A de Rham decomposition type theorem for contact sub-Riemannian manifolds

In this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point q∈M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in M$$\end{document} such that the holonomy group Ψ(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi (q)$$\end{document} acts reducibly on H(q) yielding a decomposition H(q)=H1(q)⊕⋯⊕Hm(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(q) = H_1(q)\oplus \cdots \oplus H_m(q)$$\end{document} into Ψ(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi (q)$$\end{document}-irreducible factors. Using parallel transport we obtain the decomposition H=H1⊕⋯⊕Hm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H = H_1\oplus \cdots \oplus H_m$$\end{document} of H into sub-distributions Hi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_i$$\end{document}. Unlike the Riemannian case, the distributions Hi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_i$$\end{document} are not integrable, however they induce integrable distributions Δi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _i$$\end{document} on M/ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M/\xi $$\end{document}, which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that T(U/ξ)=Δ1⊕⋯⊕Δm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T(U/\xi )=\Delta _1\oplus \cdots \oplus \Delta _m$$\end{document}, and the latter decomposition of T(U/ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T(U/\xi )$$\end{document} induces the decomposition of U/ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U/\xi $$\end{document} into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.


Introduction and statement of results
Let M be a smooth (by smooth we mean of class C ∞ ) 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 This is a principle bundle with structure group O(2n). Moreover we will assume that H and T M are oriented, so the structure group can be reduced to SO(2n). Let ξ be the Reeb vector field which is well defined in such a situation. We will assume that ξ is an infinitesimal isometry. Now it can be proved [13] that there exists a unique connection on O H ,g (M) which is torsion-free (the definition of the torsion in our case is presented below). In the usual way defines the covariant differentiation where we use the following notation: if E −→ M is a vector bundle then by Sec(E) we denote the C ∞ (M)-module of sections of E. Having a connection on the bundle O H ,g (M) we can consider its holonomy group (q) at a point q ∈ M. Since M is connected the groups (q 1 ) and (q 2 ) are isomorphic for any two points q 1 , q 2 ∈ M. The holonomy group (q) naturally acts on H (q) (for H is an associated vector bundle to O H ,g (M) with typical fiber R 2n ). Suppose that the action of (q) on H (q) is reducible. Then H (q) decomposes into (q)-irreducible factors H (q) = H 1 (q) ⊕ · · · ⊕ H m (q) (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 H = H 1 ⊕ · · · ⊕ H m . (1.2) Next let us consider the set M/ξ of orbits of ξ . It is locally a smooth manifold of dimension 2n.
If we fix an arbitrary point q 0 ∈ M and a neighborhood U of q 0 such that U /ξ is a connected smooth manifold, then we can canonically identify U /ξ with a regular 2n-dimensional submanifold B of M with q 0 ∈ 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 ∇ induces a connection ∇ B on B which turns out to be the Levi-Civita connection with respect to g B . Moreover, if we denote by p : U −→ B the projection in the direction of ξ , then is a linear isometry. Using this projection, the decomposition (1.2) induces a decomposition 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 : be an isometry and let ξ be the Reeb vector field on (M 1 , H 1 , g 1 ). Then for every sufficiently small open set U ⊂ M 1 , which is convex with respect to ξ (that is to say every trajectory of ξ intersects U in a connected set), the Riemannian manifolds U /ξ and f (U )/ f * ξ are isometric.
Using the results from [16] we can generalized the above theorem to contact subpseudo-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 (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 " (q) acts reducibly on H (q)" is replaced with " (q) acts nondegenerately and reducibly on H (q)", are satisfied. Then every point in M has a neighborhood U such that the manifold U /ξ is isometric to the product (B 1 , g 1 ) ×· · ·×(B m , g m ) of pseudo-Riemannian manifolds, where B i is of dimension rank H i and, as above, may be identified with a maximal integrable manifold of the distribution i , i = 1, . . . , 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 α the normalized contact 1-from (see Sect. 2 for details). Then we can define the operator J : H −→ H by dα(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 ±ib j , j = 1, . . . , 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 −→ R is a smooth function. We say that the structure (H , g) is strongly nondegenerate at a point q ∈ M, if b 1 (q) < · · · < b n (q) under suitable numeration (cf. [1] where the numbers b 1 (q), . . . , 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 ξ is an infinitesimal isometry. Denote by the unique torsion-free connection on O H ,g (M). Suppose next that (ii) the operator J is parallel with respect to . If J is strongly nondegenerate at a point q and U is a sufficiently small neighborhood of q, then U /ξ is isometric to a product of 2dimensional Riemannian manifolds. Consequently, the conformal type of U /ξ 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 /ξ 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.

