A note on flat nonholonomic Riemannian structures on three-dimensional Lie groups

Abstract

We consider flat nonholonomic Riemannian manifolds, i.e., those whose associated parallel transport (induced by the nonholonomic connection) is path-independent. We first characterize flatness for structures on three-dimensional manifolds, and hence classify the flat left-invariant structures on simply connected Lie groups.

1 Introduction

The equivalence problem for three-dimensional nonholonomic Riemannian manifolds has been considered in Barrett et al. (2016). In particular, the left-invariant nonholonomic Riemannian structures on the three-dimensional simply connected Lie groups were classified, up to nonholonomic isometry and rescaling. Moreover, the equivalence classes were described in terms of isometric invariants. In this paper we consider the flat three-dimensional nonholonomic Riemannian manifolds, i.e., those whose associated parallel transport is path-independent. We first characterize flatness in three dimensions. Hence, by making use of the classification in Barrett et al. (2016), we are able to classify the flat left-invariant structures (on the three-dimensional simply connected Lie groups).

The paper is organized as follows. In Sect. 2 we present the necessary elements of nonholonomic Riemannian geometry. Specifically, in Sect. 2.1 we recall the fundamental existence and uniqueness result for the nonholonomic connection, as well as the definitions of nonholonomic geodesics, nonholonomic isometries and the exterior covariant derivative associated to the nonholonomic connection. We also consider left-invariant nonholonomic Riemannian structures on Lie groups. Section 2.2 recalls the Schouten curvature tensor (which is canonically associated to every nonholonomic Riemannian structure) and some related tensors (obtained by contracting the Schouten tensor), whereas Sect. 2.3 introduces the parallel transport map associated to the nonholonomic connection and defines the notion of a flat nonholonomic Riemannian structure. In Sect. 3 we specialize to the case of three-dimensional nonholonomic Riemannian manifolds. Specifically, we recall the salient results of Barrett et al. (2016): the isometric invariants (Sect. 3.1) and the classification of left-invariant structures (Sect. 3.2). Section 4 contains the main results of the paper. In the first part (Sect. 4.1) we characterize, by means of the exterior covariant derivative, the flat nonholonomic Riemannian structures on three-dimensional manifolds. In the second part (Sect. 4.2) we use the foregoing characterization, together with the classification in Barrett et al. (2016), to obtain a classification of the flat left-invariant structures.

Throughout, we follow the summation convention on repeated indices. Unless stated otherwise, the following ranges on indices are used: $i,j,k = 1,\ldots ,n$ (or $i,j,k = 0,1,2$ in Sect. 4) and $a,b,c = 1,\ldots ,r$. We also assume that all manifolds, functions, vector fields, etc. are smooth, i.e., of class $\mathcal {C}^{\infty }$.

2 Nonholonomic Riemannian structures

2.1 Basic concepts

A nonholonomic Riemannian structure is a quadruple $\mathfrak {S}= (\mathsf {M},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$, where $\mathsf {M}$ is an n-dimensional (connected) manifold, $\mathcal {D}$ is a rank r nonintegrable distribution on $\mathsf {M}$, $\mathcal {D}^{\perp }$ is a distribution complementary to $\mathcal {D}$ and $\mathbf {g}$ is a (positive definite) fiber metric on $\mathcal {D}$. We shall assume that $\mathcal {D}$ is completely nonholonomic, i.e., if $\mathcal {D}^1 \subseteq \mathcal {D}^2 \subseteq \cdots$ is the flag of $\mathcal {D}$, where

$$\begin{aligned} \mathcal {D}^1 = \mathcal {D}, \quad \mathcal {D}^{i+1} = \mathcal {D}^i + [\mathcal {D}^i,\mathcal {D}^i]\quad \hbox { for}\ i \ge 1, \end{aligned}$$

then there exists $N \ge 2$ such that $\mathcal {D}^{N-1} \subsetneq T\mathsf {M}$ and $\mathcal {D}^N = T\mathsf {M}$. If $N = 2$, then $\mathcal {D}$ is called strongly nonholonomic. Every nonintegrable distribution on a three-dimensional manifold is strongly nonholonomic. It is well known that, if $\mathcal {D}$ is completely nonholonomic, then any two points in $\mathsf {M}$ can be joined by a $\mathcal {D}$-curve (i.e., an integral curve of $\mathcal {D}$). Let $\mathscr {P}$ be the projection onto $\mathcal {D}$ along $\mathcal {D}^{\perp }$ and let $\mathscr {Q}$ be the complementary projection. For convenience, we shall denote the projected Lie bracket $\mathscr {P}([\,\cdot \,,\,\cdot \,])$ by $\llbracket \,\cdot \,,\,\cdot \,\rrbracket$.

Associated to $\mathfrak {S}$ is a unique affine connection $\nabla : \varGamma (\mathcal {D}) \times \varGamma (\mathcal {D}) \rightarrow \varGamma (\mathcal {D})$ (where $\varGamma (\mathcal {D})$ denotes the space of sections of $\mathcal {D}$), called the nonholonomic connection. Here “affine connection” means that $\nabla$ is tensorial in its first argument and a derivation in the second. Like the Levi-Civita connection, the nonholonomic connection may be characterized as the unique connection that is both metric and torsion free, where the torsion of $\nabla$ is the (1, 2)-tensor $T : \varGamma (\mathcal {D}) \times \varGamma (\mathcal {D}) \rightarrow \varGamma (\mathcal {D})$ given by

$$\begin{aligned} T(X,Y) = \nabla _XY - \nabla _YX - \llbracket X,Y \rrbracket ,\qquad X,Y \in \varGamma (\mathcal {D}). \end{aligned}$$

Specifically, we have the following result (see, e.g., Langerock 2001).

Proposition 1

Let $\mathfrak {S}= (\mathsf {M},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$ be a nonholonomic Riemannian structure. There exists a unique affine connection $\nabla : \varGamma (\mathcal {D}) \times \varGamma (\mathcal {D}) \rightarrow \varGamma (\mathcal {D})$ such that $\nabla \mathbf {g} \equiv 0$ and $T \equiv 0$, i.e.,

$$\begin{aligned} Z[\mathbf {g}(X,Y)] = \mathbf {g}(\nabla _ZX,Y) + \mathbf {g}(X,\nabla _ZY) \quad \text {and}\quad \nabla _XY - \nabla _YX = \llbracket X,Y \rrbracket \end{aligned}$$

for every $X,Y,Z \in \varGamma (\mathcal {D})$. Furthermore, $\nabla$ is characterized by Koszul’s formula:

$$\begin{aligned} 2\,\mathbf {g}(\nabla _XY,Z)&= X[\mathbf {g}(Y,Z)] + Y[\mathbf {g}(X,Z)] - Z[\mathbf {g}(X,Y)]\nonumber \\&\quad + \mathbf {g}(\llbracket X,Y \rrbracket ,Z) - \mathbf {g}(\llbracket X,Z \rrbracket ,Y) - \mathbf {g}(\llbracket Y,Z \rrbracket ,X) \end{aligned}$$

(1)

for every $X,Y,Z \in \varGamma (\mathcal {D})$.

A $\mathcal {D}$-curve $\gamma$ is called a nonholonomic geodesic of $\mathfrak {S}$ if it is a geodesic of the nonholonomic connection $\nabla$, i.e., $\nabla _{{\dot{\gamma }}}{\dot{\gamma }} = 0$.

Associated to $\nabla$ is an exterior derivative. Let $\varOmega ^k(\mathcal {D},\mathcal {D})$ be the space of $\mathcal {D}$-valued k-forms on $\mathcal {D}$. The $\mathscr {P}$-exterior covariant derivative, denoted $d^{\nabla }_{\mathscr {P}} : \varOmega ^k(\mathcal {D},\mathcal {D}) \rightarrow \varOmega ^{k+1}(\mathcal {D},\mathcal {D})$, is defined as follows:

• (i) If $U \in \varOmega ^0(\mathcal {D},\mathcal {D}) = \varGamma (\mathcal {D})$, then $d^{\nabla }_{\mathscr {P}}U(X) = \nabla _XU$ for every $X \in \varGamma (\mathcal {D})$.

