1 Introduction and statement of results

Let M be a smooth (by smooth we mean of class \(C^{\infty }\)) connected manifold. Suppose that H is a smooth bracket generating distribution on M of constant rank and g is a smooth Riemannian metric on H. The pair (H, g) is called a sub-Riemannian metric or a sub-Riemannian structure on M. The triple (M, H, g) is referred to as a sub-Riemannian manifold. Sub-Riemannian manifolds appear in many mathematical as well as physical problems and have been studied by many authors—see for instance [1,2,3,4,5, 8, 12, 14] and the reference sections therein. Various problems in sub-Riemannian geometry like for instance the behavior of sub-Riemannian geodesics and their minimizing properties, conjugate and cut loci, sub-Riemannian spheres, isometries and conformal mappings, nilpotent approximations, differential properties of the sub-Riemannian distance etc. have been investigated in detail. In this paper we deal with holonomy determined by a class of connections introduced in [13] for contact sub-Riemannian manifolds, and prove a theorem that can be considered as a version of de Rham decomposition theorem for Riemannian manifolds. Different approaches to sub-Riemannian holonomy and some other problems involving it are treated, e.g., in [7, 9].

By a contact sub-Riemannian manifolds we mean a sub-Riemannian manifold (M, H, g), where \(\dim M=2n+1\), and H is a contact distribution on M. Given a contact connected sub-Riemannian manifold (M, H, g) it is natural to consider the bundle of orthonormal horizontal frames \(O_{H,g}(M)\) associated with it:

$$\begin{aligned} O_{H,g}(M) = \{(q;v_1,\dots ,v_{2n}):\;v_1,\dots ,v_{2n}\;\text {is an orthonormal basis of }H(q),\,q\in M\}. \end{aligned}$$

This is a principle bundle with structure group O(2n). Moreover we will assume that H and TM are oriented, so the structure group can be reduced to SO(2n). Let \(\xi \) be the Reeb vector field which is well defined in such a situation. We will assume that \(\xi \) is an infinitesimal isometry. Now it can be proved [13] that there exists a unique connection \(\Gamma \) on \(O_{H,g}(M)\) which is torsion-free (the definition of the torsion in our case is presented below). In the usual way \(\Gamma \) defines the covariant differentiation

$$\begin{aligned} \nabla : \mathrm {Sec}(TM)\times \mathrm {Sec}(H)\longrightarrow \mathrm {Sec}(H), \end{aligned}$$

where we use the following notation: if \(E\longrightarrow M\) is a vector bundle then by \(\mathrm {Sec}(E)\) we denote the \(C^{\infty }(M)\)-module of sections of E. Having a connection on the bundle \(O_{H,g}(M)\) we can consider its holonomy group \(\Psi (q)\) at a point \(q\in M\). Since M is connected the groups \(\Psi (q_1)\) and \(\Psi (q_2)\) are isomorphic for any two points \(q_1,q_2 \in M\). The holonomy group \(\Psi (q)\) naturally acts on H(q) (for H is an associated vector bundle to \(O_{H,g}(M)\) with typical fiber \(\mathbb {R}^{2n}\)). Suppose that the action of \(\Psi (q)\) on H(q) is reducible. Then H(q) decomposes into \(\Psi (q)\)-irreducible factors

$$\begin{aligned} H(q) = H_1(q) \oplus \cdots \oplus H_m(q) \end{aligned}$$
(1.1)

which are mutually orthogonal with respect to g. By use of parallel translations we extend \(H_i(q)\) to distributions \(H_i\) on M resulting in a global decomposition

$$\begin{aligned} H = H_1 \oplus \cdots \oplus H_m. \end{aligned}$$
(1.2)

Next let us consider the set \(M/\xi \) of orbits of \(\xi \). It is locally a smooth manifold of dimension 2n.

If we fix an arbitrary point \(q_0\in M\) and a neighborhood U of \(q_0\) such that \(U/\xi \) is a connected smooth manifold, then we can canonically identify \(U/\xi \) with a regular 2n-dimensional submanifold B of M with \(q_0\in B\) (the details can be found below). Then the sub-Riemannian metric \((H_{|U},g_{|U})\) induces a natural Riemannian metric \(g_B\) on B, and the connection \(\nabla \) induces a connection \(\nabla ^B\) on B which turns out to be the Levi-Civita connection with respect to \(g_B\). Moreover, if we denote by \(p:U\longrightarrow B\) the projection in the direction of \(\xi \), then

$$\begin{aligned} d_qp_{|H(q)}:H(q)\longrightarrow T_{\pi (q)}B \end{aligned}$$

is a linear isometry. Using this projection, the decomposition (1.2) induces a decomposition

$$\begin{aligned} TB=\Delta _1\oplus \cdots \oplus \Delta _m \end{aligned}$$
(1.3)

of TB into the Whitney sum of mutually orthogonal distributions. It is proved that \(\Delta _i\) are integrable and parallel with respect to \(\nabla ^B\), so in turn (1.3) induces a decomposition of B. The main theorem may be stated as follows.

Theorem 1.1

Suppose that (M, H, g) is a contact oriented sub-Riemannian manifold such that the Reeb vector field \(\xi \) is an infinitesimal isometry. Denote by \(\Gamma \) the unique torsion-free connection on \(O_{H,g}(M)\) and suppose that there exists a point \(q\in M\) such that the holonomy group \(\Psi (q)\) of \(\Gamma \) acts reducibly on H(q) inducing the decomposition (1.1). Then every point in M has a neighborhood U such that the manifold \(U/\xi \) is isometric to the product \((B_1,g_1)\times \cdots \times (B_m,g_m)\) of Riemannian manifolds, where \(B_i\) is of dimension \(\mathrm {rank}\,H_i\), \(i=1,\dots ,m\). More precisely, each \(B_i\) may be identified with a maximal integrable manifold for the distribution \(\Delta _i\), \(i=1,\dots ,m\).

In particular, suppose that \((M_1,H_1,g_1)\), \((M_2,H_2,g_2)\) are two sub-Riemannian manifolds satisfying the above assumptions. Let \(f:(M_1,H_1,g_1)\longrightarrow (M_2,H_2,g_2)\) be an isometry and let \(\xi \) be the Reeb vector field on \((M_1,H_1,g_1)\). Then for every sufficiently small open set \(U\subset M_1\), which is convex with respect to \(\xi \) (that is to say every trajectory of \(\xi \) intersects U in a connected set), the Riemannian manifolds \(U/\xi \) and \(f(U)/f_*\xi \) are isometric.

Using the results from [16] we can generalized the above theorem to contact sub-pseudo-Riemannian manifolds (e.g. sub-Lorentzian manifolds), i.e., when the metric g on H is not necessarily positive definite. We need only to assume that \(\Psi (q)\) acts nondegenerately and reducibly on H(q) which means that the decomposition (1.1) consists of subspaces \(H_i(q)\) nondegenerate with respect to g.

Theorem 1.2

Suppose that the assumptions of Theorem 1.1, where “sub-Riemannian manifold” is replaced with “sub-pseudo-Riemannian manifold” and “\(\Psi (q)\) acts reducibly on H(q)” is replaced with “\(\Psi (q)\) acts nondegenerately and reducibly on H(q)”, are satisfied. Then every point in M has a neighborhood U such that the manifold \(U/\xi \) is isometric to the product \((B_1,g_1)\times \cdots \times (B_m,g_m)\) of pseudo-Riemannian manifolds, where \(B_i\) is of dimension \({\mathrm {rank}}\,H_i\) and, as above, may be identified with a maximal integrable manifold of the distribution \(\Delta _i\), \(i=1,\dots ,m\).

