Invariant Distances on Legendrian Spaces - Pierre-Alexandre Arlove

What is discussed, and when

1. 0:00 – 10:00: contact distribution, contacttomorphism, contact geometry, Daru theorem
2. 10:00 – 20:00: contacttomorphism, contact distribution, Vansula and Shila Sandon, Elashburg and Poltervic
3. 20:00 – 30:00: contacttomorphism, contact geometry, discriminant distance, one jet bundle
4. 30:00 – 40:00: contacttomorphism, leandrean isotopy, contact distribution, Elashburg and Poltervic
5. 40:00 – 50:00: contacttomorphism, one jet bundle, B and B prime
6. 50:00 – 1:00:00: contacttomorphism, contact distribution, one jet bundle
7. 1:00:00 – 1:10:00: contacttomorphism, open book decomposition, Patrick Maso and Silylvan
8. 1:10:00 – 1:11:41: contacttomorphism

Notable quotes from "Invariant Distances on Legendrian Spaces - Pierre-Alexandre Arlove"

• Manifold are always of odd dimension for linear algebra reason. So this dou theorem says that in contact geometry you don't have any interesting local invariance locally…

  about contact geometry · at 1:17

• See it inside the cn plus1. So the space of complex number to the power n +1. And then the contact distribution here is given as the maximum sub space of the tangent space of…

  about contact distribution · at 6:52

• Today is the periodic onejet bundle of manifold X. So what I mean by periodic wet bundle, you take the cotangent bundle of X and then instead of taking the product with the…

  about one jet bundle · at 8:11

• Isotopic class then this space is unbounded. So it carries an unbounded invariant distance and they also show that the same is true for the zero section inside the periodic one jet bundle. So here you have some rigidity.

  about Vansula and Shila Sandon · at 10:22

• Leandrean were not contractable and more precisely Elashburg and Poltervic introduce some notion and they called this kind of leandrean spaces orderable. So what it means to be orderable it means that positive loops of leandrean. So again a.

  about Elashburg and Poltervic · at 16:45

• General results in some sense is a results of Razer Pterovich and Rosen where they construct bin variant distance that are unbounded for any closed manifolds that have admits…

  about Fraser Pirovich and Rosen · at 27:22

• So it is antisymmetric. So if you suppose that the distance between lambda and lambda prime is equal to zero, it means that lambda prime is less than lambda and lambda is less than lambda prime. So by anti-ymmetry lambda has to.

  about leandrean isotopy · at 41:01

• And then by the first step by combining the two proposition of the first step you can find a contacttomorphism that sends B to B prime and that fix lambda.

  about contacttomorphism · at 51:24

Who and what is mentioned in this episode?

People

• Vansula and Shila Sandon — Person · Referenced experts (authors of cited result) · 1 mention · first at 10:04
  Credited for a 2012 result establishing unbounded invariant distances on certain Legendrian isotopy-class universal covers (e.g., RP^n in RP^{2n+1}, and the zero section in the periodic 1-jet bundle).
• Fraser Pirovich and Rosen — Person · Referenced experts (authors of general results) · 4 mentions
  Credited for general results on unbounded bi-invariant distances on contactomorphism groups under hypotheses like existence of periodic Reeb flow and absence of contractible positive loops; also cited for discreteness phenomena and for flexibility/boundedness statements on the standard contact sphere.
• Elashburg and Poltervic — Person · Referenced experts (introduced key concept) · 2 mentions · first at 16:50
  Credited with introducing the notion of (Legendrian) orderability, used as a rigidity condition (no contractible positive loops) underpinning unboundedness constructions.
• Shakanov — Person · Referenced expert (prior work in symplectic geometry) · 1 mention · first at 25:50
  Credited with introducing a relative (Lagrangian) analogue of Hofer-type metrics—an invariant distance on spaces of closed Lagrangians—used as historical context for analogous constructions in contact/Legendrian settings.
  And Then [clears throat] this of a distance on the group was in some sense generalized by Shakanov to the relative case. So not on the group of amonian simpletomorphism but on closed lrangen.
• Patrick Maso and Silylvan — Person · Referenced experts (boundedness results) · 1 mention · first at 1:05:16
  Cited for results on boundedness of the contactomorphism group under geometric flexibility assumptions (e.g., via open book decompositions with flexible pages), providing contrast to unboundedness results.
  So this is yeah for unboundedness and for boundedness actually there is a beautiful paper of Patrick Maso and Silylvan so about boundedness also on the group of contacttomorphism where they I don't remember exactly I.