• (ii) If $\varphi \in \varOmega ^k(\mathcal {D},\mathcal {D})$, $k \ge 1$, then

  $$\begin{aligned} d^{\nabla }_{\mathscr {P}}\varphi (X_0,\ldots ,X_k)&= \sum _{i=0}^k(-1)^i\nabla _{X_i}\varphi (X_0,\ldots ,{\widehat{X}}_i,\ldots ,X_k)\\&\quad + \sum _{0 \le i < j \le k} (-1)^{i+j}\varphi (\llbracket X_i,X_j \rrbracket ,X_0,\ldots ,{\widehat{X}}_i,\ldots ,{\widehat{X}}_j,\ldots ,X_k) \end{aligned}$$

  for every $X_0,\ldots ,X_k \in \varGamma (\mathcal {D})$. Here ${\widehat{X}}_i$ indicates the omission of that element.

In particular, for a $\mathcal {D}$-valued 1-form $\varphi$, we have

$$\begin{aligned} d^{\nabla }_{\mathscr {P}}\varphi (X,Y) = \nabla _X\varphi (Y) - \nabla _Y\varphi (X) - \varphi (\llbracket X,Y \rrbracket ), \end{aligned}$$

where $X,Y \in \varGamma (\mathcal {D})$. Note that the torsion of $\nabla$ is exactly the $\mathscr {P}$-exterior covariant derivative of the identity map $\text {id}_{\mathcal {D}} : \mathcal {D} \rightarrow \mathcal {D}$.

Two nonholonomic Riemannian structures $\mathfrak {S}= (\mathsf {M},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$ and $\mathfrak {S}' = (\mathsf {M}',\mathcal {D}',{\mathcal {D}'}^{\perp },\mathbf {g}')$ are said to be NH-isometric if there exists a diffeomorphism $\phi : \mathsf {M} \rightarrow \mathsf {M}'$ such that

$$\begin{aligned} \phi _*\mathcal {D} = \mathcal {D}',\quad \phi _*\mathcal {D}^{\perp } = {\mathcal {D}'}^{\perp }\quad \text {and}\quad \mathbf {g} = \phi ^*\mathbf {g}'. \end{aligned}$$

A map satisfying the above properties is called an NH-isometry. If $\phi$ is an NH-isometry, then $\nabla = \phi ^*\nabla '$; consequently, $\phi$ establishes a one-to-one correspondence between the nonholonomic geodesics of $\mathfrak {S}$ and $\mathfrak {S}'$. Furthermore, $\phi$ preserves the projection operators: $\phi _*\mathscr {P}(X) = \mathscr {P}'(\phi _*X)$ and $\phi _*\mathscr {Q}(X) = \mathscr {Q}'(\phi _*X)$ for every $X \in \varGamma (T\mathsf {M})$.

Let $\mathsf {G}$ be a Lie group with Lie algebra $\mathfrak {g} = T_{\mathbf {1}}\mathsf {G}$. A nonholonomic Riemannian structure $\mathfrak {S}= (\mathsf {G},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$ is said to be left invariant if every left translation $L_g$, $g \in \mathsf {G}$ is an NH-isometry. The simplest left-invariant structures are those whose nonholonomic geodesics are exactly the (left cosets of) one-parameter subgroups $t \mapsto g_0\exp (t X_0)$, $X_0 \in \mathcal {D}_{\mathbf {1}}$, $g_0 \in \mathsf {G}$. These are the nonholonomic Riemannian analogues of bi-invariant Riemannian metrics. For such a structure, the nonholonomic connection $\nabla$ is called a Cartan–Schouten connection. We have the following characterization of structures with Cartan–Schouten connections.

Proposition 2

(Barrett et al. 2016) Let $\mathfrak {S}= (\mathsf {G},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$ be a left-invariant nonholonomic Riemannian structure, with associated nonholonomic connection $\nabla$. The following statements are equivalent:

• (i) $\nabla$ is Cartan–Schouten.

• (ii) $\nabla _XX = 0$ for every left-invariant $X \in \varGamma (\mathcal {D})$.

• (iii) $\nabla _XY = \frac{1}{2}\llbracket X,Y \rrbracket$ for all left-invariant $X,Y \in \varGamma (\mathcal {D})$.

2.2 The Schouten curvature tensor

As $\mathcal {D}$ is nonintegrable, the usual (Riemannian) curvature tensor cannot be defined for $\nabla$. Instead, associated to every nonholonomic Riemannian structure is the (1, 3)-tensor field

$$\begin{aligned}&K : \varGamma (\mathcal {D}) \times \varGamma (\mathcal {D}) \times \varGamma (\mathcal {D}) \rightarrow \varGamma (\mathcal {D}),\\&\quad K(X,Y)Z = [\nabla _X,\nabla _Y]Z - \nabla _{\llbracket X,Y \rrbracket }Z - \llbracket \mathscr {Q}([X,Y]),Z\rrbracket , \end{aligned}$$

called the Schouten curvature tensor (see, e.g., Dragović and Gajić 2003). Let ${\widehat{K}}$ be the associated (0, 4)-tensor ${\widehat{K}}(W,X,Y,Z) = \mathbf {g}(K(W,X)Y,Z)$. We decompose ${\widehat{K}}$ into two components ${\widehat{R}}$ and ${\widehat{C}}$:

$$\begin{aligned} {\widehat{R}}(W,X,Y,Z) = \frac{1}{2}\big [{\widehat{K}}(W,X,Y,Z) - {\widehat{K}}(W,X,Z,Y)\big ], \quad {\widehat{C}} = {\widehat{K}} - {\widehat{R}}. \end{aligned}$$

${\widehat{R}}$ behaves analogously to the Riemannian tensor, in that it satisfies the same symmetries (i.e., ${\widehat{R}}$ is skew-symmetric in the first pair and last pair of arguments, is symmetric if one swaps the first pair of arguments with the last pair, and satisfies the first Bianchi identity). Accordingly, we can define a (0, 2)-tensor, also called the Ricci tensor, as follows:

$$\begin{aligned} {{\mathrm{Ric}}}: \varGamma (\mathcal {D}) \times \varGamma (\mathcal {D}) \rightarrow \mathcal {C}^{\infty }(\mathsf {M}),\quad {{\mathrm{Ric}}}(X,Y) = \sum _a {\widehat{R}}(X_a,X,Y,X_a). \end{aligned}$$

Here $(X_a)$ is an orthonormal frame for $\mathcal {D}$. Likewise, let A be the (0, 2)-tensor

$$\begin{aligned} A : \varGamma (\mathcal {D}) \times \varGamma (\mathcal {D}) \rightarrow \mathcal {C}^{\infty }(\mathsf {M}),\quad A(X,Y) = \sum _a {\widehat{C}}(X_a,X,Y,X_a). \end{aligned}$$

Although ${{\mathrm{Ric}}}$ is symmetric, the tensor A is generally not; accordingly, let $A_{sym}$ and $A_{skew}$ be the symmetric and skew-symmetric parts of A, respectively. Since NH-isometries preserve $\nabla$ and the projection operators $\mathscr {P}$ and $\mathscr {Q}$, it follows that K, ${\widehat{K}}$, ${\widehat{R}}$, ${\widehat{C}}$, ${{\mathrm{Ric}}}$, $A_{sym}$ and $A_{skew}$ are all preserved under NH-isometry.

2.3 Parallel transport

The nonholonomic connection $\nabla$ induces a parallel transport along $\mathcal {D}$-curves. A section V of $\mathcal {D}$ along a $\mathcal {D}$-curve $\gamma$ is parallel along$\gamma$ if $\nabla _{{\dot{\gamma }}}V = 0$. A vector field $X \in \varGamma (\mathcal {D})$ is parallel if $X\circ \gamma$ is parallel along $\gamma$ for every $\mathcal {D}$-curve $\gamma$; clearly, X is parallel if and only if $\nabla X \equiv 0$.

Proposition 3

Let $\gamma : [0,1] \rightarrow \mathsf {M}$ be a $\mathcal {D}$-curve and let $V_0 \in \mathcal {D}_{\gamma (0)}$. There exists a unique parallel section V of $\mathcal {D}$ along $\gamma$ such that $V(0) = V_0$. (V is called the parallel translate of$V_0$ along $\gamma$.)

Let $\gamma : [0,1] \rightarrow \mathsf {M}$ be an $\mathcal {D}$-curve. The parallel translation$\Pi _{\gamma }^t : \mathcal {D}_{\gamma (0)} \rightarrow \mathcal {D}_{\gamma (t)}$, $t \in [0,1]$ is specified by setting $\Pi ^t_{\gamma }(V_0) = V(t)$, where V is the parallel translate of $V_0 \in \mathcal {D}_{\gamma (0)}$ along $\gamma$.