Of course the remark made after the statement of Theorem 1.1 remains true with obvious modifications.

Finally, we state the last theorem that we prove in the present paper. If (M, H, g) is a given contact and oriented sub-Riemannian manifold, \(\dim M=2n+1\), denote by \(\alpha \) the normalized contact 1-from (see Sect. 2 for details). Then we can define the operator \(J:H\longrightarrow H\) by \(d\alpha (X,Y)=g(X,J(Y))\). The operator J is a vector bundle morphism covering the identity. Furthermore, J is nondegenerate and antisymmetric with respect to g, therefore it has purely imaginary eigenvalues \(\pm ib_j\), \(j=1,\dots ,n\) (see [11] for further properties of J in the indefinite case). If the \(b_j\)’s are pointwise mutually distinct, then each \(b_j:M\longrightarrow \mathbb {R}\) is a smooth function. We say that the structure (H, g) is strongly nondegenerate at a point \(q\in M\), if \(b_1(q)<\cdots <b_n(q)\) under suitable numeration (cf. [1] where the numbers \(b_1(q),\dots ,b_n(q)\) are called fundamental frequencies).

Theorem 1.3

Suppose that (M, H, g) is a contact oriented sub-Riemannian manifold. Suppose that (i) the Reeb vector field \(\xi \) is an infinitesimal isometry. Denote by \(\Gamma \) the unique torsion-free connection on \(O_{H,g}(M)\). Suppose next that (ii) the operator J is parallel with respect to \(\Gamma \). If J is strongly nondegenerate at a point q and U is a sufficiently small neighborhood of q, then \(U/\xi \) is isometric to a product of 2-dimensional Riemannian manifolds. Consequently, the conformal type of \(U/\xi \) depends neither on the choice of a metric g satisfying (i) and (ii) nor on a point q at which J is strongly nondegenerate.

As the reader can see, all above theorems concern a decomposition of the quotient manifold \(U/\xi \) into a product of (pseudo-)Riemannian manifolds, provided that U is a sufficiently small neighborhood of a fixed point. However, it would be very interesting to know if the set U itself admits a decomposition into a product of sub-(pseudo-)Riemannian manifolds. In the sub-Riemannian case, for instance, the set U is a so-called geodesic metric space. Then we know [10] that U admits a decomposition into a product of metric spaces. Such a decomposition is unique (up to a permutation of factors) and it would be of high importance to explicate if the factors in the mentioned decomposition carry some natural sub-Riemannian structure.

1.1 Content of the paper

In Sect. 2 we recall basic notions from contact sub-Riemannian geometry. In particular, we present the theory of connections on H introduced in the paper [13]. In Sect. 3 we prove the theorems.

Throughout the paper we adopt the following convention. A vector \(v\in TM\) which belongs to H will be called horizontal. On the other hand, if \(\Gamma \) is a distribution on \(O_{H,g}(M)\), e.g., a connection, then a vector \(V\in TO_{H,g}(M)\) belonging to \(\Gamma \) will be referred to as a \(\Gamma \)-horizontal vector.

2 Contact sub-Riemannian geometry

Suppose that (M, H, g) is a contact sub-Riemannian manifold, \(\dim M=2n+1\). We assume M to be connected. Let us suppose that M is oriented as a contact manifold which means that the vector bundles TM and H are oriented. This is equivalent to the existence of a globally defined contact form, i.e., a 1-form \(\alpha \) on M with the property that \(H=\ker \alpha \) (see [6, 11]). In such a situation, i.e., when there exists a globally defined contact form, we will say that the sub-Riemannian manifold (M, H, g) is oriented. Such a contact form is not unique, so we normalize it as follows: we suppose that

$$\begin{aligned} \underbrace{d\alpha \wedge \cdots \wedge d\alpha }_{n\; \mathrm {factors}}(X_1,\dots ,X_{2n}) = 1, \end{aligned}$$
(2.1)

where \(X_1,\dots ,X_{2n}\) is a fixed local positively oriented orthonormal frame for H. For n even we have two such forms \(\alpha \) defined up to a sign, so we choose either of them. A form satisfying (2.1) will be referred to as the normalized contact form.

If \(\alpha \) is the normalized contact form then we define the Reeb vector field \(\xi \) on (M, H, g) as the solution to the system of equations

$$\begin{aligned} d\alpha (\xi ,\cdot )=0,\; \alpha (\xi )=1. \end{aligned}$$

Such a field has the property that \([\xi ,X]\in \mathrm {Sec}(H)\) whenever \(X\in \mathrm {Sec}(H)\). In particular the (local) flow \(\varphi ^t\) of \(\xi \) preserves the distribution H. Moreover, \(\xi \) defines a canonical decomposition \(TM=H\oplus \mathrm {Span}\{\xi \}\). The projection defined by this decomposition will be denoted by

$$\begin{aligned} P:TM\longrightarrow H. \end{aligned}$$
(2.2)

2.1 Geodesics

Suppose that \(X_1,\dots ,X_{2n}\) be an orthonormal frame defined on an open set \(U\subset M\). Let \({\mathcal {H}}:T^*M_{|U}\longrightarrow \mathbb {R}\) be defined by

$$\begin{aligned} {\mathcal {H}}(q,\lambda ) = \frac{1}{2}\sum _{i=1}^{2n}\langle \lambda ,X_i(q)\rangle ^2. \end{aligned}$$
(2.3)

Clearly, the value of (2.3) does not depend on the choice of an orthonormal basis, so \({\mathcal {H}}\) is in fact defined on the whole cotangent bundle: \({\mathcal {H}}:T^*M\longrightarrow \mathbb {R}\). We call \({\mathcal {H}}\) the geodesic Hamiltonian. By a normal or Hamiltonian geodesic we mean any curve being a projection onto M of the trajectory of the Hamiltonian vector field \(\overrightarrow{{\mathcal {H}}}\). In other words, a curve \(\sigma :[a,b]\longrightarrow M\) is a Hamiltonian geodesic if there exists \(\lambda :[a,b]\longrightarrow T^*M\) such that

$$\begin{aligned} \lambda (t)\in T^*_{\sigma (t)}M\; \text { and } \;\; ({\dot{\sigma }}(t),{\dot{\lambda }}(t)) = \overrightarrow{{\mathcal {H}}}. \end{aligned}$$
(2.4)

It can be proved that in the contact case, every geodesic, i.e., a curve which locally minimizes the sub-Riemannian distance, is a Hamiltonian geodesic. However we will not use this fact. Let \(\sigma :[a,b]\longrightarrow M\) be a Hamiltonian geodesic and let \(\lambda (t)\) be its lift to \(T^*M\) as in (2.4). Suppose that S is a submanifold in M such that \(\sigma (a)\in S\). We say that \(\sigma \) satisfies the (Pontryagin) transversality condition with respect to S if \(\lambda (a)_{|T_{\sigma (a)}S}=0\).

For a point \(q\in M\), denote by \({\mathcal {D}}_q\) the set of all covectors \(\lambda \in T_q^*M\) such that the Hamiltonian geodesic with initial condition \((q,\lambda )\) exists on the interval [0, 1]. Then we define the exponential mapping with pole at q as follows:

$$\begin{aligned} \exp _q:{\mathcal {D}}_q\longrightarrow M, \;\;\exp _q(\lambda ) = \sigma (1), \end{aligned}$$

where \(\sigma (t)\) is the Hamiltonian geodesic with initial condition \((q,\lambda )\). One proves that \(\exp _q\) is smooth.

