By Elisabetta Barletta

ISBN-10: 0821843044

ISBN-13: 9780821843048

The authors research the connection among foliation conception and differential geometry and research on Cauchy-Riemann (CR) manifolds. the most items of research are transversally and tangentially CR foliations, Levi foliations of CR manifolds, strategies of the Yang-Mills equations, tangentially Monge-AmpГѓВ©re foliations, the transverse Beltrami equations, and CR orbifolds. the newness of the authors' strategy is composed within the total use of the tools of foliation thought and selection of particular functions. Examples of such functions are Rea's holomorphic extension of Levi foliations, Stanton's holomorphic degeneracy, Boas and Straube's nearly commuting vector fields process for the learn of worldwide regularity of Neumann operators and Bergman projections in multi-dimensional advanced research in different complicated variables, in addition to numerous purposes to differential geometry. Many open difficulties proposed within the monograph may possibly allure the mathematical group and bring about additional functions of foliation idea in advanced research and geometry of Cauchy-Riemann manifolds.

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**Additional resources for Foliations in Cauchy-Riemann Geometry (Mathematical Surveys and Monographs)**

**Example text**

25. 16) is direct. F. F) 9 T(C(M)) = Ker(a) ® Ker(d7r) 2. FOLIATED CR MANIFOLDS 30 then V = XT + fS for some X E T(M) and f E Coo(C(M)), where XT := 0 X. e. F)T + Ker(drr). 16) is proved. F). e. F). F). Then WH = 0. Since n _ 1 n+2{dy+rr`rlo } (for some 1-form rf on M, determined in terms of 9) and (dy)S = 1, it follows that rl(Wv) = 1/(n + 2). e. 19) G9((dir)VH, (dir)WH) + 9((d7r)WH)tl(Vv) = 0. Let us set W = V. e. (dir)VH = 0 hence Vii E Ker(drr) f1 Ker(q) = (0). F). e. Vv = 0. e. F). F) hence there F9 has signature (2n + 1 - q, 1).

Let F be a codimension q foliation of M. e. G9(X,T) = 0, X E T(M). e. do is a degenerate metric. We start by studying the geometry of F in (M, Go). The following concepts and terminology should be kept in mind as to degenerate metrics (cf. [97]). Let E --+ M be a real vector bundle of rank q (q > 2) over a C°° manifold M. 3A commutator of the form [X, YJ has length 2. 2. 7. By a (bundle) metric in E we intend a C°° section g : x E M - gx E Ey OR Ex* in E' ® E' such that gx is symmetric and has constant index ind(gx) = a, for any x E M.

The opposite inclusion may be proved in a similar manner. F)H(Af)]1) = (Y - 0(Y)T)1 = ao(HIY), for anyYET(M). 2. F) is degenerate in (T(M), G9). However, the pullback of F to the (total space of the) principal S'-bundle C(M) := [K(M) \ {zero section}]/IR+ turns out to be nondegenerate in (C(M), Fe), where Fe is the Fefferman metric of (M, 0). One may see C. M. Lee, [168], or the monograph [89] for a detailed description of the Fefferman metric. Nevertheless, to facilitate reading we collect a few notions and results below.

### Foliations in Cauchy-Riemann Geometry (Mathematical Surveys and Monographs) by Elisabetta Barletta

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