A parallel frame$(X_a)$ for $\mathcal {D}$ is an orthonormal frame for $\mathcal {D}$ such that each $X_a$ is parallel. The existence of a parallel frame is not guaranteed: it imposes quite severe restrictions on the structure. We say that a nonholonomic Riemannian structure $\mathfrak {S}$ is flat on$\mathcal {U}$ (where $\mathcal {U} \subseteq \mathsf {M}$ is open) if there exists a parallel frame for $\mathcal {D}$ defined on $\mathcal {U}$; if $\mathcal {U} = \mathsf {M}$, then we simply say $\mathfrak {S}$ is flat. Given an NH-isometry $\phi$ between structures $\mathfrak {S}$ and $\mathfrak {S}'$, if $\mathfrak {S}$ is flat on $\mathcal {U}$, then $\mathfrak {S}'$ is flat on $\phi (\mathcal {U})$. The converse does not hold: in three dimensions, there are many non-NH-isometric flat structures (see Sect. 4.2).

Proposition 4

An orthonormal frame $(X_a)$ for $\mathcal {D}$ is parallel if and only if $\llbracket X_a,X_b \rrbracket = 0$ for every $a,b = 1,\ldots ,r$.

Proof

One implication is immediate, since $\nabla$ is torsion free; the other follows from Koszul’s formula (1).$\square$

The following characterization of flatness is a straightforward generalization of a standard result.

Proposition 5

$\mathfrak {S}$ is flat on $\mathcal {U} \subseteq \mathsf {M}$ if and only if for any two points $p,q \in \mathcal {U}$ and for any $\mathcal {D}$-curve $\gamma : [0,1] \rightarrow \mathcal {U}$ joining p to q, the parallel translation $\Pi ^1_{\gamma } : \mathcal {D}_p \rightarrow \mathcal {D}_q$ does not depend on $\gamma$.

For Riemannian manifolds, flatness is also characterized by the vanishing of the Riemannian curvature tensor. By contrast, the situation is more complicated in the nonholonomic Riemannian setting. The Schouten curvature tensor K does not generally characterize flatness (see, e.g., Dragović and Gajić 2003; however, in Sect. 4.1 we show that $K \equiv 0$ is a sufficient condition for flatness in three dimensions). Rather, one can define the Wagner curvature tensor (Dragović and Gajić 2003; Wagner 1935), the vanishing of which does characterize flatness. The construction of this tensor is quite sophisticated, and relies on the flag of the distribution. However, the construction is generally not intrinsic: it relies not only on the data $(\mathsf {M},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$, but also on some additional assumptions. Nonetheless, if the distribution is strongly nonholonomic, then these additional assumptions are automatically satisfied.

We briefly describe Wagner’s approach. Let $\mathfrak {S}= (\mathsf {M},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$ be a nonholonomic Riemannian structure and let $\mathcal {D} = \mathcal {D}^1 \subsetneq \mathcal {D}^2 \subsetneq \cdots \subsetneq \mathcal {D}^{N-1} \subsetneq \mathcal {D}^N = T\mathsf {M}$, $N \ge 2$ be the flag of $\mathcal {D}$. The nonholonomic connection $\nabla ^1 = \nabla$ induces a parallel transport along $\mathcal {D}^1$-curves. For each component $\mathcal {D}^i$, $i = 2,\ldots ,N$ of the flag, Wagner constructs a connection $\nabla ^i : \varGamma (\mathcal {D}^i) \times \varGamma (\mathcal {D}) \rightarrow \varGamma (\mathcal {D})$. Such a connection induces a parallel transport along $\mathcal {D}^i$-curves. Furthermore, $\nabla ^i$ is defined in such a way that it extends $\nabla ^{i-1}$ and the set of parallel tensors of $\nabla ^i$ coincides with that of $\nabla ^{i-1}$. Finally, one gets a vector bundle connection $\nabla ^N$ on $\mathcal {D}$ (whose corresponding parallel transport is along any curve in $\mathsf {M}$) with an associated curvature tensor $K^N$; this is the Wagner curvature tensor. The vanishing of $K^N$ characterizes the flatness of $\nabla ^N$, and hence (by construction of $\nabla ^2,\ldots ,\nabla ^{N-1}$) the flatness of $\mathfrak {S}$. In this paper, we do not employ the Wagner curvature tensor, preferring a more direct approach: we characterize flatness by finding necessary and sufficient conditions for the existence of a rotation taking an arbitrary orthonormal frame for $\mathcal {D}$ to a parallel frame. The relation between this characterization and the Wagner tensor will be explored elsewhere. It is worth mentioning that Wagner also characterized the flatness of three-dimensional nonholonomic Riemannian manifolds (Wagner 1938); however, he stopped short of classifying these structures.

3 Nonholonomic Riemannian structures in three dimensions

Let $\mathsf {M}$ be a three-dimensional manifold and $\mathcal {D}$ a rank 2 completely nonholonomic distribution on $\mathsf {M}$. There exists (locally) a contact form $\omega$ such that $\mathcal {D} = \ker \omega$. Clearly, $\omega$ is unique only up to a multiple by a nonvanishing function. Let $(Y_0,Y_1,Y_2)$ be a (local) frame on $\mathsf {M}$ such that $(Y_1,Y_2)$ is an orthonormal frame for $\mathcal {D}$ and $Y_0$ spans $\mathcal {D}^{\perp }$. Let $c_{ij}^k$ be the functions (structure constants) given by $[Y_i,Y_j] = c^k_{ij}Y_k$. We assume, without loss of generality, that $c_{21}^0 = 1$. By imposing the normalization condition $|d\omega (Y_1,Y_2)| = 1$, we may fix the contact form $\omega$ up to sign. Evidently, the value of $|d\omega (Y_1,Y_2)|$ is independent of the choice of $Y_1$ and $Y_2$. Let $Z \in \varGamma (T\mathsf {M})$ denote the Reeb vector field of $\omega$, i.e., Z is the unique vector field such that $\omega (Z) = 1$ and $d\omega (Z,\,\cdot \,) = 0$. The normalized contact form and the Reeb vector field depend only on the data $(\mathsf {M},\mathcal {D},\mathbf {g})$, hence are preserved (up to sign) under NH-isometry.

3.1 Isometric invariants

Several isometric invariants for nonholonomic Riemannian structures in three dimensions were introduced in Barrett et al. (2016). The first invariant $\vartheta \in \mathcal {C}^{\infty }(\mathsf {M})$ is defined as $\vartheta = \Vert \mathscr {P}(Z)\Vert ^2$. Evidently, we have $\vartheta = 0$ exactly when $\mathcal {D}^{\perp } = {{\mathrm{span}}}\{Z\}$. Three curvature invariants ($\kappa$, $\chi _1$ and $\chi _2$) were also introduced. The first is defined as $\kappa = \frac{1}{2}{{\mathrm{tr}}}(\mathbf {g}^{\sharp }\circ {{\mathrm{Ric}}}^{\flat })$; this invariant can be interpreted as the sectional curvature of $\mathcal {D}$. The second two invariants are defined to be the positive eigenvalue of $\mathbf {g}^{\sharp }\circ A^{\flat }_{sym}$ and the absolute value of the Pfaffian of $\mathbf {g}^{\sharp }\circ A^{\flat }_{skew}$, respectively; we have

$$\begin{aligned} \chi _1 = \sqrt{-\det (\mathbf {g}^{\sharp }\circ A^{\flat }_{sym})} \quad \text {and}\quad \chi _2 = \sqrt{\det (\mathbf {g}^{\sharp }\circ A^{\flat }_{skew})}. \end{aligned}$$

In terms of the structure constants, the invariants take the form

$$\begin{aligned} \vartheta&= (c_{10}^0)^2 + (c_{20}^0)^2,\quad \kappa = \frac{1}{2}(c^2_{10}-c^1_{20}) - (c_{21}^1)^2 - (c_{21}^2)^2 - Y_1[c_{21}^2] + Y_2[c_{21}^1],\\ \chi _1&= \frac{1}{2}\sqrt{(c_{10}^2 + c_{20}^1)^2 + (c_{10}^1 - c_{20}^2)^2},\quad \chi _2 = \frac{1}{2}|c_{10}^1 + c_{20}^2|. \end{aligned}$$

If $\mathfrak {S}$ is a left-invariant structure on a Lie group, then we may take $Y_0$, $Y_1$ and $Y_2$ to be left invariant. The contact form $\omega$ and Reeb vector field Z are also left invariant. In this case, the structure constants are in fact constant, as are the invariants $\vartheta$, $\kappa$, $\chi _1$ and $\chi _2$.