2.2 Isometries and infinitesimal isometries

Given two contact sub-Riemannian manifolds \((M_1,H_1,g_1)\), \((M_2,H_2,g_2)\), a diffeomorphism \(f:M_1\longrightarrow M_2\) is called an isometry if \(d_qf(H_1(q))\subset H_2(f(q))\) and \(d_qf:H_1(q)\longrightarrow H_2(f(q))\) is a linear isometry for every \(q\in M\). In other words \(g_2(d_qf(v),d_qf(w))=g_1(v,w)\) for all \(q\in M\) and \(v,w\in H_1(q)\). If the manifolds \((M_i,H_i,g_i)\), \(i=1,2\), are oriented and \(f:M_1\longrightarrow M_2\) is an isometry, then \(f^*\alpha _2=\pm \alpha _1\), as well as \(f_*\xi _1=\pm \xi _2\), where \(\alpha _i\) is the normalized contact form and \(\xi _i\) is the Reeb vector field on \(M_i\), \(i=1,2\). It can be also proved that isometries preserve Hamiltonian geodesics. More precisely, if f is an isometry and \(\sigma :[a,b]\longrightarrow M\) is a Hamiltonian geodesic satisfying the transversality condition with respect to a submanifold S, then \(f\circ \sigma \) is a Hamiltonian geodesic satisfying the transversality condition with respect to f(S).

A vector field Z on a sub-Riemannian manifold (M, H, g) is called an infinitesimal isometry if its (local) flow consists of isometries. It can be shown that Z is an infinitesimal isometry if and only if (i) \([Z,Y]\in \mathrm {Sec}(H)\) and (ii) \(Z(g(X,Y)) = g([Z,X],Y)+g(X,[Z,Y])\) for every \(X,Y\in \mathrm {Sec}(H)\).

2.3 Connection on the bundle of horizontal frames

In this subsection we present the construction of the connection which agrees with a given sub-Riemannian structure. Details are described in [13]. Note that [13, Proposition 7.1] is not true (one needs to impose stronger assumptions).

Let (M, H, g) be an oriented contact sub-Riemannian manifold. Consider the bundle of horizontal frames determined by it:

$$\begin{aligned} L_H(M) = \{(q;v_1,\dots ,v_{2n}):\; q\in M,\,H(q) = \mathrm {Span}\{v_1,\dots ,v_{2n}\}\}\text {;} \end{aligned}$$

by \(\pi :L_H(M)\longrightarrow M\) we denote its projection, i.e., \(\pi (q;v_1,\dots ,v_{2n})=q\). This is a principle bundle with the structure group GL(2n). Indeed, we have a natural right action: \((q;v_1,\dots ,v_{2n}).a = (q;a^i_1 v_i,\dots ,a^i_{2n} v_i)\), \(a\in GL(2n)\) (here and below we use the Einstein summation convention). Moreover, if \(X_1,\dots ,X_{2n}\) is a basis of sections of H defined on an open set \(U\subset M\) then the local trivialization \(\psi :\pi ^{-1}(U)\longrightarrow U\times GL(2n)\) of \(L_H(M)\) acts as follows. If \(l=(q;v_1,\dots ,v_{2n})\) then

$$\begin{aligned} \psi (l) = (q,a(l)), \end{aligned}$$

where \(a(l)\in GL(2n)\) is such that \(v_i = a^j_i(l)X_j(q)\). The metric g reduces \(L_H(M)\) to the bundle

$$\begin{aligned} O_{H,g}(M) = \{(q;v_1,\dots ,v_{2n})\in L_H(M):\; g(v_i,v_j)=\delta _{ij},i,j=1,\dots ,2n\} \end{aligned}$$

of orthonormal horizontal frames. This is a principle O(2n)-bundle. Every \(l=(q;v_1,\dots ,v_{2n})\in O_{H,g}(M)\) defines the linear isomorphism \(l:\mathbb {R}^{2n}\longrightarrow H(\pi (l))=H(q)\) which is given by

$$\begin{aligned} l(r) = r^iv_i. \end{aligned}$$
(2.5)

As usual, by a connection on \(O_{H,g}(M)\) we mean a distribution \(\Gamma \subset TO_{H,g}(M)\) such that \(TO_{H,g}(M) = \Gamma \oplus V\) and which is O(2n)-invariant, i.e., \(d_lR_a(\Gamma _l) = \Gamma _{l.a}\) for every \(a\in O(2n)\) and \(l\in O_{H,g}(M)\). Here V stands for the vertical distribution on \(O_{H,g}(M)\): \(V_l = \ker d_l\pi \), and \(R_a:O_{H,g}(M)\longrightarrow O_{H,g}(M)\) is the right action of O(2n). Note that if \(\Gamma \) is a connection on \(O_{H,g}(M)\) then we have a natural splitting

$$\begin{aligned} \Gamma = \Gamma ^H\oplus \Gamma ^{\xi }, \end{aligned}$$
(2.6)

where \(\Gamma ^H = (d\pi )^{-1}(H)\cap \Gamma \) and \(\Gamma ^{\xi } = (d\pi )^{-1}(\mathrm {Span}\{\xi \})\cap \Gamma \); as above \(\xi \) stands for the Reeb vector field.

Given a connection \(\Gamma \) on \(O_{H,g}(M)\) we want to define its torsion. First of all we need to specify the counter part of the canonical 1-form from the theory of linear frame bundles. We do it as follows. For every \(l\in O_{H,g}(M)\) we define

$$\begin{aligned} \theta (l) = l^{-1}\circ P\circ d_l\pi :T_lO_{H,g}(M)\longrightarrow \mathbb {R}^{2n}, \end{aligned}$$
(2.7)

where P is defined in (2.2). The object \(\theta \) is a 1-form on \(O_{H,g}(M)\) with values in \(\mathbb {R}^{2n}\) and will be called the canonical 1-form on \(O_{H,g}(M)\). Now by the torsion form of \(\Gamma \) we mean the 2-form \(\Theta \) which is given by

$$\begin{aligned} \Theta = d\theta \circ (\mathrm {pr},\mathrm {pr}), \end{aligned}$$
(2.8)

where \(\mathrm {pr}:TO_{H,g}(M)=\Gamma \oplus V\longrightarrow \Gamma \) stands for the projection. Due to the splitting (2.6), the torsion can be decomposed into the horizontal torsion and vertical torsion (see [13]). It can be proved [13] that there always exist connections on \(O_{H,g}(M)\) with vanishing horizontal torsion. The class of connections with vanishing horizontal torsion is determined by a canonical choice of \(\Gamma ^H\). To be more precise, various connections with vanishing horizontal torsion have the same component \(\Gamma ^H\), while the component \(\Gamma ^{\xi }\) may be different.

Suppose further that the Reeb field is an infinitesimal isometry. Under such assumptions one can prove [13] that there exist a unique connection on \(O_{H,g}(M)\) which is torsion-free. In other words, under the mentioned assumptions, there exists a unique torsion-free and metric connection associated with the structure (H, g). Such a connection induces the covariant derivation \(\nabla :\mathrm {Sec}(TM)\times \mathrm {Sec}(H)\longrightarrow \mathrm {Sec}(H)\). Being metric means that

$$\begin{aligned} Z(g(X,Y)) = g(\nabla _ZX,Y) + g(X,\nabla _ZY), \end{aligned}$$

moreover, the vanishing of the horizontal torsion means that

$$\begin{aligned} \nabla _XY-\nabla _YX = P([X,Y]), \end{aligned}$$
(2.9)

whereas the vanishing of the vertical torsion is expressed by

$$\begin{aligned} \nabla _{\xi }X=[\xi ,X], \end{aligned}$$
(2.10)

whenever \(Z\in \mathrm {Sec}(TM)\), \(X,Y\in \mathrm {Sec}(H)\)—cf. [13].

At the end of this section let us note that if \(\Gamma \) is the mentioned torsion-free connection on \(O_{H,g}(M)\), then the component \(\Gamma ^{\xi }\) in the splitting (2.6) is given by \(\Gamma ^{\xi }=\mathrm {Span}\{\xi ^*\}\), where the vector field \(\xi ^*\) (being the \(\Gamma \)-horizontal lift of \(\xi \)) is defined as follows. Take \(q\in M\) and let \(\varphi ^t:U\longrightarrow M\) be the (local) flow of \(\xi \), where U is a neighborhood of q. Then we can lift \(\varphi ^t\) to the mapping \(\Phi ^t:\pi ^{-1}(U)\longrightarrow O_{H,g}(M)\), \(\Phi ^t(q;v_1,\dots ,v_{2n}) = (\varphi ^t(q);d_q\varphi ^t(v_1),\dots ,d_q\varphi ^t(v_{2n}))\), and for \(l\in \pi ^{-1}(q)\) we set

$$\begin{aligned} \xi ^*(l) = \frac{d}{dt}\Big \vert _{t=0}\Phi ^t(l). \end{aligned}$$

2.4 Holonomy

Given a connection \(\Gamma \) on \(O_{H,g}(M)\), we can define parallel displacement along curves on M and the holonomy group in the standard manner (see [15]). Consider a piecewise smooth curve \(\gamma :[a,b]\longrightarrow M\). The curve \(\gamma \) induces the parallel displacement of fibers

$$\begin{aligned} \tau _{\gamma }:\pi ^{-1}(\gamma (a))\longrightarrow \pi ^{-1}(\gamma (b)) \end{aligned}$$

which is defined as follows. Take \(l\in \pi ^{-1}(\gamma (a))\) and let \(\gamma ^*:[a,b]\longrightarrow O_{H,g}(M)\) be the \(\Gamma \)-horizontal lift of \(\gamma \) initiating at l, i.e., \(\pi \circ \gamma ^*=\gamma \), \(\frac{d}{dt}\gamma ^*(t)\in \Gamma _{\gamma ^*(t)}\) whenever the derivative exists, and \(\gamma ^*(a) = l\). Then \(\tau _{\gamma }(l)=\gamma ^*(b)\). Moreover, if we are given two piecewise smooth curves \(\gamma _i:[a_i,b_i]\longrightarrow M\), \(i=1,2\), such that \(\gamma _1(b_1) = \gamma _2(a_2)\), we have \(\tau _{\gamma _2\cdot \gamma _1} = \tau _{\gamma _2}\circ \tau _{\gamma _1}\), where \(\gamma _2\cdot \gamma _1\) is the concatenation of \(\gamma _1\) and \(\gamma _2\). In particular, if C(q) denotes the set of all piecewise smooth loops at a point \(q\in M\), then

$$\begin{aligned} \Psi (q) = \{\tau _{\gamma }:\; \gamma \in C(q)\} \end{aligned}$$

is a Lie group which is called the holonomy group at q and is denoted by \(\Psi (q)\). Such a group can be realized as a subgroup \(\Psi (l)\), \(l\in \pi ^{-1}(q)\), of the structure group O(2n): if \(\gamma \in C(q)\) then l and \(\tau _{\gamma }(l)\) belong to the same fiber of \(\pi :O_{H,g}(M)\longrightarrow M\), hence \(\gamma \) determines a unique element \(a_{\gamma }\in O(2n)\) such that \(\tau _{\gamma }(l) = l.a_{\gamma }\). In this way \(\Psi (l)=\{a_{\gamma }:\;\gamma \in C(q)\}\) is a subgroup of O(2n). It is proved that if M is connected then holonomy groups at any two points are isomorphic.

The connection \(\Gamma \) induces also the parallel displacement in every vector bundle associated with \(O_{H,g}(M)\), so in particular in H. More precisely, if \(\gamma :[a,b]\longrightarrow M\) is a curve then we can define the parallel displacement or translation along \(\gamma \) (we use the same notation as above)

$$\begin{aligned} \tau _{\gamma }:H(\gamma (a))\longrightarrow H(\gamma (b)) \end{aligned}$$

as \(\tau (v) = \gamma ^*(b)(r)\), where \(\gamma ^*:[a,b]\longrightarrow O_{H,g}(M)\) is a \(\Gamma \)-horizontal lift of \(\gamma \), \(\pi (\gamma ^*(a))=\gamma (a)\), and \(r\in \mathbb {R}^{2n}\) is such that \(\gamma ^*(a)(r)=v\). Notice that for every \(\gamma \) the map \(\tau _{\gamma }\) is a linear isometry. In particular, the holonomy group \(\Psi (q)\) acts on H(q).

Suppose that \(\Psi (q)\) acts reducibly on H(q), and let

$$\begin{aligned} H(q) = H_1(q)\oplus \cdots \oplus H_m(q) \end{aligned}$$
(2.11)

be the decomposition of H(q) into \(\Psi (q)\)-irreducible and \(\Psi (q)\)-invariant mutually orthogonal subspaces. It is a standard observation that the decomposition (2.11) can be extended by the parallel displacement to the decomposition

$$\begin{aligned} H = H_1\oplus \cdots \oplus H_m \end{aligned}$$
(2.12)

of the distribution H. Indeed, if \(\gamma \) is a curve starting at q then \(\tau _{\gamma }(H_i(q))\) does not depend on \(\gamma \) but only on its endpoints. To end this subsection, we note that, by the definition of the covariant derivation induced by \(\Gamma \),

$$\begin{aligned} \nabla _Z(\mathrm {Sec}(H_i))\subset \mathrm {Sec}(H_i) \end{aligned}$$
(2.13)

for every \(Z\in \mathrm {Sec}(TM)\) and \(i=1,\dots ,m\).

3 Proof of Theorems 1.1 and 1.2

In this section we assume that (M, H, g) is a fixed contact oriented and connected sub-Riemannian manifold, \(\dim M=2n+1\). Suppose that the Reeb vector field \(\xi \) is an infinitesimal isometry and denote its (local) flow by \(\varphi ^t\). Moreover, let \(\Gamma \) be the unique torsion-free connection on \(O_{H,g}(M)\). Suppose that the holonomy group acts reducibly on H and the corresponding decomposition is

$$\begin{aligned} H = H_1 \oplus \cdots \oplus H_m. \end{aligned}$$

As above \(H_i\) are constant rank distributions which are pairwise orthogonal with respect to g, \(\mathrm {rank}\,H_i>0\), \(i=1,\dots m\). By \(\nabla \) we will denote the covariant derivation induced by \(\Gamma \).

3.1 Distributions \({\widetilde{H}}_i\)

Distributions \(H_i\) need not be integrable, however their extensions are. For every \(i=1,\dots ,m\) let us define

$$\begin{aligned} {\widetilde{H}}_i = H_i\oplus \mathrm {Span}\{\xi \}. \end{aligned}$$

Proposition 3.1

The distribution \({\widetilde{H}}_i\) is integrable, \(i = 1,\dots ,m\).

Proof

Indeed by (2.13) it follows that

$$\begin{aligned} \nabla _XY\in \mathrm {Sec}(H_i) \;\; \mathrm {and}\;\;\nabla _{\xi }X=[\xi ,X]\in \mathrm {Sec}(H_i) \end{aligned}$$

for every \(X,Y\in \mathrm {Sec}(H_i)\). Consequently,

$$\begin{aligned} P([X,Y]) = \nabla _XY - \nabla _YX \in \mathrm {Sec}(H_i) \end{aligned}$$

which in turn implies \([X,Y]\in \mathrm {Sec}({\widetilde{H}}_i)\). \(\square \)

In particular we see that the distributions \(H_i\), as well as \({\widetilde{H}}_i\), are invariant by the flow of \(\xi \).

