Equivalence and embedding problems for CR-structures of any codimension.

*(English)*Zbl 1079.32022The authors discuss in this paper a new approach to the general problem of establishing the CR equivalence of CR manifolds of arbitrary codimension. Recall that a CR manifold \(M\), with analytic tangent space \(HM\), is of finite type [in the sense of T. Bloom and I. Graham, Invent. Math. 40, 217–243 (1977; Zbl 0339.32003)] if the smooth sections of \(HM\) are generators of the Lie algebra of smooth vectors fields on \(M\). Denote by \(J\) the partial complex structure on \(HM\) and set \(T^{0,1}M=\{X+iJX\mid X\in HM\}\). Let \({T^*}^0M\) and \({T^*}^{1,0}M\) be the subbundles of the complexified cotangent bundle \(\mathbb{C}T^*M\) consisting of the forms that annihilate \(HM\) and \(T^{0,1}M\), respectively. Then \({T^*}^0M\subset {T^*}^{1,0}M\) and the Lie derivative \(\mathcal{L}_{\overline{Z}}\), for \(\overline{Z}\in\mathcal{C}^\infty(M,T^{0,1}M)\), maps \(\mathcal{C}^\infty(M,{T^*}^{1,0}M)\) into itself. Then \(M\) is finitely nondegenerate at \(p\in M\) if the span of \(\mathcal{L}_{\overline{Z}_1} \circ\cdots\circ \mathcal{L}_{\overline{Z}_m}(\theta)(p)\), for \(\overline{Z}_1,\dots, \overline{Z}_m\in\mathcal{C}^\infty (M,T^{0,1}M)\), and \(\theta\in\mathcal{C}^\infty(M,{T^*}^0)\) is \({T^*}^{1,0}_pM\) [this notion was introduced by C. K. Han, Invent. Math. 73, 51–69 (1983; Zbl 0517.32007)].

For CR manifolds \(M\), \(M'\), which are at the same time of finite type and finitely nondegenerate, the Authors show that there exist a positive integer \(r\) and a smooth map \(\Phi:G^r(M,M')\to G^{r+1}(M,M')\) such that every smooth CR-diffeomorphism \(f:M\to M'\) satisfies a complete differential system of the form \(\jmath_x^{r+1}(f)=\Phi(\jmath_x^{r}(f))\). Here \(G^h(M,M')\) is the set of \(h\)-jets of smooth diffeomorphisms \(f:M\to M'\) and \(\jmath_x^h(f)\) is the \(h\)-jet of \(f\) at \(x\in M\). Moreover, if \(M\) is connected, a CR diffeomorphism is completely determined by its \(r\)-jet at any point of \(M\).

The authors also consider, and reduce to the study of complete differential systems, the problem of the CR embedding of \(M\) into some other CR manifold \(M'\) and give also interesting examples and other applications.

The main feature and the novelty of this paper in the literature of CR-equivalence is that the authors do not make any real analyticity assumption for the CR manifolds involved.

For CR manifolds \(M\), \(M'\), which are at the same time of finite type and finitely nondegenerate, the Authors show that there exist a positive integer \(r\) and a smooth map \(\Phi:G^r(M,M')\to G^{r+1}(M,M')\) such that every smooth CR-diffeomorphism \(f:M\to M'\) satisfies a complete differential system of the form \(\jmath_x^{r+1}(f)=\Phi(\jmath_x^{r}(f))\). Here \(G^h(M,M')\) is the set of \(h\)-jets of smooth diffeomorphisms \(f:M\to M'\) and \(\jmath_x^h(f)\) is the \(h\)-jet of \(f\) at \(x\in M\). Moreover, if \(M\) is connected, a CR diffeomorphism is completely determined by its \(r\)-jet at any point of \(M\).

The authors also consider, and reduce to the study of complete differential systems, the problem of the CR embedding of \(M\) into some other CR manifold \(M'\) and give also interesting examples and other applications.

The main feature and the novelty of this paper in the literature of CR-equivalence is that the authors do not make any real analyticity assumption for the CR manifolds involved.

Reviewer: Mauro Nacinovich (Roma)

##### MSC:

32V25 | Extension of functions and other analytic objects from CR manifolds |

53A40 | Other special differential geometries |

32V35 | Finite-type conditions on CR manifolds |