On the dimension of the algebras of local infinitesimal isometries of 3-dimensional special sub-Riemannian manifolds

Abstract

Suppose that we are given a contact sub-Riemannian manifold (M, H, g) of dimension 3 such that the Reeb vector field is an infinitesimal isometry (such manifolds will be referred to as special). For a point $q\in M$ denote by $\mathfrak {i}^*(q)$ the Lie algebra of germs at q of infinitesimal isometries of (M, H, g). It is proved that for a generic point $q\in M$, $\dim \mathfrak {i}^*(q)$ can only assume the values 1, 2, 4. Moreover, $\dim \mathfrak {i}^*(q) = 4$ if and only if the curvature function determined by the canonical sub-Riemannian connection is constant in a neighborhood of q. The latter case is possible if (M, H, g) is locally isometric, in a neighborhood of q, to a left-invariant sub-Riemannian structure on a 3-dimensional Lie group.

1 Introduction and statement of the results

Suppose that M is a smooth manifold. By a sub-Riemannian structure on M, we mean a couple (H, g), where H is a smooth (i.e., of class $C^{\infty }$) bracket generating distribution on M of constant rank, and g is a smooth Riemannian metric on H. The triple (M, H, g) is called a sub-Riemannian manifold.¹ If $\dim M = 2n+1$ and H is a contact distribution then (M, H, g) is called a contact sub-Riemannian manifold. Sub-Riemannian geometry arises in many mathematical and physical problems and has been intensively studied for many years—see for instance [1,2,3,4,5,6,7] and the reference sections therein. Among the main problems that have been addressed so far there are, for instance, the behavior of sub-Riemannian geodesics and their length minimizing properties, conjugate and cut loci, smoothness of sub-Riemannian spheres, isometric and conformal classification in low dimensions, nilpotent approximations, differential properties of the sub-Riemannian distance. In this paper, we deal with sub-Riemannian (infinitesimal) isometries. By an isometry of a sub-Riemannian manifold (M, H, g), we mean a diffeomorphism $G:M\longrightarrow M$ such that $d_qG(H_q) = H_{G(q)}$ and $g(d_qG(v),d_qG(w)) = g(v,w)$ for every $q\in M$, $v,w\in H_q$. An infinitesimal isometry of (M, H, g) is a smooth vector field Z on M such that around any point the local flow of Z consists of isometries.

There is quite a lot of works treating sub-Riemannian isometries in case of left-invariant structures on Lie groups, among which are [2, 8,9,10]. In this paper, however, we are interested in contact sub-Riemannian structures on manifolds which are not Lie groups. According to the author’s knowledge, the non-group case is not so well treated, and the literature devoted to it is much less numerous. Some results in this direction were obtained, e.g., in [11,12,13,14]. The set of infinitesimal isometries forms a Lie algebra over $\mathbb {R}$ whose dimension in the contact case does not exceed $(n+1)^2$, where $\dim M = 2n+1$. In fact, such algebras (apart from those determined by left-invariant structures on Lie groups) are almost not known even for small values of n.

An oriented and contact sub-Riemannian manifold (M, H, g) will be called special if the Reeb vector field (see the definition below) associated with the structure (H, g) is an infinitesimal isometry. The aim of this paper is to establish possible dimensions of Lie algebras of local infinitesimal isometries of special 3-dimensional sub-Riemannian manifolds.

To be more specific, we fix a special sub-Riemannian manifold (M, H, g). For a point, $q\in M$ denote by $\mathfrak {i}^*(q)$ the algebra of germs at q of local infinitesimal isometries of (M, H, g). A point q is called $\mathfrak {i}^*$-regular if the function $p\longrightarrow \dim \mathfrak {i}^*(p)$ is constant on a neighborhood of q. Below, we will show that the set of $\mathfrak {i}^*$-regular points is an open and dense subset of M. The main result of the paper is the following theorem.

Theorem 1.1

Let (M, H, g) be a special sub-Riemannian manifold, $\dim M = 3$. Then, for every $\mathfrak {i}^*$-regular point $q\in M$, $\dim \mathfrak {i}^*(q)$ can assume only the values 1, 2, 4.

The proof relies on the construction of the appropriate exterior differential system on a certain 7-dimensional manifold with the property that (local) infinitesimal isometries of (M, H, g) are in bijective correspondence with 3-dimensional integral manifolds of the mentioned system. The method of the proof extends some ideas used in the paper [15] in which the authors reprove the known result on a number of infinitesimal isometries of 2-dimensional Riemannian manifolds. The construction of the mentioned differential system is possible due to the existence of metric and torsion-free sub-Riemannian connections introduced in [16, 17].

Remark 1.1

If (M, H, g) is a 3-dimensional contact sub-Riemannian manifold which is not special, then the case $\dim \mathfrak {i}^*(q)=3$ is possible. One can see this, for instance, by considering the suitable left-invariant sub-Riemannian structure on the group $\widetilde{\textrm{SE}}(2)$, the universal cover of the group $\textrm{SE}(2)$ of Euclidean motions of the plane—see, e.g., [10].

Note that in the paper [14], the authors investigated the Lie algebra of infinitesimal isometries of sub-Riemannian manifolds (M, H, g) where $\dim M = 3$ and H is a Martinet distribution. In the latter situation, the possible dimensions of $\mathfrak {i}^*(q)$ are 0, 1, 2.

Content of the paper In Sect. 2, we describe basic properties of the Reeb vector field and infinitesimal isometries of contact sub-Riemannian manifolds. In Sect. 3, we recall the construction of the sub-Riemannian connection introduced in [16, 17]. In Sect. 4, we set up the suitable exterior differential system and using it we prove that the algebras of local infinitesimal isometries of a 3-dimensional special sub-Riemannian manifold cannot have dimension 3. Sect. 5 presents three examples of special sub-Riemannian manifolds where $\dim \mathfrak {i}^*(q)$ is, respectively, 4, 2 and 1. The main tool in evaluating the dimension $\dim \mathfrak {i}^*(q)$ is the curvature function K (see 4.1) of our connection. K happens to coincide with the invariant $\kappa$ introduced by Agrachev in [1, 2]. In Section 6, we give an example of a contact sub-Riemannian manifold without non-trivial isometries.

In the paper, we use the following notation. If $E\longrightarrow M$ is a vector bundle over M, then, by $\textrm{Sec}(E)$, we denote the $C^{\infty }(M)$-module of sections of E. If H is a distribution on M, then, the elements of $\textrm{Sec}(H)$ are often referred to as horizontal vector fields. Moreover, if X is a vector field on M, then, ${\mathcal {L}}_{X}$ is the Lie derivative in the direction X.

2 The Reeb vector field and Infinitesimal isometries

Fix a contact sub-Riemannian manifold (M, H, g), $\dim M = 2n+1$, which is assumed to be oriented as a contact manifold, i.e., the bundles TM and H are oriented. Equivalently, we can say that there exists a globally defined contact 1-form $\alpha$ such that $H = \text { ker }\alpha$—see [12, 18]. We will suppose that $\alpha$ is chosen so as to have

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

(2.1)

where $X_1,\dots ,X_{2n}$ is a local positively oriented orthonormal frame for H. The condition (2.1) does not depend on a choice of such a frame. If n is 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.

From now on, $\alpha$ will stand for the normalized contact 1-form. The form $\alpha$ allows one to define the so-called Reeb vector field. It will be denoted by $\xi$ and is defined as a unique solution to the system of equations $\alpha (\xi ) = 1$, $d\alpha (\xi ,\cdot ) = 0$. $\xi$ has many special properties, in particular $[\xi ,\textrm{Sec}(H)]\subset \textrm{Sec}(H)$ and ${\mathcal {L}}_{\xi }\alpha = 0$.