Concepts & topics

• contact geometry — Concept · Core topic · 4 mentions · first at 1:24
  The main mathematical field discussed in the podcast.
• contact distribution — Concept · Key concept · 11 mentions · first at 0:56
  Fundamental to the definition of contact manifolds.
• Daru theorem — Concept · Core topic · 1 mention · first at 0:43
  Central to defining contact manifolds and their properties.
  Something I forgot to say contact manifold are always of odd dimension for linear algebra reason. So this dou theorem says that in contact geometry you don't have any…
• contacttomorphism — Concept · Key concept · 43 mentions · first at 2:07
  Describes the symmetry group of a contact manifold.
• leandrean isotopy — Concept · Key concept · 4 mentions · first at 11:20
  Discussed in the context of global properties of contact structures.
• one jet bundle — Concept · Example · 7 mentions · first at 10:33
  An example of a contact manifold discussed in the context.
• discriminant distance — Concept · Key concept · 2 mentions · first at 11:11
  A type of distance discussed in the context of contact geometry.
  Least yes some flavor of the proof and construction they were giving. So in some sense the distance they construct a lot of distance but I will describe just one of them that…
• simplectic geometry — Concept · Related field · 2 mentions · first at 25:04
  A related mathematical field to contact geometry.
  Of flexibility more than in simplectic geometry because in simplectic geometry you can have fininsler type distance. Here you have too much flexibility to expect fininsler type…
• Hamiltonian simpletomorphism — Concept · Key concept · 1 mention · first at 25:13
  Discussed in the context of simplectic geometry.
  Interesting. So first the search for invariant distance started in simplectic geometry. The first one was constructed by offer. So the offer distance is a B invariant distance on the group of Hamiltonian simpletomorphism.
• B and B prime — Concept · Example · 1 mention · first at 46:32
  Used in the proof of theorem B regarding invariant distances.
  And you suppose that B intersect the zero section. Okay, these two sets are in the one jet bundle here. And you suppose that the red set the B intersect the zero section in a small chart of.
• open book decomposition — Concept · Key concept · 1 mention · first at 1:05:35
  Discussed in the context of contact manifolds.
  So this is yeah for unboundedness and for boundedness actually there is a beautiful paper of Patrick Maso and Silylvan so about boundedness also on the group of contacttomorphism.
• RPN — Concept · Example · 3 mentions
  An example discussed in the context of contact geometry.
  Antipodal action preserve the contact distribution. So when you caution the sphere by this antipodal action you get the real projective space and the contact distribution…
• closed manifold — Concept · Key concept · 1 mention · first at 27:00
  A type of manifold discussed in the context.
  And interesting here mean unbounded this is what I hear by interesting and then Zapolski also using the same kind of techniques so generating function techniques constructed…

Full transcript with timestamps

0:16 It's a smooth manifold endored with a distribution of hyperplanes which is maximally non-integraable. It means that if you have a sub manifold which is everywhere tangent to the distribution then the dimension of this sub manifold will be at most half the dimension of the distribution of hyper plane. Another way to define the contact manifold contact structure is given by Daru theorem which says more precisely that if you have a manifold endored with a distribution of hyperplanes then this distribution is maximally non-integraable if and only if locally it looks like the standard contact distribution on R2n + one given by the kernel of the one form dz minus the sum of yi dxi And here you have a picture of how it looks in R R3 for N is equal to one. So yes something I forgot to say contact manifold are always of odd dimension for linear algebra reason. So this dou theorem says that in contact geometry

1:27 you don't have any interesting local invariance locally everyone looks the same. So if we want to say something interesting, it will have to be about global properties that a contact structure confer to a contact manifold. And today we will talk about some of these global properties, interesting global properties we can deduce from a contact structure. And another thing that this DU theorem says is that contact geometry is very flexible. So more precisely if you look at the group of symmetry of a contact manifold it will be very big. So it's the group of contacttomorphism. So defomorphism that preserve the contact distribution. You can put on it some natural fresh manifold structure and turns it into an infinite dimensional uh group. And the group of interest today won't be the whole group of contacttomorphism but contacttomorphism that are contact isotopic to the identity and that I denote by cont not

