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On symmetric Cauchy-Riemann manifolds. (English) Zbl 0954.32016
In this paper the notion of symmetry of Riemannian and Hermitian manifolds is generalized to the category of CR-manifolds and CR-spaces.
An isometric CR-diffeomorphism $$\sigma$$ of a connected Hermitian CR-space $$M$$ is called a symmetry at $$a \in M$$ if $$a$$ is a fixed point (not necessarily isolated) of $$\sigma$$ and if the differential of $$\sigma$$ at $$a$$ coincides with the negative identity on the holomorphic subspace of the tangent space $$T_{a}$$ of $$M$$ at $$a$$ and on the orthogonal complement of the subspace of $$T_{a}$$ which is generated by the global holomorphic vector fields on $$M$$. The authors show that there is at most one symmetry at $$a$$ (which is necessarily involutive) and call $$M$$ a symmetric CR-space (SCR-space), if there is a symmetry $$s_{a}$$ at every $$a \in M$$. They have dropped the assumption of isolated fixed points because otherwise manifolds like the unit sphere in $$\mathbb{C}^{n}$$ would have been excluded. The group $$G$$ generated by the symmetries of an SCR-space $$M$$ operates properly and transitively on $$M$$ and is a closed subgroup with at most two connectivity components of the Lie group of all CR-diffeomorphisms of $$M$$. The isotropy subgroup $$K$$ of any fixed base point $$a \in M$$ is compact, and $$M$$ can be identified with $$G/K$$. The centralizer $$C(\sigma)$$ of $$\sigma_{a}$$ in $$G$$ contains $$K$$ and is a closed subgroup of $$G$$. The group $$L$$ generated by $$C^{0}(\sigma)$$ and $$K$$ leads to a canonical fibration $$\nu: G/K \rightarrow G/L$$ with connected typical fiber $$L/K$$. Every symmetry $$s_{b}$$ on $$M$$ can be pushed down to an involutive diffeomorphisms of $$G/L$$ with $$\nu(b)$$ as isolated fixed point. In special situations $$G/L$$ has the structure of a SCR-manifold, $$\nu$$ is a CR-map and a partial isometry, $$M$$ and $$N$$ have the same CR-dimension and the pushed down symmetries are symmetries of $$G/L$$. Then $$G/L$$ is called a symmetric reduction of $$M$$.
The authors present a method to produce examples of SCR-spaces (all of them with the structure of a generalized Heisenberg group) of arbitrary CR-dimension and arbitrary CR-codimension and arbitrary Levi form at a given point. From coverings of symmetric domains in the unit sphere of $$\mathbb{C}^{n}$$ they obtain an uncountable family of pairwise non-isomorphic SCR-manifolds. Every SCR-space can be obtained by a construction principle in terms of Lie groups which is thoroughly discussed. There are also Lie theoretic conditions given for an SCR-space to be embeddable into a complex manifold; such an embedding is not always possible. Finally the authors study the Shilov boundary $$S$$ of a bounded symmetric domain $$D$$ in $$\mathbb{C}^{n}$$, realized as a bounded circular convex domain. $$S$$ is a SCR-manifold and as a main result the authors prove that $$D$$ does not have a factor of tube type if and only if every smooth CR-function on $$S$$ extends to a continuous function on the topological closure $$\overline{D}$$ of $$D$$, holomorphic on $$D$$. Moreover this is the case if and only if the group of holomorphic automorphisms of $$D$$ coincides with the group of smooth CR-diffeomorphisms of the Shilov-boundary $$S$$.

##### MSC:
 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 32V10 CR functions 32V25 Extension of functions and other analytic objects from CR manifolds 53C35 Differential geometry of symmetric spaces
##### Keywords:
symmetric CR-spaces; Shilov boundary
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