By an isometry of (M, H, g), we mean a diffeomorphism $G:M\longrightarrow M$ such that $d_qG(H_q) = H_{G(q)}$ and $g(d_qG(v),d_qG(w)) = g(v,w)$ for every $q\in M$, $v,w\in H_q$. Clearly, if G is an orientation-preserving isometry, then, $G^*\alpha = \alpha$ and, consequently, $G^*\xi = \xi$. A smooth vector field Z on M is called an infinitesimal isometry (or a Killing vector field) if its (local) flow is composed of (local) isometries. Similarly, as in the classical case, one makes sure that the following proposition is true.

Proposition 2.1

$Z\in \textrm{Sec}(TM)$ is an infinitesimal isometry of (M, H, g) if and only, if (i) $[Z,\textrm{Sec}(H)]\subset \textrm{Sec}(H)$ (that is to say Z is a contact vector field), and (ii) $Z(g(X,Y)) = g([Z,X],Y) + g(X,[Z,Y])$ for all $X,Y\in \textrm{Sec}(H)$.

It is clear that infinitesimal isometries form a Lie algebra whose dimension over $\mathbb {R}$ is bounded from above by $(n+1)^2$ (cf. [12]). Suppose that $e_1,\dots ,e_{2n}$ is an orthonormal frame for (H, g) defined on an open set U and let $\omega ^1,\dots ,\omega ^{2n},\alpha$ be the coframe dual to the frame $e_1,\dots ,e_{2n},\xi$. Denoting by $s{\circ } t$ the symmetric product of tensor fields s and t, Proposition 2.1 can be equivalently formulated in the following way.

Proposition 2.2

A vector field Z defined on U is an infinitesimal isometry if and only if Z is a contact vector field, and moreover,

$$\begin{aligned} {\mathcal {L}}_Z \left( \omega ^1\,{\circ }\,\omega ^1 + \dots + \omega ^{2n}\,{\circ }\,\omega ^{2n} \right) = 0\text {.} \end{aligned}$$

(2.2)

Proof

Let ${\tilde{g}}$ be a Riemannian metric determined by the conditions ${\tilde{g}}_{|H\times H} = g$, ${\tilde{g}}(\xi ,\xi )=1$, and ${\tilde{g}}(\xi ,H)=0$. In other words, ${\tilde{g}} = \omega ^1\,{\circ }\,\omega ^1+\dots +\omega ^{2n}\,{\circ }\,\omega ^{2n} + \alpha \,{\circ }\,\alpha$. It is clear that Z is an infinitesimal isometry for the metric g if and only if it is, simultaneously, a contact vector field and an infinitesimal isometry of the metric ${\tilde{g}}$. Thus, if Z is an infinitesimal isometry of g, then, ${\mathcal {L}}_Z{\tilde{g}} = 0$ and ${\mathcal {L}}_Z\alpha = 0$, so (2.2) is satisfied.

Conversely, suppose that Z is a contact vector field and (2.2) holds true. Then

$$\begin{aligned} {\mathcal {L}}_Z{\tilde{g}} = 2({\mathcal {L}}_Z\alpha )\,{\circ }\,\alpha \text {.} \end{aligned}$$

Applying both sides of the last equation to a pair (X, Y), where $X,Y\in \textrm{Sec}(H)$, we obtain precisely the condition (ii) of Proposition 2.1. $\square$

If $U\subset M$ is an open subset, the algebra of infinitesimal isometries of $(U,H_{|U},g_{|U})$ will be denoted by $\mathfrak {i}(U)$. Below, we summarized some important properties of infinitesimal isometries.

Proposition 2.3

Let $U\subset M$ be an open subset. Then,

• (1) If $Z\in \mathfrak {i}(U)$, then, $[\xi ,Z]=0$, i.e., the Reeb field commutes with infinitesimal isometries;

• (2) ${\mathcal {L}}_Z\alpha = 0$ for all $Z\in \mathfrak {i}(U)$;

• (3) If $Z\in \textrm{Sec}(TM)$ is an infinitesimal isometry on an open and dense subset of M, then $Z\in \mathfrak {i}(M)$.

• (4) If $Z_1,Z_2\in \mathfrak {i}(U)$, where U is assumed to be connected, and $Z_{1|V} = Z_{2|V}$ for an open set $V\subset U$ then $Z_1 = Z_2$;

• (5) For every non-trivial isometry $Z\in \mathfrak {i}(U)$, where U is assumed to be connected, the set $\{q\in U:\;Z(q)\notin H_q\}$ is open and dense in U;

• (6) If $Z\in \mathfrak {i}(U)$ is a non-trivial isometry and $hZ\in \mathfrak {i}(U)$, where $h\in C^{\infty }(U)$, then, h is constant;

• (7) If $h\xi \in \mathfrak {i}(U)$, where $h\in C^{\infty }(U)$, then, h is constant.

Proof

(1), (2), (3) are immediate from the definition.

To prove (4), we use the observation made in the proof of Proposition 2.2, namely every isometry of (M, H, g) can be viewed as an isometry of the Riemannian metric ${\tilde{g}}$, so the result follows from classical theorems—see, e.g., [19]. (4) can also be proved in the intrinsic terms, i.e., without introducing a Riemannian metric, but the proof is longer.

In order to prove (5), let us first notice that the set $\{q\in U:\; Z(q)=0\}$ cannot be open because otherwise Z would vanish identically by (4). Suppose that $U_Z = \{q\in U:\; Z(q)\in H_q\backslash \{0\}\}$ is open. Recall that H is strongly bracket generating [7], so whenever $q\in U_Z$, we must have $H_q + [Z,\textrm{Sec}(H)]_q = T_qM$. This is, however, a contradiction with the inclusion $[Z,\textrm{Sec}(H)]\subset \textrm{Sec}(H)$.

Now, we prove (6). Take a non-trivial isometry $Z\in \mathfrak {i}(U)$. By (1)

$$\begin{aligned} 0 = [hZ,\xi ] = -\xi (h)Z\text {,} \end{aligned}$$

therefore by (5) $\xi (h)$ vanishes on a dense and open subset of M, hence vanishes identically. Moreover, for any $X\in \textrm{Sec}(H)$, the fields $[X,hZ] = X(h)Z + h[Z,X]$ and h[Z, X] are horizontal, so again by (5) X(h) vanishes on a dense and open subset in M, and consequently everywhere in M. Thus, h is a constant function.

(7) is proved analogously to (6). $\square$

Fix a point $q\in M$ and consider a sequence $\{U_m\}$ of connected neighborhoods of q such that

$$\begin{aligned} {\overline{U}}_{m+1}\subset U_m, m=1,2,\dots , \;\; \bigcap _{m=1}^{\infty }U_m = \{q\}\text {.} \end{aligned}$$

Note that by (4), Proposition 2.3, the mapping

$$\begin{aligned} \mathfrak {i}(U_m) \ni Z \longrightarrow Z_{|U_{m+1}} \in \mathfrak {i}(U_{m+1}) \end{aligned}$$

is injective for every m. It follows that $\{\dim \mathfrak {i}(U_m)\}_m$ is a non-decreasing sequence of positive integers bounded from above by $(n+1)^2$. Thus, there exists a positive integer $N\le (n+1)^2$ and a positive integer $m_0$ such that for every $m > m_0$, we have $\dim \mathfrak {i}(U_m) = N$. If $\mathfrak {i}^*(q)$ stands for the Lie algebra of germs at q of local infinitesimal isometries of (M, H, g), then, $\dim \mathfrak {i}^*(q) = \dim \mathfrak {i}(U_m)$ for every $m > m_0$. In particular, we have proved that every point q has a neighborhood U, which can be chosen to be arbitrarily small, and such that $\mathfrak {i}(U)\ni Z\longrightarrow (Z)_q \in \mathfrak {i}^*(q)$ is an isomorphism; here, $(Z)_q$ stands for the germ of Z at q. Such a neighborhood will be called $\mathfrak {i}$-special.

A point $q\in M$ is called an $\mathfrak {i}^*$-regular point if the function $p\longrightarrow \dim \mathfrak {i}^*(p)$ is constant on a neighborhood of q. The set of $\mathfrak {i}^*$-regular points of M will be denoted by $M^*$. Similarly, as in [19], we prove

Proposition 2.4

The set $M^*$ is an open and dense subset of M.

