We begin by recalling the definitions of Lie groups, of group actions, and of smooth actions, and establish some elementary properties.
Although the centre of our interest is in actions of compact (including finite) groups, the geometrical properties extend to all proper group actions.Akey step is the notion of slice. We establish the existence of slices for arbitrary proper actions. This leads at once to a local model for a proper smooth actions, which is the basis for all the subsequent results.
We show that the development of basic results in §1.1 can be parallelled in the group action situation: we have covers by coordinate neighbourhoods, partitions of unity, an approximation lemma, and invariant Riemannian metrics. There is also a theorem on the existence of an equivariant embedding in Euclidean space (with an orthogonal action), which applies when the group is compact.
We continue by defining orbit types, and the stratification of the manifold by orbit types. This stratification is locally finite and smoothly locally trivial. One consequence is that if the manifold is connected, one orbit type is dense and open: orbits of this type are called principal orbits. We give a model for a neighbourhood of a stratum, and proceed to an analysis of the case with two strata.
We conclude with examples.
Lie groups
We recall from §1.3 that a Lie group is a smooth manifold G, which is also a group, such that the group operations are smooth maps G → G, G × G → G.
Important examples are the general linear groups and of nonsingular m × m real, respectively complex, matrices, which are open submanifolds of the vector space of all matrices. We also use the notation GL(V) for the group of linear endomorphisms of the vector space V.
A Lie subgroup is a smooth submanifold which is also a subgroup. Any subgroup of a Lie group G which is a closed subset is a Lie subgroup. This result is not trivial: a proof is given, for example, in [146, Theorem 4.1] or in [148, §3.1].