2:34 and then it's universal cover that I denote denote by conta and the easiest way to think about this universal cover is to think about it as the space of contact isotopes starting at the identity and then you identify two isotopies if they are homotopic. relatively to end points. So as I said we don't have any local invariant but do interesting global uh quantities invariant global quantities exist and a first statement is that invariant quantities cannot come from functions on pupil of points and the reason is really easy it's because the group of contacttomorphism actually it acts mtransitively on your manifold for any integer little m and a a concrete example where we can apply this statement and reason is if you look at the distance function. So distance function on m is a special kind of uh functions on couple of points and any invariant distance has to be equivalent to the trivial distance. So

3:48 you cannot have interesting invariant distance on the manifold m itself. And the proof is really easy. You take two points that are different on your manifold M X not and Y not the distance between them you put it equal to C. And then for any over couple any over two points that are not the same. You can find a contacttomorphism that send the first one to X not and the second one to Y not because the group of contacttomorphism acts too transitively on this manifold. And then the distance between x and y will be the same as distance between five of x and five of y because the distance has been taken invariant and it is the distance between x not and y not so. So it will be equal to c. So the distance is constant. So it's not very interesting.

4:41 And the morality of this uh example is that if we want to construct invariant global quantities that are interesting, we will have to look at some manifolds that exhibit more contact rigidity than points. On points we have a lot of flexibility. We can send any tuple on any pupil. So we have to look here for more interesting submanifolds. And of course the sub manifold we will look at today will be the leandrean submanifolds and leandrean sub manifolds are sub manifolds that remember a lot about the contact distribution and more precisely what is a leandrean sub manifold. So it's a sub manifold which is everywhere tangent to your contact distribution and which is of maximal dimension. So it will be equal to half the dimension of your uh distribution of hyper plane. So if your contact manifold is of dimension 2n + one your leandrean will be of dimension n.

5:46 And I will look at the space of leandrean that are leandrean isotopic to a fixed leandrean that I denote by curly lambda and also to its universal cover. So again the easiest way to think about it this universal cover I think is you look at all the genre and isotopes and then you identify two of them if they are homotopic one to each other relatively to end points and you have the group of contacttomorphism acts one transitively on the space of leandre and isotopy and the universal cover of the group uh acts one transitively also on the universal cover of the the space of leandre and isotopies.

6:36 So now some examples that will be of interest today. So first I define the the first example I want to give is the the odd contact sphere. So you take the odd dimensional sphere s + one. You can see it inside the cn plus1. So the space of complex number to the power n +1. And then the contact distribution here is given as the maximum sub space of the tangent space of the sphere which is invariant by um multiplication by the number i. So this is the contact manifold and then a leandre a natural leandrean inside it is just you take the small sphere uh the small sphere sn inside s2n plus one. This mode sphere will be a legandrian and actually you have an antipodal action on the sphere given by yeah the plus minus and this antipodal action preserve the contact distribution. So when you caution the sphere by this antipodal action you get uh the real projective space and the contact distribution descends to the

7:47 real projective space and uh endo this real projective space with a contact distribution and the little sphere SN also descends to to this little RPN which will be a leandre inside RP1. A second example I will be interested in today is the periodic onejet bundle of manifold X. So what I mean by periodic wet bundle, you take the cotangent bundle of X and then instead of taking the product with the real line as usual, you take the product with the circle S1 and here you have also a standard contact distribution on this space given by the kernel of the one form DZ minus lambda where lambda is the canonical one form of the contangent. So in local co coordinated speedq. So this is the contact manifold and your lejandrean inside is just the zero section.

8:52 And finally a last example I will be interested in is you take a manifold X and you take its projectivise cotangent bundle and this manifold carry a canonical contact distribution given here by the formula at the end of the line. And if you take any submanifold S inside X and you take the projectivise co normal bundle of S, it gives you a leandrean inside the projectivise cotangent bundle of X. Okay. So these are three examples I want to yeah talk about today. And if you have no question maybe I go a bit forward. So yes, do these leandrean sub manifold exhibit more contact rigidity than points? Can we say more interesting things about the leandrean? And of course the answer will be yes. And first I introduce some terminology. So I will say that the universal cover of the leandrean isotopic class of leandrean lambda is unbounded if it admits an unbounded invariant distance.