Proof

The fact that $M^*$ is open follows immediately from the definition. Pick an arbitrary point $q\in M$ and let U be an $\mathfrak {i}$-special neighborhood of q. For any $p\in U$, it is clear that $\dim \mathfrak {i}^*(p) \ge \dim \mathfrak {i}^*(q)$. Let ${\bar{p}}\in U$ be a point such that $\dim \mathfrak {i}^*({\bar{p}}) = \max _{p\in U}\dim \mathfrak {i}^*(p)$. Denote by V an $\mathfrak {i}$-special neighborhood of ${\bar{p}}$ such that $V\subset U$. We have again $\dim \mathfrak {i}^*(p) \ge \dim \mathfrak {i}^*({\bar{p}})$ for every $p\in V$ and at the same time $\dim \mathfrak {i}^*(p) \le \dim \mathfrak {i}^*({\bar{p}})$ for every $p\in U$. It follows that $V\ni p\longrightarrow \dim \mathfrak {i}^*(p)$ is constant, and ${\bar{p}}$ is $\mathfrak {i}^*$-regular. Therefore, $M^*\cap U \ne \emptyset$, so $M^*$ is dense. $\square$

3 Metric and torsion-free Connection associated with a contact sub-Riemannian structure

In section, we present the construction of a metric and torsion-free sub-Riemannian connection which was introduced in [16, 17]. Suppose that (M, H, g) is a fixed contact sub-Riemannian manifold. We assume M to be connected. We consider the bundle $O_{H,g}(M)$ of horizontal orthonormal frames, i.e.,

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

By $\pi$, we denote the canonical projection $\pi :O_{H,g}(M)\longrightarrow M$. We have a natural right action of $\textrm{O}(2n)$ on $O_{H,g}(M)$. If $a\in \textrm{O}(2n)$ and $l=(q;v_1,\dots ,v_{2n})$, then,

$$\begin{aligned} R_al = \left( q;a^i_1v_i,\dots ,a^i_{2n}v_i \right) \text {;} \end{aligned}$$

here, and below, we use the summation convention with indices varying from 1 to 2n. In fact, $O_{H,g}(M)$ is a principle bundle with structure group $\textrm{O}(2n)$. Let us note that if $l = (q;v_1,\dots ,v_{2n})\in O_{H,g}(M)$, then, we have a linear isomorphism

$$\begin{aligned} l:\mathbb {R}^{2n}\longrightarrow H_q \end{aligned}$$

determined by $l(\zeta ) = \zeta ^iv_i$. As above, we denote by $\xi$ the Reeb vector field and let

$$\begin{aligned} P:TM\longrightarrow H \end{aligned}$$

be the projection determined by the splitting $TM = H\oplus \textrm{Span}\{\xi \}$. We define a 1-form $\theta$ with values in $\mathbb {R}^{2n}$ on the bundle $O_{H,g}(M)$ with formula

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

This is the counterpart of the canonical 1-form considered in the theory of linear frame bundles—see [20]. We check that $R_a^*\theta = a^{-1}\cdot \theta$. Suppose that $\Gamma$ is a connection on $O_{H,g}(M)$. It means that $\Gamma$ is a distribution on $O_{H,g}(M)$ which is invariant under the action of $\textrm{O}(2n)$ and such that

$$\begin{aligned} \hbox {TO}_{H,g}(M) = \Gamma \oplus V\text {;} \end{aligned}$$

(3.1)

here, V stands for the vertical distribution: if $l\in O_{H,g}(M)$ then $V_l = \text { ker }d_l\pi$. Obviously, $d_l\pi _{|\Gamma _l}:\Gamma _l\longrightarrow T_{\pi (l)}M$ is a linear isomorphism. $\Gamma$ admits a decomposition

$$\begin{aligned} \Gamma = \Gamma ^H \oplus \Gamma ^{\xi } \text {,} \end{aligned}$$

where $\Gamma ^H = (d\pi {|\Gamma })^{-1}(H)$, $\Gamma ^{\xi } = (d\pi {|\Gamma })^{-1}(\textrm{Span}\{\xi \})$. It is easy to verify that $\theta _l:\Gamma ^H_l\longrightarrow \mathbb {R}^{2n}$ is a linear isomorphism, $l\in O_{H,g}(M)$. Fix a connection $\Gamma$. $\Gamma$ induces the covariant derivation $\nabla :\textrm{Sec}(TM)\times \textrm{Sec}(H)\longrightarrow \textrm{Sec}(H)$ which is defined in the following way. For $X\in \textrm{Sec}(TM)$, $Y\in \textrm{Sec}(H)$ and $q\in M$, we set

$$\begin{aligned} (\nabla _XY)(q) = l(X_l^*(F_Y))\text {,} \end{aligned}$$

where $X^*$ is the horizontal lift of X, i.e., a vector field on $O_{H,g}(M)$ with values in $\Gamma$ such that $d\pi (X^*) = X$, $F_Y:O_{H,g}(M)\longrightarrow \mathbb {R}^{2n}$ is a smooth function defined by $F_Y(m) = m^{-1}(Y(\pi (m)))$, $l\in \pi ^{-1}(q)$.

We define the torsion 1-form of the connection $\Gamma$ as

$$\begin{aligned} \Theta = d\theta \,{\circ }\, (pr_{\Gamma },pr_{\Gamma })\text {,} \end{aligned}$$

where $pr_{\Gamma }:TO_{H,g}(M)\longrightarrow \Gamma$ is the projection determined by the splitting (3.1). This is a 1-form with values in $\mathbb {R}^{2n}$. According to the classical procedure, we determine the torsion tensor $T:\textrm{Sec}(TM)\times \textrm{Sec}(TM)\longrightarrow \textrm{Sec}(H)$: for $X,Y\in \textrm{Sec}(TM)$ and $q\in M$, we put

$$\begin{aligned} T(X,Y)(q) = l(\Theta _{l}(X^*,Y^*))\text {,} \end{aligned}$$

(3.2)

where $l\in \pi ^{-1}(q)$, $X^*,Y^*$ are horizontal lifts of X, Y, respectively. Using (3.2), we have

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

(3.3)

for $X,Y\in \textrm{Sec}(H)$,

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

(3.4)

for $X\in \textrm{Span}\{\xi \},Y\in \textrm{Sec}(H)$, and

$$\begin{aligned} T(X,Y) = 0 \end{aligned}$$

for $X,Y\in \textrm{Span}\{\xi \}$.

Let $\omega$ be the connection form corresponding to $\Gamma$, i.e., $\omega$ is a 1-form on $O_{H,g}(M)$ with values in o(2n) such that $\Gamma = \text { ker }\omega$, $R^*_a\omega = Ad_{a^{-1}}\cdot \omega$, $a\in \textrm{O}(2n)$, and $\omega (A^*) = A$, where $A\in o(2n)$ and $A^*$ is the vertical vector field defined by $A^*_l = \frac{\hbox {d}}{\hbox {d}t}\Big \vert _{t=0}l.(\exp tA)$. Let $\alpha$ be the normalized contact 1-form. Moreover, let us set $\eta = \pi ^*\alpha$. Writing $\omega = (\omega ^i_j)_{i,j=1,\dots ,2n}$ and $\theta = (\theta ^1,\dots ,\theta ^{2n})$, we see that $\omega ^i_j,\theta ^k,\eta$, $i,j,k = 1,\dots ,2n$, $i<j$, is a coframe on $O_{H,g}(M)$. Now, it is not hard to verify [16] that

$$\begin{aligned} \Theta ^i = T^i_{jk}\theta ^j\wedge \theta ^k + S^i_j\theta ^j\wedge \eta \text {,} \end{aligned}$$

(3.5)

where $\Theta = (\Theta ^1,\dots ,\Theta ^{2n})$, and $T^i_{jk},S^i_j$ are smooth functions on $O_{H,g}(M)$, $i,j,k = 1,\dots ,2n$. The first summand on the right-hand side in (3.5) is called the horizontal torsion, while the other is the vertical torsion. Using (3.5), (3.3), (3.4), the following proposition is clear.

Proposition 3.1

• (1) The horizontal torsion vanishes if and only if $\nabla _XY = \nabla _YX + P([X,Y])$ for all $X,Y\in \textrm{Sec}(H)$;

• (2) The vertical torsion vanishes if and only if $\nabla _XY = P([X,Y])$ for every $X\in \textrm{Span}\{\xi \}$, $Y\in \textrm{Sec}(H)$.

Now, similarly, as in the classical case, we make sure that the following structure equations [16]

$$\begin{aligned} \hbox {d} \theta ^i = -\omega ^i_j\wedge \theta ^j + \Theta ^i\text {,} \end{aligned}$$

(3.6)

$i,j = 1,\dots ,2n$, hold true.

Let us now assume that our (M, H, g) is a special sub-Riemannian manifold, that is to say, it is oriented as a contact manifold and the Reeb vector field $\xi$ is an infinitesimal isometry. Moreover, let the connection $\Gamma$ be chosen so as to have $\Gamma ^{\xi } = \textrm{Span}\{{\widetilde{\xi }}\}$, where $\widetilde{\xi }$ is defined in the following way. Take a point $q_0\in M$ and write $\varphi ^t$ for the (local) flow of the field $\xi$ on an neighborhood $U\subset M$ of $q_0$ and for $|t|<\varepsilon$. Consider the lift $\widetilde{\varphi }^t:\pi ^{-1}(U)\longrightarrow O_{H,g}(M)$ of $\varphi ^t$ defined by

$$\begin{aligned} \widetilde{\varphi }^t(q;v_1,\dots ,v_{2n}) = \left( \varphi ^t(q); \hbox {d}_q\varphi ^t(v_1),\dots , \hbox {d}_q\varphi ^t(v_{2n}) \right) \text {.} \end{aligned}$$

Now, for any $l\in \pi ^{-1}(U)$, we set

$$\begin{aligned} \widetilde{\xi }_l = \frac{\hbox {d}}{\hbox {d}t}\Big \vert _{t=0}\widetilde{\varphi }^t(l)\text {,} \end{aligned}$$

and repeat this construction over every $q_0\in M$. Such a choice of $\Gamma ^{\xi }$ ensures that the vertical torsion of the connection $\Gamma$ vanishes. Indeed, by (3.5) and (3.4), the vanishing of the vertical torsion is equivalent to $\nabla _{\xi }X = [\xi ,X]$ for every $X\in \textrm{Sec}(H)$, so it is enough to prove that the last formula holds true. In order to do this, we need a lemma.

Lemma 3.1

Let $G:M\longrightarrow M$ be an isometry and let $\widetilde{G}:O_{H,g}(M)\longrightarrow O_{H,g}(M)$ acts by formula $\widetilde{G}(q;v_1,\dots ,v_{2n}) = (G(q);d_qG(v_1),\dots ,d_qG(v_{2n}))$. Then, for every $X\in \textrm{Sec}(H)$

$$\begin{aligned} F_X\,{\circ }\,\widetilde{G} = F_{G^{-1}_*X}\text {.} \end{aligned}$$

(3.7)

Proof

See [16]. $\square$

Now, we are ready to compute $\nabla _{\xi }X$. Fix $q\in M$ and take $l\in \pi ^{-1}(q)$:

$$\begin{aligned}&\nabla _{\xi }X(q) = l\widetilde{\xi }_l(F_X) = l\frac{\hbox {d}}{\hbox {d}t}\Big \vert _{t=0}F_X\,{\circ }\,\widetilde{\varphi }^t(l) = l\frac{\hbox {d}}{\hbox {d}t}\Big \vert _{t=0}F_{\varphi ^{-t}_*X}(l) \\&\frac{\hbox {d}}{\hbox {d}t}\Big \vert _{t=0}(\varphi ^{-t}_*X)(q) = [\xi ,X](q)\text {.} \end{aligned}$$

Similarly, as in [16, Proposition 3.1], by suitably choosing $\Gamma ^H$, we can also get rid of the horizontal torsion. In this way, one obtains

Theorem 3.1

([16]) On every special sub-Riemannian manifold (M, H, g), there exists a canonical metric connection with vanishing torsion.

For a connection $\Gamma$ with the connection form $\omega$, we define the curvature form as

$$\begin{aligned} \Omega = \hbox {d} \omega \,{\circ }\, (pr_{\Gamma },pr_{\Gamma })\text {.} \end{aligned}$$

This is a 2-form on $O_{H,g}(M)$ with values in o(2n). Setting $\Omega = (\Omega ^i_j)_{i,j=1,\dots ,2n}$, we can derive (cf. [16]) the structure equations:

$$\begin{aligned} \hbox {d} \omega ^i_j = -\omega ^i_k\wedge \omega ^k_j + \Omega ^i_j\text {,} \end{aligned}$$

(3.8)

$i,j = 1,\dots ,2n$.

Fix a connection from Theorem 3.1. Let us have a closer look at Eqs. (3.6) and (3.8) in case $\dim M = 3$. From (3.6), we have

$$\begin{aligned} \begin{array}{l} \hbox {d} \theta ^1 = -\omega ^1_2\wedge \theta ^2\\ \hbox {d} \theta ^2 = \omega ^1_2\wedge \theta ^1 \end{array}\text {.} \end{aligned}$$

Differentiating these equations and using (3.8), we obtain

$$\begin{aligned} \Omega ^1_2 = \hbox {d} \omega ^1_2 = K\theta ^1\wedge \theta ^2\text {,} \end{aligned}$$

where K is a smooth function on $O_{H,g}(M)$. One more differentiation gives $dK\wedge \theta ^1\wedge \theta ^2 = 0$ which yields that K is, in fact, a smooth function on M.

Corollary 3.1

If (M, H, g) is a 3-dimensional special sub-Riemannian manifold, the structure Eqs. (3.6) and (3.8) for the connection as in Theorem 3.1 take the form

$$\begin{aligned} \begin{array}{l} \hbox {d}\theta ^1 = -\omega ^1_2\wedge \theta ^2 \\ \hbox {d}\theta ^2 = \omega ^1_2\wedge \theta ^1 \\ \hbox {d}\omega ^1_2 = K\theta ^1\wedge \theta ^2 . \end{array} \end{aligned}$$

(3.9)

4 Infinitesimal isometries via exterior differential systems

By an exterior differential system on a manifold S, we mean a differential ideal ${\mathcal {I}}$ in the ring of smooth differential forms on S generated by a collection of differential forms $\sigma ^1,\dots ,\sigma ^r$. To put it in another way, ${\mathcal {I}}$ is algebraically generated by $\sigma ^1,\dots ,\sigma ^r,d\sigma ^1,\dots ,d\sigma ^r$. In our case, $\sigma ^i$ will be 1-forms. A submanifold N in S is said to be an integral manifold of ${\mathcal {I}}$ if $j^*\tau = 0$ for every $\tau \in {\mathcal {I}}$, where $j:N\subset S$ in the inclusion. Let $\omega ^1,\dots ,\omega ^k$ be independent 1-forms on S such that $\omega ^i_Q\notin \{\tau _Q:\;\tau \in {\mathcal {I}}\}$ for every $Q\in S$ and $i=1,\dots ,k$. We say that $j:N\subset S$ is an integral manifold of $\{{\mathcal {I}},\,\omega ^1\wedge \dots \wedge \omega ^k\}$ if N is an integral manifold of ${\mathcal {I}}$ of dimension k such that $j^*\omega ^1\wedge \dots \wedge j^*\omega ^k\ne 0$. The condition $\omega ^1\wedge \dots \wedge \omega ^k\ne 0$ is called the independence condition. For all information concerning exterior differential systems, the reader is referred to [21], and also to [22]. Remark that although the methods for investigation of integral manifolds may be very sophisticated, we will solely use conditions based on Frobenius’ theorem applied to suitable submanifolds.

