Embeddability of real analytic Cauchy-Riemann manifolds (original) (raw)
On Cauchy-Riemann circle bundles
Kodai Mathematical Journal, 2005
Building on ideas of R. Mizner, [17]-[18], and C. Laurent-Thiébaut, [14], we study the CR geometry of real orientable hypersurfaces of a Sasakian manifold. These are shown to be CR manifolds of CR codimension two and to possess a canonical connection D (parallelizing the maximally complex distribution) similar to the Tanaka-Webster connection (cf. [21]) in pseudohermitian geometry. Examples arise as circle subbundles S 1 ! N ! p M, of the Hopf fibration, over a real hypersurface M in the complex projective space. Exploiting the relationship between the second fundamental forms of the immersions N ! S 2nþ1 and M ! CP n and a horizontal lifting technique we prove a CR extension theorem for CR functions on N. Under suitable assumptions [Ric D ðZ; ZÞ þ 2gðZ; ðI À aÞZÞ b 0, Z A T 1; 0 ðNÞ, where a is the Weingarten operator of the immersion N ! S 2nþ1 ] on the Ricci curvature Ric D of D, we show that the first Kohn-Rossi cohomology group of M vanishes. We show that whenever Ric D ðZ; W Þ À 2gðZ; W Þ ¼ ðm pÞgðZ; W Þ for some m A C y ðMÞ, M is a pseudo-Einstein manifold.
The Geometry of Tangent Bundles and Almost Complex Structures
2018
In this paper, we study the geometry of a tangent bundle of a Riemannian manifold endowed with a Sasaki metric. Using O’Neill tensors given in [7], we prove some characteristic theorems comparing the geometries of a smooth manifold and its tangent bundle. We also show that there exists an almost complex structure on a Riemannian manifold which is not holomorphic to the canonical almost complex structure of its tangent bundle.
On Certain Structures Defined on the Tangent Bundle
Rocky Mountain Journal of Mathematics, 2006
The differential geometry of tangent bundles was studied by several authors, for example: Davies [4], Yano and Davies [5], Dombrowski [6], Ledger and Yano [9] and Blair [1], among others. It is well known that an almost complex structure defined on a differentiable manifold M of class C ∞ can be lifted to the same type of structure on its tangent bundle T (M). However, when we consider an almost contact structure, we do not get the same type of structure on T (M). In this case we consider an odd dimensional base manifold while our tangent bundle remains to be even dimensional. The purpose of this paper is to examine certain structures on the base manifold M in relation to that of the tangent bundle T (M).
Complex Differential Geometry: An Introduction
Let M be a 2m-dimensional topological manifold. A coordinate atlas {(U, φ U : U → C m )} is called holomorphic if the transition functions φ U • φ −1 V are holomorphic functions between subsets of C m ; in this case the coordinate charts φ U are called local holomorphic coordinates. The manifold M is called complex if it admits a holomorphic atlas. Two holomorphic atlases are called equivalent if their union is a holomorphic atlas. An equivalence class of holomorphic atlases on M is called a complex structure.
Nonorientable Manifolds, Complex Structures, and Holomorphic Vector Bundles
Acta Applicandae Mathematicae - ACTA APPL MATH, 2001
A generalization of the notion of almost complex structure is defined on a nonorientable smooth manifold M of even dimension. It is defined by giving an isomorphism J from the tangent bundle TM to the tensor product of the tangent bundle with the orientation bundle such that J?J=-IdTM. The composition J?J is realized as an automorphism of TM using the fact that the orientation bundle is of order two. A notion of integrability of this almost complex structure is defined; also the Kähler condition has been extended. The usual notion of a complex vector bundle is generalized to the nonorientable context. It is a real vector bundle of even rank such that the almost complex structure of a fiber is given up to the sign. Such bundles have generalized Chern classes. These classes take value in the cohomology of the tensor power of the local system defined by the orientation bundle. The notion of a holomorphic vector bundle is extended to the context under consideration. Stable vector bundle...
Transactions of the American Mathematical Society, 1974
Complex analytic submanifolds and totally real submanifolds are two typical c l a s s e s among all submanifolds of an almost Hermitian manifold. In this paper, some characterizations of totally real submanifolds are given. Moreover some classifications of totally real submanifolds in complex space forms are obtained. (resp. M). We call M an totally real suJbmanifold of M if M admits an isometric CU immersion into M such that for all x, ] (Tx(M))C v X , where Tx(M)denotes the tangent space of M at x and vx the normal space at x . By a plane section we mean a 2-dimensional linear subspace of a tangent CU space. A plane section r i s called antiholomorphic if ]r i s perpendicular to 7. Proposition 2.1. Let M be a submanifold immersed in an almost Hermitian CU .-u manifold M. T h e n M i s a totally real submanifold of M if and only if every plane section of M is antiholomorphic. Proof. Let X be an arbitrary vector in Tx(M),and let e l = X I e 2 , .*, en be a basis of Tx(M). We denote by rij the plane section spannei by ei and e j . Assume that every plane section i s antiJolomorphic. Then ] r l . are perpen-
Real structures of Teichmüller spaces, Dehn twists, and moduli spaces of real curves
Mathematische Zeitschrift, 1999
An orientation reversing involution σ of a topological compact genus g, g > 2, surface Σ induces an antiholomorphic involution σ * : T g → T g of the Teichmüller space of genus g Riemann surfaces. Two such involutions σ * and τ * are conjugate in the mapping class group if and only if the corresponding orientation reversing involutions σ and τ of Σ are conjugate in the automorphism group of Σ. This is equivalent to saying that the quotient surfaces Σ/ σ and Σ/ τ are homeomorphic. Hence the Teichmüller space T g has m g = 3g+4 2 distinct antiholomorphic involutions, which are also called real structures of T g ([7]). This result is a simple fact that follows from Royden's theorem ([4]) stating that the the mapping class group is the full group of holomorphic automorphisms of the Teichmüller space(g > 2). Let σ * : T g → T g and τ * : T g → T g be two real structures that are not conjugate in the mapping class group. In this paper we construct a real analytic diffeomorphism d : T g → T g such that σ * = d −1 • τ * • d. (1) This mapping d is a product of full and half Dehn-twists around certain simple closed curves on the surface Σ. This has applications to the moduli spaces of real algebraic curves. A compact Riemann surface (Σ, X) admitting an antiholomorphic involution σ : (Σ, X) → (Σ, X) is a real algebraic This work has been partially supported by the European Union Human Capital and Mobility Project "Real Algebraic and Analytic Geometry" and by Swiss National Science Foundation, Contract 21-40742.94. The paper grew out of discussions that took place when both authors were visiting Robert Silhol in Arles. We thank Silhol for his great hospitality during that visit.
Section spaces of real analytic vector bundles and a theorem of Grothendieck and Poly
Linear and Non-Linear Theory of Generalized Functions and its Applications, 2010
The structure of the section space of a real analytic vector bundle on a real analytic manifold X is studied. This is used to improve a result of Grothendieck and Poly on the zero spaces of elliptic operators and to extend a result of Domański and the author on the nonexistence of bases to the present case.
On almost complex structures in the cotangent bundle
Turkish Journal of Mathematics
E. M. Patterson and K. Yano studied vertical and complete lifts of tensor fields and connections from a manifold Mn to its cotangent bundle T * (Mn) . Afterwards, K. Yano studied the behavior on the crosssection of the lifts of tensor fields and connections on a manifold Mn to T * (Mn) and proved that when ϕ defines an integrable almost complex structure on Mn , its complete lift C ϕ is a complex structure. The main result of the present paper is the following theorem: Let ϕ be an almost complex structure on a Riemannian manifold Mn . Then the complete lift C ϕ of ϕ , when restricted to the cross-section determined by an almost analytic 1 -form ω on Mn , is an almost complex structure.
Augmented bundles and real structures
2018
Real structures on a mathematical object equipped with a holomorphic structure are antiholomorphic involutions. Klein surfaces, that is, pairs consisting of a Riemann surface and a real structure, were rst studied by Klein and Weichold, who gave in [54] and [86] a topological classication of them. Real structures on complex Lie algebras and complex Lie groups were studied by E. Cartan, Tits, Vogan, etc. Real structures on complex vector spaces are antilinear involutions. Real structures for vector bundles are antiholomorphic involutions, which are antilinear in the bres. This notion was introduced by Atiyah in [4]. Gross and Harris gave in [43] a classication of real line bundles on Klein surfaces. A common feature of these involutions is that their xed-point subspaces are dened over the real numbers. For instance, they are real algebraic curves in the case of Klein surfaces, real vector spaces in the case of real structures on complex vector spaces, or a real Lie groups in the case of real structures on complex Lie groups. Vector bundles or principal bundles equipped with various extra structures are called generically as augmented bundles. This thesis deals with the study of three examples of augmented bundles (Higgs pairs, G-Higgs bundles and G-parabolic Higgs bundles, where G is a real reductive Lie group) in the presence of real structures on their constituent components. 0.1.1 Historical background The Hodge correspondence establishes an isomorphism between representations Hom(π 1 (X), G) of the fundamental group π 1 (X) of a compact Riemann surface X into an abelian group G and the cohomological group H 1 (X, G), that parametrizes isomorphism classes of G-bundles over X. The augmented bundles that we study have appeared as sucessive generalizations of the non-abelian Hodge correspondence, that is an extension, when the group is not abelian, of the Hodge correspondence. If G is the unitary group U(n), the non-abelian Hodge correspondence is known as the theorem of Narasimhan and Seshadri. The moduli space of stable holo-1 morphic vector bundles of rank n and degree d on a compact Riemman surface X was constructed by Mumford in [58]. Narasimhan and Seshadri proved in [57] that this moduli space, for d = 0 and genus g of X greater than one, was homeomorphic to the moduli space of irreducible representations of π 1 (X) into U(n). Seventeen years later, Donaldson gave in [30] a new proof of the theorem of Narasimhan and Seshadri using the approach of Gauge Theory. If G is a compact Lie group, the correspondence was proved by Ramanathan. He introduced in [68] and [69] a notion of stability for G-bundles and constructed the moduli space M d (G) of G-bundles of topological class d ∈ π 1 (G), for any compact Riemann surface X of genus g ≥ 2. Ramanathan proved in [67] that M 0 (G) is homeomorphic to the moduli space of representations R(G) of π 1 (X) into G. If G is a non compact reductive Lie group, Higgs bundles are required in the correspondence. They were rst introduced by Hitchin in [46], for G = SL(2, C). He constructed its moduli space M(SL(2, C)) relating Mumford's notion of stability with solutions of the Hitchin equations. Hitchin pointed out that a solution of the Hitchin equations produces a reductive at connection. The existence of a harmonic metric on a reductive at bundle was proved by Donaldson in [32], for SL(2, C) and by Corlette in [29], for a complex reductive Lie group G. The Corlette-Donaldson correspondence establishes a homeomorphism between the moduli space of reductive at connections and the moduli space of harmonic bundles, which, in turn, is in bijection with solutions of the Hitchin equations. The existence of the moduli space of G-Higgs bundles M(G), for any complex reductive Lie group G, was constructed by Simpson in [79, 80]. He generalized the non-abelian Hodge correspondence in [78] for G-Higgs bundles over compact Kähler manifolds. Gothen, Garcia-Prada and Mundet i Riera introduced in [39] the notion of Higgs pairs (E, ϕ) , which consist of a holomorphic G-bundle E over a compact Riemann surface X, where G is a connected reductive complex Lie group and a holomorphic section ϕ of the vector bundle associated to E, via a representation ρ : G → GL(V) of G on a complex vector space V, tensored by a line bundle L. Let H be a maximal compact subgroup of G, such that ρ reduce to a representation ρ : H → U(V). These authors dened α-polystability, where α is an element of the center of
Sub-Bundles of the Complexified Tangent Bundle
Transactions of the American Mathematical Society
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.
Almost complex structures on cotangent bundles and generalized geometry
Journal of Geometry and Physics, 2010
We study a class of complex structures on the generalized tangent bundle of a smooth manifold M endowed with a torsion free linear connection, ∇. We introduce the concept of ∇-integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of M induced by complex structures on the generalized tangent bundle of M.
On Some Cohomological Properties of Almost Complex Manifolds
Journal of Geometric Analysis, 2010
We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang in [16], in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by discrete subgroups. We obtain families of pure and full almost complex structures on compact nilmanifolds and solvmanifolds. Some of these families are parametrized by real 2-forms which are anti-invariant with respect to the almost complex structures.
A remark on almost complex manifold with linear connections
INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE “TECHNOLOGY IN AGRICULTURE, ENERGY AND ECOLOGY” (TAEE2022)
The complex and almost complex manifolds are enormous and very fruitful fields for differential geometry. J.A. Schouten and D. Van Dantzig were the first to try to apply the finding in differential geometry of spaces with Riemannian metric and affine connection to the situation of complex structure spaces. C. Ehresmann defined an almost complex space as an even-dimensional differentiable manifold containing a tensor field with a square root of minus unity. The present paper intended to study, some fundamental properties with linear connections of an almost complex manifold. If almost complex structure be converted from a complex structure, then the various integrable and completely integrable condition has been investigated. Furthermore, the symmetric affine connections of almost complex manifold have also been investigated.
A remark on the embedding theorem associated to complex connections of mixed type
Osaka Journal of Mathematics, 2010
Let M be a compact complex manifold and let ( L, H ) be a holomorphic Hermitian line bundle overM such that the curvature form of h is nondegenerate and splits into the difference2C 2 of two semipositive forms2C and 2 whose null spaces define mutually transverse holomorphic foliations F and FC, respectively. Then Lm admits, for sufficiently largem 2 N, C1 sections whose ratio embeds M into CP N holomorphically (resp. antiholomorphically) along FC (resp. alongF ). Introduction In the theory of complex manifolds, geometric structures de fined by the subbundles of tangent bundles are basic in analyzing submanifolds and h olomorphic maps. Foliations is one of such structures. In [6], and embedding theorem was established for those mani folds equipped with two mutually transverse holomorphic foliations. Namely, i t was proved that, given a compact complex manifoldM with holomorphic foliationsFC and F such that the tangent bundle ofM is the direct sum of those of FC andF , M is embeddable...
Pacific Journal of Mathematics, 1988
This paper is devoted to the study of a particular kind of complex manifold with non-trivial topology: holomorphic fiber bundles with fibers biholomorphic to plane annuli. 0. Introduction. In recent years, some work has been done on function theory in complex manifolds with non-trivial topology. Two different approaches have been developed, a variational one and a purely complex-theoretical one.