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in the course of last night’s events, I created a handful of stubs for some basic concepts:
We’ve already got convex space which has a section ’abstract convex sets’. Should ’convex set’ be amalgamated?
Oh, sorry, didn’t see that. Could you do that for me?
It’s not so straightforward, since we have the Boolean field as a convex space not realised as a subset of a vector space, so not compatible with the star-shaped property. I’m not sure at what level of generality I can include your convex set material into the flow of the convex space article.
It’s not so straightforward, since we have the Boolean field as a convex space not realised as a subset of a vector space, so not compatible with the star-shaped property.
Maybe I am missing something, but the entry convex space talks about the relation to the ordinary notion of convex subsets of affine spaces.
I simply put in a hyperlink under the word “convex subset”.
Also put at convext set a remark that there is a generalization of this notion called “convex space”.
But if I am missing something, please correct.
I didn’t know if terminology was settled. It’s a little odd that convex sets are kinds of space, and convex spaces are often not very space like.
For geodesic flow I am looking for a pointer to some book to the theorem that if the metric has no conjugate points, then geodesic flow is a diffeomorphism.
I am not at a library right now and have to rely on Google Books. Which is a pain.
Urs, is there a typo on the page geodesic convexity, under strongly geodesically convex: should that second $X$ be a $C$?
Also, are you sure there’s a theorem along the lines in #7? I’m getting the impression that it could be false (by examples within the context of the Cartan-Hadamard theorem, involving manifolds of negative sectional curvature). Still investigating…
Edit: look within the proof of the Cartan-Hadamard theorem in John M. Lee’s Riemannian Manifolds: An Introduction to Curvature, page 196 (available via Google books). So in particular, I believe every orientable (connected) surface of genus greater than 1 has the hyperbolic plane as universal covering space (hence carries everywhere negative curvature), and the covering map (which is a local diffeomorphism but not a diffeomorphism), and the import of the Cartan-Hadamard theorem in this context is that the exponential map $\exp: T_p(S) \to S$ is the covering map which realizes this universal covering space. But clearly I’m no expert on this.
Yes, that was a typo. I have fixed it. This statement is taken from here
I corrected boundary and added few entries at topology. Though I find nice that Urs added a section on literature under topology, I think the only reference which is now there would somewhat better fit in more special entry than in general entry on topology. It is an unfinished advanced-level compilation of Strickland of various examples of spaces appearing in constructions of algebraic topology. I think the general list for topology should list basic references (like textbooks, e.g. Munkres, Engelking...) and maybe few main advanced encyclopedic books and few online tutorials or general topological online repositories. The Strickland's book is too advanced for beginner, too unfinished to be a reference, and it does not have basic examples of interest say in general/set-theoretic topology, nor in shape theory, nor in coarse topology (for which I just wrote an entry), nor in the theory of simplicial sets, but instead has examples of interest in algebraic topology, like important CW-complexes, manifolds, moduli spaces and their building blocks. So I think it may be a bit of distraction for a newcomer to have such advanced but specialized references in top lists.
Of course, do not take this as more than a mere light suggestion, it is OK if it stays there. I also added few of the terms mentioned above to geometry.
I corrected boundary
Thanks, I said “does not have a neighbourhood diffeomorphic to a half-ball”, where of course thet “not” should not have been there!
Also, are you sure there’s a theorem along the lines in #7?
I was looking at page 72 of
Gabriel Pedro Paternain, Geodesic flows
where the author invokes such a statement. But it is hard to tell what the remaining assumptions are, since I cannot see many of the previous pages.
I’ll further look into this. I would be shocked if the geodesic flow around any point inside a little patch that itself is swept out diffeomorphically by geodesic flow for some small parameter $\epsilon$ weren’t a diffeomorphism.
Todd,
one more on
Also, are you sure there’s a theorem along the lines in #7?
Here it seems to be stated that on a closed manifold, the injectivity radius of geodesic flow (the smallest radius over which geodesic flow is diffeomorphic around any point in the manifold) is the minimal distance between two conjugate points.
So if there are no conjugate points, then I’d guess that it’s infinite. But maybe i am misunderstanding the statement.
Here is a reference that says this in a little more detail:
See around page 2.
Shouldn’t star-shaped be renamed to a noun?
Well, I am looking at Grant, and specifically corollary 1.5 on page 4, which rules out conjugate points in cases where the Riemannian manifold has negative sectional curvature. So again, I am considering a surface $S$ of genus 2 given by a suitable quotient $H/\Gamma$ of the hyperbolic plane, which ought to carry negative sectional curvature. So therefore no conjugate points, and yet the exponential map $\exp: T_p(S) \to S$ can’t possibly be a global diffeomorphism.
Am I making a stupid error?
Hi Todd,
you are right. What sounded like the statement one sees on Wikipedia is not actually stated in that article.
But I think I have found now a precise statement that does what I need it to do: in
it is shown that every paracompact manifold admits a metric with positive injectivity radius.
So pick such a metric, choose the patches in the construction at good open cover to be of geodesic diameter equal to that injectivity radius. Then clearly every geodesic flow inside such a patch is diffeomorphic and hence it follows with the argument given there that the resulting good open cover has all finite non-empty intersections diffeomorphic to a ball.
Whom should we thank/blame for your sudden surge of interest in Riemannian geometry, Urs ?
Okay, good Urs. Glad that got resolved to everyone’s satisfaction!
Todd,
I finally understand now how the statement on Wikipedia was meant: the injectivity radius is either equal to half the length of the smalled periodic geodesic, or equal to the smallest distance between two conjugate points.
So in the case you keep looking at it is the either-part that applies.
I recorded all this at geodesic flow now.
Zoran,
you ask
Whom should we thank/blame for your sudden surge of interest in Riemannian geometry, Urs ?
Am still working on the properties of the local $\infty$-topos $Sh_\infty(CartSp)$. I noticed that I needed to fill a gap in the discussion of cofibrant resolutions of paracompact manifolds. Turns out to construct the required good covers, one can use geodesic flow. So that’s what I am doing here.
Shouldn’t star-shaped be renamed to a noun?
Done.
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