1 Introduction and notation

Consider a CR manifold (M, D, J) of type (n, k). This means that M is a manifold of dimension \(2n+k\) with a rank 2n distribution D and a field of endomorphisms \(J:D\rightarrow D\) such that \(J^2=-{\text {id}}\). We assume that M is integrable, meaning that the \(-\mathrm {i}\) eigendistribution \(D^{0,1}\) of J is involutive, that is,

$$\begin{aligned}{}[\mathfrak {X}^{0,1} , \mathfrak {X}^{0,1} ]\subseteq \mathfrak {X}^{0,1} , \end{aligned}$$

where

$$\begin{aligned} \mathfrak {X}^{0,1} = \{Z\in \Gamma (D)_\mathbb {C}:Z=X+\mathrm {i}JX, X\in \Gamma (D) \}. \end{aligned}$$

As usual, \(\Gamma\) indicates a space of sections and a subscript \(\mathbb {C}\) indicates a complexification. We write \(\mathfrak {X}\) for the space of vector fields on M.

We say that \(F:M\rightarrow \tilde{M}\) is a CR embedding of a CR-manifold (M, D, J) of type (n, k) into another CR-manifold \((\tilde{M},\tilde{D},\tilde{J})\) of type \((n+\ell ,k-\ell )\) if F is a smooth embedding and \(F_*:D^{0,1} \rightarrow \tilde{D}^{0,1}\) satisfies

$$\begin{aligned} F_* J = \tilde{J} F_*. \end{aligned}$$

The case where \(\ell = k\) is of particular interest, as this corresponds to an embedding into a complex manifold.

Finding embeddings of the type envisaged above is one of the fundamental problems in CR geometry. We are going to consider the local problem only; that is, we fix a point p and look for an embedding of a neighbourhood of p. This means that we can replace M by a small open neighbourhood of p at any stage.

We mention a few contributions that are particularly relevant. It is well known that analytic CR-manifolds can always be locally embedded in complex space. Baouendi, Rothschild and Treves [1] consider the case where there is an abelian Lie algebra \(\mathfrak {g}\) of real vector fields that is transverse, in the sense that

$$\begin{aligned} \mathfrak {X} = \Gamma (D) \oplus \mathfrak {g} , \end{aligned}$$

and normalising, in the sense that

$$\begin{aligned}{}[\mathfrak {g}, \mathfrak {X}^{0,1}] \subseteq \mathfrak {X}^{0,1}, \end{aligned}$$
(1)

and construct an embedding into a complex space. Baouendi and Rothschild [2] extended this result to deal with the case where the Lie algebra \(\mathfrak {g}\) is no longer required to be abelian. Jacobowitz [4] considers the case where \(\ell =1\) and finds a condition for the existence of an embedding into \(\mathbb {C}^{n+1}\). Finally, Hill and Nacinovich [3] treat the case where there is a solvable transverse normalising Lie algebra of complex vector fields of dimension \(\ell\), and construct an embedding into a manifold of type \((n+\ell , k-\ell )\); they use solvability to extend by induction on dimension. We are going to treat the case of a finite-dimensional Lie algebra extension of \(\mathfrak {X}^{0,1}\) in \(\mathfrak {X}_{\mathbb {C}}\) by nonvanishing complex vector fields \(X_1\), ..., \(X_s\), and show that M embeds into a CR manifold \(\tilde{M}\) of type \((n+\ell , k-\ell )\) if \(\dim (\mathfrak {X}^{(0,1)} + \langle X_1, \dots , X_s\rangle ) = n+\ell\).

2 Main results

We state our theorem more precisely.

Theorem 1

Let (M, D, J) be a CR-manifold of type (n, k). Suppose that \(X_1,\dots , X_s\) are nonvanishing complex vector fields that normalise \(\mathfrak {X}^{0,1}\) (as in (1)) and satisfy

$$\begin{aligned}{}[X_\alpha ,X_\beta ]=c_{\alpha \beta }^\gamma X_\gamma \mod \mathfrak {X}^{0,1}, \end{aligned}$$

where the \(c_{\alpha \beta }^\gamma\) are constants. If

$$\begin{aligned} \dim ( \mathfrak {X}^{0,1} + {\text {span}}\{ X_1, \dots , X_s \} ) = n + \ell , \end{aligned}$$

then there is a (local) CR-embedding of M into a CR-manifold \(\tilde{M}\) of type \((n+\ell ,k-\ell )\).