10:04 And there's a striking result by Vansula and Shila Sandon in 2012 that say if you if you take the the Leandrean to be the little real projective space inside the big projective space you take the universal cover of its leandrean isotopic class then this space is unbounded. So it carries an unbounded invariant distance and they also show that the same is true for the zero section inside the periodic one jet bundle. So here you have some rigidity. You have you can construct interesting distance that are not just constant distance as on points. They can take arbitrarily large value and I will just give some u idea of the proof or at least yes some flavor of the proof and construction they were giving. So in some sense the distance they construct a lot of distance but I will describe just one of them that they call the discriminant uh distance. So this discriminant distance in some sense it's

11:13 a word metric. Okay. So the words here will be embedded leandrean isotopies. So I will say that a leandrean isotopy is embedded if it does not have self intersection. So lambda t does not intersect lambda s if t is different from s. And then the distance between two element lambda tilda and lambda prime tilda is the minimum number of embedded leandrean isotopies I need to go from lambda tilda and to reach lambda prime. Okay. So the minimum number of leandrean isotopies I need so that by concatenate concatenating lambda tilda by these leandrean isotopies I can reach lambda prime.

12:07 So in some sense what it counts this distance it counts leandrean intersection. So what lejandrean intersection we cannot get rid of. So if this distance are unbounded it means that we cannot get rid of a lot of leandrean [snorts] intersection in some sense. This is the rigidity it captures and so uh a bit more concrete here how they what is the unbounded the sequence they choose for RPN. So on uh so for the real projective space so on the real projective space the rep flow is uh given just by hop vibration so it's just the multiplication by exponential IP ip and the red flow then it's a periodic uh periodic isotopy of contacttomorphism so it come backs at the identity of the integer amount of time and if you apply the rep flow to this leandrean which is RPN then after integer amount of time this leandreen come backs to itself. So you have an intersection and again if you go two times if you iterate two times the red

13:30 flow and apply to the to this end you cross this RPN two times and actually and Shell Sandon show that this intersection you cannot get rid of them. Okay. And they use uh given tiles nonlinear mass of index to show that you cannot get rid of them. So by applying the ref flow uh to the to this leandrean RPN your distance grows uh as the number of iteration of the rep flow and a bit similar construction uh holds also for the one chat bundle here the they take also the rep flow. So here the rep flow is just the uh translation in the last coordinate and again the rep flow is periodic. So after integer amount of time it come backs to the identity and then with the same argument not the same argument but then again if you apply the red flow to the zero section after it integer amount of time it come back to itself. So you have a an intersection and they show that vans and shellendon they show that

14:43 you cannot get rid of this intersection by using uh some generating function techniques and also some spectral selectors that they deduce from existence of generating function. Actually this was tools constructed by Shila Sandon and then they use these tools to prove that you cannot get rid of this uh intersections and u so if you have no questions I just go uh to yes a bit uh more conceptually what was important in these two examples. So first there was this existence of a periodic rep flow. So [clears throat] what in other words you have loop of contacttomorphism. So a loop of contacttomorphism. So an isotopy of contacttomorphism that come back to the identity and more geometrically what is a rep flow? It just an isotopia of contacttomorphism whose speed is everywhere transverse to the to the contact distribution. So if you look at the speed of a point under the red flow then the speed of this point will be

15:57 everywhere transverse to the contact distribution. So this was one ingredient that we used. And another thing that was very important actually is that when we were applying the rep flow to the uh to a leandreen so we we were getting loop of leandreen but these loops of leandreen were not contractable because if they were contractable then the distance would not change. I I I don't know if you remember it was just a because we are looking at leandrean isotopies up to omotopy. So if we have a loop which is contractable then this loop we just don't see it. So it was very important that these loops of leandrean were not contractable and more precisely Elashburg and Poltervic introduce some notion and they called this kind of leandrean spaces orderable. So what it means to be orderable it means that positive loops of leandrean. So again a positive loop of leandrean is a lejandrian isotopy whose speed is

17:11 everywhere transverse to the contact distribution and loop it means that it come back to itself. So positive loops of landreen are not contractable and usually to show orderability it's something hard it's uh it is where the rigidity lies actually because you have to find obstruction to existence of a contractable group and this obstruction usually use generating function techniques given to nonlinear mass of index or as we will see also leandre and contactmology So it's uh yeah this is where the rigidity lies uh in in what we will say later.

