14.1Introduction

A sub-pseudo-Riemannian contact manifold (M,?, g) is an odd-dimensional manifold M endowed with a pair (?, g) where ? is a contact distribution on M and g is a pseudo-Riemannian metric on ?. In particular, if g is positive-definite then (?, g) is called contact sub-Riemannian structure, and if g has signature (−, +, . . . , +) then (?, g) is called contact sub-Lorentzian structure. A sub-pseudo-isometry (or shortly an isometry) of a sub-pseudo-Riemannian manifold (M,?, g) is a diffeomorphism f : M → M such that (i) f∗(?q) = ?f(q) for every q ∈ M and (ii) f∗|?q : ?q → ?f(q) is a linear isometry of g. In the present paper, we study local geometry of the subpseudo-Riemannian contact manifolds with respect to the action of the group of all sub-pseudo-isometries. All objects are assumed to be smooth.

It is well known that any contact manifold (M,?) of dimension 2n + 1 is locally diffeomorphic to the (2n + 1)-dimensional Heisenberg group, that is, there exist local coordinates (x1, . . . , xn, y1, . . . , yn, z) on M such that

Therefore, we can assume that there is given the Heisenberg group equipped with an additional metric on ?. If the metric is left-invariant with respect to the action of the Heisenberg group then the structure is called flat.

The contact sub-pseudo-Riemannian structures appear in control theory. For instance, many efforts have been made aiming to analyze the behavior of the subpseudo-Riemannian geodesics (see [1] for the sub-Riemannian case and [7, 14, 18] for the sub-Lorentzian case) or constructing normal forms [2, 6, 7, 23]. Other applications can be found in [12, 21]. The geometry of the structures is well understood in dimension 3 only. In this dimension, invariants have been constructed in [1, 3] for the Riemannian signature and in [10] for the Lorentzian signature. An alternative approach to the equivalence problem in dimension 3 is proposed in [11] and it uses Tanaka’s theory of graded, nilpotent Lie algebras. In the present paper, we present yet another approach and generalize it to higher dimensions. We construct a system of invariants of the contact sub-pseudo-Riemannian structures in any dimension and any signature (Theorems 4.3 and 4.5). We also show that there are connections between sub-pseudo-Riemannian and Einstein–Weyl structures [4, 13, 15] in dimension 3 (Theorems 3.2 and 3.5). The last section contains remarks on isometries of the considered structures.

The main idea of the paper is to consider extensions of g on ? to metrics on the tangent bundle TM. Actually, for a given sub-pseudo-Riemannian metric, we construct a one-parameter family of pseudo-Riemannian metrics, denoted by Gc, where c ∈ ℝ {0}, such that Gc|? = g. The key role in the construction plays the Reeb vector field on M which is transverse to ? and uniquely defined by g.We use the Levi–Civita connections of the extended metrics Gc and show that the corresponding curvature tensors decompose into homogeneous components with respect to the degree of parameter c. The leading term, denoted by κ, is the main new invariant playing the role of the curvature of the system. Additionally, we get: h and ω – a symmetric and a skewsymmetric invariant bilinear forms on ?. We show that κ, h, and ω contain all basic information of the original structure (Theorem 4.5).

In Section 14.3, we prove that under additional conditions in dimension 3 there is a unique Einstein metric in the one-parameter family Gc of the extended metrics. Further, we consider deformations of these particular Einstein metrics and propose a new method of construction of certain classical examples of the Einstein–Weyl structures studied before in [15, 22] or [5].

14.2Dimension 3

Structural functions

We start with the simplest case of dimension 3, studied before in [1–3] in the sub-Riemannian case and in [7, 10] in the sub-Lorentzian case. We present new, uniform approach to both cases. Later, in Section 14.4 our approach is generalized to the case of arbitrary dimension.

Let ? be a contact distribution on a three-dimensional manifold M and let g be a metric on ?. If g is positive-definite, we say that (?, g) is a sub-Riemannian structure on M and if g is indefinite we say that (?, g) is a sub-Lorentzian structure on M. We shall assume that there is given an orientation of ?. If it is the case then (?, g) is referred to as an oriented contact sub-pseudo-Riemannian structure.

Let us consider a local, positively oriented, orthonormal frame (X1, X 2) of ?. In the sub-Riemannian case

In the sub-Lorentzian case, we assume that X1 is unit time-like and X2 is unit spacelike, that is

Thus, the matrix of g in the basis (X1, X 2) is

or

depending on the signature.

