From: baez@math.ucr.edu (john baez) Newsgroups: sci.math.research Subject: Re: are $C^\infty$ manifolds real-analytic? Date: 7 Dec 1997 19:26:51 -0800 In article , Vladimir Pestov wrote: >Is it true that every $C^\infty$ manifold (compact or not) admits >a real-analytic model, that is, supports the structure of a >real-analytic manifold? I've been assured by various worthies that every paracompact C^infinity manifold admits a real-analytic structure which is unique up to C^infinity diffeomorphism. I haven't been able to find a reference in print for the uniqueness part, but I think existence is lurking in the published literature somewhere. You might try: H. Whitney, Differentiable manifolds, Ann. Math. 37 (1936) 648-680. Richard S. Palais, "Real algebraic differential topology", Wilmington, Del., Publish or Perish, 1981. I forget what exactly is in these references, but I remember them being relevant. If anyone answers Pestov's question by email, please pass on a copy to me! ============================================================================== From: knop@math.rutgers.edu (Friedrich Knop) Newsgroups: sci.math.research Subject: Re: are $C^\infty$ manifolds real-analytic? Date: 9 Dec 1997 17:21:24 -0500 baez@math.ucr.edu (john baez) writes: >[...] >Thanks for the hint concerning Hirsch's book, btw. The place to look >is pp. 65-66. It says here that Whitney was the first to prove >the existence of a real-analytic structure on any paracompact >C^r manifold (r >= 1). The reference given is the article I >mentioned in my previous post: >H. Whitney, Differentiable manifolds, Ann. Math. 36 (1936), 645-680. >(Ah, the long-gone days when one could write a paper with that >title!) >Unfortunately for me, he does not mention uniqueness of real-analytic >structures, so I assume this is one of those results that is only >well-known to those in the know. Uniqueness is a consequence of a Theorem of H. Grauert (Ann. Math 68, 460-472, 1958): The set of real-analytic maps between two real-analytic (paracompact) manifolds is dense with respect to the Whitney topology in the set of all C^\infty maps. Observe that being a diffeomorphism is an open property in the Whitney topology. ============================================================================== From: ilya@math.ohio-state.edu (Ilya Zakharevich) Newsgroups: sci.math.research Subject: Re: are $C^\infty$ manifolds real-analytic? Date: 9 Dec 1997 23:34:42 GMT In article <66kf1d$ob8@charity.ucr.edu>, john baez wrote: > >On the other hand, if your really meant to say that uniqueness is up to > >C^\omega diffeomorphism, I would be skeptical, since that certainly does not > >hold for the complex case. > > Indeed, this is certainly false: just take a real-analytic atlas A and > apply a smooth but not real-analytic diffeomorphism of the manifold > to it, to obtain a new atlas A'. These are not related by a real-analytic > diffeomorphism. They are (at least sometimes). Try for example a circle. I vaguely remember that this statement (in general C^1 case) *was* in Hirsh, but it may be a glitch of my memory in the years that passed... Ilya ============================================================================== From: "David L. Johnson" Newsgroups: sci.math.research Subject: Re: are $C^\infty$ manifolds real-analytic? Date: Tue, 09 Dec 1997 21:20:54 -0500 john baez wrote: > I meant that given two real-analytic atlases compatible with the smooth > structure of a smooth manifold, there is a smooth diffeomorphism > carrying one atlas to the other. I don't think this is trivial. For > example, given two smooth atlases compatible with the topological > structure of a topological manifold, there needn't be a homeomorphism > carrying one atlas to another. What? How about atlas #1 going back to the continuous manifold M? That is a homeomorphism by dint of the definition of compatibility. Now, take the corresponding homeomorphism, but in the other direction, to atlas #1. The composition of those two is a homemorphism. > Similarly, given two complex-analytic > structures compatible with the smooth structure of a smooth manifold, > there needn't be a smooth diffeomorphism carrying one to the other. > Same construction applies. The compatibility condition is that, really, the real-analytic manifold structure is smoothly diffeomorphic with the original smooth manifold. > >On the other hand, if your really meant to say that uniqueness is up to > >C^\omega diffeomorphism, I would be skeptical, since that certainly does not > >hold for the complex case. > > Indeed, this is certainly false: just take a real-analytic atlas A and > apply a smooth but not real-analytic diffeomorphism of the manifold > to it, to obtain a new atlas A'. These are not related by a real-analytic > diffeomorphism. Not necessarily true. How do you know that the non real-analytic diffeomorphism is the only way to find that relationship? There might be another real-analytic diffeomorphism between those two manifolds. I submit that that is a difficult question. > > If I'm making mistakes here I want to know about it! This stuff is > a little slippery, and people are often a bit sloppy about