17:58 So and actually we got a generalization of the theorem of vans and sanon with simuel some years ago. So if you take a contact manifold and you take a closed lantrian inside it actually the two uh hypothesis we were discussing just before are exactly enough to uh show that the space is unbounded. So more precisely if the universal cover of the leandrean isotopic class is orderable. So that positive loops of leandrean are not contractable and there exist a positive loop of contacttomorphism then elilda is unbounded. Okay. So we recover the results of Shilasendon because in their examples uh they were satisfying these assumptions. But we get some new examples of Leandrean whose universal cover isotopic leandrean isotopic class is unbounded. So for example if you take any uh sub manifold in the sphere sn with sufficiently big co- dimension then the universal cover u the universal cover of its leandrean

19:16 isotopic class. So the sorry here the lejandrien is the projectiviz bundle of this sumold s inside the projectivized cotangent bundle of the sphere sn if you take the uh its leandrean isotopic class and you take its universal cover then it is unbounded thanks to this theorem and the proof goes as follow actually shuff and nimi They prove that the projectivized conormal bundle of uh this submanifold s is orderable by using also generating function techniques and so we have the first sorry we have the first assumption of the theorem. So the space is orderable and moreover you have a positive loop of contacttomorphism and this positive loop is just giving given by the cozic flow.

20:18 Okay. So no questions. Okay. And so recently I have a second generalization. So a generalization of the theorem we got I just present with Simo which is the following. Now again we suppose that L tilda is orderable and there exist a positive loop of Leandreen. So by assumption this positive loop is not contractable and assume moreover that this loop can be extended to a loop of contacttomorphism then elilda is unbounded. So this theorem generalized the previous one because here I don't expect this loop of contacttomorphism to be itself positive. It extend a positive loop of leandreen but it does not have to be positive everywhere itself this loop of contacttomorphism.

21:18 Okay. And actually we have concrete example where we can apply this theorem a but we cannot apply the theorem before. So it's a real generalization. It's not vacuous. So the the corery is like that. So a new example of leandrean which is unbounded whose universal cover leandrean isotopic class is unbounded is given by this. You take an hypersurface taurus tn minus one inside tn. You take its projectivized conormal bundle. Then the universal cover of its landre and isotopic class satisfy all the assumption of theorem a but it does not satisfy the assumption of the previous theorems. So this space is uh unbounded and we could not deduce unboundedness uh from previous theorem and the proof goes as follow. The first thing is that in the unitary the projectivized cotangent sorry in the unitary cotangent of the Taurus shuff and nimirski they have shown actually that does not exist positive loop of contacttomorphism so this is why

22:36 we cannot apply the previous theorem here we don't have positive loop of contacttomorphism but uh actually the projectivized conormal bundle of this hypersurface Taurus is orderable. And here you can use results of colon and dimlurid cell because this leandrean is hypertite and you can use this theorem to show that this space is orderable using and they use leandrean contacttomology techniques and then the loop of leandrean the positive loop of leandrean is just given by this loop. So you take the taurus times t and then you look at t for t going to zero to one. So you get a loop of leandrean and it's easy to compute that this loop is positive and moreover this loop extends to this loop of contacttomorphism given by just the translation in the qn coordinate. So this is a loop of contacttomorphism that is of course not positive.

23:47 So maybe you have a question here. I don't know. No. Okay. Uh maybe I have a question. So why in this your theorem non-contractable is in brackets?

24:02 Uh ah it's just because u it was it's just to recall what is the definition because L tilda is orderable. So of course if I have a positive loop this positive loop cannot be contractable by assumption.

24:21 I see. I see. So it was just to remind uh to remind that this group is Yeah. Okay.

24:31 Okay. Thank you. And uh Okay. So now uh before going to the next theorem I want to present. So I want to show two theorem today. So this one and another one and before showing talking about the other one I wanted to put a bit more historical context maybe around theorem a and also why I find it uh interesting. So first the search for invariant distance started in simplectic geometry. The first one was constructed by offer. So the offer distance is a B invariant distance on the group of Hamiltonian simpletomorphism.

