Talk:Soul theorem

I changed the part under the soul conjecture heading. I've read Perelman's paper, and am looking at it now, and I think he is proving the soul theorem in its full generality. Though he mentions the conjectural statement of Gromoll and Cheeger that pointwise sectional curvature anywhere in M is enough to send S to a single point, I do not know of a proof of this anywhere. Though I think it is possibly true, I'm not sure how to go about proving it, especially using the Sharafutdinov fibers as Perelman does. Hope I'm not missing something obvious. Cypa 04:23, 4 October 2005 (UTC)

You did not state Perelman's theorem and it makes no sense this way. I will revert in a few days if nothing will happen Tosha 16:40, 4 October 2005 (UTC)

Hey Tosha, its possible you're right, but i have the proof right here. Can you explain what part of the proof refers to local curvature restrictions? From what I can tell, we are always dealing with global submersion equations onto S. If you would like to state Perelman's thereom explicitly, then I can do that. It comes in three parts, A, B, and C, and it is nothing but a slightly fleshed out version of the generalized soul theorem. Part A, for example, explicitly states a nonnegative curvature assumption, not positive curvature. I added the corollary because it is the stronger case of the statement that was originally quoted as the conjecture, the part that is known. I thought it provided some insight, but we can remove it. If I'm mistaken and Perelman is proving that any point with positive curvature collapses S to a point, then I would really appreciate knowing how. Cypa 20:16, 4 October 2005 (UTC)

There are two things: soul conjecture and perelman's submersion theorem. second one imply first. it does not have sense to mention perelman's result without stating it. Tosha 00:28, 7 October 2005 (UTC)

I don't think so. I just reread the proof. Are we talking about the same paper? I'm reading "Proof of the Soul Conjecture of Cheeger and Gromoll" by Perelman. The theorem is stated in 3 parts. Part A says, assuming nonneg K, with Sharafutdinov map $P\colon M\to S$, then for x in S and normal vector v in the normal bundle of S, then at x $P(\exp {tv})=x,\forall t\geq 0$. Part B is, for any geodesic l in S and v parallel along l, the correpsonding exponential map sends geodesics flows to minimizing geodesic flows. And C says, the map P is a continuous Riemannian submersion (in fact a submetry) which has eigenvalues bounded by 1/injrad(S). Since exp is just a surjective diffeomorphism here, it in no way follows (that i can see) that strictly positive sectional curvature sends u(0) to u(1), which is what would have to happen for your statement " but K > 0 at some point. Then soul of M has to be a point (or equivalently M is diffeomorphic to {\mathbb R}^n)" to hold ... no? In fact, the usual bound on injrad makes it clear that the submersion does not send everything to a point.

I am concerned that the statement in the article is in fact wrong right now. Can you please explain yourself or I will have to rewrite the second section. Cypa 13:43, 7 October 2005 (UTC)

I do not think you should edit the article before you understand the statement. the formulated that statement implies that if soul is not a point then for any point there is a sectional direction with zero curvature. and therefore soul conjecture follows. Tosha 20:49, 7 October 2005 (UTC)

How? I'm not sure I even understand the sentence :"the formulated that statement implies that if soul is not a point then for any point there is a sectional direction with zero curvature. and therefore soul conjecture follows. " Do you mean the formulation of that statement ...? If so (and assuming by "that statement" you mean the statement of Perelmen's proof I provided above), then I don't understand. Perelman only shows the cut locus for an arbitrary direction. Again, please, just point to the part of the proof that you are talking about, or explain yourself coherently ... if it is my mistake, then I am sorry, but I cannot see it, and I am starting to think that you can't either. If you are suggesting that $P(\exp {tv})=x,\forall t\geq 0$ implies zero curvature, I think not. All it says is that the normal bundle is diffeomorphic pointwise. Cypa 21:30, 7 October 2005 (UTC)

Look, I do not want to explain it. My point is: you should not edit page if you do not understand it. Tosha 22:45, 9 October 2005 (UTC)

