By Elisabetta Barletta
The authors learn the connection among foliation idea and differential geometry and research on Cauchy-Riemann (CR) manifolds. the most items of analysis are transversally and tangentially CR foliations, Levi foliations of CR manifolds, ideas of the Yang-Mills equations, tangentially Monge-AmpГѓВ©re foliations, the transverse Beltrami equations, and CR orbifolds. the newness of the authors' technique is composed within the total use of the tools of foliation concept 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 technique for the research of world regularity of Neumann operators and Bergman projections in multi-dimensional advanced research in different complicated variables, in addition to quite a few purposes to differential geometry. Many open difficulties proposed within the monograph may perhaps allure the mathematical neighborhood and result in additional functions of foliation idea in complicated research and geometry of Cauchy-Riemann manifolds.
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Additional info for Foliations in Cauchy-Riemann Geometry (Mathematical Surveys and Monographs)
E. one has the direct sum decomposition T(M) = H(M) e RT and it allows us to extend Go to a natural semi-Riemannian metric on M namely the Webster metric go given by go(X,Y) =Go(X,Y), 9o(X,T) =0, go(T,T) = 1, for any X, Y E H(M). The Webster metric is a semi-Riemannian metric on M of index 2s where (r, s) is the signature of Lo (r+s = n). By a fundamental result due to N. Tanaka. M. e. 8) ToJ+Jor=O, where r(X) := To (T, X) for any X E T(M). r is the pseudohermitian torsion of the Tanaka-Webster connection.
If that does happen the Lorentz manifold in question is said to be time oriented (a time oriented Lorentz manifold is commonly referred to as a space-time, cf. g. , p. 17). It turns out that the total space C(M) of the canonical circle bundle over a strictly pseudoconvex CR manifold M is time oriented. 20. e. (C(M), Fe) is a space-time. The time orientation TT - S divides the set of all nonspacelike tangent vectors into two subsets. 21. A nonspacelike vector v E TZ (C(M)) is called future directed (respectively past directed) if Fe,,,,(Tz -SZ , v) < 0 (respectively if Fe,.
2. FOLIATIONS OF CR MANIFOLDS AND THE FEFFERMAN METRIC 27 A remark is in order. The concept of a (r, 0)-form is clearly similar to that of a form of type (r, 0) on a complex n-dimensional complex manifold X. j, dz"l A . . : U - C) hence the top degree forms of type (r, 0) on a complex n-dimensional manifold are the forms of type (n, 0). The case of a CR manifold M of dimension dims M = 2n + 1 and CR dimension n is rather different in this respect. Indeed, let us assume that M is nondegenerate and fix as customary a contact form 9 and the corresponding characteristic direction T.
Foliations in Cauchy-Riemann Geometry (Mathematical Surveys and Monographs) by Elisabetta Barletta