25:17 So here for simplicity I I am assuming that my simple manifold w omega is closed and this offer distance is a fin type distance and it was constructed by offer in the early 90s for art and then but yeah it was generalized by professor ptovich and lon mdaf for every simple manifold and Then [clears throat] uh this of a distance on the group was in some sense generalized by Shakanov to the relative case. So not on the group of amonian simpletomorphism but on closed lrangen. So and shakenov distance also was an invariant distance on the space of lrangens and in contact geometry the story is a bit the same. So first uh people have constructed invariant distance on the group of contacttomorphism and then using these distances on the group uh later we deduce some distances on space of leandrean that we discussed before.

26:28 So the story in contact geometry started with Shella Sandon who constructed an interesting bin variant distance on the group of contacttomorphism of R1* S1 and interesting here mean unbounded this is what I hear by interesting and then uh Zapolski also using the same kind of techniques so generating function techniques constructed invariant B invariant distance on the group of contacttomorphism of the periodic one jet bundle of a closed manifold X then Vula and Shellendon constructed over uh distances on the universal cover of the group of contacttomorphism of R1 * S1 and also the universal cover of the real project space and let's say finally I think the most general results in some sense is a results of Razer Pterovich and Rosen where they construct bin variant distance that are unbounded for any closed manifolds that have uh admits periodic rep flow and no contractable positive loop of contacttomorphism

27:46 and uh okay so this is the story for being invariant distance on the group of contacttomorphism and then the story for invariant distance on and spaces is the story I've just discussed before. So let's say the van vanilla ascend on distance on conta is the one is was also called the discriminant distance that allows them to then define this disc discriminant distance on space of jeandreen and then the distance of razer povich Rosen is in some sense on the space of contacttomorphism in some sense is the one we considered with simal to construct our distance on the space of Leandrean.

28:34 And some some remarks here is that in contact geometry all the results about invariant distance happens when there is periodic flow or at least a positive loop of contacttomorphism. And the only results for the moment interesting result that does not make this assumption of existence of positive loop of contacttomorphism is the theorem a I just shown before. And this is why I was a bit happy to have yeah a result without about invariant distance without asking for positive loop of leandrean. So this was a first remark and the second one is that all the distances b invariant distances on the group of contactomorphism and invariant distances on spaces of leandrean they were all taking discrete values so integer values and actually frazzer povich and roen showed that it has to be the case on uh the group of contacttomorphism as we will see more precisely later and I assume that it it has to be the case

29:51 also for Leandrean and this is the next slide. Okay. So and in some sense it's a remaining of the flexibility. We had some rigidity that allows to construct unbounded distance but there's still lot of uh flexibility more than in simplectic geometry because in simplectic geometry you can have fininsler type distance. Here you have too much flexibility to expect fininsler type distance. So more precisely if you have an invariant distance on the lejandre and isotopic class of a fixed leandre and lambda then there exist a positive constant C such that the distance between lambda and lambda prime is greater than c for any lambda which is different from lambda prime. And I have a similar statement for universal covers but it's a bit more hard to state. So I'll just state the one for the Leandre and isotopic class.

30:53 And yes as I said just before there is a similar result found by Fraser Pterovich and Rosen for the uh bin variant distance on the group of contacttomorphism. So there exist a constant C such that the distance between PH and S is greater than C for any PH different from S if your distance was being variant. Okay. So if you don't have any question maybe I move to the proof. So now I I want to prove this uh to theorem or at least give some ideas of the proof and be yeah before giving the proof I want to talk a little bit more about orderability again. So we said that uh the universal cover of the leandrean isotopy class is orderable if positive loops of leandrean are not contractable.