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 ∈ T M which belongs to H will be called horizontal. On the other hand, if is a distribution on O H ,g (M), e.g., a connection, then a vector V ∈ T O H ,g (M) belonging to will be referred to as a -horizontal vector.

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 T M and H are oriented. This is equivalent to the existence of a globally defined contact form, i.e., a 1-form α on M with the property that H = ker α (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 where X 1 , . . . , X 2n is a fixed local positively oriented orthonormal frame for H . For n even we have two such forms α 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 α is the normalized contact form then we define the Reeb vector field ξ on (M, H , g) as the solution to the system of equations Such a field has the property that [ξ, X ] ∈ Sec(H ) whenever X ∈ Sec(H ). In particular the (local) flow ϕ t of ξ preserves the distribution H . Moreover, ξ defines a canonical decomposition T M = H ⊕ Span{ξ }. The projection defined by this decomposition will be denoted by (2.2)

Geodesics
Suppose that X 1 , . . . , X 2n be an orthonormal frame defined on an open set U ⊂ M.
Let H : T * M |U −→ R be defined by Clearly, the value of (2. 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 σ : [a, b] −→ M be a Hamiltonian geodesic and let λ(t) be its lift to T * M as in (2.4). Suppose that S is a submanifold in M such that σ (a) ∈ S. We say that σ satisfies the (Pontryagin) transversality condition with respect to S if λ(a) |T σ (a) S = 0. For a point q ∈ M, denote by D q the set of all covectors λ ∈ T * q M such that the Hamiltonian geodesic with initial condition (q, λ) exists on the interval [0, 1]. Then we define the exponential mapping with pole at q as follows: where σ (t) is the Hamiltonian geodesic with initial condition (q, λ). One proves that exp q is smooth.

Isometries and infinitesimal isometries
Given two contact sub-Riemannian manifolds (M 1 , where α i is the normalized contact form and ξ 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 σ : [a, b] −→ M is a Hamiltonian geodesic satisfying the transversality condition with respect to a submanifold S, then f • σ 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

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: by π : L H (M) −→ M we denote its projection, i.e., π(q; v 1 , . . . , v 2n ) = q. This is a principle bundle with the structure group G L(2n). Indeed, we have a natural right action: , a ∈ G L(2n) (here and below we use the Einstein summation convention). Moreover, if X 1 , . . . , X 2n is a basis of sections of H defined on an open set U ⊂ M then the local trivialization ψ : where H = (dπ) −1 (H ) ∩ and ξ = (dπ) −1 (Span{ξ }) ∩ ; as above ξ stands for the Reeb vector field. Given a connection 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 ∈ O H ,g (M) we define  [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 ∇ : moreover, the vanishing of the horizontal torsion means that whereas the vanishing of the vertical torsion is expressed by [13].
At the end of this section let us note that if is the mentioned torsion-free connection on O H ,g (M), then the component ξ in the splitting (2.6) is given by ξ = Span{ξ * }, where the vector field ξ * (being the -horizontal lift of ξ ) is defined as follows. Take q ∈ M and let ϕ t : U −→ M be the (local) flow of ξ , where U is a neighborhood of q. Then we can lift ϕ t to the mapping t :

Holonomy
Given a connection 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 γ : [a, b] −→ M. The curve γ induces the parallel displacement of fibers which is defined as follows. Take l ∈ π −1 (γ (a)) and let γ * : be the -horizontal lift of γ initiating at l, i.e., π •γ * = γ , d dt γ * (t) ∈ γ * (t) whenever the derivative exists, and γ * (a) = l. Then τ γ (l) = γ * (b). Moreover, if we are given two piecewise smooth curves γ i : [a i , b i ] −→ M, i = 1, 2, such that γ 1 (b 1 ) = γ 2 (a 2 ), we have τ γ 2 ·γ 1 = τ γ 2 • τ γ 1 , where γ 2 · γ 1 is the concatenation of γ 1 and γ 2 . In particular, if C(q) denotes the set of all piecewise smooth loops at a point q ∈ M, then (q) = {τ γ : γ ∈ C(q)} is a Lie group which is called the holonomy group at q and is denoted by (q). Such a group can be realized as a subgroup (l), l ∈ π −1 (q), of the structure group O(2n): if γ ∈ C(q) then l and τ γ (l) belong to the same fiber of π : O H ,g (M) −→ M, hence γ determines a unique element a γ ∈ O(2n) such that τ γ (l) = l.a γ . In this way . It is proved that if M is connected then holonomy groups at any two points are isomorphic.
The connection induces also the parallel displacement in every vector bundle associated with O H ,g (M), so in particular in H . More precisely, if γ : [a, b] −→ M is a curve then we can define the parallel displacement or translation along γ (we use the same notation as above) is a -horizontal lift of γ , π(γ * (a)) = γ (a), and r ∈ R 2n is such that γ * (a)(r ) = v. Notice that for every γ the map τ γ is a linear isometry. In particular, the holonomy group (q) acts on H (q).
Suppose that (q) acts reducibly on H (q), and let be the decomposition of H (q) into (q)-irreducible and (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 of the distribution H . Indeed, if γ is a curve starting at q then τ γ (H i (q)) does not depend on γ but only on its endpoints. To end this subsection, we note that, by the definition of the covariant derivation induced by , for every Z ∈ Sec(T M) and i = 1, . . . , m.

