Complexified Tangent Bundle Research Articles

Connections on the Total Space of a Holomorphic Lie Algebroid

The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle $$T_{\mathbb {C}}E=T'E\oplus T''E$$ and its link, via the tangent anchor map, with the complexified tangent bundle $$T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)$$ . A holomorphic Lie algebroid structure is emphasized on $$T'E$$ . A special study is made for integral curves of a spray on $$T'E$$ . Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on $$T'E$$ . In the second part of section two, we study the holomorphic prolongation $$\mathcal {T}'E$$ of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on $$T'M$$ induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from $$T'M$$ .

The complex Frobenius theorem for rough involutive structures

We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in the analysis is a precise version of the Newlander-Nirenberg theorem with parameters, for integrable almost complex structures with minimal regularity, which builds on recent work of the authors.

Generic systems of co-rank one vector distributions

This paper studies a generic class of sub-bundles of the complexified tangent bundle. Involutive, generic structures always exist and have Levi forms with only simple zeroes. For a compact, orientable three-manifold the Chern class of the sub-bundle is mod 2 2 equivalent to the Poincaré dual of the characteristic set of the associated system of linear partial differential equations.

Sub-bundles of the complexified tangent bundle

We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold. The aim is to understand implications of properties of interest in partial differential equations.

Harmonic morphisms from even-dimensional hyperbolic spaces

In this paper we give a method for constructing complex valued harmonic morphisms in some pseudo-Riemannian manifolds using a parametrization of isotropic subbundles of the complexified tangent bundle. As a result we construct the first known examples of complex valued harmonic morphisms in real hyperbolic spaces of even dimension not equal to 4 which do not have totally geodesic fibres.

Wick type deformation quantization of Fedosov manifolds

A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of the Fedosov procedure. The main ingredient of this approach is a bilinear symmetric form defined on the complexified tangent bundle of the symplectic manifold and subject to some set of algebraic and differential conditions. It is precisely the structure which describes a deviation of the Wick-type star-product from the Weyl one in the first order in the deformation parameter. The geometry of the symplectic manifolds equipped by such a bilinear form is explored and a certain analogue of the Newlander-Nirenberg theorem is presented. The 2-form is explicitly identified which cohomological class coincides with the Fedosov class of the Wick-type star-product. For the particular case of K\"ahler manifold this class is shown to be proportional to the Chern class of a complex manifold. We also show that the symbol construction admits canonical superextension, which can be thought of as the Wick-type deformation of the exterior algebra of differential forms on the base (even) manifold. Possible applications of the deformed superalgebra to the noncommutative field theory and strings are discussed.

On totally real spheres in complex space

Let M ,N be two totally real and real analytic submanifolds in Cn . We say that M and N are biholomorphically equivalent if there is a biholomorphic mapping F defined in a neighborhood of M such that F (M ) = N . As a standard fact of complexification, one knows that all totally real and real analytic embeddings of M in C n are biholomorphically equivalent if M is of maximal dimension n . However, the topology of the manifold plays a major role in the existence of totally real immersions or embeddings. For instance, R.O. Wells [23] proved that if an n-dimensional compact and orientable manifold M admits a totally real embedding in Cn , then its Euler number must vanish. It was also observed by Wells that if M is a manifold of dimension n and it admits a totally real immersion in Cn , then its complexified tangent bundle T cM = TM⊗C is trivial. Conversely, the triviality of T cM also implies the existence of totally real immersions of M in Cn . This was obtained by M.L. Gromov in [11] through the method of convex integration. A stronger result due to J.A. Lees [17] says that M also admits Lagrangian immersions in Cn . The sphere S k : x 2 1 + . . . + x 2 k+1 = 1 in R k+1 gives us a trivial totally real embedding of S k in Ck+1. On the other hand, the works of Gromov [11], AhernRudin [1] and Stout-Zame [21] tell us that S k admits a totally real and real analytic embedding in Ck if and only if k = 1, 3. Our main result is the following. Theorem 1.1. If k ≤ 4, all totally real and real analytic embeddings of S k in Cn are biholomorphically equivalent. If k ≥ 5 and nk = k + 2[ k−1 4 ], there exist totally

Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms

LetM andN be real analytic (C) CR manifolds of hypersurface type of dimensions 2m+1 and 2n+1, m ≤ n, respectively. In this paper we show under generic assumptions that a CR mapping f : M → N satisfies a certain Pfaffian system in the jet space, which implies the analyticity and rigidity of CR mappings. We use a method of prolongation for the tangential Cauchy-Riemann equations. The technique is, roughly speaking, separating the holomorphic derivatives of CR functions from their complex conjugates and applying the tangential Cauchy-Riemann operators and then counting the order of the missing-directional derivatives in the holomorphic part. The argument of this paper is purely local, thus for instance, a manifold should be understood as a germ of a manifold at a reference point and mappings are supposed to preserve the reference point. First, we recall some basic definitions. For other definitions and proofs of the facts that we do not present here the readers are refered to [Jac]. Let M be a differentiable manifold of dimension 2m+1. A CR structure on M is a subbundle V of the complexified tangent bundle TCM having the following properties : i) each fiber is of complex dimension m, ii) V ∩ V = { 0 }, iii) [ V,V ] ⊂ V (integrability). Given a CR structure V we have Levi form

Orbits and global unique continuation for systems of vector fields

SupposeM is a smooth manifold andV is a locally integrable vector subbundle of the complexified tangent bundle ofM. This paper explores the global unique continuation property of distribution solutions ofV, i.e., the distributionsu onM such thatLu = 0 wheneverL is a section ofV, and the closely related problem of the structure of the Sussmann orbits ofHV.