Note that the frame (X1, X2) is complemented to the full frame on M by the socalled Reeb vector field X0, which we are going to define now. To this end let α be a 1-form such that ? = ker α. After multiplying α by the suitable nonvanishing function, we can suppose that dα(X1, X 2) = 1. By definition, X0 is such a unique vector field on M that

It follows directly from the definition that the Lie bracket [X0, Xi] is a section of ?, for i = 1, 2, and [X1, X2] = X0 mod ?. Thus, we canwrite

and the coefficients are referred to as the structural functions of the frame (X1, X2). The triple (X1, X 2, X0) defines an orientation on M, which is induced by the original orientation of ?.

Invariants

We shall recall now the formulae for the fundamental invariant tensors (with respect to the group of sub-pseudo-Riemannian isometries) of the structure (?, g), which were defined in [1] in the sub-Riemannian case and in [10] in the sub-Lorentzian case. There are two of them. The first one, denoted by h, depends on the orientation of ?. It is a bilinear form on ? defined by the formula

where ℒX0 denotes the Lie derivative in the direction of X0. It is well defined because the flow of the Reeb field X0 preserves ?, that is, [X0,?] = ?. The matrix of h in the frame (X1, X 2) takes the form

in the sub-Riemannian case, where and the form

in the sub-Lorentzian case, where Note that h does not depend on the choice of a positively oriented orthonormal frame although its matrix does. It will be convenient to consider the endomorphism h♯ : ? → ?

instead of the bilinear form h. In the matrix notation h♯ = g−1h.

The second invariant is a function, denoted by κ, given by the formula

in the sub-Riemannian case and by the formula

in the sub-Lorentzian case.35It is known that in both cases κ can be interpreted as a curvature of the system [1, 11]. The invariant does not depend on the orientation of ?. Our approach to κ is presented in Theorem 2.2.

Remark 2.1. Note that tr is always zero independently of the signature. Indeed, expanding the iterated Lie bracket [X0, [X1, X 2]] in the frame (X0, X1, X 2) and using the Jacobi identity one sees that the coefficient next to X0 is zero on the one hand and equals on the other hand. Moreover, in the sub-Riemannian case one can easily prove that det h♯ = 0 implies that h♯ = 0 and consequently h = 0. Therefore, in the sub-Riemannian context one can use the functional invariant χ = det h♯ instead of the bilinear form h in most of the situations (cf. [1]). In the sub-Lorentzian case the situation is slightly more subtle. It is not necessarily true that det h♯ = 0 implies h = 0. However, also in the sub-Lorentzian case one can produce one functional invariant out of h, simply by considering the action of SO(1, 1) on it (see [11] for details and for the discussion concerning the problem whether and when the invariants distinguish sub-Lorentzian structures).

Canonical extensions

Let c ∈ ℝ {0} be a fixed constant. Any metric g on ? extends uniquely to a metric Gc on M such that

and

If the original metric g is indefinite then, clearly, Gc is indefinite too. If g is positivedefinite then Gc is positive definite for c > 0 and indefinite for c < 0. Moreover, the extension Gc does not depend on the orientation of ?. Indeed, a change of the orientation implies that X 0 is multiplied by −1 and this does not affect Gc. It follows that the extension Gc is canonically defined by the original structure (?, g) and the constantc. Therefore the associated Levi–Civita connection ∇c and the curvature tensor of ∇c are defined by the structure (?, g) itself.

The connection ∇c can be easily computed in terms of the structural functions. We get

where the signs depend on the signature of the metric: the upper sign is for the sub-Riemannian case and the lower sign is for the sub-Lorentzian case. As before in the sub-Riemannian case and in the sub-Lorentzian case. The formulae imply the following

Theorem 2.2. Let (?, g) be a contact sub-Riemannian or sub-Lorentzian structure on a three-dimensional manifold. Then the sectional curvature of ? with respect to the extended metric Gc equals to

Proof. Let (X1, X 2) be an orthonormal frame of ?. The formula is obtained by directcomputations of using (2.2), (2.3), the formulae for ∇provided above and the following identities:

in the sub-Riemannian case and

in the sub-Lorentzian case.