Okay, well, I've looked into a bit more, refreshed some of the research I did several years back on the topic, and have come to the conclusion that you are just wrong. I would recommend Guijarro & Walschap's paper, "The Metric projection onto the soul" for help undertsanding Perelman's proof. Perhpas you are confusing the 'soul with the pseudosoul of Yim's, see Yim, Distance nonincreasing retraction on a complete open manifold of nonnegative sectional curvature, Ann. Global Anal. Geom. 6 (1988), 191-206. Here are some other references if you'd like to discuss ANY of them, I am game, as I have read them all:

G. Walschap, The Soul at infinity in Dimension 4, Proc. Amer. Math. Soc. 112 (1991), 563-567

G. Walschap, Soul-Preserving Submersions, Michigan Math Journal, 41 (1994), 609-617

I. Belegradek & V. Kapovitch, Finiteness theorems for nonnegatively curved vector bundles, Duke Math. J., 108 (2001), 109-134

I. Belegradek, Vector bundles with infinitely many souls, Proc. Amer. Math. Soc. 131(2003), no. 7, 2217-2221

V. Berestovskii & L. Guijarro, A Metric Characterization of Riemannian Submersions, Annals of Global Analysis and Geometry 18 (2000), pp.577-588:

J. Cheeger & D. Ebin, Comparison Theorems in Riemannain Geometry, North-Holland, New York, 1975

D. Gromoll & K. Grove, The low-dimensional metric foliations of Euclidean Spheres, J. Differential Geometry 28 (1988), 143-156

L. Guijarro and G. Walschap, Twisting and nonnegative curvature metrics on vector bundles over the round sphere, J. Differential Geometry 52 (1999), 189-202

L. Guijarro and G. Walschap, The metric projection onto the soul, Trans. Amer. Math. Soc. 352 (2000), 55-69

L. Guijarro & G. Walschap, Transitive holonomy group rigidity in nonnegative curvature, Math. Zeitschrift 237 (2001), 251-257

L. Guijarro, On the Metric Structure of Open Manifolds with Nonnegative Curvature, Pacific Journal of Mathematics, 196, No. 2, 2000 S. Kobayashi & K. Nomizu, Foundations of Differential Geometry, Interscience, New York, 1963

B. O'Neill The Fundamental Equations of a Submersion, Michigan Math. J. 13 (1966), 459-469

Mathematics is a complex subject full of nuances, and the soul theorem is an unusually nuance heavy theorem. I editted the page because I feared that the information was incorrect and thought it better to be safe than sorry -- that is, in providing false information to the WWW. However, since you are incapable of defending your point of view with any facts, I feel forced to push mine, which I am fully willing to defend with many facts and a discussion of the pertinant mathematics when necessary. I am sorry if editting the page out of turn offended you. I appreciate all the work you've done here on wiki, and i admire you for staying with it. However, I am now editting the page as I firmly believe that it is making an inaccurate and false statement about the Soul conjecture. Cypa 05:38, 10 October 2005 (UTC)

So, do you object that in the paper of Perelman, he (among other things) proved the following:

Suppose, M is complete and noncompact with sectional curvature $K\geq 0$, but $K>0$ at some point. Then soul of M has to be a point (or equivalently M is diffeomorphic to ${\mathbb {R} }^{n}$).

? Tosha 06:13, 13 October 2005 (UTC)