3.2 The submanifold B: construction of the bundle over B

Fix a point \(q_0\in M\). Let \(q_0\in U\) where \(U\subset M\) is an open set that will be specified below. We construct a regular submanifold B of M, \(q_0\in B\), which can be canonically identified with \(U/\xi \).

We start by choosing a coordinate system around \(q_0\) which will be convenient for our purposes. Denote by \(\delta :(-\varepsilon , \varepsilon )\longrightarrow M\) the trajectory of the Reeb field \(\xi \) such that \(\delta (0)=q_0\). Select a local basis \(X_1,\dots ,X_{2n}\) of section of H defined near \(q_0\) and let \(g^t_i\) stand for the (local) flow of \(X_i\), \(i=1,\dots ,2n\). We can assume that each \(g^t_i\) is defined on a neighborhood of \(q_0\) and for \(|t|<\varepsilon \). By shrinking U we can suppose that the mapping

$$\begin{aligned} ({\tilde{x}}^1,\dots ,{\tilde{x}}^{2n},z) \longrightarrow g_1^{{\tilde{x}}^1}\circ \cdots \circ g_{2n}^{{\tilde{x}}^{2n}}\circ \delta (z) \end{aligned}$$

defines coordinates \(({\tilde{x}}^1,\dots ,{\tilde{x}}^{2n},z)\) on U such that \({\tilde{x}}^i(q_0)=z(q_0)=0\),

$$\begin{aligned} H_{|\delta } = \mathrm {Span}\Big \{\frac{\partial }{\partial {\tilde{x}}^1}, \dots , \frac{\partial }{\partial {\tilde{x}}^{2n}}\Big \}, \end{aligned}$$

and \(\xi _{|\delta } = \frac{\partial }{\partial z}\). Let \(({\tilde{x}}^1,\dots ,{\tilde{x}}^{2n},z,{\tilde{p}}_1,\dots ,{\tilde{p}}_{2n},r)\) be the Darboux coordinates on \(T^*M_{|U}\) and let us set

$$\begin{aligned} A = \{(0,\dots ,0,z,{\tilde{p}}_1,\dots ,{\tilde{p}}_{2n},0):\;|z|,|{\tilde{p}}_1|,\dots ,|{\tilde{p}}_{2n}| < \varepsilon \}\subset T^*M_{|U}. \end{aligned}$$

The set A can be regarded as the set of initial conditions for sub-Riemannian geodesics satisfying the Pontryagin transversality conditions with respect to \(\delta \) (cf. [1]). Now, the assignment

$$\begin{aligned} (x^1,\dots ,x^{2n},z) \longrightarrow \exp _{(0,\dots ,0,z)}(x^1,\dots ,x^{2n},0) \end{aligned}$$
(3.1)

defines the desired coordinates \((x^1,\dots ,x^{2n},z)\) around \(q_0\). We can suppose that they are defined on U (shrinking U again if necessary). Let us notice that

$$\begin{aligned} H_{|\delta } = \mathrm {Span}\Big \{\frac{\partial }{\partial x^1}, \dots , \frac{\partial }{\partial x^{2n}}\Big \}, \end{aligned}$$

\(\xi _{|\delta } = \frac{\partial }{\partial z}\), and straight lines \(t\longrightarrow (tv,z)\) in these coordinates, where \(v\in \mathbb {R}^{2n}\) and \(\sum _{i=1}^nv_i^2=1\), are sub-Riemannian geodesics parameterized by arc length, which satisfy the transversality conditions with respect to \(\delta \). We define the following family of hypersurfaces

$$\begin{aligned} S_w = \{q\in U:\; z(q) = w\} \end{aligned}$$

transverse to \(\delta \). We will identify U with an open subset of \(\mathbb {R}^{2n+1}\) with coordinates (x, z), so we will also write \(S_w = \{(x,z):\; z = w\}\).

Proposition 3.2

\(\varphi ^t(S_w) = S_{w+t}\).

Proof

\(S_w\) is the union of geodesics which in our coordinates have the form \(\sigma (s) = (sv,w)\), \(|v|=1\). As we said above these are are exactly the geodesics that start from \(\delta \) and satisfy the Pontryagin transversality conditions with respect to \(\delta \). Fix such a geodesic \(\sigma (s) = (sv,w)\). Since \(\varphi ^t\) is an isometry preserving \(\delta \), the curve \(s\longrightarrow \varphi ^t(\sigma (s))\) is again a geodesic that starts from \(\delta \) and satisfies the transversality condition with respect to \(\delta \). Thus it must be of the form \(s\longrightarrow (s{\tilde{v}},{\tilde{w}})\). Because \(\varphi ^t(\sigma (0)) = (0,w+t)\) in (x, z)-coordinates, \({\tilde{w}} = w+t\) which ends the proof. \(\square \)

Now, let us set \(B=S_0\). Remark that B (or more precisely, its germ at \(q_0\)) is defined canonically and does not depend on coordinates. We define the projection

$$\begin{aligned} p:U\longrightarrow B \end{aligned}$$

as the projection onto B in the direction of \(\xi \):

$$\begin{aligned} p(q) = \varphi ^{t(q)}(q), \end{aligned}$$

where t(q) is such that \(\varphi ^{t(q)}(q)\in B\). By construction, \(\xi \) is transverse to B, and t(q) depends smoothly on q by the implicit function theorem. Obviously \(t(q)=0\) for \(q\in B\) and, moreover,

$$\begin{aligned} p\circ \varphi ^t = p. \end{aligned}$$
(3.2)

We see that \(p:U\longrightarrow B\) is a fiber bundle with fibers being trajectories of \(\xi \).

3.3 Induced metric and connection on B

Our next aim is to endow B with a suitably induced Riemannian metric and a connection. Suppose that \(X\in \mathrm {Sec}(TB)\). First we construct the canonical ’lift’ of X to the field \({\widetilde{X}}\in \mathrm {Sec}(H)\) on U by formula

$$\begin{aligned} {\widetilde{X}}(\varphi ^t(q)) = d_q\varphi ^t(X(q) - \alpha _q(X)\xi (q)) \end{aligned}$$
(3.3)

for every \(q\in B\) and every t for which the above expression is defined. Recall that \(\alpha \) stands for the normalized conatct form.

Proposition 3.3

Suppose that \(X\in \mathrm {Sec}(TB)\) and let \({\widetilde{X}}\in \mathrm {Sec}(H)\) be the horizontal lift defined above. For every \(q\in U\)

$$\begin{aligned} d_q p({\widetilde{X}}(q)) = X(p(q)). \end{aligned}$$

Proof

Let \(q = \varphi ^t({\bar{q}})\), where \({\bar{q}}\in B\). Then using (3.3) and (3.2) we have

$$\begin{aligned} d_q p({\widetilde{X}}(q)) = d_{\varphi ^t({\bar{q}})} p\circ d_{{\bar{q}}}\varphi ^t(X({\bar{q}}) - \alpha _{{\bar{q}}}(X)\xi ({\bar{q}}))= d_{{\bar{q}}} p(X({\bar{q}}) - \alpha _{{\bar{q}}}(X)\xi ({\bar{q}})) \end{aligned}$$

and it suffices to notice that \(d_{{\bar{q}}}p(X({\bar{q}})) = X({\bar{q}})\) and \(d_{{\bar{q}}} p(\xi ) = 0\). The first equality follows from \(p_{|B}=\mathrm {id}\) and the other from the definition of p. \(\square \)

Now we define the announced Riemannian metric on B. For \(q\in B\) and \(X,Y\in \mathrm {Sec}(TB)\) we set

$$\begin{aligned} g_B(X(q),Y(q)) = g\big (X(q) - \alpha _q(X)\xi (q),Y(q) - \alpha _q(Y)\xi (q)\big ). \end{aligned}$$
(3.4)

The last equation can be rewritten as

$$\begin{aligned} g_B(X(q),Y(q)) = g_B(d_q p({\widetilde{X}}(q)),d_q p({\widetilde{Y}}(q))) = g({\widetilde{X}}(q),{\widetilde{X}}(q)). \end{aligned}$$
(3.5)

Note that if \(X_1,\dots ,X_{2n}\) is a basis of \(T_qB\), then \({\widetilde{X}}_1 = X_1-\alpha (X_1)\xi (q),\dots ,{\widetilde{X}}_{2n} = X_{2n}-\alpha (X_{2n})\xi (q)\) is a basis of \(H_q\) and \(d_q p({\widetilde{X}}_i) = X_i\) for every i. Remembering (3.5) we obtain the following statement.

