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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
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