3.2 Classification of invariant structures

The left-invariant nonholonomic Riemannian structures on the three-dimensional simply connected Lie groups were recently classified (Barrett et al. 2016). We shall recall the salient aspects of this classification below.

Consider first the case when $\vartheta = 0$; then $\mathcal {D}^{\perp } = {{\mathrm{span}}}\{Z\}$. In this case the nonholonomic Riemannian structure $(\mathsf {M},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$ is completely specified by the associated sub-Riemannian structure $(\mathsf {M},\mathcal {D},\mathbf {g})$. The isometric invariants $\kappa$ and $\chi _1$ form a complete set of differential invariants for such structures (there are also discrete invariants; see Agrachev and Barilari 2012). Furthermore, by rescaling the metric, we may normalize $(\kappa ,\chi _1)$ such that $\kappa = \chi _1 = 0$ or $\kappa ^2 + \chi _1^2 = 1$. We recall below the classification of left-invariant structures (with $\vartheta = 0$) on three-dimensional simply connected Lie groups; see also Fig. 1. The complete list of three-dimensional Lie algebras and their associated simply connected Lie groups is given in the “Appendix”.

Theorem 1

( Agrachev and Barilari 2012; Barrett et al. 2016) Let $\mathfrak {S}= (\mathsf {G},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$ and $\mathfrak {S}' = (\mathsf {G}',\mathcal {D}',{\mathcal {D}'}^{\perp },\mathbf {g}')$ be left-invariant nonholonomic Riemannian structures on three-dimensional simply connected Lie groups such that $\vartheta = \vartheta ' = 0$, $\kappa = \kappa '$ and $\chi _1 = \chi _1'$.

• (i) If $\kappa = \chi _1 = 0$, then $\mathfrak {S}$ is NH-isometric to (any structure on) the Heisenberg group $\mathsf {H}_3$.

• (ii) If $\chi _1 \ne 0$, or $\chi _1 = 0$ and $\kappa \ge 0$, then $\mathfrak {S}$ is NH-isometric to $\mathfrak {S}'$ if and only if $\mathfrak {g}$ is isomorphic to $\mathfrak {g}'$.

• (iii) If $\chi _1 = 0$ and $\kappa < 0$, then $\mathfrak {S}$ is NH-isometric to the structure on $\widetilde{\mathsf {SL}}(2,\mathbb {R})$ with elliptic-type distribution and metric (at identity) being (a rescaling of) the restriction of the Killing form to $\mathcal {D}_{\mathbf {1}}$.

Remark 1

There is a single equivalence class of structures with $\vartheta = 0$ on the affine group ${{\mathrm{\mathsf {Aff}}}}(\mathbb {R})_0 \times \mathbb {R}$. Remarkably, these are all NH-isometric to the structure on $\widetilde{\mathsf {SL}}(2,\mathbb {R})_{ell}$ in item (iii).

Fig. 1

Normalized invariants for left-invariant nonholonomic Riemannian structures with vanishing $\vartheta$

Consider now the case when $\vartheta > 0$. Define a canonical frame $(X_0,X_1,X_2)$ on $\mathsf {M}$ as follows: $X_0 = \mathscr {Q}(Z)$, $X_1 = \mathscr {P}(Z) / \Vert \mathscr {P}(Z)\Vert$ and $X_2 \in \varGamma (\mathcal {D})$ is the unique unit vector field orthogonal to $X_1$ such that $d\omega (X_1,X_2) = 1$ (here $\Vert \cdot \Vert$ is the norm on $\mathcal {D}$ induced by $\mathbf {g}$). Since $\omega$ and Z are specified up to sign, so are $X_0$ and $X_1$; on the other hand, $X_2$ is uniquely specified. Accordingly, if $\phi$ is an NH-isometry between nonholonomic Riemannian structures $\mathfrak {S}$ and $\mathfrak {S}'$, then $\phi _*X_0 = \pm X_0'$, $\phi _*X_1 = \pm X_1'$ and $\phi _*X_2 = X_2'$. The commutator relations of $(X_0,X_1,X_2)$ are given by

$$\begin{aligned} \left\{ \begin{aligned}{}[] [X_1,X_0]&= c_{10}^1X_1 + c_{10}^2X_2\\ [X_2,X_0]&= c_{20}^0 X_0 + c_{20}^1X_1 + c_{20}^2X_2\\ [X_2,X_1]&= X_0 + c_{21}^1X_1 + c_{21}^2X_2. \end{aligned} \right. \end{aligned}$$

(2)

The commutator relations (2) uniquely determine (up to sign) the nonholonomic Riemannian structure. Let $\mathbf {C}$ be the matrix-valued function on $\mathsf {M}$ given by

$$\begin{aligned} \mathbf {C} = \begin{bmatrix} 0&c_{10}^1&c_{10}^2\\ c_{20}^0&c_{20}^1&c_{20}^2\\ 1&c_{21}^1&c_{21}^2 \end{bmatrix}. \end{aligned}$$

In the case of a left-invariant structure, we have that the canonical frame is left invariant. It follows that the structure constants $c^k_{ij}$, and hence $\mathbf {C}$, are constant. Furthermore, it turns out that NH-isometries must preserve the group structure:

Proposition 6

Let $\mathfrak {S}$ and $\mathfrak {S}'$ be left-invariant nonholonomic Riemannian structures on three-dimensional simply connected Lie groups $\mathsf {G}$ and $\mathsf {G}'$, respectively; assume that $\vartheta ,\vartheta ' > 0$. If $\mathfrak {S}$ is NH-isometric to $\mathfrak {S}'$ with respect to an NH-isometry $\phi : \mathsf {G} \rightarrow \mathsf {G}'$, then $\phi = L_{\phi (\mathbf {1})}\circ \phi '$, where $L_{\phi (\mathbf {1})}$ is a left translation and $\phi ' : \mathsf {G} \rightarrow \mathsf {G}'$ is a Lie group isomorphism.

Table 1 Left-invariant nonholonomic Riemannian structures (with $\vartheta = 1$) on the unimodular Lie groups

It follows from Proposition 6 that NH-isometries preserve the Killing form $\mathcal {K}$. Hence we can introduce three further invariants $\varrho _0$, $\varrho _1$ and $\varrho _2$, defined as $\varrho _i = -\frac{1}{2}\mathcal {K}(X_i(\mathbf {1}),X_i(\mathbf {1}))$. In terms of the structure constants, we have

$$\begin{aligned} \varrho _0&= -\frac{1}{2}\left[ (c_{10}^1)^2 + 2c_{20}^1c_{10}^2 + (c_{20}^2)^2\right] , \quad \varrho _1 = c_{10}^2 - \frac{1}{2}(c_{21}^2)^2,\\ \varrho _2&= -\frac{1}{2}\left[ (c_{20}^0)^2+2c_{20}^1+(c_{21}^1)^2\right] . \end{aligned}$$

Table 2 Left-invariant nonholonomic Riemannian structures (with $\vartheta = 1$) on the non-unimodular Lie groups

For structures on the unimodular Lie groups, $\vartheta$, $\varrho _0$, $\varrho _1$ and $\varrho _2$ form a complete set of invariants. On the non-unimodular Lie groups (except for $\mathsf {G}^h_{3.5}$, $h = 1$) there exist at most two structures with the same invariants $\vartheta$, $\varrho _0$, $\varrho _1$ and $\varrho _2$. On the other hand, there are infinitely many structures on $\mathsf {G}^h_{3.5}$, $h = 1$ with the same invariants $\vartheta$, $\varrho _0$, $\varrho _1$ and $\varrho _2$ (and at most two with the same invariants $\vartheta$, $\kappa$ and $\chi _2$).

Theorem 2

(Barrett et al. 2016) Let $\mathfrak {S}$ be a left-invariant nonholonomic Riemannian structure on a three-dimensional simply connected Lie group, rescaled such that $\vartheta = 1$. Then $\mathfrak {S}$ is NH-isometric to exactly one of the equivalence class representatives listed in Tables 1 and 2.