Remark 2.3. The formula for also holds if c is not a constant but a function onM. Indeed, if c depends on a point in M then and where i = 1, 2, aremodified by a term of the form However, the modification does not affect In particular, if

then (2.4) reduces to

Remark 2.4. The invariant κ was defined in [10] by formula (2.3) as an invariant of a time- and space-oriented sub-Lorentzian structure. Theorem 2.2 permits to regard κ as an invariant of a sub-Lorentzian structure without any orientation. Indeed, our choice of the orientation does not affect neither the metric Gc nor the sectional curvature

On the other hand, the invariant h defined by (2.1) depends essentially on X0 and thus on the orientation of ?. More precisely, the change of the orientation results in the change of the sign of the bilinear form h. In particular it follows that the condition h = 0 is independent of the choice of the orientation and depends only on the structure (?, g).

Remark 2.5. Observe that no matter the value of c is, the trajectories of the Reeb fieldare geodesics for the metric Gc. Indeed, we have

Symmetric case

Assume that h = 0 as a bilinear form on ? (cf. Remark 2.4). It means that the Reeb vector field is an infinitesimal isometry of g, that is, the metric g is preserved by the flow of X0. Therefore, one can consider (at least locally) the quotient manifold

and there is unique metric g ̃ on N such that its pullback to ? on M coincides withg. The metric g ̃ will be referred to as the projection of g. As pointed out above, the condition h = 0 is independent of the orientation of ?. Similarly, N and g ̃ do not depend on the sign of X0.

The following theorem and its corollaries slightly generalize the results obtainedin [1, 10].

Theorem 2.6. Let (?, g) be a contact sub-Riemannian or sub-Lorentzian structure on a three-dimensional manifold M. If h = 0 then the projection of g to the quotient manifold N determines the structure (?, g) uniquely.

Proof. Let N = M /X0 be the quotient manifold equipped with g̃. Let (X̃1, X̃ 2) be an orthonormal frame on N. Any vector field on N lifts uniquely to a vector field on M, tangent to ?. In particular one can consider lift of the frame (X̃1, X ̃ 2) and get a frame (X1, X 2) of ?. The frame (X1, X 2) is orthonormal for g, as follows from the definition of g̃. The structure functions of (X1, X 2) are special. Indeed

because X1 and are lifts of vector fields on N and is tangent to the fibers of X 2 X0M → N. Moreover and are constant along X0 and are actually defined by

Now, assume that there are given two structures (?1, g1) and (?2, g2) such that h1 = h 2 = 0 and the corresponding metrics g̃1 and g̃2 are isomorphic. Then one can choose orthonormal frames for g̃1 and g̃2 such that their structural functions coincide. It follows that the lifted frames also share the structural functions. The theorem follows from the result of E. Cartan on the equivalence of frames.

Corollary 2.7. If h = 0 then the Gauss curvature of the metric g ̃ on the quotient manifold N equals to κ.

Proof. We use an orthonormal frame (X1, X 2) of ? defined by the lift of (X̃1, X ̃2) as inthe proof of Theorem 2.6 above. Then, there is no term involving in κ and the formula reduces to the Gauss curvature of g ̃ computed in terms of the structural functions of (X̃1, X̃ 2).

Corollary 2.8. The structure (?, g) is locally equivalent to the flat structure on the Heisenberg group if and only if h = 0 and κ = 0.

14.3Einstein–Weyl geometry

Motivations

In the previous section, we have assigned a one-parameter family Gc, c ∈ ℝ {0}, of pseudo-Riemannian metrics on a manifold M to a given sub-pseudo-Riemannian structure (?, g) on M. Our goal in the present section is to pick a particular metric in the family that is special. We succeed only partially. Namely,we assumethat h = 0 and κ is constant and prove that under these conditions there is a unique Einstein metric in the family of extended metrics Gc.

The results in this section apply to a relatively small class of sub-pseudo-Rieman-nian structures due to restrictive assumptions, which imply that the structures are left-invariant on Lie groups (see [1, 3, 10]). However, the results are interesting from the point of view of the Einstein–Weyl geometry in dimension 3. Indeed, the Einstein metrics of the form Gc can be deformed and give rise to families of Einstein–Weyl structures. In this way, we propose a new method of constructing Einstein–Weyl structures. In particular, we rediscover classical examples of the Einstein–Weyl structures defined originally in [15, 22], where the authors reduce metrics on four-dimensional manifolds to get the Einstein–Weyl structures on three-dimensionalmanifolds. In our approach, in order to get a three-dimensional Einstein–Weyl structure, we extend a metric defined originally on a two-dimensional manifold as in Theorem 2.6.