From now on, we assume that (M, H, g) is a special sub-Riemannian manifold of dimension 3. Below, we will set up an exterior differential system whose integral manifolds correspond to local infinitesimal isometries of (M, H, g).

Fix an $\mathfrak {i}^*$-regular point $q_0\in M$ and let U be an $\mathfrak {i}$-special neighborhood of $q_0$ such that the function $U\ni p\longrightarrow \dim \mathfrak {i}^*(p)$ is constant. From now on, we restrict our considerations to the subset U about which we will assume that is as small as we wish. We will need some preparation results. Denote by $e_1,e_2$ a positively oriented orthonormal frame for H on U. Let $s:U\longrightarrow O_{H,g}(M)$, $s(q)=(q;e_1(q),e_2(q))$, be the corresponding (local) section of $O_{H,g}(M)$. If $\theta$ is the canonical form on $O_{H,g}(M)$, then, define

$$\begin{aligned} \omega ^i = s^*\theta ^i \text {.} \end{aligned}$$

Recall that by $\alpha$, we denote the normalized contact form while $\xi$ is the Reeb vector field. Then, clearly, $\omega ^1,\omega ^2,\alpha$ is a coframe defined on U dual to the frame $e_1,e_2,\xi$. We introduce the metric and torsion-free connection as it is explained in Sect. 3 with the connection form $\omega = (\omega ^i_j)$. After pulling back the structure Eqs. (3.9) for this connection by the section s we obtain

$$\begin{aligned} \begin{array}{l} \hbox {d}\omega ^1 = -\varphi \wedge \omega ^2 \\ \hbox {d}\omega ^2 = \varphi \wedge \omega ^1 \\ \hbox {d}\varphi = K\omega ^1\wedge \omega ^2 \end{array}\text {,} \end{aligned}$$

(4.1)

on U, where by $\varphi$ we denote $s^*\omega ^1_2$. Moreover we have $d\alpha = \omega ^1\wedge \omega ^2$. Remark that K is a smooth function which is defined on the whole M and does not depend on the choice of the frame $e_1,e_2$, as it follows from the considerations conducted in the previous section. We will find an explicit expression for K which will be used in Sect. 5 to estimate the dimension of the algebra of local infinitesimal isometries. Let us introduce the following notation:

$$\begin{aligned}{}[e_1,e_2] = C^1_{12}e_1 + C^2_{12}e_2 + C^3_{12}\xi \end{aligned}$$

(4.2)

and

$$\begin{aligned} {[}\xi ,e_1] = \gamma e_2, \; [\xi ,e_2] = -\gamma e_1 \end{aligned}$$

(4.3)

(cf. Proposition 2.1), where $C^i_{12}, \gamma$, $i=1,2,3$, are smooth functions on U. Using $d\alpha = \omega ^1\wedge \omega ^2$, we find $C^3_{12} = -1$.

Remark 4.1

Using the above notation, the following formula holds true:

$$\begin{aligned} K = e_1 \left( C^2_{12} \right) - e_2 \left( C^1_{12} \right) - \left( C^1_{12} \right) ^2 - \left( C^2_{12} \right) ^2 - \gamma \text {.} \end{aligned}$$

(4.4)

Proof

Indeed, $K = d\varphi (e_1,e_2)$ and the formula follows from the relations $\varphi (e_1) = C^1_{12}$, $\varphi (e_2) = C^2_{12}$, $\varphi (\xi ) = -\gamma$ which can be obtained from (4.1). $\square$

Note that K agrees with the invariant introduced by Agrachev in [1], and our formula (4.4) coincides with formula (14) for $\kappa$ in [2]. There is also a relation between (4.4) and the sectional curvature in the direction H of a certain connection on M as it is described by formula (2.4) in [13].

Let Z be a non-trivial infinitesimal isometry. We write

$$\begin{aligned} Z = Z^1e_1 + Z^2e_2 + Z^3\xi \end{aligned}$$

and we will calculate the differentials $dZ^i$. We start from $Z^3 = \alpha (Z)$. Since $\alpha$ is preserved by isometries, we have

$$\begin{aligned} 0 = {\mathcal {L}}_Z\alpha = \hbox {d} (\alpha (Z)) + i_Z\,{\circ }\,\hbox {d} \alpha = \hbox {d} Z^3 + i_Z \left( \omega ^1\wedge \omega ^2 \right) \end{aligned}$$

which finally gives

$$\begin{aligned} \hbox {d}Z^3 = Z^2\omega ^1 - Z^1\omega ^2\text {.} \end{aligned}$$

(4.5)

In order to calculate $dZ^i$, $i=1,2$, we use

$$\begin{aligned} {\mathcal {L}}_Z \left( \omega ^1\,{\circ }\,\omega ^1 + \omega ^2\,{\circ }\,\omega ^2 \right) = 0 \end{aligned}$$

(4.6)

(cf. Proposition 2.2) and proceed similarly as in [15]. So

$$\begin{aligned} {\mathcal {L}}_Z\omega ^1 = \hbox {d} Z^1 -\varphi (Z)\omega ^2 + Z^2\varphi , \;\;{\mathcal {L}}_Z\omega ^2 = \hbox {d} Z^2 +\varphi (Z)\omega ^1 - Z^1\varphi \text {,} \end{aligned}$$

and consequently

$$\begin{aligned} \frac{1}{2} {\mathcal {L}}_Z \left( \omega ^1\,{\circ }\,\omega ^1 + \omega ^2\,{\circ }\omega ^2 \right) = \left( \hbox {d} Z^1 + Z^2\varphi \right) \,{\circ }\omega ^1 + \left( \hbox {d} Z^2 - Z^1\varphi \right) \,{\circ }\,\omega ^2\text {.} \end{aligned}$$

This leads us to define functions $Z^i_j$ with formulas

$$\begin{aligned} \begin{array}{l} \hbox {d} Z^1 + Z^2\varphi = Z^1_1\omega ^1 + Z^1_2\omega ^2 + Z^1_3\alpha \\ \hbox {d} Z^2 - Z^1\varphi = Z^2_1\omega ^1 + Z^2_2\omega ^2 + Z^2_3\alpha \end{array}\text {.} \end{aligned}$$

Now, the condition (4.6) takes the form

$$\begin{aligned} Z^1_1\omega ^1\,{\circ }\,\omega ^1 + Z^2_2\omega ^2\,{\circ }\,\omega ^2 + \left( Z^1_2 + Z^2_1 \right) \omega ^1\,{\circ }\,\omega ^2 + Z^1_3\omega ^1\,{\circ }\,\alpha + Z^2_3\omega ^2\,{\circ }\,\alpha = 0 \end{aligned}$$

which implies

$$\begin{aligned} Z^1_1 = 0, \;\; Z^2_2 = 0, \;\; Z^1_2 + Z^2_1 = 0, \;\; Z^1_3 = Z^2_3 = 0\text {,} \end{aligned}$$

and consequently

$$\begin{aligned} \begin{array}{l} \hbox {d} Z^1 = -Z^2\varphi + Z^1_2\omega ^2 \\ \hbox {d} Z^2 = Z^1\varphi - Z^1_2\omega ^1 \end{array}\text {.} \end{aligned}$$

Differentiating two last equations and using (4.1), we obtain

$$\begin{aligned} \hbox {d} Z^1_2 = KZ^2\omega ^1 - KZ^1\omega ^2\text {.} \end{aligned}$$