31:50 But why elashburg and ptovich called it orderability? It's because actually it induces some partial order and here I want to present this partial order. So so if you take uh yeah a cooriented contact structure so what it means a co oriented contract structure one one way to see it is to say that you have uh one form such that the kernel of the one form uh is exactly your contact distribution. then I will say that an element so a contacttomorphism so here I see contacttomorphism just for simplicity but contacttomorphism here it's an element in the universal cover so I take a contacttomorphism I say that it is greater that than the identity if and only if there exist a contact isotopy that represent it such that the speed of this isotopy is non- negative okay so this is the definition of this binary relation on the group conta and then you can define the same binary relation on L tilda by saying that two

33:04 elements lambda tilda is less than lambda prime tilda if and only if there exist five tilda which is greater than the identity a contacttomorphism greater than the identity that sends lambda tilda on lambda prime ta okay so this is an easy lima actually this binary relation so here I'm talking about the binary relation on l tilda so the binary relation on l tilda is reflexive it is transitive and it is invariant by contactomorphism so what it means to be invariant by contactomorphism so ph lambda is less than fi lambda prime if and only if lambda is less than lambda prime.

33:54 So this is uh easy to check. What is interesting and hard to check is that antiymmetry of this binary relation is not always uh true. And if it is anti-Symmetric then it is a partial order. And Elashburg and Pterovich and also Chennov and Murovski they show that this binary relation is a partial order if and only if Elilda is orderable. So if and only if positive loops are not contractable. Okay. So this was just again some other ability that we will yeah because we will use it in our construction this binary relation.

34:38 Okay. So now just uh idea of the proof of theorem A. So theorem A is the construction of an unbounded distance. So just recall it's the hypothesis of theorem A or some of them at least. So you take a positive loop of leandrean lambda t and then you take a loop of contacttomorphism that extend it. So this loop of contacttomorphism does not have to be positive everywhere but yeah it exists. You take a loop that extended. Then some notation I will denote by lambda tilda s the lift of the leandrean isotopy lambda st. So s here is a fixed real number and t goes from 0 to one.

35:28 Okay. So you take lambda st and then you lift it to the universal cover and f tilda s is the same. You take you take a real number s and then f tilda s is the leaf of 5 s and t goes from 0 to one. And just to make sure that everyone agrees with the notation. So f tilda s applies to lambda 0 gives you lambda s. This is just some notations. Okay. And then a first lema which I won't prove but it's not very hard to prove. So for any uh element of the universal cover lambda tilda there exist an integer n such that lambda tilda is in between lambda n tilda and lambda minus n tilda. Okay, I won't prove it but the idea of the proof is really easy. It just come from the fact that lambda t is positive. So if you iterate it enough time, it will get more and more positive. So that at some point it will be greater than lambda tilda and the same is true for lambda minus n.

36:44 Since it is positive if you do it backwards it will get more and more negative so that at the end it will be less than lambda tilda. Okay so this was a lema and the second one which is not very much harder but I I'll try to give a a proof for this one at least. So [clears throat] for all lambda and lambda prime there exist an integer n such that lambda tilda is in between 5 n apply to lambda prime and 5 - n apply to lambda prime. So here the little more difficulty comes from the fact that 5t was not a positive loop. So by iterating it more and more times I'm not getting more and more positive. So it was just extending a positive loop of leandrean but it's not hard to get rid of this difficulty.

37:46 So yeah proof of this second lema. So first yeah you fix lambda tilda and lambda tilda prime from the lema and then you find a contactorphism that sends the time zero of your positive leandrean isotopy to lambda prime and then by the previous lema there exist an integer n such that s minus one lambda is in between lambda n and lambda minus n but lambda n and lambda minus n from the notation we uh introduce it's just 5 - n applied to lambda 0 and 5 n applied to lambda 0 okay and now what we do we just uh multiply all this inequality by s tilda okay and we don't change the sense of inequalities because this binary relation is invariant and uh and now we use the fact actually that fit is a loop of contacttomorphism.

38:57 So it come back to the identity. So it means that filda n for an integer n it commutes with all the elements of the universal cover. Okay, because ft is a loop, it compacted the identity. It commutes with all the elements of the universal cover of the group of contacttomorphism. So then I use the fact that it commutes with S to get the last uh inequality. Okay, since it commutes with S and S lambda 0 is equal to lambda prime, we get uh the proof of the lema. Okay. So now uh if everyone is agrees with that I go to yeah proof of theorem a. So now I assume moreover that lilda is orderable. Then the map uh which takes uh two leandre lambda and lambda prime and associate to it the smallest n such that lambda is in between 5 n lambda prime and fi minus n lambda prime.