Corollary 3.1

For every \(q\in B\)

$$\begin{aligned} d_q p_{|H_q}:H_q\longrightarrow T_qB \end{aligned}$$

is a linear isometry.

Notice that B carries a natural orientation determined by the orientation of H.

Remark 3.1

Let us remark that if we apply the same procedure to define the Riemannian metric on \(S_w\), \(w\ne 0\), then the resulting Riemannian manifold will be isometric to \((B,g_B)\).

We proceed to define a connection on B. Denote by O(B) the bundle of orthonormal frames of B. Let \(\pi _B:O(B)\longrightarrow B\) be the corresponding projection and \(V_B = \ker d\pi _B\) be the vertical distribution. By Corollary 3.1 we have the natural mapping \({\hat{p}}:O_{H,g}(U) \longrightarrow O(B)\), \({\hat{p}}(q;v_1,\dots ,v_{2n}) = (p(q);d_q p(v_1),\dots ,d_q p(v_{2n}))\). Of course the diagram

(3.6)

is commutative. Recall that we have the decomposition \(\Gamma = \Gamma ^H\oplus \Gamma ^{\xi }\), where \(\Gamma ^{\xi } = \mathrm {Span}\{\xi ^*\}\). Note that \({\hat{p}}\circ \Phi ^t = {\hat{p}}\) (where \(\Phi ^t\) is the local flow of \(\xi ^*\)). Indeed,

$$\begin{aligned}&{\hat{p}}\circ \Phi ^t(q;v_1,\dots ,v_{2n}) = (p(\varphi ^t(q));d_q(p\circ \varphi ^t)(v_1),\dots ,d_q(p\circ \varphi ^t)(v_{2n})) \\&\quad =(p(q);d_q p(v_1),\dots ,d_q p(v_{2n})). \end{aligned}$$

Proposition 3.4

The mapping \({\hat{p}}\) defined above is a surjective submersion.

Proof

Evidently \({\hat{p}}\) is onto B. Fix \(l=(q;v_1,\dots ,v_{2n})\in O_{H,g}(U)\), \(q\in B\), and take \(w\in T_{{\hat{p}}(l)}O(B)\). Then \(w={\bar{\sigma }}^{\centerdot }(0)\), where \({\bar{\sigma }}:[-\varepsilon ,\varepsilon ]\longrightarrow O(B)\) is a suitable smooth curve. Clearly, \({\bar{\sigma }}(t)=(\sigma (t);w_1(t),\dots ,w_{2n}(t))\), \({\bar{\sigma }}(0)={\hat{p}}(l)\), so in particular \(w_i(0)=d_qp(v_i)\). For the curve \(\sigma :[-\varepsilon ,\varepsilon ]\longrightarrow B\), \(\sigma (0)=q\), let us construct its lift to a horizontal curve \({\widetilde{\sigma }}:[-\varepsilon ,\varepsilon ]\longrightarrow U\), \({\widetilde{\sigma }}(0) = \sigma (0)\), as follows. Supposing that \(\varepsilon >0\) is sufficiently small and \(\sigma \) is contained in a coordinate chart V, extend the field \({\dot{\sigma }}(t)\) to a vector field \(Z\in \mathrm {Sec}(TB_{|V})\). Using (3.3), we obtain the field \({\widetilde{Z}}\in \mathrm {Sec}(H_{|p^{-1}(V)})\) and as \({\widetilde{\sigma }}:[-\varepsilon ,\varepsilon ]\longrightarrow U\) we simply take the trajectory of \({\widetilde{Z}}\) starting from \(\sigma (0)\). By construction, \(p\circ {\widetilde{\sigma }}=\sigma \). Now define

$$\begin{aligned} v_i(t) = (d_{{\widetilde{\sigma }}(t)}p_{|H({\widetilde{\sigma }}(t))})^{-1}w_i(t),\;\; i=1,\dots ,2n, \end{aligned}$$

and set

$$\begin{aligned} c(t) = ({\widetilde{\sigma }}(t);v_1(t),\dots ,v_{2n}(t)). \end{aligned}$$

It is easy to check that \(d_l{\hat{p}}({\dot{c}}(0)) = w\).

\(\square \)

Proposition 3.5

(a) \(d{\hat{p}}(\Gamma ^{\xi }) = {0}\);

(b) \(d{\hat{p}}(V) = V_B\);

(c) The distribution \(\Gamma ^B = d{\hat{p}}(\Gamma ^H)\) is a connection on O(B).

Proof

Part (a) and the inclusion \(d{\hat{p}}(V)\subset V_B\) follow from the equation before Proposition 3.4, diagram (3.6) and Proposition 3.4. Now take \(w\in V_B\). Then \(w=d{\hat{p}}(v)\), \(v\in TO_{H,g}(M)\), and since \(dp(d\pi (v)) = d\pi _B(w) = 0\), it must be \(v=\lambda \xi ^*+v'\), \(\lambda \in \mathbb {R}\), \(v'\in V\). Consequently \(w=d{\hat{p}}(v')\in d{\hat{p}}(V)\).

We will prove (c). First notice that

$$\begin{aligned} {\hat{p}}\circ R_a = R_a\circ {\hat{p}}. \end{aligned}$$
(3.7)

Next, since \(d_lR_a(\Gamma _l^H) = \Gamma _{l.a}^H\),

$$\begin{aligned} d_{{\hat{p}}(l)}R_a(\Gamma _{{\hat{p}}(l)}^B) = d_{{\hat{p}}(l)}R_a\circ d_l{\hat{p}}(\Gamma _l^H) = d_{l.a}{\hat{p}}\circ d_lR_a(\Gamma _l^H)=d_{l.a}{\hat{p}}\circ (\Gamma _{l.a}^H)=\Gamma _{{\hat{p}}(l).a}^B. \end{aligned}$$

Moreover, \(d\pi _B(\Gamma ^B) = d\pi _B\circ d{\hat{p}}(\Gamma ^H)=dp\circ d\pi (\Gamma ^H) = dp(H)=TB\), so

$$\begin{aligned} TO(B)=\Gamma ^B\oplus V_B. \end{aligned}$$
(3.8)

\(\square \)

Next we will compute the torsion of \(\Gamma ^B\). To this end denote by \(\mathrm {pr}_B:TO(B)\longrightarrow \Gamma ^B\) the projection corresponding to the decomposition (3.8).

Corollary 3.2

\(\mathrm {pr}_B\circ d{\hat{p}} = d{\hat{p}}\circ \mathrm {pr}\).

Denote by \(\theta _B\) the canonical 1-form on O(B).