Isotropic vector fields on spheres, spinor structures on spheres and projective spaces

In a previous paper “Fields of totally isotropic subspaces and almost complex structures” [1c], we constructed on the sphere S 4, in the complexified tangent bundle, a global field of totally isotropic planes to obtain a counterexample nullifying a statement about some manifolds and vector bundles; we proved that if the complexified tangent bundle T c( M) of a 2 r-dimensional C ∞ manifold M is a Withney sum T c(M) = η ⊕ η′ , where η, η′ are r-complex subbundles such that for any x belonging to M, η′ x = η macr; x , M does not have necessarily an almost complex structure. This article has two aims. First, we construct on any sphere global fields of isotropic vectors, nowhere zero, cross-sections of the complexified tangent bundle. We give for n ≥ 3 two independent methods, one with continuous hypothesis only, the second method with smooth conditions, valid for any dimension n ≥ 2. Our developments bring a new proof of the statement given in [1c] and a more precise and new result: the C ∞ complexified tangent bundle of any sphere is trivializable. Our second aim is to give an elementary approach to the existence conditions for spinor structures on spheres and projective spaces, using naturally our methods developed in numerous papers, particularly in [1d] and [1e]. However, the major part of the paper deals with the isotropic vector field problem on spheres and the reader unfamilar with spinors will be able to read the Secs. I, II and III, except some details at the end. We think that this paper will contribute to explaining some delicate questions and remove some errors or sophisms and also point out the role of the elementary geometry in spinor structures for particularly simple cases.

Canonical gravity from a variation principle in a copy of a tangent bundle

The new Hamiltonian formulation of complex general relativity first given by Ashtekar (1986) is derived from an action principle for connections in a bundle isomorphic with the complexified tangent bundle. On introducing spinors or orthonormal frames one obtains the theory in the spinorial or triad form respectively, as given by Jacobson and Smolin (1987), and Goldberg (1988). The more usual spinorial or triad Hamiltonian formalisms for real Lorentzian gravity are also recovered. In the present derivation stress is put on geometrical structures involved. Spinors are introduced in two different from standard ways, which causes some structural simplifications to occur in the general complex case. The role of reality conditions is expounded.

Embedding Compact Three-Dimensional CR Manifolds of Finite Type in C n

An abstract three-dimensional CR manifold X is a smooth manifold, without boundary, of real dimension three, equipped with a smooth one-dimensional complex sub-bundle T0, of its complexified tangent bundle which for each x E X satisfies TxOL n Txo, 1= {O}. Canonically associated to the CR structure is a first-order differential operator, db, which maps functions on X to sections of the bundle B 1 dual to TO, and is defined by

Pseudo-Chern classes of an almost pseudo-Hermitian manifold

For an almost pseudo-Hermitian manifold, pseudo-Chern classes are defined on its complexified tangent bundle with the pseudo-Hermitian structure as represented by certain ad ( U ( p , q ) ) {\text {ad}}(U(p,\,q)) -invariant forms on the manifold. It is shown that such a manifold always admits an almost Hermitian structure, and hence that Chern classes are also defined on the complexified tangent bundle with such an almost Hermitian structure. A relation between the pseudo-Chern classes and the Chern classes is established. From the relation, the pseudo-Chern classes are considered as the characteristic classes which measure how the almost pseudo-Hermitian structure deviates from an almost Hermitian structure.

Flat partial connections and holomorphic structures in 𝐶^{∞} vector bundles

The notion of a flat partial connection D in a C ∞ {C^\infty } vector bundle E, defined on an integrable subbundle F of the complexified tangent bundle of a manifold X is defined. It is shown that E can be trivialized by local sections s satisfying D s = 0 Ds = 0 . The sheaf of germs of sections s of E satisfying D s = 0 Ds = 0 has a natural fine resolution, giving the de Rham and Dolbeault resolutions as special cases. If X is a complex manifold and F the tangents of type (0, 1), the flat partial connections in a C ∞ {C^\infty } vector bundle E are put in correspondence with the holomorphic structures in E. If X, E are homogeneous and F invariant, then invariant flat connections in E can be characterized as extensions of the representation of the isotropic subgroup to which E is associated, extending results of Tirao and Wolf in the holomorphic case.