Definitions

A metric G is Einstein if the Ricci tensor of the Levi–Civita connection is proportional to G. The Einstein–Weyl structures are conformal counterparts of the Einstein metrics. We shall briefly recall a definition of the Einstein–Weyl structures in dimension 3 and refer to [4, 13, 15, 20, 22] for more information on the subject. Let M be a three-dimen-sionalmanifold equipped with a conformal metric [G]. We say that a linear connection ∇ on M is a Weyl connection for [G] if

for some one-form η. The one-form is not defined uniquely by [G] but depends on the representative G. Indeed, if one takes a different representative, given in the form ϕ2G, then η is modified by a closed form 2d(ln ϕ). A Weyl structure on M is a pair ([G], ∇), where ∇ is a Weyl connection. Note that, as a particular example, one can take as ∇ the Levi–Civita connection for G. In this case η = 0. In general, the connection ∇ is uniquely determined by a representative G and the corresponding η. Therefore, we will sometimes write that the Weyl structure is given by a pair (G, η).

A Weyl structure is called Einstein–Weyl if it satisfies the following conformal Einstein equation:

where Ric(∇)sym is the symmetric part of the Ricci tensor and RG is the scalar curvature of ∇ with respect to G.

Ricci curvature of sub-pseudo-Riemannian structures

Let us now consider a three-dimensional sub-Riemannian or sub-Lorentzian structure(?, g) and assume that the Reeb vector field X0 is an infinitesimal isometry. We havethe following

Proposition 3.1. If h = 0 then the Ricci tensor of ∇c in the frame (X1, X2, X0) is represented by the matrix

in the sub-Riemannian case, and by the matrix

in the sub-Lorentzian case.

Proof. The proof is based on computations. We shall show the details in the sub-Lorentzian case only. Let us recall that the condition h = 0 in terms of the structural functions is equivalent to

Using this and applying the Jacobi identity to vector fields X1, X2, and X0, we get and By direct calculations, we compute

Gc(R(Xi , Xj)Xk , Xi) = 0

provided that j ≠ k. It follows that Ric(∇c ) is diagonal. In order to get the diagonal terms, we show that

and the rest follows from Theorem 2.2 and symmetries of the Riemann tensor.

As a corollary, we get the following result.

Theorem 3.2. Let (?, g) be a contact sub-Riemannian or sub-Lorentzian structure on a three-dimensional manifold M. Assume that h = 0 and κ is constant and nonzero. Then the metric Gκ is the unique Einstein metric in the family Gc of extended metrics. The structure has the Riemannian signature if g is sub-Riemannian and κ > 0 and otherwise it has the Lorentzian signature.

Proof. Follows directly from Proposition 3.1, where we substitute c = κ.

Remark 3.3. Note that all structures ([Gκ], ∇κ), considered as Einstein–Weyl structures, are locally equivalent up to the sign of κ and signatures of g and Gc. Indeed a simple rescaling of Gκ by a constant function gives an equivalence of ([Gκ], ∇κ) and ([G1], ∇1) or ([G−1], ∇−1), depending on the sign of κ.

Deformations

We shall show now that the Einstein structures defined above in Theorem 3.2 can be deformed to one parameter families of Einstein–Weyl structures. Let α be the one-form on M annihilating ? and such that

where X0 is the Reeb vector field, as before. We will consider Weyl structures defined by pairs (Gc, 2ϵcα), where Gc is an extension of g, c ∈ ℝ {0} and ϵ ∈ ℝ. All these structures are canonically defined by the original sub-pseudo-Riemannian structure. The Weyl connection defined by (Gc, 2ϵcα) will be denoted by We have

Proposition 3.4. If h = 0 then the symmetric part of the Ricci tensor of in the frame (X1, X 2, X0) is represented by the matrix

in the sub-Riemannian case, and by the matrix

in the sub-Lorentzian case.

Proof. We have ∇cα = −cα1 ∧ α2, where Therefore ∇jciα is skew-symmetric and it follows from the formulae for the Ricci tensor of in terms of the Levi–Civita connection of Gc and the one-form η defining the Weyl structure (Gc , η) that only diagonal terms appear in Ric sym (see [4, 15]). We have computed these terms directly.