Proof

Fix \(p \in M\). Without loss of generality we assume that each \(X_\alpha (p)\) is not purely imaginary. The \(c_{\alpha \beta }^\gamma\) are the structure constants of the Lie algebra \(\mathfrak {g}\) defined by

$$\begin{aligned} \mathfrak {g} := ({\text {span}}\{ X_1, \dots , X_s \} + \mathfrak {X}^{0,1})/\mathfrak {X}^{0,1}. \end{aligned}$$

By renumbering the vector fields and passing to a submanifold of M containing p if necessary, we may suppose that

$$\begin{aligned} \mathfrak {X}^{0,1} + {\text {span}}\{ X_1, \dots , X_s \} = \mathfrak {X}^{0,1} \oplus {\text {span}}\{ X_1, \dots , X_\ell \} , \end{aligned}$$

where \(\ell \le s\). We shall construct complex vector fields

$$\begin{aligned} Y_\alpha =\lambda _\alpha ^\gamma (t) X_\gamma + \mathrm {i}\partial _{\alpha }, \end{aligned}$$

where \(\alpha =1,\dots , s\) on a neighbourhood of (p, 0) in \(M\times \mathbb {R}^s\) such that

$$\begin{aligned}{}[Y_\alpha , Y_\beta ] = 0 \mod \mathfrak {X}^{0,1}; \end{aligned}$$
(2)

here \(t^1,\dots ,t^s\) are coordinates in \(\mathbb {R}^s\) and \(\partial _{\alpha }\) means \({\partial }/{\partial t^\alpha }\). Then we shall show that the functions \(\lambda _\alpha ^\gamma\) can be chosen in such a way that

$$\begin{aligned} \lambda _\alpha ^\gamma (t^1, \dots , t^s) = \lambda _\alpha ^\gamma (t^1, \dots , t^\ell ) \end{aligned}$$
(3)

when \(\alpha \le \ell\). It will then follow quickly that the vector fields \(Y_\alpha\) with \(\alpha \le \ell\) define a CR structure on \(M\times V\), where V is a neighbourhood of 0 in \(\mathbb {R}^\ell\), and there is a CR-embedding of M in \(M\times V\).

To show that (2) holds, we observe that the \(Y_\alpha\) preserve the (lifted) \(\mathfrak {X}^{0,1}\), and choose the functions \(\lambda _{\alpha }^\gamma (t)\) such that \(\lambda _\alpha ^\gamma (0) = \delta _\alpha ^\gamma\) and the \(Y_\alpha\) commute modulo \(\mathfrak {X}^{0,1}\). Equivalently,

$$\begin{aligned} \partial _\alpha \lambda ^\gamma _\beta -\partial _\beta \lambda ^\gamma _\alpha =\mathrm {i}\lambda _\alpha ^\mu \lambda _\beta ^\nu c_{\mu \nu }^\gamma . \end{aligned}$$
(4)

Consider this system of PDE. Let \(\{\xi _\gamma \}\) be a basis of an abstract copy of the Lie algebra \(\mathfrak {g}\) and \((t^1,\dots , t^s)\) be coordinates of \(\mathfrak {g}\) with respect to this basis. Then \(\Lambda := \lambda ^\gamma _\alpha \xi _\gamma dt^\alpha\) is a Lie algebra valued 1-form, and the system (4) may be rewritten as

$$\begin{aligned} d\Lambda =\frac{\mathrm {i}}{2} [\Lambda ,\Lambda ], \end{aligned}$$
(5)

where d is the exterior derivative with respect to the t variables. This nonlinear autonomous system of PDE is similar to the structure equation of the Maurer–Cartan form, and this similarity allows us to solve the system (5). Let \(\Omega\) be the left-invariant Maurer–Cartan form on the simply connected Lie group G with Lie algebra \(\mathfrak {g}\). Then \(\Omega\) satisfies the Maurer–Cartan equation

$$\begin{aligned} d\Omega =-\frac{1}{2} [\Omega ,\Omega ]. \end{aligned}$$

Let t be real-analytic local coordinates on a neighbourhood of the identity e in G such that 0 corresponds to e and define \(\Omega :=\omega ^\gamma _\alpha (t) \xi _\gamma dt^\alpha\). Then \(\Omega (0)=\xi _\alpha dt^\alpha\). Let

$$\begin{aligned} \lambda ^\gamma _\alpha (t)=\omega ^\gamma _\alpha (-\mathrm {i}t). \end{aligned}$$

This is well defined since the \(\omega ^\gamma _\alpha\) are real-analytic, and the \(\lambda ^\gamma _\alpha\) defined in this way satisfy the equations (4) and \(\omega ^\gamma _\alpha (0)=\delta ^\gamma _\alpha\).