We have the following characterization of structures with a Cartan–Schouten connection.

Proposition 7

Let $\mathfrak {S}$ be a left-invariant nonholonomic Riemannian structure on a three-dimensional Lie group $\mathsf {G}$.

• (i) Suppose $\mathsf {G}$ is unimodular; then $\nabla$ is Cartan–Schouten if and only if $\vartheta = 0$. (When $\mathsf {G}$ is non-unimodular and $\nabla$ is Cartan–Schouten, we have $\vartheta > 0$.)

• (ii) Suppose $\mathsf {G}$ is non-unimodular; then $\nabla$ is Cartan–Schouten if and only if $\varrho _0 = \varrho _1 = \chi _2 = 0$ and $\varrho _2 + \frac{1}{2}\vartheta = 2\kappa$.

Accordingly, every equivalence class of structures on a unimodular Lie group in Fig. 1 has a Cartan–Schouten connection. On the other hand, for structures on a non-unimodular group, the equivalence classes $\mathfrak {S}^{{{\mathrm{\mathsf {Aff}}}}(\mathbb {R})_0\times \mathbb {R}}_{0,1}$, $\mathfrak {S}^{\mathsf {G}_{3.2}}_1$, $\mathfrak {S}^{\mathsf {G}^h_{3.4}}_1$ and $\mathfrak {S}^{\mathsf {G}^h_{3.5}}_1$ (see Table 2) are exactly those with a Cartan–Schouten connection.

4 Flat structures

In this section we consider the flat nonholonomic Riemannian structures on three-dimensional manifolds. We first characterize flatness in three dimensions; this is followed by a classification of the flat left-invariant structures.

4.1 Characterization

Let $\mathfrak {S}= (\mathsf {M},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$ be a nonholonomic Riemannian structure on a three-dimensional manifold $\mathsf {M}$. Let $(X_0,X_1,X_2)$ be a local frame defined on an open neighbourhood $\mathcal {U} \subseteq \mathsf {M}$ such that $X_0$ spans $\mathcal {D}^{\perp }$ and $(X_1,X_2)$ is an orthonormal frame for $\mathcal {D}$. The structure constants of $(X_0,X_1,X_2)$ are denoted by $c^k_{ij}$; we may assume that $c^0_{21} = 1$. The dual frame $(\nu ^0,\nu ^1,\nu ^2)$ satisfies the structure equations $d\nu ^k = \sum _{0 \le i < j \le 2} c^k_{ij}\nu ^j \wedge \nu ^i$.

Lemma 1

$\mathfrak {S}$ is locally flat on $\mathcal {U}$ if and only if there exists $\theta \in \mathcal {C}^{\infty }(\mathcal {U})$ such that

$$\begin{aligned} X_1[\theta ] = c^1_{21} \quad \text {and}\quad X_2[\theta ] = c^2_{21}. \end{aligned}$$

If such a function $\theta$ exists, then the rotated frame $(\cos \theta X_1 - \sin \theta X_2,\sin \theta X_1 + \cos \theta X_2)$ is parallel.

Proof

Let $(Y_1,Y_2)$ be another orthonormal frame for $\mathcal {D}$. There exists an orthogonal transformation taking $(X_1,X_2)$ to $(Y_1,Y_2)$. That is, there exist $\sigma \in \{-1,1\}$ and $\theta \in \mathcal {C}^{\infty }(\mathcal {U})$ such that

$$\begin{aligned} \left\{ \begin{aligned} Y_1&= \sigma \cos \theta X_1 - \sin \theta X_2\\ Y_2&= \sigma \sin \theta X_1 + \cos \theta X_2. \end{aligned} \right. \end{aligned}$$

By Proposition 4, the frame $(Y_1,Y_2)$ is a parallel frame for $\mathcal {D}$ if and only if $\llbracket Y_2,Y_1 \rrbracket = 0$. We have $\llbracket Y_2,Y_1 \rrbracket = (X_1[\theta ]-\sigma c^1_{21})X_1 + (X_2[\theta ]-\sigma c^2_{21})X_2$, and so $(Y_1,Y_2)$ is parallel exactly when $X_1[\theta ] = \sigma c^1_{21}$ and $X_2[\theta ] = \sigma c^2_{21}$. Changing the sign of $\theta$ if necessary, we may take $\sigma = 1$.$\square$

Lemma 2

There exists a rotation $(Y_1,Y_2)$ of $(X_1,X_2)$ such that $\llbracket Y_2,Y_1 \rrbracket = 0$ if and only if the following equations hold:

$$\begin{aligned} \left\{ \begin{aligned}&(c^1_{10}-c^2_{20})c^1_{21}+(c^2_{10}+c^1_{20})c^2_{21}+c^0_{20}c^1_{10}-\tfrac{1}{2}c^0_{10}(c^2_{10}+c^1_{20}) + c^0_{10}\kappa \\&\quad = -\tfrac{1}{2}X_1[c^2_{10}+c^1_{20}]+X_1[\kappa ]+X_2[c^1_{10}]\\&(c^2_{10}+c^1_{20})c^1_{21}-(c^1_{10}-c^2_{20})c^2_{21}-c^0_{10}c^2_{20}+\tfrac{1}{2}c^0_{20}(c^2_{10}+c^1_{20})+c^0_{20}\kappa \\&\quad = \phantom {-}\tfrac{1}{2}X_2[c^2_{10}+c^1_{20}]+X_2[\kappa ]-X_1[c^2_{20}]. \end{aligned} \right. \end{aligned}$$

(3)

Proof

Let $(Y_1,Y_2) = (\cos \theta X_1 - \sin \theta X_2,\sin \theta X_1 + \cos \theta X_2)$ be a rotation of $(X_1,X_2)$, where $\theta \in \mathcal {C}^{\infty }(\mathcal {U})$. By Lemma 1, we have that $\llbracket Y_2,Y_1 \rrbracket = 0$ if and only if $X_1[\theta ] = c^1_{21}$ and $X_2[\theta ] = c^2_{21}$. We claim that there exists $\theta \in \mathcal {C}^{\infty }(\mathcal {U})$ satisfying the conditions $X_1[\theta ] = c^1_{21}$, $X_2[\theta ] = c^2_{21}$ if and only if (3) hold. If such a function $\theta$ exists, then

$$\begin{aligned} d\theta&= X_0[\theta ]\nu ^0+c^1_{21}\nu ^1+c^2_{21}\nu ^2\\&= ([X_2,X_1][\theta ]-c^1_{21}X_1[\theta ]-c^2_{21}X_2[\theta ])\nu ^0 + c^1_{21}\nu ^1 + c^2_{21}\nu ^2\\&= (X_2[c^1_{21}]-X_1[c^2_{21}]-(c^1_{21})^2-(c^2_{21})^2)\nu ^0 + c^1_{21}\nu ^1 + c^2_{21}\nu ^2\\&= (\kappa -\tfrac{1}{2}(c^2_{10}-c^1_{20}))\nu ^0 + c^1_{21}\nu ^1 + c^2_{21}\nu ^2. \end{aligned}$$

The right-hand side is independent of $\theta$; accordingly, let $\varpi = (\kappa -\tfrac{1}{2}(c^2_{10}-c^1_{20}))\nu ^0 + c^1_{21}\nu ^1 + c^2_{21}\nu ^2$. Then $d\varpi = f_{01}\,\nu ^0\wedge \nu ^1+f_{02}\,\nu ^0\wedge \nu ^2$, where

$$\begin{aligned} \left\{ \begin{aligned} f_{01}&= c^1_{10}c^1_{21}+c^2_{10}c^2_{21}-c^0_{10}(\tfrac{1}{2}(c^2_{10}-c^1_{20})-\kappa )\\&\quad + X_1[\tfrac{1}{2}(c^2_{10}-c^1_{20})-\kappa ]+X_0[c^1_{21}]\\ f_{02}&= c^1_{20}c^1_{21}+c^2_{20}c^2_{21}-c^0_{20}(\tfrac{1}{2}(c^2_{10}-c^1_{20})-\kappa )\\&\quad + X_2[\tfrac{1}{2}(c^2_{10}-c^1_{20})-\kappa ]+X_0[c^2_{21}]. \end{aligned} \right. \end{aligned}$$