Our main result in this section is the following

Theorem 3.5. Let (?, g) be an oriented contact sub-Riemannian or sub-Lorentzian structure on a three-dimensional manifold M. Assume that h = 0 and κ is constant. If κ = 0 then the pair (Gc, 2ϵcα) defines an Einstein–Weyl structure on M if and only if g is sub-Lorentzian, ϵ2 = 1 and c ∈ ℝ {0} is arbitrary. If κ ≠ 0 then the pair (Gc , 2ϵcα) defines an Einstein– Weyl structure on M if and only if

in the Riemannian signature, or

in the Lorentzian signature, provided that ϵ2 ≠ 1.

Proof. We use Proposition 3.4 and deduce that the symmetric part of the Ricci tensorof is proportional to the metric if and only if the following algebraic equation:

is satisfied, where the sign next to ϵ2 depends on the signature. Thus, the Einstein equation is reduced to c(1 + ϵ2) = κ in the Riemannian case, or c(1 − ϵ2) = κ in the Lorentzian case. Therefore, if κ = 0 then g has to be Lorentzian and ϵ2 = 1 because c ≠ 0. If κ ≠ 0 then depending on the signature.

If κ = 0 then the family (Gc, 2cα) defined by Theorem 3.5 in the sub-Lorentzian case is the family of Nil Einstein–Weyl structures defined in [5]. It follows that the structures are of hyper-CR type [4, 5]. The family is a deformation of the flat Lorentzian Einstein– Weyl structure. Indeed, if c tends to 0 then the structure tends to the flat one.

If κ ≠ 0 then by rescaling, we can assume κ = 1 or κ = −1, depending onthe sign of κ. The corresponding families for positive κ and for negative κ, defined in Theorem 3.5, are deformations of the structures defined in Theorem 3.2. Indeed, taking ϵ = 0 we get that ηϵ = 0 and the corresponding Weyl connection is the Levi–Civita connection of G1 or G−1, respectively. Summarizing, there are four families, depending on the sign of κ and signature of g.

The structures can be easily write down in coordinates. For instance, in the Lorentzian signature and for κ = 1 the family is given by (Gϵ , ηϵ), where

This is the Lorentzian counterpart of the Einstein–Weyl metric on the Berger sphere written in nonspherical coordinates [22, page 96]. Our family in the Riemannian signature with κ = 1 coincides with [15, eq. (5.10)].

14.4Dimension 2n + 1

Structural functions

Let M be a manifold of dimension 2n + 1 with a contact distribution ? equipped with a metric g of arbitrary signature. We assume that there is given an orientation of ?, and, as in dimension 3, we choose a local, positively oriented, orthonormal frame (X1, . . . , X 2n) of ?. Moreover,we assume that Mis oriented. We have that g(Xi , X j) = 0 for i ≠ j and

where si ∈ {−1, 1} depends on the signature. The frame (X1, . . . , X2n) is complemented to a full frame on M by the Reeb vector field, denoted by X 0. In order to define X0, we consider a one-form α annihilating ?. It is given up to a multiplication by anonvanishing function. However α can be normalized by the condition

which gives α up to sign. The Reeb vector field of a contact one-form α is uniquelydefined by

Thus, a priori, there are two Reeb vector fields that differ by sign and are defined by the two one-forms normalized via (4.1).We chose the one for which (X1, . . . , X 2m, X0) is a positively oriented frame on M.

It follows from the definition that the flow of X0 preserves ?, that is, [X0,?] = 0.Therefore

for some functions Moreover, we can write

and in this way we define structural functions of the frame (X1, . . . , X2n). Note that

Remark 4.1. There are more subtle notions of orientation of sub-pseudo-Riemannian structures based on the so-called casual decomposition of ? into its space-like and time-like subspaces [8]. However, we shall not use them, and only consider an orientations of M and ? needed in order to define the Reeb vector field. Actually, if n is odd then M is canonically oriented by (dα)n ∧ α and if n is even then ? is canonically oriented by (dα)n. See [9] for details.

Canonical extension and invariatns

The one-form α, used above in the definition of the Reeb field, defines an invariantskew-symmetric form ω = −dα|? on ?, that is

where X and Y are sections of ?. If n = 1 then ω is just the volume form on ?defining the orientation.