Yeah, I don't think he is proving that, and in fact I don't think it's even true (though it 'was' weakly conjectured by proxy in 1994) - due in part to recent results in the field (since 1998). So, the first example that is usually given in the soul theorem/conjecture is to think of a parabola P, take its umbilic; then because of the diffeomorphism of its injrad on the ambient space, the point S has no finite cut locus as its an extremal point in P. But now consider the hyperboloid of revolution H based at the origin in the upper half plane. Now, the diffeomorphism that covers the space is a compact set, namely the set of poles in H. Well, recall that there is volume bound on the metric by the injrad construction in Riemannian geometry. So a compact set, in particular, is not a point by the deformation retract and the corresponding constraints on the metric. So even though the curvature is strictly positive by construction at a collection of points, S is not a point. Now, this is purely intuitive, as much more needs to be said. For example, we may be able to use deformations and the splitting theorem to show that standard degree four polynomial embedded in ${\mathbb {R} }^{2}$) can be reduced to a parabola, and thus has S=point. However, there is a well characterized volume growth bound on any M from above, with respect to the holonomy group of S. So for an arbitrary degree n polynomial, it takes some work to show if S=point, and in general, by induction, is false. The work of Perelman's is just an argument using the standard techniques - and skipping more than several steps - that shows that for an arbitry nonneg complete riemann manifold, the soul exists and is well characterized. Namely, it is a C^1 Riemannian submersion. Subsequent work has improved this to (1,1) Holder continuous in the general case, and C^2 in the special case. The reason Perelman's work is noteworthy and exciting, is because it reduces the extremely complex soul theorem to a relatively transparent collection of basic algebraic tools. It is however somewhat vague about the nuances and details of the larger idea. Cypa 20:56, 15 October 2005 (UTC)

That is enough, as I stated before, I do not want to explain you anything, and from what you say it is clear that you are not able understand any mathematics. It would be better for everybody if you will NOT edit mathemetics in the future. Tosha 22:48, 15 October 2005 (UTC)

I am not a mathemetician but I have just read this talk page and am struck by the following. User:Cypa has gone to enormous lengths to discuss the article citing evidence, etc. User:Tosha is responding using the fallacy of argumentum ad verecundiam {argument from authority}. This isn't good enough, please try to resolve the dispute by properly explaining your position on this talk page. Majts 23:53, 15 October 2005 (UTC)

Tosha, please stop being ridiculous. Surely there is SOMETHING you can say if you have such a refined understanding of the topic. Or perhaps you can explain where I am misunderstanding? MAthematics is based on evidence and facts. I am going to send this page to RFC. Cypa 23:01, 15 October 2005 (UTC)

Here Tosha, please see this paper: VOLUME GROWTH AND HOLONOMY IN NONNEGATIVE CURVATURE. I think if you read through this, this should clear up why the statement above is false. Very vaguely, the reason for this bound, in an intuitive sense, is that the lipschitz condition along with the injrad condition sets bounds on how fast the surjective curvature in S can grow; so when you have alot of poles on a space, they aren't all necessarily going to collapse to a single S, since S can only take on curvature so fast. Does that make sense? Cypa 23:33, 15 October 2005 (UTC)

I'm ready to explain anything to anybody who can unerstand something, but Cypa is not one of those. If you are doing math it should be clear from nearly any of his sentence. Tosha 05:23, 16 October 2005 (UTC)

That's not an adequate response. If it's that obvious, it will be easy for you to quickly state why any of Cypa's arguments are false. That would be good enough for me, but until then I will help keep the article reverted to Cypa's version despite the fact I have no understanding of the subject Majts 05:41, 16 October 2005 (UTC)

Majts, there is no arguments in Cypa's writing, and again please do not edit article which you do not undersand. read the papers then come back. Tosha 00:12, 17 October 2005 (UTC)

Tosha, what are you talking about? I have explained the parabola/hyperboloid example vaguely, because it is the first argument in the paper by Gromoll and Cheeger, and so I assumed you had some familiarity with it.

It is NOT an argument!

I am a mathematician by training, and these insults are merely ridiculous. I know quite alot about Riemannian geometry, differential geometry and algebraic topology - which is all that is really required to understand this theorem ... along with some elementary analysis.

Maybe you know enough, but you do not understand

Though initially I thought you perhaps had some insight into this subject that I might have missed, I am becomer more and more convinced that you don't even have an elementary understanding of what the soul theorem is.

Trust me I do

What is so strange is this hubris that seems to accompany it. I can assure you however that all of my sentences are based on direct referencing.