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 ξ is an infinitesimal isometry and denote its (local) flow by ϕ t . Moreover, let 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 As above H i are constant rank distributions which are pairwise orthogonal with respect to g, rank H i > 0, i = 1, . . . m. By ∇ we will denote the covariant derivation induced by .

Distributions H i
Distributions H i need not be integrable, however their extensions are. For every i = 1, . . . , m let us define

Proposition 3.1 The distribution H i is integrable
Proof Indeed by (2.13) it follows that for every X , Y ∈ Sec(H i ). Consequently, In particular we see that the distributions H i , as well as H i , are invariant by the flow of ξ .

The submanifold B: construction of the bundle over B
Fix a point q 0 ∈ M. Let q 0 ∈ U where U ⊂ M is an open set that will be specified below. We construct a regular submanifold B of M, q 0 ∈ B, which can be canonically identified with U /ξ . We start by choosing a coordinate system around q 0 which will be convenient for our purposes. Denote by δ : (−ε, ε) −→ M the trajectory of the Reeb field ξ such that δ(0) = q 0 . Select a local basis X 1 , . . . , 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, . . . , 2n. We can assume that each g t i is defined on a neighborhood of q 0 and for |t| < ε. By shrinking U we can suppose that the mapping and ξ |δ = ∂ ∂z . Let (x 1 , . . . ,x 2n , z,p 1 , . . . ,p 2n , r ) be the Darboux coordinates on T * M |U and let us set . . . , 0, z,p 1 , . . . ,p 2n , 0) : |z|, |p 1 |, . . . , |p 2n | < ε} ⊂ T * M |U .
The set A can be regarded as the set of initial conditions for sub-Riemannian geodesics satisfying the Pontryagin transversality conditions with respect to δ (cf. [1]). Now, the assignment defines the desired coordinates (x 1 , . . . , x 2n , z) around q 0 . We can suppose that they are defined on U (shrinking U again if necessary). Let us notice that ∂z , and straight lines t −→ (tv, z) in these coordinates, where v ∈ R 2n and n i=1 v 2 i = 1, are sub-Riemannian geodesics parameterized by arc length, which satisfy the transversality conditions with respect to δ. We define the following family of hypersurfaces S w = {q ∈ U : z(q) = w} transverse to δ. We will identify U with an open subset of R 2n+1 with coordinates (x, z), so we will also write S w = {(x, z) : z = w}.

Proposition 3.2 ϕ t (S w ) = S w+t .
Proof S w is the union of geodesics which in our coordinates have the form σ (s) = (sv, w), |v| = 1. As we said above these are are exactly the geodesics that start from δ and satisfy the Pontryagin transversality conditions with respect to δ. Fix such a geodesic σ (s) = (sv, w). Since ϕ t is an isometry preserving δ, the curve s −→ ϕ t (σ (s)) is again a geodesic that starts from δ and satisfies the transversality condition with respect to δ. Thus it must be of the form s −→ (sṽ,w). Because ϕ t (σ (0)) = (0, w + t) in (x, z)-coordinates,w = w + t which ends the proof. 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 as the projection onto B in the direction of ξ : where t(q) is such that ϕ t(q) (q) ∈ B. By construction, ξ is transverse to B, and t(q) depends smoothly on q by the implicit function theorem. Obviously t(q) = 0 for q ∈ B and, moreover, We see that p : U −→ B is a fiber bundle with fibers being trajectories of ξ .

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 ∈ Sec(T B). First we construct the canonical 'lift' of X to the field X ∈ Sec(H ) on U by formula for every q ∈ B and every t for which the above expression is defined. Recall that α stands for the normalized conatct form.