In order to construct additional invariant tensors (with respect to the group of sub-pseudo-Riemannian isometries) one proceeds similarly to the case of dimension3. First, one defines an invariant bilinear symmetric form h by

Then one considers extended metrics Gc defined by

and

and gets the Levi–Civita connections ∇c and the corresponding curvature tensors Rc. The part of Rc restricted to ? will be denoted by whereas the corresponding sectional curvature of a plane span{X, Y} ⊂ ? will be denoted by As in dimension 3, we will use h♯ : ? → ? defined by

Additionally, we will extend g to a metric on exterior powers ⋀k ?in the standard way.

Remark 4.2. Note that h and ω depend on the orientation of ? and M. On the other hand the extended metrics Gc as well as ∇c and the corresponding curvatures depend on the sub-pseudo-Riemannian structure (?, g) only. Indeed, the conditions (4.4) and (4.5) do not depend on the sign of X0.

In terms of the structural functions of a frame (X1, . . . , X 2n) we have

and

where as before si ∈ {−1, 1} depends on the signature of g. Moreover, we have

and

Additionally The following result generalizes Theorem 2.2.

Theorem 4.3. Let (?, g) be a contact sub-pseudo-Riemannian structure on a (2n + 1)-dimensional manifold. Then the sectional curvature decomposes as follows:

where (X1, . . . , X2n) is an orthonormal frame of (?, g) and κ?(Xi , Xj) is a quantity independent of the chosen constant c. In terms of the structural functions

Proof. The proof is reduced to computations generalising three-dimensional case. In particular (4.6) generalizes (2.4), and (4.7) generalizes (2.2) and (2.3).

Remark 4.4. Note that the sectional curvature determines completely theRiemann tensor by the well-known formula [16, Lemma 3.3.3, page 144]. By the same formula applied to κ?(Xi , Xj) we can define a (3, 1)-tensor, denoted by R?, independent of the choice of c.

Symmetric case

Assume that h = 0.We consider the quotient manifold

with the unique metric g̃, called projection of g to N, such that its pullback to ? on M coincides with g. Similarly, if ℒX0ω = 0 then there is the unique 2-form ω̃ on N, called projection of ω, such that its pullback to ? on M coincides with ω. As in the case of dimension 3, the condition h = 0, and similarly ℒX0ω = 0, is independent of the orientation of ?.We have the following generalization of Theorem 2.6.

Theorem 4.5. Let (?, g) be a sub-pseudo-Riemannian structure on a (2n + 1)-dimensional manifold M. If h = 0 and ℒX0ω = 0 then the projections of g and ω to the quotient manifold N determine the structure (?, g) uniquely.

Proof. The proof is a repetition of the proof in dimension 3. We shall consider an orthonormal frame (X̃1, . . . , X̃ 2n) on N and its lift (X1, . . . , X2n) on M. Then the frame (X1, . . . , X 2n) is an orthonormal frame of ?and it is easy to show that the corresponding structural functions are determined by the structural functions of the original frame on N and by the projection of ω. The theorem follows from the result of E. Cartan on the equivalence of frames.

Corollary 4.6. If h = 0 then the pullback to ? of the Riemann curvature tensor of the metric g ̃ coincides with R?.

Proof. We use an orthonormal frame (X1, . . . , X 2n) of ?defined by the lift of the frame (X̃1, . . . , X̃ 2n) as in the proof of Theorem 4.5 above. Then, there is no term involving in κD(Xi , Xj) and the formula reduces to the sectional curvature of g ̃ computed in terms of the structural functions of (X̃1, . . . , X̃2n).

14.5Contact sub-pseudo-Riemannian symmetries

Sub-pseudo-Riemannian isometries

Suppose that (M,?, g) is a contact (2n + 1)-dimensional sub-pseudo-Riemannian manifold. Recall that a diffeomorphism f : M → M is an isometry (or a sub-pseudoisometry) of (M,?, g) if (i) f∗(?q) = ?f(q) for every q ∈ M, (ii) f∗|?q : ?q → ?f(q) is alinear isometry of g. Of course, the set of all isometries of (M,?, g) forms a group.

