By Akira Fujiki (auth.), Junjiro Noguchi, Takeo Ohsawa (eds.)
In the Teichmüller concept of Riemann surfaces, in addition to the classical thought of quasi-conformal mappings, vari- ous techniques from differential geometry and algebraic geometry have merged lately. therefore the significant topic of "Complex constitution" was once a well timed selection for the joint conferences in Katata and Kyoto in 1989. The invited members exchanged rules on varied ways to similar subject matters in complicated geometry and mapped out the customers for the following couple of years of research.
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Extra resources for Prospects in Complex Geometry: Proceedings of the 25th Taniguchi International Symposium held in Katata, and the Conference held in Kyoto, July 31–August 9, 1989
Example text
Carnot groups have their own concept of differentiation. In order to present this concept introduced in [Pansu (1989)], we first have to define a concept of linear maps between two Carnot groups. 41 SDE's and Carnot Groups Let Gi and G2 be two Carnot groups with Lie algebras fli and g2. A Lie group morphism 4> : Gi —> G2 is said to be a Carnot group morphism if for any t > 0, g £ G\, where A Gl (resp. AG2) denote the canonical dilations on Gi (resp. G2). In the same way, a Lie algebra morphism a : gi —> 92 is said to be a Carnot algebra morphism if for any t > 0, x e gi, Cxi V* dj \ — u* ix\djy, where 5Gl (resp.
The number M(e) of these balls has the lower bound: We deduce that for every s > 0, which shows that if s < D then the Hausdorff s-dimensional measure of • B5(0,1) is +00. 6 Observe, and this is typical in sub-Riemannian geometry, that the Hausdorff dimension of G is therefore strictly greater than the topological dimension. Carnot groups have their own concept of differentiation. In order to present this concept introduced in [Pansu (1989)], we first have to define a concept of linear maps between two Carnot groups.
These groups called the free Carnot groups are introduced and their geometries are discussed. When the Lie algebra £ is not nilpotent, this representation does not hold anymore but provides a good approximation for Xx° in small times. We conclude the chapter with an introduction to the rough paths theory of [Lyons (1998)], in which Carnot groups also play a fundamental role. e. that [V;,Vj] = 0 for 1 < i,ji < d. This is therefore the simplest possible case: it has been first studied by [Doss (1977)] and [Siissmann (1978)].