Proposition 3.3 Suppose that X ∈ Sec(T B) and let X ∈ Sec(H ) be the horizontal lift defined above. For every q ∈ U d q p( X (q)) = X ( p(q)).
Proof Let q = ϕ t (q), whereq ∈ B. Then using (3.3) and (3.2) we have and it suffices to notice that dq p(X (q)) = X (q) and dq p(ξ ) = 0. The first equality follows from p |B = id and the other from the definition of p.
Now we define the announced Riemannian metric on B. For q ∈ B and X , Y ∈ Sec(T B) we set The last equation can be rewritten as Note that if X 1 , . . . , X 2n is a basis of T q B, then X 1 = X 1 − α(X 1 )ξ(q), . . . , X 2n = X 2n − α(X 2n )ξ(q) is a basis of H q and d q p( X i ) = X i for every i. Remembering (3.5) we obtain the following statement.

Corollary 3.1 For every q
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 = 0, then the resulting Riemannian manifold will be isometric to (B, g B ). We is commutative. Recall that we have the decomposition = H ⊕ ξ , where ξ = Span{ξ * }. Note thatp • t =p (where t is the local flow of ξ * ). Indeed,

Proposition 3.4 The mappingp defined above is a surjective submersion.
Proof Supposing that ε > 0 is sufficiently small and σ is contained in a coordinate chart V , extend the fieldσ (t) to a vector field Z ∈ Sec(T B |V ). Using (3.3), we obtain the field Z ∈ Sec(H | p −1 (V ) ) and as σ : [−ε, ε] −→ U we simply take the trajectory of Z starting from σ (0). By construction, and set It is easy to check that d lp (ċ(0)) = w. Proposition 3.5 (a) dp( ξ ) = 0; a connection on O(B).
Proof Part (a) and the inclusion dp(V ) ⊂ V B follow from the equation before Proposition 3.4, diagram (3.6) and Proposition 3.4. Now take w ∈ V B . Then w = dp(v), v ∈ T O H ,g (M), and since dp(dπ We will prove (c). First notice that and recalling (2.7) it is enough to prove that Then v = dp(v), w = dp(ŵ) forv,ŵ ∈ H ⊂ T O H ,g (M), and where is the torsion form of (see (2.8)). We proved the following proposition.
where X, Y are defined according to formula (3.3).
We postpone the proof of this proposition until the appendix.

Distributions 1 i and decomposition of TB
The decomposition of H induces the decomposition of T B into the distributions on B which are defined as i = dp(H i ) = dp( H i ),   and v ∈ H ( γ (a)) such that dp( v) = v. Suppose that γ * : is a -horizontal lift of γ . Choose r ∈ R 2n such that γ * (a)(r ) = v. Then, by definition of the parallel transport, τ γ ( v) = γ * (b)(r ). Now the curve (again by definition) is parallel in T B, projects onto γ and initiates at v, therefore This proves the commutativity of the above diagram. Proof Fix an index i and take a piecewise smooth curve γ : [a, b] −→ B. Pick numbers a = a 0 < a 1 < · · · < a m = b such that each γ j = γ |[a j−1 ,a j ] admits a lift to a horizontal curve γ j : [a j−1 , a j ] −→ U , γ j (a j−1 ) = γ (a j−1 ), as it is described in the proof of Proposition 3.4. Obviously τ B γ = τ B γ m • · · · • τ B γ 1 and by Lemma 3.2 each τ B γ j preserves the distribution i . It follows that τ B γ ( i (γ (a)) = i (γ (b)) as desired. Fix now a point q 0 ∈ B and suppose that we have a decomposition Therefore there exists a nonzerov ∈ H (1) i (q 0 ) and a horizontal curve γ : [0, 1] −→ M such that γ (0) = γ (1) = q 0 and τ γ (v) ∈ H (2) i (q 0 ). Let γ = p • γ . Now dp(v) ∈ (1) i (q 0 ) and by Lemma 3.2 τ B γ (dp(v)) ∈ (2) i (q 0 ) which proves that In particular it follows that ∇ B X (Sec( i )) ⊂ Sec( i ) for every X ∈ Sec(T B) and, . 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 i can be also proved in the following way. For a point q ∈ B denote by M i the maximal integral manifold of H i passing through q. Using, e.g., Corollary 3.1 we deduce that p |M i : M i −→ B is of constant rank and hence p(M i ) is an integral manifold of 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 where k is the index of a metric g, and by an orthonormal frame we mean every frame X 1 , . . . , X 2n such that g(X i , 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].