Suppose that, as in the previous section, ?and M are oriented. Let (X1, . . . , X 2n) be an orthonormal positively oriented frame of (?, g), and let α be the contact one-form normalized as in (4.1) used in the definition of the Reeb vector field X0. Finally let (α1, . . . , α2n, α) be a coframe dual to (X1, . . . , X2n, X0). If f is an isometry then, clearly,

for some smooth functions ai, and λ. The normalization condition reads

which gives λ = ±1 and consequently f ∗α = ±α. Using this, it is easy to show that α(f∗X0) = ±1 and dα(f∗X0, ⋅) = 0 proving that f∗X0 = ±X0 (note that dα has one-dimensional kernel). Consequently, if we extend g to the pseudo-Riemannian metric G = G1 by setting G(X0, X0) = 1, then any sub-pseudo-Riemannian isometry automatically becomes an isometry of the metric G. This observation is independent of the choice of an orientation on ?. In this way (cf. [17]) we are led to the following

Theorem 5.1. The set I(M,?, g) of all isometries of (M,?, g) is a Lie group with respect to the open-compact topology. Moreover, in the sub-Riemannian case the isotropic subgroup Iq(M,?, g) of any point q ∈ M is compact.

Proof. Indeed, I(M,?, g) is a closed subgroup in the group of isometries of the pseudo-Riemannian manifold (M, G).

We also obtain

Proposition 5.2. Any contact sub-pseudo-Riemannian isometry f is uniquely determined by two values: f(q0) and f∗(q0), where q0 ∈ M is an arbitrarily fixed point.

Proof. The result follows from known properties of isometries in the pseudo-Rieman-nian geometry.

Fix an isometry f of (M,?, g). Examining (5.1) in more detail it is easy to check that a1 = . . . = a2n = 0. Moreover it is clear that where l is the index of g. It follows that sub-pseudo-Riemannian structures (?, g) on M are in a one-to-one correspondence with ?-structures on M where

Indeed, any such ?-structure can be realized as the bundle of horizontal orthonormal frames O?,g(M) → M with

The component of identity I0(M,?, g) in the group of isometries I(M,?, g) can be identified now with the group of all fiber-preserving mappings F : O?,g(M) → O?,g(M) such that F ∗θ = θ, where θ stands for the restriction to O?,g(M) of the canonical form on the bundle of linear frames on M.

Fix an element (q; υ1, . . . , υ2n , X0(q)) ∈ O?,g(M). Thanks to Proposition 5.2, we have the embedding

Note that the embedding can be used to proof Theorem 5.1.

Symplectic structure

If f is an orientation-preserving isometry of (M,?, g) then, clearly, f ∗dα = dα. In particular f ∗ω = ω, where ω = −dα|?, defined by (4.2), may be regarded as a symplectic form on ?q for every q ∈ M. This leads to the following idea. If the two structures on ?: g and ω are compatible, meaning that there exists a symplectic basis for dα|?q, whichis orthonormal for g, then and such sub-pseudo-Riemannian structures (?, g) on M may be viewed as reductions of ?-structures with ? defined in (5.2) to ℋ-structures on M with

This is the case e.g., for the Heisenberg group with the left-invariant sub-pseudo-Riemannian structure. More precisely, the natural left-invariant distribution ? on the Heisenberg group is, in the exponential coordinates q = (x1, . . . , xn, y1, . . . , yn, z), spanned by the following fields

The natural left-invariant metric g on ? is defined by declaring the fields Xi and Yi to be orthonormal with g(Xi , Xi) = si, g(Yi , Yi) = ti where si , ti ∈ {−1, 1} depending on the signature of g, i = 1, . . . , n. Here

so the symplectic form on ? q is

and ω(Xi , Yj) = δij, ω(Xi , Xj) = ω(Yi , Yj) = 0, i, j = 1, . . . , n.

Now we shall focus on the general sub-Riemannian case only and leave other signatures to future works [9]. If g and ω are compatible then the operator J : ? → ? defined by

for all X, Y ∈ ?, is a complex structure in every distribution plane ?q, q ∈ M, and the group ℋ from (5.4) is isomorphic to the unitary group U(n).We get the following

Theorem 5.3. Let (M,?, g) be a contact sub-Riemannianmanifold of dimension 2n+1.Then dim I(M,?, g) ≤ (n + 1)2.