Using $d^2 = 0$ on the structure equations for $d\nu ^1$ and $d\nu ^2$, we get

$$\begin{aligned} \left\{ \begin{aligned} X_0[c^1_{21}]&= -c^1_{20}c^0_{10}+c^1_{10}c^0_{20}-c^2_{20}c^1_{21}+c^1_{20}c^2_{21}+X_1[c^1_{20}]-X_2[c^1_{10}]\\ X_0[c^2_{21}]&= -c^2_{20}c^0_{10}+c^2_{10}c^0_{20}+c^2_{10}c^1_{21}-c^1_{10}c^2_{21}+X_1[c^2_{20}]-X_2[c^2_{10}]. \end{aligned} \right. \end{aligned}$$

Hence

$$\begin{aligned} \left\{ \begin{aligned} f_{01}&= (c^1_{10}-c^2_{20})c^1_{21}+(c^2_{10}+c^1_{20})c^2_{21}+c^0_{20}c^1_{10}-\tfrac{1}{2}c^0_{10}(c^2_{10}+c^1_{20}) + c^0_{10}\kappa \\&\quad + \tfrac{1}{2}X_1[c^2_{10}+c^1_{20}]-X_1[\kappa ]-X_2[c^1_{10}]\\ f_{02}&= (c^2_{10}+c^1_{20})c^1_{21}-(c^1_{10}-c^2_{20})c^2_{21}-c^2_{20}c^0_{10}+\tfrac{1}{2}c^0_{20}(c^2_{10}+c^1_{20})+c^0_{20}\kappa \\&\quad - \tfrac{1}{2}X_2[c^2_{10}+c^1_{20}]-X_2[\kappa ]+X_1[c^2_{20}]. \end{aligned} \right. \end{aligned}$$

Suppose $\theta$ exists, so that $d\varpi = d^2\theta = 0$; then $f_{01} = f_{02} = 0$, which yields the equations (3). Conversely, if (3) hold, then $d\varpi = 0$, i.e., $\varpi$ is closed. Hence $\varpi$ is locally exact: there exists an open neighbourhood $\mathcal {U}' \subseteq \mathcal {U}$ and $\theta \in \mathcal {C}^{\infty }(\mathcal {U}')$ such that $\varpi = d\theta$. The rotated frame $(Y_1,Y_2)$ then satisfies $\llbracket Y_2,Y_1 \rrbracket = 0$.$\square$

Using the condition for flatness in Lemma 2 (which depends on the choice of an orthonormal frame for $\mathcal {D}$), we shall derive an invariant characterization of the flat structures. We have the decomposition $T\mathsf {M} = \mathcal {D} \oplus {{\mathrm{span}}}\{Z\}$, where Z is the Reeb vector field of the normalized contact form $\omega$. Let $\mathscr {R} : T\mathsf {M} \rightarrow {{\mathrm{span}}}\{Z\}$ be the projection onto the distribution spanned by Z. In particular, we have $\mathscr {R}([X_2,X_1]) = d\omega (X_1,X_2)Z$.

Theorem 3

$\mathfrak {S}$ is locally flat on $\mathcal {U}$ if and only if

$$\begin{aligned} d^{\nabla }_{\mathscr {P}}F = F\circ \rho \quad \hbox { on}\ \mathcal {U}. \end{aligned}$$

(4)

Here $F = \mathbf {g}^{\sharp }\circ ({{\mathrm{Ric}}}^{\flat }+A^{\flat }_{sym}+A^{\flat }_{skew})$ and $\rho \in \varOmega ^2(\mathcal {D},\mathcal {D})$ is given by $\rho (X_1,X_2) = \mathscr {P}(\mathscr {R}([X_2,X_1]))$.

Proof

Since $(\mathbf {g}^{\sharp }\circ {{\mathrm{Ric}}}^{\flat })(X_a) = {{\mathrm{Ric}}}(X_a,X_1)X_1 + {{\mathrm{Ric}}}(X_a,X_2)X_2$, ${{\mathrm{Ric}}}(X_a,X_a) = \kappa$ and ${{\mathrm{Ric}}}(X_a,X_b) = 0$ for $a \ne b$, we have

$$\begin{aligned} d^{\nabla }_{\mathscr {P}}(\mathbf {g}^{\sharp }\circ {{\mathrm{Ric}}}^{\flat })(X_1,X_2)&= \nabla _{X_1}(\mathbf {g}^{\sharp }\circ {{\mathrm{Ric}}}^{\flat })(X_2) - \nabla _{X_2}(\mathbf {g}^{\sharp }\circ {{\mathrm{Ric}}}^{\flat })(X_1)\\&\quad - (\mathbf {g}^{\sharp }\circ {{\mathrm{Ric}}}^{\flat })(\llbracket X_1,X_2 \rrbracket )\\&= \nabla _{X_1}(\kappa X_2) - \nabla _{X_2}(\kappa X_1) + (c^1_{21}\kappa X_1 + c^2_{21}\kappa X_2). \end{aligned}$$

It is not difficult to show that $\nabla _{X_1}X_2 = -c^1_{21}X_1$ and $\nabla _{X_2}X_1 = c^2_{21}X_2$. Hence

$$\begin{aligned} d^{\nabla }_{\mathscr {P}}(\mathbf {g}^{\sharp }\circ {{\mathrm{Ric}}}^{\flat })(X_1,X_2)&= X_1[\kappa ]X_2 - \kappa c^1_{21}X_1 - X_2[\kappa ]X_1 - \kappa c^2_{21}X_2\\&\quad + (c^1_{21}\kappa X_1 + c^2_{21}\kappa X_2)\\&= -X_2[\kappa ]X_1 + X_1[\kappa ]X_2. \end{aligned}$$

Similar calculations yield

$$\begin{aligned} d^{\nabla }_{\mathscr {P}}(\mathbf {g}^{\sharp }\circ A^{\flat })(X_1,X_2)&= (X_1[c^2_{20}]-\tfrac{1}{2}X_2[c^2_{10}+c^1_{20}])X_1 + (X_2[c^1_{10}]\\&\quad - \tfrac{1}{2}X_1[c^2_{10}+c^1_{20}])X_2 + 2\,(\mathbf {g}^{\sharp }\circ A^{\flat }_{sym})(\llbracket X_2,X_1 \rrbracket ). \end{aligned}$$

Here $A = A_{sym} + A_{skew}$. In addition, we have $(\mathbf {g}^{\sharp }\circ {{\mathrm{Ric}}}^{\flat }\circ \,\rho )(X_1,X_2) = -c^0_{20}\kappa X_1 + c^0_{10}\kappa X_2$ and

$$\begin{aligned} (\mathbf {g}^{\sharp }\circ A^{\flat }\circ \rho )(X_1,X_2)&= -(c^0_{10}c^1_{10} + \tfrac{1}{2}c^0_{20}(c^2_{10}+c^1_{20}))X_1 \\&\quad - (c^0_{20}c^2_{20} + \tfrac{1}{2}c^0_{10}(c^2_{10}+c^1_{20}))X_2. \end{aligned}$$

Let $f_{01}$ and $f_{02}$ be defined as in the proof of Lemma 2. There exists a parallel frame for $\mathcal {D}$ if and only if $f_{01} = f_{02} = 0$. From the foregoing calculations, it follows that

$$\begin{aligned} (d^{\nabla }_{\mathscr {P}}F - F\circ \rho )(X_1,X_2) = f_{02}X_1 - f_{01}X_2. \end{aligned}$$

(As $d^{\nabla }_{\mathscr {P}}F - F\circ \rho$ is skew-symmetric, it is fully determined by its evaluation of $X_1 \wedge X_2$.) This proves the result.$\square$

Corollary 1

Suppose $\vartheta = 0$; then $\mathfrak {S}$ is locally flat on $\mathcal {U}$ if and only if $d^{\nabla }_{\mathscr {P}}F \equiv 0$ on $\mathcal {U}$.

Proof

If $\vartheta = 0$, then $\mathscr {P}(Z) = 0$, and so $\rho \equiv 0$. Hence $\mathfrak {S}$ is flat exactly when $d^{\nabla }_{\mathscr {P}}F$ vanishes.$\square$

Corollary 2

If $K \equiv 0$ on $\mathcal {U}$, then $\mathfrak {S}$ is locally flat on $\mathcal {U}$.