Which all unrelated

I am not writing from memory Tosha, I am looking at the proofs that sit right before me, and paraphrasing. So if my statements are wrong, then so are the theorems.

The fact that you have a paper in hand does not mean you understand it, I write from memory, but I do remember all prrofs

Another reason I am comfortable with my statements is because I have discussed many of these with my friends and colleges - professional mathematicians - on a daily basis.

professional mathematicians? hmmm

But all it takes is for you to point one issue out.

What do you want me to point out?

I will defend it with explanation and source material.

source is good only if you uderstand it

I could even have another mathematician or two enter into this discussion if you'd like. Really Sir, what is all this about? Why do you have no desire to reach an understanding? Cypa 01:15, 17 October 2005 (UTC)

Once more my points:

1. Disussion page is NOT for getting help in math
2. you should not edit page if you do not understand it

I claim, that my edition is correct, and your corrections make it more foggy and not that useful.

BTW, here is citation from MathSciNet (I hope you trust it)

MR1285534 (95d:53037) Perelman, G. (RS-AOS2) Proof of the soul conjecture of Cheeger and Gromoll. J. Differential Geom. 40 (1994), no. 1, 209 212.

Let M denote a complete noncompact manifold of nonnegative sectional curvature with soul S. It is well known that if u is a vector tangent to S, and v is orthogonal to S, then the plane spanned by u and v has zero curvature. In a short geometric argument based on the second Rauch theorem, Perel0man shows that this is also true of the parallel translate of this plane along the geodesic in direction v. As a fundamental consequence, he obtains the existence of a Riemannian submersion from the ambient space onto S. This answers in particular a question asked some twenty years earlier by Cheeger and Gromoll: If the curvature is strictly positive at some point, then M is diffeomorphic to Euclidean space.

Tosha 02:48, 17 October 2005 (UTC)

Alright, well, it seems clear that you are going to be unwilling to enter into a dialogue of any sort. I agree with Walschap's review paragraph. The difference between what he has said, and the one sentence statement of Perelman's proof that you have provided, is one demonstrates an understanding of the larger field of statements, while the other is such a closed worded simplification of the idea, that it is misleading and approaches being just wrong. I could try to explain this to you, but I am busy with other work, and this conversation is becoming redundant. At least you provided a non-vacuous response this last time (by providing a quote of someone elses), so we're moving in the right direction. You may not know this Tosha, but I have no way of knowing what your background is, or how much you know about the theorem or Riemannian geometry. That IS the purpose of the discussion page, so that people can use logic and reason to reach mutual understandings. I'm sorry you feel you're so talented that you don't need to explain yourself beyond insults and "because I'm right and I know it." This approach is both childish and impossible to make sense of. From the sum total of what you've said here, I can only conclude that English is clearly not your first language, that you think your very smart, and that you have very strong intransigence towards trying to understand any point of view that doesn't completely parallel your own. Talk about content free dialogue. Statements like: Trust me I do? Okay Tosha, I trust you, you are the source of all mathematics on the web, and I'll just take your word for it. But nonetheless, your going to have to flesh out the statement of Perelman's proof to the point where it demonstrates some understanding of the current state of the Soul Theorem, or I'm going to continue to revert to a more paultry, though less "foggy" version of the article. Cypa 16:39, 17 October 2005 (UTC)

I have to say that I am bewildered by the discussion here. The discussion page is meant to improve the article. If somebody thinks that the article is wrong, this is the place to challenge it, and if the question is not satisfactorily answered, then it is all right to change the article. It is not enough to say that the article is correct, one should be prepared to give arguments for it. Of course, it would be too much to explain it to somebody who doesn't know any mathematics, but I've seen no evidence that Cypa would be somebody like that.

On the other hand, it is also not clear to me what Cypa's problem with the page is. Is the problem:

1. Perelman's article does not claim to prove that "if M is complete and noncompact with sectional curvature K ≥ 0, but K > 0 at some point, then the soul of M has to be a point (or equivalently, M is diffeomorphic to Rⁿ)," OR
2. the article does claim to prove it but the proof is incorrect, OR
3. the article does prove it correctly, but it is misleading to summarize the statement like that?