Let S be a 7-dimensional manifold with coordinates $x,y,z,Z^1,Z^2,Z^3,Z^1_2$, where x, y, z are coordinates on U chosen in such a way that $x(q_0) = y(q_0) = z(q_0) = 0$, and $|Z^i|,|Z^1_2|$ can be supposed to be arbitrarily small. Let us define the following 1-forms

$$\begin{aligned} \begin{array}{l} \sigma ^1 = \hbox {d} Z^1 + Z^2\varphi -Z^1_2\omega ^2 \\ \sigma ^2 = \hbox {d} Z^2 - Z^1\varphi + Z^1_2\omega ^1\\ \sigma ^3 = \hbox {d} Z^3 - Z^2\omega ^1 + Z^1\omega ^2 \\ \sigma ^4 = \hbox {d} Z^1_2 - KZ^2\omega ^1 + KZ^1\omega ^2 \end{array} \end{aligned}$$

(4.7)

on S and consider the exterior differential system ${\mathcal {I}}$ generated by $\sigma ^1,\dots ,\sigma ^4$ with independence condition

$$\begin{aligned} \hbox {d} x\wedge \hbox {d} y\wedge \hbox {d} z \ne 0 \end{aligned}$$

or equivalently

$$\begin{aligned} \omega ^1\wedge \omega ^2\wedge \alpha \ne 0\text {.} \end{aligned}$$

(4.8)

From now on, we study the system

$$\begin{aligned} \{{\mathcal {I}},\,\omega ^1\wedge \omega ^2\wedge \alpha \} \end{aligned}$$

(4.9)

and its integral manifolds. We point out here that the condition (4.8) is in fact superfluous. Actually, one can choose the basis of the distribution $\bigcap _{i=1}^4\text { ker }\sigma ^i$ in the form

$$\begin{aligned} \begin{array}{l} X_1 = e_1 - C^1_{12}Z^2\frac{\partial }{\partial Z^1} + \left( C^1_{12}Z^1 - Z^1_2 \right) \frac{\partial }{\partial Z^2} + Z^2\frac{\partial }{\partial Z^3} + KZ^2\frac{\partial }{\partial Z^1_2} \\ X_2 = e_2 + \left( Z^1_2 - C^2_{12}Z^2 \right) \frac{\partial }{\partial Z^1} + C^2_{12}Z^1\frac{\partial }{\partial Z^2} - Z^1\frac{\partial }{\partial Z^3} - KZ^1\frac{\partial }{\partial Z^1_2} \\ X_3 = \xi + \gamma Z^2\frac{\partial }{\partial Z^1} - \gamma Z^1\frac{\partial }{\partial Z^2} \end{array}\text {;} \end{aligned}$$

(4.10)

see (4.2), (4.3)

The key observation in this section is included the following

Proposition 4.1

Local infinitesimal isometries of the manifold $(U,H_{|U},g_{|U})$ are in one-to-one correspondence with integral manifolds of the system (4.9).

Proof

Indeed, suppose that an integral manifold, say of class $C^1$, of (4.9) is given by

$$\begin{aligned} U\ni (x,y,z)\longrightarrow (x,y,z,Z^1(x,y,z),Z^3(x,y,z),Z^3(x,y,z),Z^1_2(x,y,z)) \text {,} \end{aligned}$$

(4.11)

where we identify U with the ball $\{(x,y,z):\; x^2+y^2+z^2<\varepsilon \}$ for small $\varepsilon >0$. To begin with let us notice that since the distribution $\bigcap _{i=1}^4\text { ker }\sigma ^i$ is smooth, the functions $Z^i$ are also smooth. Let us write

$$\begin{aligned} Z = Z^1e_1 + Z^2e_2 + Z^3\xi \text {.} \end{aligned}$$

(4.12)

Then, for instance,

$$\begin{aligned} \begin{array}{l} [Z,e_1] \equiv \left( -\hbox {d} Z^3(e_1) - Z^2C^3_{12} \right) \xi \;(\mathrm {mod\;} H) \\ {[Z,e_2]} \equiv \left( -\hbox {d} Z^3(e_2) + Z^1C^3_{12} \right) \xi \;(\mathrm {mod\;} H) \end{array}\text {.} \end{aligned}$$

(4.13)

Remembering that $C^3_{12} = -1$, (4.13) and (4.5) imply $[Z,e_i]\in \textrm{Sec}(H)$, so Z satisfies condition (i) of Proposition 2.1. Similarly, using the relations stated in the proof of Remark 4.1, we can prove that condition (ii) is also fulfilled.

Conversely, if (4.12) is an infinitesimal isometry, then, obviously (4.11) is an integral manifold of (4.9). $\square$

Now, we aim to estimate the dimension of $\mathfrak {i}^*(q_0)$. We start from computing the exterior derivatives of $\sigma ^i$’s. To this end let us notice that from the third equation in (4.1), it follows that $dK\wedge \omega ^1\wedge \omega ^2 = 0$, and thus, the differential of K can be written as follows:

$$\begin{aligned} dK = K_1\omega ^1 + K_2\omega ^2\text {,} \end{aligned}$$

(4.14)

where $K_1,K_2$ are smooth functions on U. After computations, we obtain

$$\begin{aligned} \begin{array}{l} d\sigma ^1 = \sigma ^2\wedge \varphi - \sigma ^4\wedge \omega ^2 \\ d\sigma ^2 = -\sigma ^1\wedge \varphi + \sigma ^4\wedge \omega ^1 \\ d\sigma ^3 = \sigma ^1\wedge \omega ^2 - \sigma ^2\wedge \omega ^1 \\ d\sigma ^4 = K\sigma ^1\wedge \omega ^2 - K\sigma ^2\wedge \omega ^1 + \left( K_1Z^1 + K_2Z^2 \right) \omega ^1\wedge \omega ^2 \end{array}\text {.} \end{aligned}$$

(4.15)

Let

$$\begin{aligned} \Phi = K_1Z^1 + K_2Z^2\text {,} \end{aligned}$$

and let us set $S' = \{\Phi = 0\}\subset S$. Below, we will need the following lemma.

Lemma 4.1

Suppose that L is an integral manifold of (4.9). Then, $L\subset S'$.

Proof

Indeed, the form $d\sigma ^4 = K\sigma ^1\wedge \omega ^2 - K\sigma ^2\wedge \omega ^1 +\Phi \,\omega ^1\wedge \omega ^2$ as well as $\sigma ^i$’s vanish on L, hence $\Phi = 0$ on L. $\square$

Suppose at first that $\Phi$ vanishes identically on S. This means that $K_i = 0$ identically, $i=1,2$, and consequently, K is constant. In such a case (4.9) satisfies the Frobenius condition, hence, there exists a 4-parameter family of integral manifolds of (4.9), that is to say $\dim \mathfrak {i}(U) = \dim \mathfrak {i}^*(q_0) = 4$.

Next suppose that $\Phi$ does not vanish identically. Clearly, $N = \{Z^1 = Z^2 = 0\}\subset S'$. Since $K_i$ does not depend on $Z^1,Z^2$, there are points belonging to $N\cap S$ at which at least one of $K_1,K_2$ does not vanish and hence, $d\Phi$ does not vanish. Shrinking S suitably (and moving slightly $q_0$ if necessary), we can assume that $d\Phi$ does not vanish on S. Then, $S'=\{\Phi = 0\}$ is a smooth submanifold in S. We compute $d\Phi$. Differentiating (4.14) and using (4.1), we have

$$\begin{aligned} (\hbox {d} K_1 + K_2\varphi )\wedge \omega ^1 + (\hbox {d} K_2 - K_1\varphi )\wedge \omega ^2 = 0\text {,} \end{aligned}$$

so by Cartan’s lemma, we can write

$$\begin{aligned} \begin{array}{l} \hbox {d} K_1 = K_{11}\omega ^1 + K_{12}\omega ^2 - K_2\varphi \\ \hbox {d} K_2 = K_{12}\omega ^1 + K_{22}\omega ^2 + K_1\varphi \end{array} \end{aligned}$$

for some smooth functions $K_{ij}:U\longrightarrow \mathbb {R}$. Now,

$$\begin{aligned} \hbox {d} \Phi&= Z^1\hbox {d} K_1 + Z^2\hbox {d} K_2 + K_1\sigma ^1 + K_1 \left( -Z^2\varphi +Z^1_2\omega ^2 \right) \\&\quad + K_2\sigma ^2 + K_2 \left( Z^1\varphi -Z^1_2\omega ^1 \right) \end{aligned}$$

which gives

$$\begin{aligned} d\Phi = \Phi _1\omega ^1 + \Phi _2\omega ^2 + K_1\sigma ^1 + K_2\sigma ^2 \end{aligned}$$

