Date: Tue, 2 May 95 15:34:57 CDT From: rusin (Dave Rusin) To: nms36@rz.uni-kiel.d400.de Subject: Re: "INVERSE FUNCTION THEOREM >I have started to look through the literature on the Nash-Moser-Implicit >Function Theorem, but at the moment I have only found the smooth case of >this theorem. But there is still some hope that I will also found a >reference for the analytic case. Your comments got me interested enough to look on MathSciDisc and see what matches to "embedding[s]", "analytic" and "manifold[s]" I could find. Most of the 700 or so hits were useless but I did see some interesting references you can try. There is a 1958 paper in Ann. Math. by Morrey entitled "Analytic Embeddings of abstract real analytic manifolds", which certainly seems like the analytic version of the Nash embedding theorem. Griffiths had a 1966 paper (Extension problems in Complex Analysis) which seems appropriate too, but better, perhaps, is a 1983 paper (Duke math. J) he wrote with Bryant and Yang; the review in Math Reviews mentions the "Janet-Cartan" theorem, which apparently states that an analytic manifold of dimension n can be embedded in R^N if N>= n(n+1)/2. I don't know if this is quite what you wanted either, and I certainly haven't read the articles, but I found this to be an interesting addition to a collection of embedding theorems. dave ============================================================================== Date: Tue, 9 May 1995 18:11:57 +0200 From: nms71 To: RUSIN@math.niu.edu Subject: "RE: INVERSE FUNCTION THEOREM Many thanks for your mail last week. I have checked the references that you have pointed out to me. Since I am not an expert in this field and the papers are long, I am only able to understand these papers in a very superficially way. Nevertheless, I will give you my impressions from these articles. Hopefully, you are interested in these matters. 1. (C. B. Morrey, The analytic embedding of abstract real-analytic manifolds, Ann. of Math. 68 (1958), 159-201) It is proved that a compact real-analytic manifold of dimension n can be embedded analytically in the (2n+1)-dimensional euclidean space. Bochner has proved this result in 1937, if the manifold posesses an analytic Riemannian metric, and Morrey proves the existence of such a metric. The latter construction is reduced to the contruction of some analytic functions on the manifold, and this is done by Hilbert space methods. 2. (Bryant, Griffiths, Yang, Characteristics and existence of isometric embeddings, Duke Math. J. 50 (1983), 893-994) The main problem of this paper is the local isometric embedding of 3-dimensional C^\infty-manifolds in the 6-dimensional euclidean space. In section IV, there is a discussion of Nash's implicit function theorem, but they do not mention the analytic case. 3. (H. Jacobowitz, Local isometric embeddings, Ann. of Math. Studies 102, 381-393, Princeton 1982 (Ed. Shing-Tung Yau)) This article contains the outline of a proof of the Cartan-Janet Theorem. This theorem seems to be much simpler than the Nash Embedding Theorem. The statement of the theorem, i.e. the existence of an analytic isometric embedding, is stronger, but, on the other hand, the analyticity of the Riemannian metric is a strong condition. The proof is based on the Cauchy-Kowalewski Theorem (a theorem on the existence of analytic solutions to analytic differential equations), and, in particular, no implicit function theorem is used. Interestingly, the Cauchy-Kowalewski Theorem can also be used to prove the analytic inverse function theorem in R^n (see e.g. Krantz, Parks, A primer of real analytic functions, Birkhauser 1992, section 1.8). In the last week I have also found two potential references. The first one is the article Pisanelli, Domingos: The proof of the inversion mapping theorem in a Banach scale, in Complex analysis, functional analysis and approximation theory (Campinas 1984), 281-285 (North Holland Math. Studies 125 (1986)), but this is not in our library, hence it will take some time until I know exactly what this paper contains. The second reference is the article Hauser, Mueller: A rank theorem for analytic maps between power series spaces, Publicationes Mathematiques Institut des Hautes Etudes Scientifiques 80 (1994), 95-115. In this paper there is a proof of an analytic inverse function theorem for something called a Banach scale, but this should be more general than one single Banach space. Mysteriously, the review of the Pisanelli-Domingos article states a general theorem of this type, but the second paper contains also a full proof. I hope this long mail was not to uninteresting for you. P.S. Since the last week, I use a new e-mail number (nms71@... instead of nms36@...), but the old number will also work. Hauke