quotient manifold theorem

Statement

Suppose is a Lie group with an action on a smooth manifold that is smooth, free, and proper. Then the orbit space is a topological manifold of dimension equal to , and has a unique structure with the property that the projection map is a smooth submersion.

Necessary Lemmas

Convergence of subsequence in

Let be a manifold and let be a Lie group acting continuously and properly on . If is a sequence in and is a sequence in such that and converge, then a subsequence of converges.

Proof

The proof can be found in @lee2013 on page 543.

Orbits as submanifolds

Statement

Suppose is a Lie group that acts smoothly and properly on a smooth manifold . For any point , the orbit map is a proper map, the orbit is closed in . Additionally, if the stabilizer , then the orbit map is a smooth embedding and the orbit is a properly embedded submanifold.

Therefore, the orbits of a free Lie group action on a manifold will be properly embedded submanifolds of .

Link to original

Proof

todo The proof of the theorem is given in @lee2013 on pages 544-547. Here I will only give a small sketch of the main ideas and how each of the hypotheses is used in the theorem.

The main idea of the proof is to show that you can find charts centered at , such that each orbit of the action of on intersects as either the empty set or as a slice chart. These charts are called adapted charts (they are adapted to the action).

This condition is a sensible thing to want as the orbits are submanifolds by the second lemma above. This would mean that on these coordinate charts the action is limited to moving the first coordinates, so on the “rest” of the manifold it doesn’t do anything as acting by would just move along the slice chart which is constant in the last coordinates.

To prove that these slice charts exists (though not that the orbits intersect in a single slice) @lee2013 used Frobenius theorem.

Consider the distribution of tangent spaces to the orbits, i.e.

It can be shown that this is involutive, so each orbit is a leaf of the resulting foliation, i.e. the orbits are integral submanifolds of the distribution (somewhat obviously).

Then, we want to show that we can shrink these charts from the foliation small enough, so each orbit only intersects as one slice. Assuming the opposite for contradiction, we can look at subsets of the open sets from the charts of the foliation centered at such that the “width” of the open set is less than or equal to in any coordinate direction of the center, .

Since we assumed that each orbit intersects each chart in more than just a single slice, we can pick a sequence of points that are in the two different slices of each (increasingly smaller) open subset. Each open subset obviously shrinks towards the center point , so both and .

Here is where we need the proper assumption, to use the first lemma above, we can see that since are in the same orbits that

Therefore, some subsesequence of converges (lets say to ), and we can use this subsequence:

And here is where we need the assumption that the action is free, so we know that . This is useful, because we then can use a restricted version of the orbit map, which is a local diffeomorphism (and thus injective) to show that the two points we assumed (for contradiction) were distinct must be the same.

Now that we know these adapted charts exist, we can look at the image under the projection , which when using adapted charts has a coordinate representation . In other words, it throws out the first coordinates, as that is where the action “lives”.

After this it only remains to show that the new coordinate charts of which come from follow the definition of a manifold, and that is a smooth submersion. The rest can be found as cited in @lee2013.

Notes

The open sets are easy to understand in . It is constructed so that is a quotient map so in the quotient topology, a set is open if is open.

Also, if is compact, then the proper condition is unnecessary.