The article says "Cheeger and Gromoll conjectured that the same conclusion [the soul is a single point] can be obtained under the weaker assumption that M contains a point where all sectional curvatures are positive", this is the only conjecture that I can find in the article so I assume that this is meant by the "Soul Conjecture", and Perelman claims that this is "an immediate consequence of Part (B)". The fact that this is accepted by the journal's referees and Walschap, writing for MathSciNet, is enough for me to believe that the proof is correct, at least unless arguments for the contrary are given.

So, what is going on here? -- Jitse Niesen (talk) 18:04, 17 October 2005 (UTC)

Thanks Jitse. Yeah, your right, what is going on here? Anyway, I'm going to rescind my complaint. There is nothing being accomplished here (and that is my own fault). Sorry for reducing myself to drivel. To vaguely clarify the original POV, I was trying to state something along the lines of option 3 above. Tosha the page is all yours. You've earned it. Cypa 23:40, 17 October 2005 (UTC)

Who proved what?

I am puzzled by this page, which I came to from a link on the Perelman page. That page states that he proved the Soul Conjecture. But this page states that Cheeger and Gromoll proved it. Whereas the Perelman citation on this page indicates that Cheeger and Gromoll conjectured it and that Perelman proved it. What is the correct story here???

Also, not having even heard of the Soul Conjecture before, I don't pretend to know anything about its proof. But I am having trouble seeing how the Soul Conjecture -- as stated on this page -- can hold for the simple case of, say, the surface z = x^2 + y^2 in R^3. I do not think this paraboloid P contains -any- compact submanifold whose normal bundle is diffeomorphic to P (not to mention a compact totally convex, totally geodesic submanifold whose normal bundle is diffeomorphic to P).

But if the condition of compactness is dropped, then it is easy to see that the modified statement would be true for P (using any complete geodesic through the vertex of P as the submanifold).Daqu 10:14, 5 March 2006 (UTC)

OOPS -- please disregard the last two paragraphs; I confess I did not think of a single point, as I should have.Daqu 10:26, 5 March 2006 (UTC)

• • • I wrote, however, *three* paragraphs and the first one is still unresolved:

"I am puzzled by this page, which I came to from a link on the Perelman page. That page states that he proved the Soul Conjecture. But this page states that Cheeger and Gromoll proved it. Whereas the Perelman citation on this page indicates that Cheeger and Gromoll conjectured it and that Perelman proved it. What is the correct story here???"Daqu 03:41, 22 March 2006 (UTC)

There's a soul theorem and a soul conjecture. The article currently states the theorem was proven by Cheeger and Gromoll who conjectured the conjecture, which was proven by Perelman. --C S (Talk) 04:50, 28 March 2006 (UTC)

Question

Why is it called the Soul Theorem? --Zhang Guo Lao 13:46, 24 August 2006 (UTC)

Minor error

In the paraboloid example, please remove the part "Not every point x of M is a soul of M, since there may be geodesic loops based at x", because it is obviously wrong: The normal bundle at any point-submanifold is diffeomorph to R^2, (which is diffeomorph to the paraboloid). 188.174.106.216 (talk) 12:43, 13 January 2014 (UTC)

Which part is incorrect? There are geodesic loops on the paraboloid (Clairaut's relation tells us that a geodesic travelling at an angle down the paraboloid will 'turn around' as the paraboloid narrows, then wind back upwards and meet itself eventually). Whether a point is a soul or not comes down (by definition) to whether it arises from the process of choosing a point, considering rays coming from it, constructing compact totally convex regions and then shrinking those down inductively. Points that are not souls can still have their normal bundle diffeomorphic to the whole space. It's just a matter of convention and definition that 'soul' means subspaces from this construction, not subspaces with the nice property of their normal bundles. 128.91.40.39 (talk) 15:46, 30 March 2016 (UTC)