(4.16)

with

$$\begin{aligned} \begin{array}{l} \Phi _1 = K_{11}Z^1 + K_{12}Z^2 - K_2Z^1_2 \\ \Phi _2 = K_{12}Z^1 + K_{22}Z^2 + K_1Z^1_2 \end{array}\text {.} \end{aligned}$$

We will show that $\Phi _1$ and $\Phi _2$ cannot be both identically zero on $S'$. Indeed, if $\Phi _1 = \Phi _2 = 0$ on $S'$, then, $K_1Z^1_2 = K_2Z^1_2 = 0$ on $S\cap N$, and therefore, $K_1 = K_2 = 0$ on $S\cap N$ which contradicts $d\Phi \ne 0$ on S.

Lemma 4.2

Every integral manifold of (4.9) is contained in $S''=\{Q\in S:\;\Phi (Q) = \Phi _1(Q) = \Phi _2(Q) = 0\}$.

Proof

Denote by $L\subset S$ an integral manifold of (4.9). By Lemma 4.1, we already know that $L\subset \{Q\in S:\; \Phi (Q) = 0\}$. Let $Q\in L$ and $v\in T_QL$. Then, $\sigma ^i(v) = 0$, $d_Q\Phi (v) = 0$; hence, $\Phi _1(Q)\omega ^1(v) + \Phi _2(Q)\omega ^2(v) = 0$, and this is so for every $v\in T_QL$. Since the pullbacks of $\omega ^1,\omega ^2,\alpha$ to L are pointwise linearly independent, we infer that $\Phi _1$ and $\Phi _2$ vanish on L. $\square$

Now, the remaining procedure is easy. If

$$\begin{aligned} (x,y,z)\longrightarrow \left( x,y,z, Z^1(x,y,z),Z^2(x,y,z),Z^3(x,y,z),Z^1_2(x,y,z) \right) \end{aligned}$$

is an integral manifold, then in particular the system of equations

$$\begin{aligned} \begin{array}{l} K_1Z^1 + K_2Z^2 = 0 \\ K_{11}Z^1 + K_{12}Z^2 -K_2Z^1_2 = 0 \\ K_{12}Z^1 + K_{22}Z^2 + K_1Z^1_2 = 0. \end{array} \end{aligned}$$

(4.17)

must be satisfied. Let us investigate the rank of the matrix

$$\begin{aligned} J = \begin{pmatrix} K_1 &{} \quad K_2 &{} \quad 0 \\ K_{11} &{} \quad K_{12} &{} \quad -K_2 \\ K_{12} &{} \quad K_{22} &{} \quad K_1 \end{pmatrix}\text {.} \end{aligned}$$

If J has rank 3 at a point of U, then, we can suppose that so it is on the whole U. Indeed, we agreed to assume U to be as small as we wish. Moreover, we can slightly shift $q_0$ without changing the number $\dim \mathfrak {i}^*(q_0)$. Then, (4.17) has only the trivial solution which corresponds to the Reeb field and hence, $\dim \mathfrak {i}^*(q_0) = 1$. Next, suppose $\mathrm {rank\,}J < 3$ on U. If J has rank 2 at a point of U, then, again we can assume that so it is on the whole of U. Consequently, (4.17) has the trivial solution (which again corresponds to the Reeb field) and a 1-parameter family of non-trivial solutions, which may or may not give rise to an isometry (cf. (6) Proposition 2.3). Therefore, $\dim \mathfrak {i}^*(q_0)$ equals 1 or 2 in this case. Finally, $\mathrm {rank\,}J$ cannot be equal to either 0 nor 1 because then $K_1^2 + K_2^2 = 0$ contradicting our assumptions.

In this way, we excluded 3 as the possible value for $\dim \mathfrak {i}(U) = \dim \mathfrak {i}^*(q_0)$. During the proof, we also established the following corollary.

Corollary 4.1

Let (M, H, g) be a special 3-dimensional sub-Riemannian manifold and let $q_0$ be an $\mathfrak {i}^*$-regular point. Then, $\dim \mathfrak {i}^*(q_0) = 4$ if and only if the function K appearing in (4.1) is constant in a neighborhood of $q_0$.

Using the paper [2], we can draw more specific conclusions from the above considerations.

Corollary 4.2

Let (M, H, g) be a special 3-dimensional sub-Riemannian manifold and let $q_0$ be an $\mathfrak {i}^*$-regular point. Then, $\dim \mathfrak {i}^*(q_0) = 4$ if and only if (M, H, g) is locally isometric in a neighborhood of $q_0$ (modulo dilations) to either the Heisenberg group, $\textrm{SU}(2)$ or $\textrm{SL}(2)$ with the suitable left-invariant sub-Riemannian structure.

5 Examples

In the previous section, we showed that $\mathfrak {i}^*(q)$ cannot have dimension 3 for an $\mathfrak {i}^*$-regular point q. In order to finish the proof of Theorem 1.1, we give examples of special sub-Riemannian manifolds (M, H, g) of dimension 3 with $\dim \mathfrak {i}^*(q) = 1,2,4$, respectively. In all examples, below, $H = \textrm{Span}\{e_1,e_2\}$ and $e_1,e_2$ is a g-orthonormal basis for H.

5.1 Dimension 4

Let $M = \mathbb {R}^3$ with coordinates x, y, z, $e_1 = \frac{\partial }{\partial x} + \frac{1}{2} y\frac{\partial }{\partial z}$, $e_2 = \frac{\partial }{\partial y} - \frac{1}{2} y\frac{\partial }{\partial z}$. The normalized contact form is

$$\begin{aligned} \alpha = 2\hbox {d}z - y\hbox {d}x + y\hbox {d}y\text {,} \end{aligned}$$

the Reeb field is equal to $\frac{1}{2}\frac{\partial }{\partial z}$. Let us write $Z = P(x,y,z)\frac{\partial }{\partial x} + Q(x,y,z)\frac{\partial }{\partial y} + R(x,y,z)\frac{\partial }{\partial z}$. In order to compute infinitesimal isometries, we use Proposition 2.1 which reduces the problem to solving a system of five linear PDS’s of first order for three unknown functions P, Q, R. These equations are

$$\begin{aligned} \begin{array}{l} \alpha ([Z,e_1]) = 0 \\ \alpha ([Z,e_2]) = 0 \\ g([Z,e_1],e_1) = 0 \\ g([Z,e_2],e_2) = 0 \\ g([Z,e_1],e_2) + g(e_1,[Z,e_2]) = 0 \end{array}\text {.} \end{aligned}$$

(5.1)

By Proposition 2.3, the Reeb field commutes with isometries, so P, Q, R are, in fact, functions of two variables x and y. Equation (5.1) take the form

$$\begin{aligned} yP_x - yQ_x + Q - 2R_x = 0\text {,} \end{aligned}$$

(5.2a)

$$\begin{aligned} yP_y - yQ_y - Q - 2R_y = 0\text {,} \end{aligned}$$

(5.2b)

$$\begin{aligned} P_x = 0\text {,} \end{aligned}$$

(5.2c)

$$\begin{aligned} Q_y = 0\text {,} \end{aligned}$$

(5.2d)

$$\begin{aligned} Q_x + P_y = 0\text {.} \end{aligned}$$

(5.2e)

