Actions Of Compact Lie Groups Research Articles

Combinatorial geometry and actions of compact Lie groups

Combinatorial geometry and actions of compact Lie groups

Resolving actions of compact Lie groups

A general process for the desingularization of smooth actions of compact Lie groups is described. If G is a compact Lie group, it is shown that there is naturally associated to any compact G manifold M a compact G × (Z/2)p manifold on which G acts principally. Here Z/2 denotes the cyclic group of order two and p + 1 is the number of orbit types of the G action on M.

Continuous Actions of Compact Lie Groups on Riemannian Manifolds

Continuous Actions of Compact Lie Groups on Riemannian Manifolds

Continuous actions of compact Lie groups on Riemannian manifolds

M. H. A. Newman proved that if M is a connected topological manifold with metric d, there exists a number E > 0, depending only upon M and d, such that every compact Lie group acting effectively on M has at least one orbit of diameter at least e. In this paper the authors consider the case where M is a Riemannian manifold and d is the distance function on M arising from the Riemannian metric. They obtain estimates for e in terms of convexity and curvature invariants of M.

FORMAL GROUPS AND THE ATIYAH-HIRZEBRUCH FORMULA

In this article, manifolds with actions of compact Lie groups are considered. For each rational Hirzebruch genus , an equivariant genus , a homomorphism from the bordism ring of -manifolds to the ring , is constructed. With the aid of the language of formal groups, for some genera it is proved that for a connected compact Lie group , the image of belongs to the subring . As a consequence, extremely simple relations between the values of these genera on bordism classes of -manifolds and submanifolds of its fixed points are found. In particular, a new proof of the Atiyah-Hirzebruch formula is obtained.Bibliography: 10 items.

Differentiable Actions of Compact Connected Lie Groups: III

Introduction .220 Chapter I. A fixed point theorem for finite diffeomorphism groups generated by reflections 221 Chapter II. Geometric weight system and a fixed point theorem for compact connected Lie groups.... 231 1. Actions of compact simple Lie groups of rank 1.232 2. Basic properties of the geometric weight system.237 3. A fixed point theorem for compact connected Lie groups.240 Chapter III. Classification of the types of principal isotropy subgroups for actions of compact connected Lie groups on acyclic manifolds .242 1. Classification of principal orbit types for actions of simple compact connected Lie groups on acyclic manifolds .243 2. Classification of principal orbit types for actions of general compact connected Lie groups .247 References ...... 255

Extending equivariant maps for compact Lie group actions

Extending equivariant maps for compact Lie group actions

Foliations and compact lie group actions

Foliations and compact lie group actions

C 1 Actions of Compact Lie Groups on Compact Manifolds are C 1 -Equivalent to C ∞ Actions

Introduction. Let C. and C2 be two categories and let F: C,---O#C2 be a weakening of structure ( forgetful ) functor, i. e. for each pair of objects X1 and X2 of C, the map of Morph (X1, 2X) to Morph(F(X1), F(X2)) is injective. Let 01 and 02 denote the equivalence classes of objects of C, and C2 defined by the equivalence relations of isomorphisms in the respective categories. Then F induces a map F: 01O2 and the question of whether X is injective, surjective, or bijective is often of interest. For example for O ? kc ?oo let Mank denote the category of Ck paracompact manifolds and Ck maps, and for k > I let FZk: Mank Manz be the obvious forgetful functor. A Then for I = 0 Fok: Mank -Man0 is neither surjective nor injective, i.e. there exist topological manifolds which admit no Uk structure [3] and there exist topological manifolds with non-isomorphic Ck structures [5]. However if I> 0 then Flk is always bijective, i.e. every CL manifold admits a compatible Ck structure [10] and if two Ck manifolds are CL diffeomorphic they are Ck diffeomorphic (the latter is trivial from standard approximation theorems).

Remarks on Differentiable Actions of Non-Compact Semi-Simple Lie Groups on Euclidean Spaces

Introduction. It is well known that differentiable actions of compact Lie groups are far more regular than those of the non-compact Lie groups and the behavior of compact Lie group actions are comparatively much more well understood than those of the non-compact Lie groups. Of course, one of the most basic and important cases of non-compact Lie group actions are those Rl-actions which are naturally associated with ordinary differential equations [6 ]. So far, much important research has been done in that direction. However, knowledge about R'-actions still seems rather inadequate. Since the group R1 is a simply connected one-dimensional Lie group and it contains no non-trivial compact subgroups, neither the theory of Lie groups nor knowledge about compact Lie group actions are of any help in the study of R'-actions. In this note, we try to look at the other extreme case of actions of large semi-simple non-compact Lie groups. The situation here is quite different from the study of RI-actions. Roughly speaking, the problem is more complicated on the one hand, and has much more built in structures on the other hand. For example, one naturally expects that the profound theory of semisimple Lie groups and the knowledge of compact Lie grfoup actions might be useful at least for some special but natural cases. Due to the fact that one knows so little about the behavior of differentiable actions of non-compact Lie groups in general, it seems to be worthwhile to investigate some special but natural cases such as semi-simple Lie group actions on euclidean spaces. Let G be a semi-simple non-compact Lie group. It is well known that there exists a maximal compact subgroup H which is unique up to conjugacy. For a given differentiable manifold M'4, let r(GMm]Ii4) and r (H, Mm) be the set of equivalence classes of differentiable actions of G and H on Ml', respectively. To restrict those G-actions on MIlm to H induces a natural luap i !: r (G, Mm) r (H, Mm) which is independent of the choice of the maximal compact subgroup H. A reasonable way to study G-actions on Mm is to investigate the image and the kernel of the above map. To be more precise,

Differentiable Actions of Compact Abelian Lie Groups on S n

Differentiable Actions of Compact Abelian Lie Groups on S n

Differentiable actions of compact abelian Lie Groups on 𝑆ⁿ

Differentiable actions of compact abelian Lie Groups on 𝑆ⁿ