Proof

If $K \equiv 0$, then ${{\mathrm{Ric}}}$, $A_{sym}$ and $A_{skew}$ all vanish identically. Hence $F \equiv 0$, and the condition $d^{\nabla }_{\mathscr {P}}F = F\circ \rho$ is trivially satisfied.$\square$

4.2 Classification

Let $\mathfrak {S}= (\mathsf {G},\mathcal {D},\mathcal {D}^{\perp },\mathbf {g})$ be a left-invariant nonholonomic Riemannian structure on a three-dimensional simply connected Lie group $\mathsf {G}$. Suppose that $(X_0,X_1,X_2)$ is a left-invariant frame on $\mathsf {G}$ such that $X_0$ spans $\mathcal {D}^{\perp }$ and $(X_1,X_2)$ is an orthonormal frame for $\mathcal {D}$. Let $c^k_{ij}$ be the structure constants of the frame; as before, we take $c^0_{21} = 1$. With respect to $(X_1,X_2)$, we have

$$\begin{aligned} d^{\nabla }_{\mathscr {P}}F(X_1,X_2) = 2\,(\mathbf {g}^{\sharp }\circ A^{\flat }_{sym})(\llbracket X_2,X_1 \rrbracket ) = \begin{bmatrix} c^2_{10}+c^1_{20}&-(c^1_{10}-c^2_{20})\\ c^2_{20}-c^1_{10}&-(c^2_{10}+c^1_{20}) \end{bmatrix} \begin{bmatrix} c^1_{21} \\ c^2_{21} \end{bmatrix} \end{aligned}$$

(5)

and

$$\begin{aligned} (F\circ \rho )(X_1,X_2) = \begin{bmatrix} \kappa + \tfrac{1}{2}(c^2_{10}+c^1_{20})&c^2_{20}\\ -c^1_{10}&\kappa - \tfrac{1}{2}(c^2_{10}+c^1_{20}) \end{bmatrix} \begin{bmatrix} -c^0_{20} \\ \phantom {-}c^0_{10} \end{bmatrix}. \end{aligned}$$

(6)

We shall consider the following cases: (I) $\vartheta = 0$; (II-a) $\vartheta > 0$ and $\mathsf {G}$ is unimodular; (II-b) $\vartheta > 0$ and $\mathsf {G}$ is non-unimodular.

Theorem 4

(Case I: $\vartheta = 0)$$\mathfrak {S}$ is flat if and only if $\mathsf {G}$ is unimodular (hence $\nabla$ is a Cartan–Schouten connection) or $\mathsf {G} = {{\mathrm{\mathsf {Aff}}}}(\mathbb {R})_0\times \mathbb {R}$.

Proof

By Corollary 1, $\mathfrak {S}$ is flat if and only if $d^{\nabla }_{\mathscr {P}}F$ vanishes. From (5), this happens exactly when $A^{\flat }_{sym}(\llbracket X_2,X_1 \rrbracket ) = 0$. Suppose $\chi _1 > 0$. Then $A^{\flat }_{sym}$ is invertible, and $\llbracket X_2,X_1 \rrbracket = 0$ implies that every left-invariant frame for $\mathcal {D}$ is parallel; that $c^1_{21} = c^2_{21} = 0$; and that $\nabla$ is a Cartan–Schouten connection. Moreover, $\mathsf {G}$ must be unimodular. Conversely, every such structure is clearly flat.

On the other hand, suppose $\chi _1 = 0$. Then $A^{\flat }_{sym} \equiv 0$, and so every such structure in this case is flat. From Theorem 1 (see also Fig. 1), these are the following structures (up to NH-isometry and rescaling):

• the structure on $\mathsf {SU}(2)$ with metric (at identity) being the Killing form restricted to $\mathcal {D}_{\mathbf {1}}$ (when $\kappa > 0$);

• any structure on the Heisenberg group $\mathsf {H}_3$ (when $\kappa = 0$);

• the structure on $\widetilde{\mathsf {SL}}(2,\mathbb {R})$ with elliptic-type distribution and metric (at identity) being the Killing form restricted to $\mathcal {D}_{\mathbf {1}}$ (when $\kappa < 0$); also, any structure on ${{\mathrm{\mathsf {Aff}}}}(\mathbb {R})_0\times \mathbb {R}$ (see Remark 1).

With the exception of those on ${{\mathrm{\mathsf {Aff}}}}(\mathbb {R})_0 \times \mathbb {R}$, these are all structures on unimodular Lie groups with Cartan–Schouten connections.$\square$

Remark 2

The case when $\chi _1 = 0$ was essentially proved in Agrachev and Barilari (2012). Indeed, it was shown that for a sub-Riemannian structure $(\mathsf {M},\mathcal {D},\mathbf {g})$ on a three-dimensional simply connected manifold (which may be viewed as a nonholonomic Riemannian structure with $\vartheta = 0$) with constant $\kappa$ and $\chi _1 = 0$, there exists a rotation $({\widehat{X}}_1,{\widehat{X}}_2)$ of the frame $(X_1,X_2)$ such that $[{\widehat{X}}_1,Z] = \kappa {\widehat{X}}_2$, $[{\widehat{X}}_2,Z] = -\kappa {\widehat{X}}_1$ and $[{\widehat{X}}_2,{\widehat{X}}_1] = Z$, i.e., such that $({\widehat{X}}_1,{\widehat{X}}_2)$ is parallel.

Theorem 5

(Case II-a:   $\vartheta > 0$, $\mathsf {G}$ unimodular) $\mathfrak {S}$ is flat if and only if $\varrho _0 = \vartheta \varrho _1$ and $\varrho _2 = 0$ (in which case $\kappa ^2 = \chi _1^2$). Consequently, any flat structure is NH-isometric up to rescaling to exactly one of the following structures:

$$\begin{aligned}&\mathfrak {S}^{\mathsf {SE}(1,1)}_{\alpha _1,0} \quad (\text {when }\varrho _0 < 0),&\mathfrak {S}^{{\mathsf {H}_3}} \quad (\text {when }\varrho _0 = 0),&\mathfrak {S}^{\widetilde{\mathsf {SE}}(2)}_{\alpha _1,0} \quad (\text {when }\varrho _0 > 0). \end{aligned}$$

Proof

We take $(X_0,X_1,X_2)$ to be the canonical frame (described in Sect. 3.2). Since $\mathsf {G}$ is unimodular, we have $c^2_{21} = 0$ and $c^1_{21} = -c^0_{20} = \Vert \mathscr {P}(Z)\Vert$, whence $\llbracket X_2,X_1 \rrbracket = d\omega (X_1,X_2)\mathscr {P}(Z) = \rho (X_1,X_2)$. Furthermore, we have $\chi _2 = 0$, i.e., $A_{skew} \equiv 0$. The condition (4) becomes

$$\begin{aligned}&2\,(\mathbf {g}^{\sharp }\circ A^{\flat }_{sym})(\llbracket X_2,X_1 \rrbracket ) = (\mathbf {g}^{\sharp }\circ ({{\mathrm{Ric}}}^{\flat } + A^{\flat }_{sym}))(\llbracket X_2,X_1 \rrbracket )\\&\quad \Longleftrightarrow \quad {{\mathrm{Ric}}}^{\flat }(\llbracket X_2,X_1 \rrbracket ) = A^{\flat }_{sym}(\llbracket X_2,X_1 \rrbracket )\\&\quad \Longleftrightarrow \quad {{\mathrm{Ric}}}^{\flat }(X_1) = A^{\flat }_{sym}(X_1)\\&\quad \Longleftrightarrow \quad A_{sym}(X_1,X_1) = \kappa \text { and }A_{sym}(X_1,X_2) = 0. \end{aligned}$$

In terms of the structure constants, this is equivalent to the conditions $c^1_{10} = 0$, $c^1_{20} = -(c^0_{20})^2$, which are in turn equivalent to $\varrho _0 = \vartheta \varrho _1$ and $\varrho _2 = 0$. Furthermore, this implies that $\kappa ^2 = \chi _1^2$. In terms of the classification in Theorem 2, these are the following structures:

• the equivalence classes $\mathfrak {S}^{\mathsf {SE}(1,1)}_{\alpha _1,\alpha _2}$ with $\alpha _2 = 0$ (when $\varrho _0 < 0$);

• the equivalence class $\mathfrak {S}^{{\mathsf {H}_3}}$ (when $\varrho _0 = 0$);

• the equivalence classes $\mathfrak {S}^{\widetilde{\mathsf {SE}}(2)}_{\alpha _1,\alpha _2}$ with $\alpha _2 = 0$ (when $\varrho _0 > 0$).