Lemma 3.1

Let \(\theta \) be the canonical 1-form on \(O_{H,g}(M)\) defined in Sect. 2.3. Then

$$\begin{aligned} {\hat{p}}^*\theta _B = \theta . \end{aligned}$$
(3.9)

Proof

Take \(l\in O_{H,g}(M)\). We have

$$\begin{aligned} ({\hat{p}}^*\theta _B)(l) = \theta _B({\hat{p}}(l))\circ d_l{\hat{p}} = {\hat{p}}(l)^{-1}\circ d_{{\hat{p}}(l)}\pi _B\circ d_l{\hat{p}} = {\hat{p}}(l)^{-1}\circ d_{\pi (l)}p\circ d_l\pi \end{aligned}$$

and recalling (2.7) it is enough to prove that

$$\begin{aligned} {\hat{p}}(l)^{-1}\circ d_{\pi (l)}p = l^{-1}\circ P. \end{aligned}$$
(3.10)

Let \(l=(q;v_1,\dots ,v_{2n})\) and \(v\in TM\). Then \({\hat{p}}(l)^{-1}\circ d_q p(v)=r\) if and only if \(d_q p(v) = d_qp(r^iv_i)\), which in turn is equivalent to \(v = r^iv_i+\lambda \xi (q)\) for a certain \(\lambda \in \mathbb {R}\). Now \(l^{-1}\circ P(v)=r\) and (3.10) is proved. \(\square \)

The torsion of \(\Gamma ^B\) is equal to \(\Theta _B = d\theta _B\circ (\mathrm {pr}_B,\mathrm {pr}_B)\). Take two vectors \(v,w\in TO(B)\). Then \(v=d{\hat{p}}({\hat{v}})\), \(w=d{\hat{p}}({\hat{w}})\) for \({\hat{v}},{\hat{w}}\in \Gamma ^H\subset TO_{H,g}(M)\), and

$$\begin{aligned}&\Theta _B(v,w) = d\theta _B(\mathrm {pr}_B\circ d{\hat{p}}({\hat{v}}),\mathrm {pr}_B\circ d{\hat{p}}({\hat{w}})) = d\theta _B(d{\hat{p}}\circ \mathrm {pr}({\hat{v}}),d{\hat{p}}\circ \mathrm {pr}({\hat{w}})) \\&\quad ={\hat{p}}^*(d\theta _B)(\mathrm {pr}({\hat{v}}),\mathrm {pr}({\hat{w}})) = \Theta ({\hat{v}},{\hat{w}}) = 0, \end{aligned}$$

where \(\Theta \) is the torsion form of \(\Gamma \) (see (2.8)). We proved the following proposition.

Proposition 3.6

\(\Gamma ^B\) is the Levi-Civita connection with respect to the metric \(g_B\).

Denote by \(\nabla ^B:\mathrm {Sec}(TB)\times \mathrm {Sec}(TB)\longrightarrow \mathrm {Sec}(TB)\) the covariant derivation induced by \(\Gamma ^B\). We have an explicit formula for \(\nabla ^B\).

Proposition 3.7

For every \(X,Y\in \mathrm {Sec}(TB)\) and \(q\in B\)

$$\begin{aligned} (\nabla ^B_X Y)(q) = d_q p(\nabla _{{\widetilde{X}}} {\widetilde{Y}})(q), \end{aligned}$$
(3.11)

where \({\widetilde{X}}\), \({\widetilde{Y}}\) are defined according to formula (3.3).

We postpone the proof of this proposition until the appendix.

3.4 Distributions \(\Delta _i\) and decomposition of TB

The decomposition of H induces the decomposition of TB into the distributions on B which are defined as

$$\begin{aligned} \Delta _i = d p (H_i) = d p ({\widetilde{H}}_i), \end{aligned}$$
(3.12)

\(i=1,\dots ,m\). By Proposition 3.3 such a definition is correct.

For a curve \(\gamma :[a,b]\longrightarrow B\) denote by \(\tau ^B_{\gamma }:H(\gamma (a))\longrightarrow H(\gamma (b))\) the parallel translation along \(\gamma \) determined by the connection \(\Gamma ^B\). We prove the following lemma.

Lemma 3.2

Suppose that \(\gamma :[a,b]\longrightarrow B\) and \({\widetilde{\gamma }}:[a,b]\longrightarrow M\) are piecewise smooth curves such that \({\widetilde{\gamma }}\) is horizontal and \(p\circ {\widetilde{\gamma }}=\gamma \). Then the diagram

is commutative.

Proof

Take \(v\in T_{\gamma (a)}B\) and \({\widetilde{v}}\in H({\widetilde{\gamma }}(a))\) such that \(dp({\widetilde{v}})=v\). Suppose that \({\widetilde{\gamma }}^*:[a,b]\longrightarrow O_{H,g}(M)\) is a \(\Gamma \)-horizontal lift of \({\widetilde{\gamma }}\). Choose \(r\in \mathbb {R}^{2n}\) such that \({\widetilde{\gamma }}^*(a)(r)={\widetilde{v}}\). Then, by definition of the parallel transport, \(\tau _{\gamma }({\widetilde{v}})={\widetilde{\gamma }}^*(b)(r)\). Now the curve

$$\begin{aligned} t\longrightarrow {\hat{p}}({\widetilde{\gamma }}^*(t))(r)=d_{\gamma (t)}p\big ({\widetilde{\gamma }}^*(t)(r)\big ) \end{aligned}$$

(again by definition) is parallel in TB, projects onto \(\gamma \) and initiates at v, therefore

$$\begin{aligned} \tau ^B_\gamma (v)=d_{\gamma (b)}p\big ({\widetilde{\gamma }}^*(b)(r)\big ). \end{aligned}$$

This proves the commutativity of the above diagram. \(\square \)

Proposition 3.8

The distributions \(\Delta _i\) are parallel with respect to the connection \(\Gamma ^B\), i.e., for every point \(q\in B\), each \(\Delta _i\) can be obtained from \(\Delta _i(q)\) by parallel transport. Moreover, \(\Delta _i\) are irreducible with respect to the holonomy group of \(\Gamma ^B\).

Proof

Fix an index i and take a piecewise smooth curve \(\gamma :[a,b]\longrightarrow B\). Pick numbers \(a=a_0<a_1<\cdots <a_m=b\) such that each \(\gamma _j = \gamma _{|[a_{j-1},a_j]}\) admits a lift to a horizontal curve \({\widetilde{\gamma }}_j:[a_{j-1},a_j]\longrightarrow U\), \({\widetilde{\gamma }}_j(a_{j-1}) = \gamma (a_{j-1})\), as it is described in the proof of Proposition 3.4. Obviously \(\tau ^B_{\gamma } = \tau ^B_{\gamma _m} \circ \cdots \circ \tau ^B_{\gamma _1}\) and by Lemma 3.2 each \(\tau ^B_{\gamma _j}\) preserves the distribution \(\Delta _i\). It follows that \(\tau ^B_{\gamma }(\Delta _i(\gamma (a)) = \Delta _i(\gamma (b))\) as desired.