it when > they say things like "the 7-sphere has 28 distinct smooth structures". > Smooth structures compatible with what? Distinct up to what? (In > this case, it's up to *orientation-preserving* homeomorphism.) No. In that case all the smooth structures are homeomorphic with S^7, thus with each other. Orientation-preserving or reversing is not an issue, since there is an orientation-reversing diffeomorphism. The compatibility is with the underlying topological structure that the standard 7-sphere has. The distinction is that there is no diffeomorhism bewteen them. -- David L. Johnson dlj0@lehigh.edu, dlj0@netaxs.com Department of Mathematics http://www.lehigh.edu/~dlj0/dlj0.html Lehigh University 14 E. Packer Avenue (610) 758-3759 Bethlehem, PA 18015-3174 ============================================================================== From: David Epstein Newsgroups: sci.math.research Subject: Re: are $C^\infty$ manifolds real-analytic? Date: Thu, 18 Dec 1997 12:54:33 +0000 john baez wrote: > (I'll use "real-analytic" instead of "C^\omega" and "smooth" instead > of "C^\infty", since I'm fond of English.) > > I meant that given two real-analytic atlases compatible with the smooth > structure of a smooth manifold, there is a smooth diffeomorphism > carrying one atlas to the other. I don't think this is trivial. For > example, given two smooth atlases compatible with the topological > structure of a topological manifold, there needn't be a homeomorphism > carrying one atlas to another. Similarly, given two complex-analytic > structures compatible with the smooth structure of a smooth manifold, > there needn't be a smooth diffeomorphism carrying one to the other. I was hoping that someone more knowledgeable than I am would respond authoritatively on this topic. Since no-one has, I will respond, but not authoritatively. First of all, let's stick to paracompact manifolds. Non-paracompact manifolds can be horrible, and I have no idea what is true for them, probably not much. The basic result in this theory is that every C^r function from a real analytic manifold to the reals can be approximated in the fine (sometimes called strong) C^r topology by a real analytic function. This is a hard result, and, as far as I know, can only be proved using complex function theory. It was first proved for compact manifolds by Morrey and then in generality by Grauert (both papers in Annals 1958). Grauert's proof is based on embedding the given analytic manifold in a Stein manifold and then approximating functions defined on a Stein manifold. Most of the theory can be read in Hormander's book on complex analysis in several variables. This is a lovely book, easy to read, which contains all of this theory except for the connection with real analytic manifolds. Treatments of this connection in the literature are horrible (complexifying a real analytic manifold), even though the result is actually not too hard to prove. The approximation theorem will need most of Hormander's book even without the stuff about complexifying a real analytic manifold, so it's a long haul if you really want to do it all. So how do you apply Morrey/Grauert to deal with the questions raised in this thread? First note that if X and Y are two analytic manifolds, we can approximate maps from X to Y by analytic maps. (This is done by embedding Y smoothly in a euclidean space, and then approximating that embedding by an analytic embedding. After that we can use the tubular nhd thm to give an analytic retraction of a nhd of Y in the euclidean space onto Y. Then combine Morrey/Grauert with the analytic retraction.) In particular this shows that if X and Y are analytic manifolds which are diffeomorphic, then they are real analytically isomorphic. (But, of course, as John Baez has already pointed out, if you have two distinct analytic structures on the same smooth manifold, then the analytic isomorphism between them is not the identity, even though it can be made arbitrarily near the identity in the fine smooth topology.) Now we turn to our smooth (=C^\infinity) manifold M. How can we show it has an analytic structure? We partially order open subsets of M with analytic structures, using inclusion and whether the structures are identical, to give the order. Take a maximal such open set U. This must be the whole of M. For otherwise there is some point x in the boundary of U. Take a chart C around x. Think of C and U as analytic manifolds whose smooth structures coincide on their intersection. The identity map on the intersection can be approximated by an analytic isomorphism between the two analytic structures on the intersection. By using the fine topology, we can make the isomorphism closer and closer to the identity as we get towards the boundary of the intersection. As a result, when we extend this isomorphism by the identity on C\U, we get a smooth diffeo of C with itself. (This is the kind of strange and unfamiliar trick that one can pull using the fine topology.) Composing the original chart on C with this smooth diffeo, we get a new smooth chart on C, but now the overlap with U is analytic, contradicting the maximality of U. David Epstein