Proof. Indeed, if (?, g) is the left-invariant sub-Riemannian structure on the Heisenberg group as above, then it is known that I(ℝ2n+1,?, g) = ℝ2n+1 ⋉ U(n) (see e.g., [24]). In particular dimI(ℝ2n+1,?, g) = 2n + 1 + n2 = (n + 1)2 and it follows that dim I(M,?, g) ≤ (n + 1)2 for all contact sub-Riemannian structures such that g and ω are compatible. In the general case, when g and ω are not compatible, the bundle O?,g(M) reduces to the bundle of orthonormal frames such that where bi are certain smooth functions on M, called infundamental frequencies in [1] (±ibi(q) are eigenvalues of Jq). Let us assume first that bi = const, i = 1, . . . , n and let m be a number of different values of bi’s. Then we can decompose ? = ?1 ⊕ . . . ⊕ ?m where ?j, j = 1, . . . , m, are sub-distributions of ? corresponding to different values of frequencies. It follows that the reduced frame bundle is a principal bundle with the group U(n1) ⊕⋅ ⋅⋅ ⊕ U(njmm) where and Now, it is easy to see (e.g computing the Tanaka prolongation)that dim The case of nonconstant bi follows from [19].

14.6Appendix: Isometries in dimension 5

In this appendix, we will compute explicitly the group of isometries for structures(ℝ5,?, g) defined by vector fields (5.5) in dimension 5, where (X1, Y1, X2, Y2) is a g-orthonormal basis of ?.We consider three cases: (1) g is Riemannian, (2) g(X1 , X1) = −1, g(Y1, Y1) = g(X2, X2) = g(Y2, Y2) = 1, (3) g(X1, X1) = g(X2, X2) = −1, g(Y1, Y1) = g(Y2, Y2) = 1. In all cases the metric structure is compatible with the symplectic structure as it is explained in Section 14.5. Therefore the embedding (5.3) allows to compute the corresponding isometry groups in the explicit form. The three structures are left invariant, so the corresponding group of isometries contains five-dimensional subgroup of left translations. In the exponential coordinates it can be represented as follows. Let denote the Reeb field. The Baker–Campbell–Hausdorff formula

gives that the isometries coming from the left translations can be written as

where (t1, t2, t3, t4, t5) ∈ ℝ5.

In case (1) the structure group ℋ in (5.4) is the unitary group Sp(4) ∩ O(4) ≃ U(2) whose dimension is equal to 4. Using suitable representation of U(2) as a subgroup of GL(4, ℝ), every σ ∈ U(2) induces an isometry (x1, y1, x2, y2, z) ⟼ (σ(x1, y1, x2, y2), z) of the Heisenberg group. In this way, we obtain the following four-parameter family of isometries:

(x1, y1, x2, y2, z) ⟼ (x2 cos θ1 − y2 sin θ1, x2 sin θ1 + y2 cos θ1, x1, y1, z) ,

(x1, y1, x2, y2, z) ⟼ (x2, y2, x1 cos θ4 − y1 sin θ4, x2 sin θ4 + y2 cos θ4, z) .

Thus, in this case dim I(ℝ5,?, g) = 9.

Next, in the case (2) the structure group ℋ = Sp(4) ∩ O(1, 3) is two-dimensional and, in addition to left translations, we have the following two-parameter family of isometries:

(x1, y1, x2, y2, z) ⟼ (x1 cosh θ1 + y1 sinh θ1, x1 sinh θ1 + y1 cosh θ1, x2, y2, z)

(x1, y1, x2, y2, z) ⟼ (x1, y1, x2 cos θ2 − y2 sin θ2, x2 sin θ2 + y2 cos θ2, z) .

Thus, in this case dim I(ℝ5,?, g) = 7.

Finally, in the case (3) the structure group ℋ = Sp(4)∩O(2, 2) is four-dimensional and, in addition to left translations, we have the following four-parameter family of isometries:

(x1, y1, x2, y2, z) ⟼ (x1 cosh θ1 + y1 sinh θ1, x1 sinh θ1 + y1 cosh θ1, x2, y2, z)

(x1, y1, x2, y2, z) ⟼ (x2, y2, x1 cosh θ4 + y1 sinh θ4, x1 sinh θ4 + y1 cosh θ4, z) .

Thus, as in the sub-Riemannian case dim I(ℝ5,?, g) = 9.

Note that in all three cases

where ℋ is the corresponding structure group.

Acknowledgment: The work of Wojciech Kryński has been partially supported by thePolish National Science Centre grant DEC-2011/03/D/ST1/03902.