$\square$

Theorem 6

(Case II-b: $\vartheta > 0$, $\mathsf {G}$ non-unimodular)

• (i) If $\mathsf {G} = {{\mathrm{\mathsf {Aff}}}}(\mathbb {R})_0 \times \mathbb {R}$, then $\mathfrak {S}$ is flat.

• (ii) Suppose $\chi _2 = 0$; $\mathfrak {S}$ is flat if and only if $\nabla$ is a Cartan–Schouten connection.

• (iii) Suppose $\chi _2 > 0$; any flat structure is NH-isometric up to rescaling to exactly one of the following structures:

  $$\begin{aligned}&\mathfrak {S}^{\mathsf {G}_{3.2}}_{\alpha ,\beta }&\text {with}\quad \alpha = -\frac{\beta }{8}\big (1\pm \sqrt{1-4\beta ^2}\big ),\;{-\tfrac{1}{2}} \le \beta< 0\\&\mathfrak {S}^{\mathsf {G}^h_{3.4}}_{\alpha ,\beta },\; 0< h< 1&\text {with}\quad \alpha = -\frac{(h^2-1)\beta }{8h^2}\big (1\pm \sqrt{1-4\beta ^2}\big ),\;0< \beta \le \tfrac{1}{2}\\&\mathfrak {S}^{\mathsf {G}^h_{3.4}}_{\alpha ,\beta },\; 1< h&\text {with}\quad \alpha = -\frac{(h^2-1)\beta }{8h^2}\big (1\pm \sqrt{1-4\beta ^2}\big ),\;{-\tfrac{1}{2}} \le \beta< 0\\&\mathfrak {S}^{\mathsf {G}^h_{3.5}}_{\alpha ,\beta }&\text {with}\quad \alpha = -\frac{(h^2+1)\beta }{8h^2}\big (1\pm \sqrt{1-4\beta ^2}\big ),\;{-\tfrac{1}{2}} \le \beta < 0. \end{aligned}$$

Proof

Considering the equivalence class representatives in Table 2, a direct (but tedious) calculation, using the condition that (5) equals (6), yields the result. We illustrate with the case of $\mathsf {G}_{3.2}$. Take $(X_0,X_1,X_2)$ to be the canonical frame. Consider first the family of equivalence classes $\mathfrak {S}^{\mathsf {G}_{3.2}}_{\beta }$; we have

$$\begin{aligned}&(d^{\nabla }_{\mathscr {P}}F-F\circ \rho )(X_1,X_2)\\&\quad = \begin{bmatrix} -\tfrac{1}{4}(\beta -2)^2&0\\ 0&\tfrac{1}{4}(\beta -2)^2 \end{bmatrix} \begin{bmatrix} 1-\beta \\ 0 \end{bmatrix} - \begin{bmatrix} -(\beta -1)^2&0\\ 0&-\tfrac{3}{4}\beta (\beta -\tfrac{4}{3}) \end{bmatrix} \begin{bmatrix} 1\\ 0 \end{bmatrix}\\&\quad = \begin{bmatrix} \tfrac{1}{4}\beta ^2(\beta -1)\\ 0 \end{bmatrix}. \end{aligned}$$

Hence a member of the equivalence class $\mathfrak {S}^{\mathsf {G}_{3.2}}_{\beta }$ is flat exactly when $\beta = 1$. From Sect. 3.2, this is exactly the structure on $\mathsf {G}_{3.2}$ with a Cartan–Schouten connection. On the other hand, consider the family of equivalence classes $\mathfrak {S}^{\mathsf {G}_{3.2}}_{\alpha ,\beta }$; then

$$\begin{aligned} (d^{\nabla }_{\mathscr {P}}F-F\circ \rho )(X_1,X_2) = \begin{bmatrix} -\frac{\alpha }{\beta ^3}(16\alpha ^2 + 4\alpha \beta + \beta ^4)\\ \phantom {-}\frac{1}{4\beta }(16\alpha ^2 + 4\alpha \beta + \beta ^4) \end{bmatrix}. \end{aligned}$$

Thus the structure is flat if and only if $16\alpha ^2 + 4\alpha \beta + \beta ^4 = 0$. It is not difficult to show that this occurs exactly when $\alpha = -\tfrac{\beta }{8}(1 \pm \sqrt{1-4\beta ^2})$ and $-\tfrac{1}{2} \le \beta < 0$.$\square$

Remark 3

Every left-invariant nonholonomic Riemannian structure on $\mathsf {H}_3$ and ${{\mathrm{\mathsf {Aff}}}}(\mathbb {R})_0 \times \mathbb {R}$ is flat. On the other hand, apart from those with Cartan–Schouten connections, there are no flat structures on the semisimple groups $\mathsf {SU}(2)$ and $\widetilde{\mathsf {SL}}(2,\mathbb {R})$. Every other group (apart from $\mathsf {H}_3$ and ${{\mathrm{\mathsf {Aff}}}}(\mathbb {R})_0 \times \mathbb {R}$) admits a one-parameter family of flat structures, up to equivalence.

Remark 4

The classical Chaplygin problem (see, e.g., Fedorov, Y., García-Naranjo 2010; Barrett et al. 2016) may be modelled by a left-invariant nonholonomic Riemannian structure on the Euclidean group $\mathsf {SE}(2)$. The reduced dynamics on the Lie algebra $\mathfrak {se}(2)$ exhibit three qualitatively different cases (of increasing analytical complexity): the “Chaplygin skate” (when $\vartheta = 0$); the “Chaplygin sleigh” (when $\vartheta > 0$ and $\kappa ^2 = \chi _1^2$); and the “hydrodynamic Chaplygin sleigh” (when $\vartheta > 0$ and $\kappa ^2 \ne \chi _1^2$). The equivalence classes of structures corresponding to the Chaplygin sleigh are $\mathfrak {S}^{\widetilde{\mathsf {SE}}(2)}_{\alpha _1,0}$. Accordingly, the flat structures for the Chaplygin problem correspond precisely to the case of the Chaplygin skate and the Chaplygin sleigh.

Appendix Real Lie algebras of dimension three

The classification of real three-dimensional Lie algebras is well known. Our preference is for the Bianchi–Behr enumeration (MacCallum 1999; Krasiński et al. 2003; Mubarakzyanov 1963). In terms of an appropriate ordered basis $(E_1,E_2,E_3)$, the commutator relations are given by

$$\begin{aligned} \left\{ \begin{aligned}{}[] [E_2,E_3]&= n_1E_1 - a E_2\\ [E_3,E_1]&= aE_1 + n_2 E_2\\ [E_1,E_2]&= n_3 E_3. \end{aligned} \right. \end{aligned}$$

The coefficients a, $n_1$, $n_2$ and $n_3$ for each type of algebra may be found in Table 3, together with the (unique) simply connected Lie group corresponding to each algebra.

Table 3 Bianchi–Behr classification of real three-dimensional Lie algebras

Remark 5

The Abelian group $\mathbb {R}^3$ and the group $\mathsf {G}_{3.3}$ do not admit completely nonholonomic left-invariant distributions. Accordingly, there do not exist left-invariant nonholonomic Riemannian structures on these groups.

Remark 6

Apart from $\widetilde{\mathsf {SL}}(2,\mathbb {R})$ (the universal cover of $\mathsf {SL}(2,\mathbb {R})$) there exists, up to Lie group automorphism, at most one completely nonholonomic left-invariant distribution on each three-dimensional simply connected Lie group. On $\widetilde{\mathsf {SL}}(2,\mathbb {R})$ there exist exactly two such distributions up to automorphism, according as whether the Killing form restricted to the distribution (at identity) is definite or indefinite. Following Agrachev and Barilari (2012), if the Killing form is definite on a given distribution, we shall say that the distribution is of elliptic type, and denote the group as $\widetilde{\mathsf {SL}}(2,\mathbb {R})_{ell}$. On the other hand, when the Killing form is indefinite on the distribution, we shall say that it is of hyperbolic type, and write $\widetilde{\mathsf {SL}}(2,\mathbb {R})_{hyp}$ for the group.