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A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n.

Is there a citeable reference for a proof of this result? For the sake of being definite, let's say that “citeable” means indexed by Mathematical Reviews or Zentralblatt, or available on arXiv.

The answer https://mathoverflow.net/a/4516 gives two references for this theorem, neither of which is citeable in the above sense: online notes and an obscure book, impossible to locate. There are many citeable sources that state this result without a proof.

Dmitri Pavlov
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    Why doesn't someone write a short note, stick it on the arXiv? And then get it published in one of American Mathematical Monthly, Expositiones Mathematicae (Elsevier, though!), Confluentes Mathematici etc (sourced from http://mathoverflow.net/questions/15366/which-journals-publish-expository-work). Also, some relevant discussion/references is in http://ncatlab.org/nlab/show/ball – David Roberts Mar 02 '15 at 22:39
  • @DavidRoberts: Good suggestion, although the note might not be as short as one might expect it to be: the proof in online notes occupies 3 pages. – Dmitri Pavlov Mar 02 '15 at 23:09
  • all the better, since a one-pager might be considered too trifling. From my perspective, a proof that includes the relevant material on existence of good open covers for manifolds would be best (if one but had time for such a thing). The road to that proof goes via estimates on convexity radii coming from estimates on curvature, going back to old work of Whitehead, and which could do with a modern interpretation. – David Roberts Mar 03 '15 at 00:09
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    @DavidRoberts: Actually I was discussing the existence of differentiable good open covers with Urs Schreiber when he visited Göttingen, and the result of the discussion was that one doesn't need any curvature or convexity radius estimates. Here's the proposed proof: to construct a differentiable good open cover of a smooth manifold use partitions of unity to construct a Riemannian metric and take the set of all geodesically convex subsets. They are closed under intersection and the inverse of the geodesic flow transforms any geodesically convex open set into a star-shaped open set. QED – Dmitri Pavlov Mar 03 '15 at 00:16
  • Are there any issues with convexity radii not being bounded away from zero on a non-compact manifold? In any case, perhaps this short argument would make the case for including the proof in a note explaining the original question even stronger... – David Roberts Mar 03 '15 at 00:35
  • Ah, existence of geodesically convex neighbourhoods for any point can be found in eg do Carmo Riemannian geometry, Proposition 4.2. Cool! I hope "someone" writes this note up. ;-) – David Roberts Mar 03 '15 at 05:13
  • I wound up down this same rabbit hole while writing my book (still in progress). If a $C^1$ version starting with convex sets (rather than star-shaped) is good enough for you, the reference I'm using for that is: M. Gromov, Convex sets and Kähler manifolds, Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, 1990, pp. 1–38, Section, I.4.C1 – Greg Friedman Mar 09 '15 at 06:08
  • @GregFriedman: Yes, I am aware of Gromov's proof, but sometimes a C^∞-diffeomorphism is needed, e.g., to construct differentiable good open covers on a C^∞-manifold, which is the application that I had in mind when posting this question. (Does it make sense to include a proof for the C^∞-case in your book? A lot of people would be grateful if you did this…) – Dmitri Pavlov Mar 09 '15 at 11:46
  • @DmitriPavlov: Hmm, I'm not sure I know how to prove this from scratch, but maybe I can incorporate the Born/Ferus proof in an appendix. Of course better would be convincing Born to publish it as a paper, and that way it would be more thoroughly refereed. Has anyone asked him to do this? – Greg Friedman Mar 09 '15 at 20:38
  • @GregFriedman: No, as far as I am aware, nobody tried to contact them. Georges Elencwajg claims in the answer that I link to in my post that Gonnord and Tosel also have a proof in their book (http://www.worldcat.org/title/calcul-differentiel/oclc/40241391), apparently simpler than that of Born's, but I was unable to locate the book, although a paper copy seems to be present in my city library. – Dmitri Pavlov Mar 09 '15 at 21:59
  • I also tried to get ahold of that book through interlibrary loan, but the library didn't manage to get it for me. Maybe we can get someone to e-mail us a scan of the proof. – Greg Friedman Mar 10 '15 at 23:06
  • @GregFriedman: Meanwhile I wrote up an easier proof that came up in a lunch discussion: http://nforum.mathforge.org/discussion/1947/ball/?Focus=52585#Comment_52585 – Dmitri Pavlov Mar 20 '15 at 18:09
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    @GregFriedman have a look at this more recent answer (http://mathoverflow.net/a/212595/11211) to the MO question linked by the OP with a link to a manuscript copy done by Erwann Aubry. In any case, I managed to buy a used copy of the book "Calcul Différentiel" by Gonnord and Tosel, and the proof there is indeed much neater than the one in Dirk Ferus's lecture notes. Unfortunately, they provide no references for it, which is a bit strange (for they do so for other results in the book) and also begs the question of whether this proof is actually due to the authors themselves or not... – Pedro Lauridsen Ribeiro Jun 11 '16 at 03:17
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    @DavidRoberts the theorem of existence of geodesically convex neighborhoods is due to J.H.C. Whitehead (Convex Regions in the Geometry of Paths, Quart. J. Math. 3 (1932) 33-42). A proof valid for any manifold with an affine connection (not just Riemannian) may be found in the charming (although unfortunately out-of-print) little book of Noel J. Hicks (the same from the Cartan-Ambrose-Hicks theorem), Notes on Differential Geometry (Van Nostrand, 1965), Section 9.4, pp. 134-136. – Pedro Lauridsen Ribeiro Jun 11 '16 at 03:32
  • The books by Hicks is available here: http://libgen.io/book/index.php?md5=FDE69DDB5FB0E16D44FD9CA73E10C8C0 – Dmitri Pavlov Jun 11 '16 at 09:15
  • @PedroLauridsenRibeiro thanks! I knew about the Whitehead reference, but it's pretty hard and buried in the midst of many other things. – David Roberts Jun 12 '16 at 03:11
  • @DmitriPavlov: I was curious to find the proof that you have written on MathForge, but I got "Error 1001: DNS resolution error" from Cloudfare (which seems to host the domain). There is no copy of the page on the WayBack Machine either. Please consider posting it on a more stable site. – Alex M. Jun 19 '18 at 14:52
  • @AlexM.: nForum moved, mathforge should be replaced with ncatlab (it is quite stable). The new link is https://nforum.ncatlab.org/discussion/1947/ball/?Focus=52585#Comment_52585. See also this proof: https://mathoverflow.net/questions/4468/what-are-the-open-subsets-of-mathbbrn-that-are-diffeomorphic-to-mathbb/212595#212595. An elegant proof of the existence of good open covers was given by Demailly: https://nforum.ncatlab.org/discussion/1947/ball/?Focus=67203#Comment_67203. – Dmitri Pavlov Jun 19 '18 at 18:40
  • @DmitriPavlov: Why don't you add your proof to https://mathoverflow.net/q/4468? It would fit naturally in that discussion, and would also get significantly more visibility (and would spare the readers the pain of hunting for it on the internet). – Alex M. Jun 19 '18 at 18:53
  • @AlexM.: My old proof on the nForum is essentially the proof of Gonnord and Tosel explained in this answer: https://mathoverflow.net/questions/4468/what-are-the-open-subsets-of-mathbbrn-that-are-diffeomorphic-to-mathbb/212595#212595, except that they make additional simplifications. – Dmitri Pavlov Jun 20 '18 at 00:22
  • @AlexM.: The proof is also present as Theorem 2.6 in the nLab article "ball": https://ncatlab.org/nlab/show/ball – Dmitri Pavlov Jun 20 '18 at 02:22
  • I gave this as a (hard) homework problem this semester in a graduate-level course Intro to Smooth Manifolds. One student came up with a really good proof. – Tom Goodwillie Nov 18 '22 at 17:35
  • @TomGoodwillie: Is the proof similar to the one presented in https://mathoverflow.net/questions/4468/what-are-the-open-subsets-of-mathbbrn-that-are-diffeomorphic-to-mathbbr/212595#212595? If not, it may be beneficial to post it there. – Dmitri Pavlov Nov 18 '22 at 17:37
  • @Dmitri Pavlov: Yes, it's similar to that one. – Tom Goodwillie Nov 18 '22 at 22:33
  • After seven years I propose the answer is "no". – mme Nov 19 '22 at 03:15