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 ±ib j of J are smooth functions on M. For any point q ∈ M there exists a neighborhood U of q and an orthonormal frame with typical fiber equal to R 2n , (R 2n ) * , (R 2n ) * ⊗ R 2n , respectively, then l acts as linear isomorphisms (cf. [15]) here v * j ∈ H * q is the covector dual to v j ∈ H q . Now fix a point q ∈ M. Choose an orthonormal basis v 1 , . . . , v 2n of H (q) such that Let us define a 2n × 2n-matrix (A i j ) by Since σ 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. To end the proof we proceed exactly as above which results in a decomposition (B 1 , g 1 )×· · ·×(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.
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Conflicts of interest
The author declares that he has no conflict of interest.
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We define the geodesic Hamiltonian H : T * M −→ R. First we do so locally. Suppose that U ⊂ M is an open subset such that there exists an orthonormal frame X 1 , . . . , X 2n for H |U . Then we set on T * M |U . Next we notice that such a definition is independent of the choice of an orthonormal frame. Indeed, if X 1 , . . . , X 2n is any other orthonormal frame for H |U , as claimed. It follows that H is well defined on the whole T * M. Recall that we have the canonical symplectic structure on T * M which we will denote by ϒ (ϒ is the exterior differential of the Liouville 1-form on T * M). Thus the geodesic Hamiltonian determines the Hamiltonian vector field − → H on T * M. Now by a Hamiltonian geodesic on the given sub-pseudo-Riemannian manifold we mean every curve on M which is a projection of a trajectory of the field − → H . Once we have the notion of Hamiltonian geodesics, we can define the exponential mapping with pole at a given point q 0 ∈ M exactly as it is done in the sub-Riemannian case.
A diffeomorphism f : (M 1 , H 1 , g 1 ) −→ (M 2 , H 2 , g 2 ) of two sub-pseudo-Riemannian manifolds is called an isometry if d q f (H 1 (q)) ⊂ H 2 ( f (q)) and d q f : ) is a linear isometry for every q ∈ M. In particular the two metrics g 1 , g 2 have the same index. Proof Fix (q, λ) ∈ T * M 1 and let X 1 , . . . , X 2n be an orthonormal frame for H 2 defined in a neighborhood of f (q). Then since (d q f ) −1 X 1 , . . . , (d q f ) −1 X 2n is an orthonormal frame for H 1 around q.
In order to prove (A.2) we note that in addition to (A.1) we also havef * ϒ 2 = ϒ 1 .
Consequently, we have finished the proof of the following proposition.

Proposition A.1 Isometries preserve Hamiltonian geodesics.
Appendix B: Proof of Proposition 3.7 In this section we prove that the operator where X , Y ∈ Sec(T B) and q ∈ B, is a Levi-Civita connection for the metric g B . As above X , Y are defined according to formula (3.3).
For a function f ∈ C ∞ (B) let us define f ∈ C ∞ (U ) by f (q) = f ( p(q)) and observe that f X = f X (B.1) whenever X ∈ Sec(T B). Further, we have Lemma B.1 Under the above notation, for X ∈ Sec(T B), f ∈ C ∞ (B), q ∈ B in particular, X ( f )(q) = X ( f )(q). Moreover, Proof The second formula is obvious since f is constant along the trajectories of ξ .
To prove the first part, fix an arbitrary point belonging to U . Such a point is of the form ϕ s (q) where q ∈ B and s ∈ R. We have ϕ s (q)).
We make sure that D defined above is indeed a connection. To this end take X , Y ∈ Sec(T B), f ∈ C ∞ (B) and q ∈ B. Then and Fix X , Y , Z ∈ Sec(T B) and q ∈ B. At first we will compute the torsion of D.
where the last equality follows from the fact that dp(ξ ) = 0. Now, for any f ∈ C ∞ (B) It follows that and D is torsion-free. Next we prove that D is a metric connection. Recall that In order to evaluate the left-hand side of (B.2) let us notice that for every s (for which it makes sense) g( X , Y )(ϕ s (q)) = g d q ϕ s X (q) − α q (X )ξ(q) , d q ϕ s Y (q) − α q (Y )ξ(q) = g X (q) − α q (X )ξ(q), Y (q) − α q (Y )ξ(q) = g B (X (q), Y (q)) = g B (X , Y )( p • ϕ s (q)) = g B (X , Y ) ¬ (ϕ s (q)).
Now the first summand on the right-hand side of (B.2) evaluated at q is (we use (3.5) here) and similarly for the second summand. Hence which ends the proof of Proposition 3.7.