec{V}}_I\) the subspace spanned by \({\varvec{v}}_i\), \(i\in I\), and by \(F^I_{\alpha \beta }\) the restriction of \(F_{\alpha \beta }\) to \({\varvec{V}}_I\), i.e.,
We denote by \(F^I\) the \(r\times r\) skew-symmetric matrix with entries \((F^I_{\alpha \beta })_{1\le \alpha ,\beta \le r}\). Note that for any subset \(I\subset \{1,\cdots , m\}\) of cardinality \(r\) we have
This shows that the computation of the Pfaffians reduces to the case when \(\dim {\varvec{V}}= r\). This is what we will assume in the remainder of this section. We fix an orthonormal basis \({\varvec{v}}_1,\cdots ,{\varvec{v}}_r\) of \({\varvec{V}}\) and we denote by \({\varvec{v}}^1,\cdots ,{\varvec{v}}^r\) the dual basis of \({\varvec{V}}^*\).
To proceed further we need to introduce some more terminology. A double-form on the above Euclidean space \({\varvec{V}}\) is, by definition, an element of the vector space
We have an associative product \(\wedge : \Lambda ^{p,q}{\varvec{V}}^*\times \Lambda ^{p',q'}{\varvec{V}}^*\rightarrow \Lambda ^{p+p',q+q'}{\varvec{V}}^*\) given by
for any \(\omega \in \Lambda ^p{\varvec{V}}^*\), \(\eta \in \Lambda ^q{\varvec{V}}^*\), \(\omega '\in \Lambda ^{p'}{\varvec{V}}^*\), \(\eta '\in \Lambda ^{q'}{\varvec{V}}^*\). Observe that the metric on \({\varvec{V}}\) produces an isomorphism
and thus we have a well defined trace
Observe that an endomorphism \(T\) of \({\varvec{V}}\) can be identified with the \((1,1)\)-double-form
We then have the equality
Let us specialize (5.2) to the case when \(E={\varvec{V}}\) and \( e^\alpha ={\varvec{v}}^\alpha \). In particular, this implies that \({\varvec{V}}\) is oriented by the volume form
If we write
then we observe that \(\Omega _F\in \Lambda ^{2,2}{\varvec{V}}^{*,*}\), and that the equality (5.3) can be rewritten in the more compact form
As explained in [1], §12.3], the formalism of double-forms and Pfaffians makes its appearance in certain Gaussian computation. Suppose that \(S\) is a random Gaussian endomorphism of \({\varvec{V}}\) with entries
centered Gaussian random variables with covariances
We regard \(S\) as \((1,1)\)-double-form
and we get a random \((r,r)\)-double-form \(S^{\wedge r}\in \Lambda ^{r,r} {\varvec{V}}^*\). Its expectation can be given a very compact description. Define the \((2,2)\)-double-form
where \(\varvec{\Xi }_{\alpha \beta |ij}:=\bigl ( K_{\alpha i|\beta j}- K_{\alpha j|\beta i}\,\bigr )\), \(\forall \alpha ,\beta ,i,j\). Using [1], Lemma 12.3.1], the case \(\mu =0\), we deduce
More generally, if \(\mu \in \Lambda ^{1,1} V^*\) is a fixed (deterministic) \((1,1)\)-double form, then [1], Lemma 12.3.1] shows that
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Nicolaescu, L.I. A stochastic Gauss–Bonnet–Chern formula. Probab. Theory Relat. Fields 165, 235–265 (2016). https://doi.org/10.1007/s00440-015-0630-z
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DOI: https://doi.org/10.1007/s00440-015-0630-z
Keywords
- Connections
- Curvature
- Euler form
- Gauss–Bonnet–Chern theorem
- Currents
- Random sections
- Gaussian measures
- Kac-Rice formula
Mathematics Subject Classification
- Primary 35P20
- 53C65
- 58J35
- 58J40
- 58J50
- 60D05