To arrange that (3) holds, we suppose that \(t^1\), ..., \(t^s\) are exponential coordinates of the second kind in some neighbourhood of e in G, that is,

$$\begin{aligned} g=\exp (t^{s}\xi _{s})\dots \exp (t^{1}\xi _1). \end{aligned}$$

We observe that the \(dt^\alpha\) component of the Maurer–Cartan form depends on \(t^1\), ..., \(t^{\alpha -1}\) only. Indeed, the (left-invariant) Maurer–Cartan form is

$$\begin{aligned} dL_{g^{-1}} dg \end{aligned}$$

and the \(dt^\alpha\) component is

$$\begin{aligned} dL_{g^{-1}} \frac{\partial g}{\partial t^\alpha }&=dL_{\exp (-t_1\xi _1)}\dots dL_{\exp (-t^{s}\xi _{s})} \\&\qquad \qquad \frac{\partial }{\partial t_\alpha } L_{\exp (t^{s}\xi _{s})}\dots L_{\exp (t^{\alpha +1}\xi _{\alpha +1})} R_{\exp (t^{1}\xi _{1})}\dots R_{\exp (t^{\alpha -1}\xi _{\alpha -1}) } \exp (t^{\alpha }\xi _{\alpha })\\&=dL_{\exp (-t^1\xi _1)}\dots dL_{\exp (-t^{\alpha -1}\xi _{\alpha -1})}dR_{\exp (t^{1}\xi _{1})}\dots dR_{\exp (t^{\alpha -1}\xi _{\alpha -1}) }\xi _\alpha \\&={\text {Ad}}_{\exp (-t^1\xi _1)}\dots {\text {Ad}}_{\exp (-t^{\alpha -1}\xi _{\alpha -1})}\xi _\alpha . \end{aligned}$$

Here L and R denote left and right translations. Therefore the functions \(\lambda _{\alpha }^\gamma\) do indeed depend only on the variables \(t^\mu\) with \(\mu <\alpha\).

It follows that \(\mathfrak {X}^{0,1} \oplus \langle Y_1,\dots , Y_\ell \rangle\) is well defined on \(M\times V\), where V is a suitable neighbourhood of 0 in \(\mathbb {R}^\ell\). It remains to show that \(\mathfrak {Y}^{0,1}\), the span of (the lift of) \(\mathfrak {X}^{0,1}\) and the vector fields \(Y_1,\dots ,Y_\ell\) defines a CR structure on \(\tilde{M} = M \times V\), that is,

$$\begin{aligned} \mathfrak {Y}^{0,1}\cap \overline{\mathfrak {Y}^{0,1}} = \{0\}. \end{aligned}$$

Suppose that \(V+a^j Y_j \in \mathfrak {Y}^{0,1}\) and \(W+b^k \overline{Y}_k \in \overline{\mathfrak {Y}^{0,1}}\). If

$$\begin{aligned} V+a^j Y_j=W+b^k \overline{Y}_k, \end{aligned}$$

then

$$\begin{aligned} W-V=a^j Y_j - b^k \overline{Y}_k=0, \end{aligned}$$

that is \(V=W=0\), and also

$$\begin{aligned} a^j (X_j+\mathrm {i}\partial _j)-b^j (\overline{X}_j-\mathrm {i}\partial _j)=0. \end{aligned}$$

Therefore \(a_j=-b_j\) and hence

$$\begin{aligned} a_j (X_j+\overline{X_j})=0. \end{aligned}$$

Since \(X_j(p)\) is not purely imaginary and the \(X_j(p)\) are linearly independent, \(X_j + \overline{X}_j \ne 0\) in a neighbourhood of p, and so (again passing to a submanifold if necessary) \(a_j=0\) is the only solution. \(\square\)

It may be worth remarking that if the CR manifold (M, D, J) admits a CR embedding into a complex space, then in fact the conditions of Baouendi, Rothschild and Treves [1] are satisfied, and a fortiori those of Hill and Nacinovich [3], and ours too. It is less clear what happens when there is a CR embedding into another CR manifold that is not a complex space.

It may also be helpful to note that in the special case where the vector fields \(X_1\), ..., \(X_s\) are real, then they generate (local) flows that preserve the CR structure; if they also generate a Lie algebra, then this generates a (local) group of transformations that preserves the structure. Further, even in the more special case where there is a transverse normalising Lie algebra of real vector fields, then our result extends that of [1]; in this case, the use of exponential coordinates of the second kind is not necessary.

Here is a corollary of the proof of our theorem.

Corollary 1

Let (G, D, J) be a left-invariant CR structure on a Lie group G. Then G can be locally embedded into complex space.

Proof

Let \(\{X_1,\dots ,X_{s}\}\) be a basis of right-invariant vector fields such that \(X_1\), ..., \(X_\ell\) are transverse to D at \(e\in G\).

As before we can find (complex) functions \(\lambda _{\alpha \beta }(t)\) such that the vector fields \(Y_\alpha\), given by

$$\begin{aligned} Y_\alpha := \sum \lambda _{\alpha \beta }(t) X_\beta + \mathrm {i}\frac{\partial }{\partial t_\alpha }, \end{aligned}$$

commute, and let \(t^1,\dots ,t^s\) be exponential coordinates of the second kind on G. Then \(\mathfrak {X}^{0,1} +\langle Y_1,\dots , Y_\ell \rangle\) determines an integrable complex structure on (a neighbourhood of \(e \times 0\) in) \(G\times \mathbb {R}^\ell\). \(\square\)

Of course, this was already known, as everything is analytic in this case, but arguably this proof is simpler.