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I know volume preserving diffeomorphisms of a sphere^2 make a group sdiff(S2). I would to know if it is a Lie group, which I assume if it is that makes interpolation easier (like with rotations).

So that is one question, is it a Lie group?

Also is the group path connected? If so, how can I interpolate between two elements in the group?

These are not subjects I know very little about. I apologize if Im phrasing it in some way that sounds ridiculous.

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Your second question can be phrased: "Is this group path-connected?" The answer to the first question is probably no; my guess is that it's infinite-dimensional. (I have my doubts about the second question too. Is there a path between the identity and the negation x -> -x?) –  Qiaochu Yuan Jul 29 '10 at 8:56
    
Thanks, seriously any time I say something that can be phrased better let me know. I guess that would be an inversion of a sphere, right? if so it does seem pretty constrained already so I would agree with you on that. I wonder if there is subgroup that is nontrivial and path connected? Maybe if you exclude self penetration? –  Jonathan Fischoff Jul 29 '10 at 9:10

2 Answers 2

up vote 3 down vote accepted

First, as others have pointed out, the group of volume preserving diffeomorphisms will be infinite dimensional.

For the second question, there is a beautiful technique known as Moser's trick which answers it. Moser's trick, in fancy language, says that if (M,w) and (M,w') are two symplectic structures on the same manifold, and if [w] = [w'] in H^2(M;R) (de Rham cohomology), then there is a family of diffeomorphism f_t:M->M with f_0 = Id and such that f_1 pulls w' back to w.

For a 2-dimensional compact, oriented manifold (like the sphere), we have H^2(M;R) = R (the real numbers), and a symplectic form is nothing but a nonzero element in R (which can be interpreted as the total volume). Since in this setting, [w] = [w'] iff they both give the same signed volume, it follows from Moser's trick that the group of (signed) volume preserving maps is connected.

If we consider unsigned volume, there will be 2 components to the group diffeomorphisms preserving the unsigned volume. This is because, as others have pointed out, one has the notion of "degree" which shows the map x-> -x is not homotopic to Id, even through just continuous maps (not neccesarily volume preserving). This shows there are AT LEAST two componenets. There are at most two components because every volume preserving diffeo can be connected to Id or (x -> -x) by Moser's trick again.


Edit: I misspoke a little bit. Moser's trick says that if you have a family w_t of symplectic forms, then there is a family of diffeomorphisms as I described above. It's not clear to me that what I said (that it's enough to have [w] = [w'] in H^2) is enough to guarantee that there is a family of symplectic forms connecting them. Further, it seems that Moser's trick only guarantees you have a path of diffeos which starts and ends at a volume preserving diffeo, but may not preserve volume for all time.

However, in the case of S^2 (or any closed, orientable 2-manifold), I can patch things up. Given w and w', volume forms, with [w] = [w'] (i.e, they have the same volume), then the form w_t = tw + (1-t)w' is a path of symplectic forms which connects them. For a fixed t, the form w_t is closed since it's a sum of closed forms (or, even easier, because it has top degree), and is nondegenerate because it's a volume form (integration shows the volume given is that of w). The fact that the volume is constant for each w_t implies that the path of diffeos preserves volume for all time.

(I wasn't able to immediately convince myself that in general, the convex sum of symplectic forms was nondegenerate, hence my initial hesitation. In fact, I think that it need not be nondegenerate.)

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Thanks for the references. I am still digesting the fancy language, but your explanation is leading to many fruitful google searches. I plan on accepting this answer, I just want to wrap my head around it more. –  Jonathan Fischoff Jul 29 '10 at 18:45

The term "volume preserving" sounds a bit ambiguous to me: do you mean that your map preserves the total volume or do you mean that its differential at every point preserves volume (i.e. has determinant 1)? The former is weaker than the latter, and gives you more room for interpolation.

In any case, there is a famous invariant of continuous maps $S^2\to S^2$ called the degree. Any two maps with the same degree are homotopic to each other. Being volume preserving (in the former sense) implies that the degree is $1$ (taking orientation into account!), so you can interpolate between any two volume preserving maps. However, the intermediate maps in this line of reasoning are only continuous, not necessarily diffeomorphisms. I'm confident that with a standard argument "approximate continuous functions by differentiable ones" you can get them to be differentiable, but I don't about "is diffeomorphism" and "is locally volume preserving" parts.

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The idea is to construct an interpolation consisting of volume preserving maps (and by that one usually means that the pullback of the volume form is the volume form) –  Mariano Suárez-Alvarez Jul 29 '10 at 15:34

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