Fix now a point \(q_0\in B\) and suppose that we have a decomposition

$$\begin{aligned} \Delta _i(q_0) = \Delta ^{(1)}_i(q_0)\oplus \Delta ^{(2)}_i(q_0) \end{aligned}$$
(3.13)

into nontrivial components. Let \(H^{(j)}_i(q_0) = (d_{q_0}p)^{-1}\big (\Delta ^{(j)}_i(q_0)\big )\cap H_i(q_0)\), \(j=1,2\). Clearly, \(H_i(q_0)=H^{(1)}_i(q_0)\oplus H^{(2)}_i(q_0)\) and \(H^{(j)}_i(q_0)\) are not \(\Psi (q_0)\)-invariant. Therefore there exists a nonzero \({\hat{v}}\in H^{(1)}_i(q_0)\) and a horizontal curve \({\widetilde{\gamma }}:[0,1]\longrightarrow M\) such that \({\widetilde{\gamma }}(0)={\widetilde{\gamma }}(1)=q_0\) and \(\tau _{{\widetilde{\gamma }}}({\hat{v}})\in H^{(2)}_i(q_0)\). Let \(\gamma = p\circ {\widetilde{\gamma }}\). Now \(dp({\hat{v}})\in \Delta ^{(1)}_i(q_0)\) and by Lemma 3.2\(\tau ^B_{\gamma }(dp({\hat{v}}))\in \Delta ^{(2)}_i(q_0)\) which proves that \(\Delta ^{(j)}_i(q_0)\) are not invariant with respect to the holonomy group of \(\Gamma ^B\). \(\square \)

In particular it follows that \(\nabla ^B_X(\mathrm {Sec}(\Delta _i))\subset \mathrm {Sec}(\Delta _i)\) for every \(X\in \mathrm {Sec}(TB)\) and, consequently, \(\Delta _i\) are integrable. Indeed, if \(X,Y\in \mathrm {Sec}(\Delta _i)\) then \(\nabla ^B_XY - \nabla ^B_YX=[X,Y]\in \mathrm {Sec}(\Delta _i)\). Now, to finish the prove of Theorem 1.1 we just use de Rham decomposition theorem [15]. Let us note that the integrability of the distributions \(\Delta _i\) can be also proved in the following way. For a point \(q\in B\) denote by \(M_i\) the maximal integral manifold of \({\widetilde{H}}_i\) passing through q. Using, e.g., Corollary 3.1 we deduce that \(p_{|M_i}:M_i\longrightarrow B\) is of constant rank and hence \(p(M_i)\) is an integral manifold of \(\Delta _i\) passing through q.

In the sub-pseudo-Riemannian case the proof goes along the same lines. The only difference is that the structure group of the bundle \(O_{H,g}(M)\) is now \(O(k,2n-k)\) where k is the index of a metric g, and by an orthonormal frame we mean every frame \(X_1,\dots ,X_{2n}\) such that \(g(X_i,X_j)=0\), \(i\ne j\), \(g(X_i,X_i)=-1\), \(1\le i\le k\), \(g(X_j,X_j)=1\), \(k+1\le j\le 2n\) . Moreover, a few words more about Hamiltonian geodesics in the indefinite case should be added, and we do it in the appendix. At the end we use the version of de Rham Theorem proved in [16].

4 Proof of Theorem 1.3

Replacing M with an open subset, if needed, we can suppose that our structure is strongly nondegenerate on M. Then the eigenvalues \(\pm ib_j\) of J are smooth functions on M. For any point \(q\in M\) there exists a neighborhood U of q and an orthonormal frame \(X_1,\dots ,X_{2n}\in \mathrm {Sec}(H_{|U})\) such that \(J(X_{2j-1}) = -b_jX_{2j}\) and \(J(X_{2j}) = b_jX_{2j-1}\) on U, \(j=1,\dots ,n\). Let us define

$$\begin{aligned} H_{j|U} = \mathrm {Span}\{X_{2j-1},X_{2j}\}. \end{aligned}$$
(4.1)

Of course \(H_{j|U}\) glue together to globally defined distributions on M and we obtain the decomposition of H

$$\begin{aligned} H = H_1\oplus \cdots \oplus H_n \end{aligned}$$
(4.2)

into the Whitney sum of pairwise orthogonal rank 2 sub-distributions.

Let \(l=(q;v_1,\dots ,v_{2n})\in O_{H,g}(M)\). Denote by \(e_1,\dots ,e_{2n}\) the standard basis of \(\mathbb {R}^{2n}\) and by \(f^1,\dots ,f^{2n}\) the dual basis of \((\mathbb {R}^{2n})^*\). Let us recall that since H, \(H^*\) and \(\mathrm {Hom}(H,H)\) are vector bundles associated with \(O_{H,g}(M)\) with typical fiber equal to \(\mathbb {R}^{2n}\), \((\mathbb {R}^{2n})^*\), \((\mathbb {R}^{2n})^*\otimes \mathbb {R}^{2n}\), respectively, then l acts as linear isomorphisms (cf. [15]) \(l:\mathbb {R}^{2n}\longrightarrow H(q)\), \(l:(\mathbb {R}^{2n})^*\longrightarrow H(q)^*\), \(l:(\mathbb {R}^{2n})^*\otimes \mathbb {R}^{2n}\longrightarrow \mathrm {Hom}(H(q),H(q))\) which are respectively defined by \(l(e_j)=v_j\), \(l(f^j)=v^{*j}\), \(l(f^j\otimes e_k) = v^{*j}\otimes v_k\); here \(v^{*j}\in H^*_q\) is the covector dual to \(v_j\in H_q\).

Now fix a point \(q\in M\). Choose an orthonormal basis \(v_1,\dots ,v_{2n}\) of H(q) such that

$$\begin{aligned} J_q(v_{2j-1}) = -b_j(q)v_{2j}, \;\; J_q(v_{2j}) = b_j(q)v_{2j-1},\;\;j=1,\dots ,n. \end{aligned}$$

Let us define a \(2n\times 2n\)-matrix \((A^i_j)\) by

$$\begin{aligned} A^i_jf^j\otimes e_i = \sum _{j=1}^n\big [-b_j(q)f^{2j-1}\otimes e_{2j} + b_j(q)f^{2j}\otimes e_{2j-1}\big ]. \end{aligned}$$

If \(l=(q;v_1,\dots ,v_{2n})\) then clearly \(l(A^i_jf^j\otimes e_i)=J_q\). Further take an arbitrary smooth horizontal curve \(\sigma :[0,1]\longrightarrow M\) such that \(\sigma (0)=q\). Denote by \(\sigma ^*:[0,1]\longrightarrow O_{H,g}(M)\) the \(\Gamma \)-horizontal lift of \(\sigma \) which satisfies \(\sigma ^*(0)=l\). Then \(\sigma ^*(0)(A^i_jf^j\otimes e_i)=J_q\). By assumption the operator J is parallel, therefore

$$\begin{aligned} \sigma ^*(t)(A^i_jf^j\otimes e_i)=J_{\sigma (t)} \end{aligned}$$
(4.3)

for \(t\in [0,1]\).Footnote 1 Equation (4.3) means that if \(\sigma ^*(t) = (\sigma (t);v_1(t),\dots ,v_{2n}(t))\) then for every t

$$\begin{aligned} J_{\sigma (t)}(v_{2j-1}(t)) = -b_j(q)v_{2j}(t), \;\; J_q(v_{2j}(t)) = b_j(q)v_{2j-1}(t),\;\;j=1,\dots ,n. \end{aligned}$$

Since \(\sigma \) is an arbitrary horizontal curve, and any two points of M can be joined by a horizontal curve, this ends the proof of the following proposition.

Proposition 4.1

Under the assumptions of Theorem 1.3, \(b_j = const\), \(j=1,\dots ,n\), in a neighborhood of every point at which the structure is strongly nondegenerate. Moreover, the distributions \(H_1,\dots ,H_n\) from (4.2) are parallel on such a neighborhood.

To end the proof we proceed exactly as above which results in a decomposition \((B_1,g_1)\times \cdots \times (B_n,g_n)\) of B into the product of 2-dimensional Riemannian manifolds. It remains to recall the classical result saying that any two Riemannian manifolds of dimension 2 are locally conformally equivalent.