# zbMATH — the first resource for mathematics

Equivalence and embedding problems for CR-structures of any codimension. (English) Zbl 1079.32022
The 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.

##### MSC:
 32V25 Extension of functions and other analytic objects from CR manifolds 53A40 Other special differential geometries 32V35 Finite-type conditions on CR manifolds
Full Text: