Compact Lie groups are ‘perfect’ entities in modern mathematics because: 

(a)

They are differentiable (and even analytical) manifolds of finite dimension and therefore can be analysed with the help of the most powerful tool of mathematics: ordinary and partial differential operators (equations);




(b)

Denoting by £(G) or L(G) = T e (G) the tangent space to G in unity e, we find that L(G) can be identified with the set of left-invariant vector fields on G (or with the set of appropriate differential operators of order one). Thus L(G) naturally becomes a Lie algebra: on L(G) there exists a structure [·, ·] which satisfies the axioms of Lie algebra (see below);




(c)

This rich algebraic structure, in turn, makes it possible to apply the powerful algebraic tools (theory of Lie algebras);




(d)

An abstract (finite dimensional) Lie algebra g defines (up to an isomorphism) a simply connected Lie group G such that £(G) = g. Therefore investigations of the Lie group G can be to large extent replaced by investigations of its Lie algebra g;




(e)

Compactness of a Lie group G makes it possible to use the theory of representations of compact groups developed by Weyl and Peter. This theory is extraordinarily beautiful and simple: it reduces to large extend to the spectral theory of compact operators;




(f)

And finally every complex, compact (connected) commutative Lie group is a torus. Moreover, as it was shown by E. Cartan, every compact Lie group contains a maximal torus T (any other such a torus is conjugated T 1 = gTg −1 ); and any point g 0 of the group G belongs to some maximal torus: G = U g gTg −1. This fundamental Cartan theorem makes it possible to apply to T (and thus to the whole of G) the theory of representations of abelian groups: every representation of the torus T decomposes into the finite (and orthogonal) sum of unit representations (over ℂ), the so called weights. Thus the knowledge of these weights is the fundamental element of the theory of representations of compact Lie groups.

Representations of Compact Lie Groups

Tangent and cotangent bundles : differential geometry

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Transformation Groups In Differential Geometry

We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversally conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group. We prove that either there exists a measure invariant under all the elements of the group (or the pseudo-group), or almost surely a long composition of maps contracts a ball exponentially. We deduce some results about the unique ergodicity.

Random Conformal Dynamical Systems

0. Hyperbolic surfaces and Fuchsian groups: summary 1. Hyperbolic 3-manifolds 2. The basis of Kleinian group theory 3. Geometrically finite Kleinian groups 4. Finitely generated Kleinian groups 5. The sphere at infinity 6. Infinite ends of hyperbolic manifolds 7. Algebraic and geometric convergences Appendix References

Hyperbolic manifolds and Kleinian groups

Several examples of transversely holomorphic foliations with non-trivial Godbillon-Vey class are given. It is shown that if the complex codimension is odd, then there are at least two foliations which are distinct as real foliations. It is also shown that the Godbillon-Vey class is rigid under deformations in the category of transversely holomorphic foliations.

Godbillon-Vey Class of Transversely Holomorphic Foliations

A Fatou-Julia decomposition of transversally holomorphic fo- liations of complex codimension one was given by Ghys, Gomez-Mont and Saludes. In this paper, we propose another decomposition in terms of nor- mal families. Two decompositions have common properties as well as cer- tain dierences. It will be shown that the Fatou sets in our sense always contain the Fatou sets in the sense of Ghys, Gomez-Mont and Saludes and the inclusion is strict in some examples. This property is important when discussing a version of Duminy's theorem in relation to secondary charac- teristic classes. The structure of Fatou sets is studied in detail, and some properties of Julia sets are discussed. Some similarities and dierences be- tween the Julia sets of foliations and those of mapping iterations will be shown. An application to the study of the transversal Kobayashi metrics is also given. R esum e. Une decomposition de Fatou-Julia de feuilletages transversale- ment holomorphes de codimension complexe un est donee par Ghys, Gomez- Mont et Saludes. Dans cet article, nous proposons une autre decomposition en utilisant des familles normales. Deux decompositions ont des proprietes communesdi´ erences certaines. Il est montre que l'ensembles de Fatounotre sense contiennent toujours ceux au sense de Ghys, Gomez- Mont et Saludes, et aussi que l'inclusion peutstricte dans quelques exemples. Cette propriete est importante en discutant une version du theoreme de Duminy relie aux classes caracteristiques secondaires. Quelques similitudes et di´ erences entre les ensembles de Julia de feuilletages et ceux d'iterations d'applications sont presentees. Une application auxde la metriques transversale de Kobayashi est aussi donee.

/pdf/a-fatou-julia-decomposition-of-transversally-holomorphic-4buudqc5me.pdf

A fatou-julia decomposition of transversally holomorphic foliations

Localization and residue of the Bott class

The Bott class for transversely holomorphic foliations and the Futaki invariant for complex manifolds are discussed. The main tools are a generalization of Heitsch's residue and the integration of Cech-de Rham cochains introduced by Lehmann. It is shown that the Bott class admits a natural decomposition as a differential character in the sense of Cheeger and Simons. The decomposition implies that the real and the complex parts of the Futaki invariant have a certain difference from the viewpoint of dynamical systems. Introduction. In the theory of characteristic classes, so-called Chern-Simons classes are very important. In this paper, the Futaki invariant and more generally, the Bott class of transversely holomorphic foliations are studied. The Bott class is the most fundamental characteristic class for transversely holomorphic foliations. Here a foliation is said to be transversely holomorphic if its holonomy is written in terms of biholomorphic diffeomorphisms. The Bott class is a Chern-Simons class or CheegerSimons class for the complex normal bundle of the foliation. Under an additional assumption on the complex normal bundle, the definition of the Bott class becomes completely parallel to that of the Godbillon-Vey class. In this sense, the Bott class is a counterpart of the Godbillon-Vey class in the complex category. However, its properties are not yet well-understood. One of difficulties is that the Bott class is an element of C/Z-coefficient cohomology class. Indeed, the imaginary part can be written by a closed form and is easier to study, while the study of the real part often needs some additional techniques or notions, e.g. 77-invariants. In this paper, first we introduce a modified version of Cech-de Rham complex and show that a certain representative of the Bott class can be treated as a differential form (Definition 2.13 and Theorem 2.19). This is based on the Chern-Weil theory on the Cech-de Rham complex found in [23]. There have been already some approaches to consider the de Rham cohomology with coefficients other than C or R [11], [21], where simplicial complexes are used. Under our formulation, the integration of Cech-de Rham cochains introduced by Lehmann [19] plays an essential role and it allows the construction to be elementary. One of the main results is a generalization of Heitsch's residue of the Bott class (Theorem 3.4). Combined with a structure theorem of complex codimension-one foliations due to Ghys-Gomez-Mont-Saludes, this residue gives a generalization of Dummy's theorem in a weak form (Corollary 5.4). As an application of the residue to higher codimensional foliations, the Futaki invariant is discussed. It is a group character defined on the groups of biholomorphic automorphisms of a complex manifold and is valued in C/Z. Given an automorphism a of a complex manifold M, one can naturally construct a foliated manifold M^ called the suspension, and the Futaki invariant is essentially the Bott class of this foliation. Calculations of the Futaki invariant often involve the 77-invariant, which need some additional structures. The calculations * Received March 10, 2003; accepted for publication May 30, 2003. + Department of Mathematics (Yoshida-Minami branch), Kyoto University, Kyoto 606-8501, Japan (asuke@math.kyoto-u.ac.jp). Supported by Ministry of Education, Culture, Sports, Science and Technology, Grant No. 13740042, 15740036.

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/2003/0007/0002/AJM-2003-0007-0002-a006.pdf

Residues of the Bott class and an application to the Futaki invariant

Dans cet article nous etudions les classes caracteristiques secondaires reelles de feuilletages transversalement holomorphes. Nous definissons un homomorphisme de l'espace H * (WO 2q ) des classes secondaires reelles vers l'espace H * (WU q ) des classes secondaires complexes qui correspond a oublier la structure transversalement holomorphe. En utilisant cet homomorphisme nous montrons, par exemple, la decomposition de la classe de Godbillon-Vey en la partie imaginaire de la classe de Bott et la premiere classe de Chern du fibre normal complexe du feuilletage. Nous montrons aussi que des exemples de Heitsch n'admettent pas de structure transversalement holomorphe.

/pdf/on-the-real-secondary-classes-of-transversely-holomorphic-2n61xvg877.pdf

Taro Asuke

Papers

Godbillon-Vey Class of Transversely Holomorphic Foliations

A fatou-julia decomposition of transversally holomorphic foliations

Localization and residue of the Bott class

Residues of the Bott class and an application to the Futaki invariant

On the real secondary classes of transversely holomorphic foliations