2 Answers2

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The result in the following paper implies that open star-shaped domainin $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$. But, in your case, a diffeomorphism can be obtained along the same lines.

K. W. Kwun. “Uniqueness of the open cone neighborhood”. Proc. Amer. Math. Soc. 15 (1964), pp. 476–479.

Postscript. Instead of a reference one could write the following lines:

Any star-shaped open set $\Omega$ is a union of a nested sequence of star-shaped open regions $\Omega_0\subset \Omega_1\subset\dots$ such that $\partial\Omega_n$ is a graph of a smooth Lipschitz function in the polar coordinates. We can assume that $\Omega_0$ is a disc; in particular $\Omega_0$ is diffeomorphic to $\mathbb{R}^n$. Observe that for each $n$ there is a diffeomorphism $\phi_n\colon\bar \Omega_{n-1}\to\bar\Omega_n$; moreover we can assume that $\phi_n$ fix all points away from a tiny neightborhood of $\partial\Omega_n$. In particular, it can be arranged that for any $x_0\in \Omega_0$, the sequence defined by $x_n=\phi_n(x_{n-1})$ stabilizes after finitely many steps. Define $f(x_0)=x_n$ for all large $n$, and observe that $f\colon \Omega_0\to\Omega$ is a diffeomorphism.

Anton Petrunin
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    Kwun's proof heavily uses gluing of piecewise continuous maps along closed subsets. Such constructions cannot be adapted to the setting of diffeomorphisms without considerable modifications, e.g., using smooth bump functions. While this probably can be done, it would require one to write a proof that is longer than the one presented in https://mathoverflow.net/questions/4468/what-are-the-open-subsets-of-mathbbrn-that-are-diffeomorphic-to-mathbbr/212595#212595, which defeats the very purpose of citing a result instead of writing it up. – Dmitri Pavlov Nov 18 '22 at 17:35
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    However, for $n\neq 4$ homeomorphism implies diffeomorphism :) – Marco Golla Nov 19 '22 at 12:23
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This result is stated as Exercise 7 in Chapter 8 of Bröcker and Jänich's book Introduction to Differential Topology (p. 86 of the English translation). This may or may not count as a citeable reference.

Dan Ramras
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  • (It's certainly citeable. But one might hope for a published proof rather than just a brief indication of how a proof might go.) – Dan Ramras Nov 18 '22 at 19:30
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    I meant a citeable reference with a proof, since there are many known citeable sources that state the result without a proof, e.g., the Bott–Tu book states it, and several others. – Dmitri Pavlov Nov 18 '22 at 21:51
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    Fair enough. My recollection, from graduate school, was that someone pointed me to this reference saying that it was the closest thing to a proof in the literature. – Dan Ramras Nov 19 '22 at 04:15