Foliations in Cauchy-Riemann geometry.

*(English)*Zbl 1122.53015
Mathematical Surveys and Monographs 140. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4304-8/hbk). x, 256 p. (2007).

The present book is a monograph in which the authors try to impact the theory of foliations in the geometry and analysis on CR manifolds. The monograph consists of eleven chapters. Chapter 1 is a short introduction to the theory of foliations which is based on the book of P. Molino [Riemannian foliations. Progress in Mathematics, Vol. 73. Boston-Basel: Birkhäuser (1988; Zbl 0633.53001)]. The next seven chapters are the main part of the book.

Chapter 2 is devoted to foliated CR manifolds. The fundamental part of this chapter formally imitates the book of Ph. Tondeur [Foliations on Riemannian manifolds. Universitext. New York etc.: Springer-Verlag (1988; Zbl 0643.53024)] which follows from the analogy between foliations on Riemannian manifolds and foliations on CR manifolds in the nondegenerate case.

In Chapter 3 the Levi foliations and their holomorphic extendibility are considered. The main theorem of this chapter is Rea’s theorem: Let \(X\) be an open subset of some complex manifold and \(M=\{ z\in X :\rho\left( z\right) =0\}\) be a Levi flat real hypersurface in \(X\) (\(\rho\in C^{\infty}\left( X,\mathbb{R}\right)\) and \(d\rho\neq 0\) in any point of \(M\)). If \(M\) is real analytic then there exist an open neighbourhood \(\Omega\) of \(M\) in \(X\) and a holomorphic foliation \(\mathcal{F}\) on \(\Omega\) of complex codimension 1 such that \(M\) is a saturated subset of \(\Omega\). Chapter 4 reports on the known results on nonexistence of Levi flat CR submanifolds in complex projective spaces. Chapter 5 is about foliations with tangential CR structure, i.e., about foliations for which any leaf \(L\) is a CR manifold and the inclusion \(\iota :L\hookrightarrow M\) is the CR map. The authors study foliations by level hypersurfaces of some smooth function on a strongly pseudoconvex domain in \(\mathbb{C}^n\) and give a new axiomatic description of the canonical connection in that case (the Graham-Lee connection). This connection is used for looking on the boundary values of a Yang-Mills field in a Hermitian holomorphic vector bundle over a smoothly bounded strongly pseudoconvex domain in \(\mathbb {C}^n\).

Chapter 6 gives basics of the theory of foliations with transverse CR structure. Chapters 7 and 8 present two main applications of such foliations. The first of them considers relations between the Gigante and Tomassini theory of CR Lie algebras and Fedida’s \(\mathcal{G}\)-Lie foliations and contains a homotopic classification of transverse \(f\)-structures. The second one is devoted to the solution of the transverse Beltrami equation. The last three chapters are devoted to the structure of orbifolds. The last of these chapters motivates the need of the theory of CR orbifolds and gives some open problems. The book also contains three appendices: on the holomorphic bisectional curvature, on the partition of unity on orbifolds, and on pseudodifferential operators on \(\mathbb{R}^n\).

Chapter 2 is devoted to foliated CR manifolds. The fundamental part of this chapter formally imitates the book of Ph. Tondeur [Foliations on Riemannian manifolds. Universitext. New York etc.: Springer-Verlag (1988; Zbl 0643.53024)] which follows from the analogy between foliations on Riemannian manifolds and foliations on CR manifolds in the nondegenerate case.

In Chapter 3 the Levi foliations and their holomorphic extendibility are considered. The main theorem of this chapter is Rea’s theorem: Let \(X\) be an open subset of some complex manifold and \(M=\{ z\in X :\rho\left( z\right) =0\}\) be a Levi flat real hypersurface in \(X\) (\(\rho\in C^{\infty}\left( X,\mathbb{R}\right)\) and \(d\rho\neq 0\) in any point of \(M\)). If \(M\) is real analytic then there exist an open neighbourhood \(\Omega\) of \(M\) in \(X\) and a holomorphic foliation \(\mathcal{F}\) on \(\Omega\) of complex codimension 1 such that \(M\) is a saturated subset of \(\Omega\). Chapter 4 reports on the known results on nonexistence of Levi flat CR submanifolds in complex projective spaces. Chapter 5 is about foliations with tangential CR structure, i.e., about foliations for which any leaf \(L\) is a CR manifold and the inclusion \(\iota :L\hookrightarrow M\) is the CR map. The authors study foliations by level hypersurfaces of some smooth function on a strongly pseudoconvex domain in \(\mathbb{C}^n\) and give a new axiomatic description of the canonical connection in that case (the Graham-Lee connection). This connection is used for looking on the boundary values of a Yang-Mills field in a Hermitian holomorphic vector bundle over a smoothly bounded strongly pseudoconvex domain in \(\mathbb {C}^n\).

Chapter 6 gives basics of the theory of foliations with transverse CR structure. Chapters 7 and 8 present two main applications of such foliations. The first of them considers relations between the Gigante and Tomassini theory of CR Lie algebras and Fedida’s \(\mathcal{G}\)-Lie foliations and contains a homotopic classification of transverse \(f\)-structures. The second one is devoted to the solution of the transverse Beltrami equation. The last three chapters are devoted to the structure of orbifolds. The last of these chapters motivates the need of the theory of CR orbifolds and gives some open problems. The book also contains three appendices: on the holomorphic bisectional curvature, on the partition of unity on orbifolds, and on pseudodifferential operators on \(\mathbb{R}^n\).

Reviewer: Andrzej Piatkowski (Łódź)

##### MSC:

53C12 | Foliations (differential geometric aspects) |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |

32T15 | Strongly pseudoconvex domains |

32V05 | CR structures, CR operators, and generalizations |

32V15 | CR manifolds as boundaries of domains |

32V20 | Analysis on CR manifolds |

32V30 | Embeddings of CR manifolds |