By (5.2c) and (5.2d), $P = P(y)$ and $Q = Q(x)$. By (5.2e) $P_{yy} = 0$ and $Q_{xx} = 0$, hence, $P = C_1 + C_2y$ and $Q = C_3 - C_2x$. Now, using (5.2a), (5.2b), we make sure that all the derivatives of R of order greater than or equal to 3 vanish. The same equations permit us to compute the first and second-order derivatives of R at (0, 0). This results in $R = C_4 + \frac{1}{2}C_3(x-y) + \frac{1}{4}C_2(y^2 + 2xy - x^2)$. Consequently, each point $q\in M$ is $\mathfrak {i}^*$-regular, $\dim \mathfrak {i}^*(q) = 4$, and $\mathfrak {i}^*(q)$ is generated, for instance, by the germs at q of the fields

$$\begin{aligned} \frac{\partial }{\partial x},\;\; y\frac{\partial }{\partial x} - x\frac{\partial }{\partial y} + \frac{1}{4} \left( y^2 + 2xy - x^2 \right) \frac{\partial }{\partial z}, \;\; \frac{\partial }{\partial y} + \frac{1}{2}(x-y)\frac{\partial }{\partial z}, \;\; \frac{\partial }{\partial z} \text {.} \end{aligned}$$

Note that if we are interested only in the dimension $\dim \mathfrak {i}^*(q)$, then, it is enough to compute the curvature function K, which in this case is $K = 0$, and use Corollary 4.1.

5.2 Dimension 2

$M = \{(x,y,z)\in \mathbb {R}^3:\; x > 0\}$, $e_1 = \frac{\partial }{\partial x} + x^2\frac{\partial }{\partial y}$, $e_2 = -x^2\frac{\partial }{\partial y} + x\frac{\partial }{\partial z}$. The normalized contact form is

$$\begin{aligned} \alpha = \hbox {d} z - x\hbox {d} x + \frac{1}{x} \hbox {d} y, \end{aligned}$$

and the Reeb is $\xi = \frac{\partial }{\partial z}$. Instead of solving Eq. (5.1), we will calculate K. To this end notice that $[e_1,e_2] = \frac{2}{x} e_2 - \xi$, $[\xi ,e_i] = 0$, therefore $C^1_{12} = 0$, $C^2_{12} = \frac{2}{x}$, $\gamma = 0$, and the formula (4.4) gives

$$\begin{aligned} K = -\frac{6}{x^2} \text {.} \end{aligned}$$

It follows that K is not constant on M, and at the same time, it is seen that $\frac{\partial }{\partial y}$ and $\frac{\partial }{\partial z}$ are infinitesimal isometries. Considerations from the previous section ensure that every point $q\in M$ is $\mathfrak {i}^*$-regular and $\dim \mathfrak {i}^*(q) = 2$.

5.3 Dimension 1

$M = \mathbb {R}^3$, $e_1 = e^{-x}\frac{\partial }{\partial x} + y^2\frac{\partial }{\partial y} + e^{-x}y\frac{\partial }{\partial z}$, $e_2 = e^x\frac{\partial }{\partial y}$. The normalized contact form and the Reeb vector field are, respectively,

$$\begin{aligned} \alpha = \hbox {d}z - y\hbox {d}x \end{aligned}$$

and $\xi = \frac{\partial }{\partial z}$, and it is clear that $\xi$ is an infinitesimal isometry. Again, we will not solve Eq. (5.1), but we will calculate K. We start from $[e_1,e_1] = (e^{-x} - 2y)e_2 - \xi$ which yields $C^1_{12} = 0$, $C^2_{12} = e^{-x} - 2y$. Next $[\xi ,e_i] = 0$ from which $\gamma = 0$. Moreover, using (4.14), we see that $dK(\xi ) = 0$ and hence, $K = K(x,y)$. Thus, in our case, the formula (4.4) turns to be

$$\begin{aligned} K = -2e^{-2x} - 6y^2 + 4e^{-x}y\text {.} \end{aligned}$$

Now, we compute $K_i$, $K_{ij}$, $i,j=1,2$, using the relations $K_i = dK(e_i)$, $K_{11} = dK_1(e_1) + K_2C^1_{12}$, $K_{12} = dK_1(e_2) + K_2C^2_{12}$, $K_{22} = dK_2(e_2) - K_1C^2_{12}$. Finally, we obtain

$$\begin{aligned} \det \begin{pmatrix} K_1(0,0) &{} \quad K_2(0,0) &{} \quad 0 \\ K_{11}(0,0) &{} \quad K_{12}(0,0) &{} \quad -K_2(0,0) \\ K_{12}(0,0) &{} \quad K_{22}(0,0) &{} \quad K_1(0,0) \end{pmatrix} = \det \begin{pmatrix} 4 &{} \quad 4 &{} \quad 0 \\ -12 &{} \quad 0 &{} \quad -4 \\ 0 &{} \quad -16 &{} \quad 4 \end{pmatrix} = -64 \ne 0 \end{aligned}$$

which, according to the above considerations, proves that (0, 0, 0) is an $\mathfrak {i}^*$-regular point and $\dim \mathfrak {i}^*(0,0,0) = 1$.

Remark 5.1

In the case, where K is not constant, but

$$\begin{aligned} \det \begin{pmatrix} K_1 &{} \quad K_2 &{} \quad 0 \\ K_{11} &{} \quad K_{12} &{} \quad -K_2 \\ K_{12} &{} \quad K_{22} &{} \quad K_1 \end{pmatrix} = 0 \end{aligned}$$

in a neighborhood of a point q, there are two possibilities: either $\dim \mathfrak {i}^*(q) = 1$ or $\dim \mathfrak {i}^*(q) = 2$. Making one more differentiation, we can express $dK_{ij}$ via the certain functions $K_{ijk}$ and than write out an explicit condition in terms of $K_{ijk}$’s under which $\dim \mathfrak {i}^*(q) = 1$ (respectively, $\dim \mathfrak {i}^*(q) = 2$ $)$. However, we will not do it in this paper.

6 Structures without non-trivial isometries

Theorem 1.1 concerns only special sub-Riemannian manifolds. Of course not every contact sub-Riemannian manifold is special. In fact, it is not hard to construct contact sub-Riemannian manifold without non-trivial isometries, similarly as above:

In order to produce such an example, we use [14]. Consider a sub-Riemannian structure (H, g) on M, where M is a neighborhood of the origin in $\mathbb {R}^3$, given by an orthonormal basis in the form $e_1 = (1+yA)\frac{\partial }{\partial x} + y^2B\frac{\partial }{\partial z}$, $e_2 = \frac{\partial }{\partial y}$, A and B are smooth functions, $B(0,0,z) = 1$. The distribution $H = \textrm{Span}\{e_1,e_2\}$ is a Martinet distribution with the Martinet surface $\{y = 0\}$, where coordinates x, y, z are normal in the sense [14]. According to the mentioned paper, infinitesimal isometries have the form

$$\begin{aligned} P(z)\frac{\partial }{\partial x} + Q(z)\frac{\partial }{\partial z}\text {,} \end{aligned}$$

where P and Q must satisfy the system of equations

$$\begin{aligned} \begin{array}{l} A_xP + A_z Q = yBP' \\ A_{xx}P + A_{xz}Q = yB_xP \end{array}\text {,} \end{aligned}$$

where the second equation is obtained by differentiating the first equation (15) in [14]. Therefore, if A and B are such that

$$\begin{aligned} A_{xx}(0,0,0)A_z(0,0,0) - A_{xz}(0,0,0)A_x(0,0,0) \ne 0 \end{aligned}$$

(6.1)

then $P = Q = 0$ (one can assume that $y = 0$), so non-trivial isometries do not exist.

Now, we let $M_0 = \{(x,y,z)\in M:\; y>0\}$ and denote by $H_0$ (respectively, $g_0$) the restriction of H (respectively, g) to $M_0$. $(M_0,H_0,g_0)$ is a contact manifold. By [14] every infinitesimal isometry of $(M_0,H_0,g_0)$ extends to an infinitesimal isometry of (M, H, g); therefore, $\mathfrak {i}(M_0) = \{0\}$.