40:06 actually it is an unbounded invariant distance and this is the invariant unbounded invariant distance I was yeah talking about. So for invariance it use more or less only the fact that uh fn since f is a loop fn commutes with elements of the universal cover. So it's uh more or less only that and also the fact that the binary relation is invariant. So with these two ingredients you prove invariance of this distance. Then non nondeneracy of this distance follows from the fact that the binary relation is antiymmetric because we have assumed that L tilda is orderable. So the binary relation is a partial order.

41:01 So it is antisymmetric. So if you suppose that the distance between lambda and lambda prime is equal to zero, it means that lambda prime is is less than lambda and lambda is less than lambda prime. So by anti-ymmetry lambda has to be equal to lambda prime. So this is uh how you deduce non- degeneracy. And then uh triangle inequality follows from transitivity and invariance of this uh binary relation. And unboundedness for unboundedness the sequence you can take is just you take the positive l the positive leandrean isotopes. So it's easy to see that the distance between lambda zero and lambda n is just equal to n. n here is an integer. And here it use also anti-ymmetry.

41:57 So this is the proof of theorem a or at least the idea of the proof and uh some um applications to study over distances. So yes maybe I talk a bit quickly about we can use this distance to study over distances. So for example with simal and also this distance was uh independently constructed by Lucas Nakamura under this following assumption. So here you take L tilda to be orderable. So you just assume orderability of L tilda. You take a contact form and you take its red flow. Then you can define absolute distance that we call with similarly the spectral distance because it has some geometric property also which is the smallest t such that lambda is in between f alpha t lambda prime and f alpha f alpha minus t lambda prime. So it gives you absolut distance but this absurdo distance is unbounded but it is not invariant by contacttomorphism and an easy way to make it invariant is just by taking the supreum uh with all

43:13 contact. So the distance between lambda lambda prime is the supreum of distance between s lambda s lambda prime for all contactorphism s. But then by doing so maybe you just get plus infinity and actually you if moreover you assume that you are in the hypothesis of theorem A. So if there exist a positive loop of leandrean and loop of contacttomorphism extending it actually this uh distance is bounded. So the this distance here is bounded and it is actually less than the distance I've just constructed. So with the hypothesis of theorem A this map is finite. So maybe it will be interesting to see that if without this assumption of theorem A uh does this map is still finite or not.

44:16 So it will mean that orderability only orderability of L would be enough to expect existence of unbounded invariant distance on space of leandre but this I I don't know for the moment and uh okay and I wanted to to talk a little bit also how we can use this distance to maybe construct norms on group of contacttomorphism but maybe I I skip it and if we have time at the end I talk about it. So now maybe I I go to the proof of theorem B. So theorem B was a discretetness of the invariant distance. So uh you remember so discretetness mean that if you have an invariant distance there exist a constant C such that the distance between two lejandreans that are not the same has to be greater than than this constant C.

45:15 And again here uh we will have two proposition uh to show this uh theorem. The first one is vinstein neighborhood theorem and it says the following. So if you have a closed leandrean inside a manifold then you can find a neighborhood of this leandrean that is contacttomorphic to the one jet bundle of this leandrean. And moreover, this contacttomorphism fixes the um the zero section. So it send so it sends the it sends your Leandrean lambda on lambda. And here I identify the second lambda with the zero section. And let's say the classic vinstein neighborhood theorem says that this contacttomorphism is just between a neighborhood of the leandrean lambda and a neighborhood of the your zero section.

46:14 But then you can use the contacttomorphism that is in the title of the slide to extend this u neighborhood of the zero section to the whole uh one jet bundle in some sense. Okay. So this is the first proposition. A second one is the following. So you take two sets B and B prime. Here B I've colored it in red, B prime in orange. And you suppose that B intersect the zero section. Okay, these two sets are in the one jet bundle here. And you suppose that the red set the B intersect the zero section in a small chart of lambda. So in an open ball in the zero section and B prime you just assume it to intersect lambda and if B uh lies above the intersection with the zero section. So if B is in this green cylinder and B is bounded then there exist a compactly supported contacttomorphism that sends B inside B prime and moreover this contacttomorphism it fixes the zero section okay and the proof is not so hard and

47:41 it's pretty much easy even so uh you take this uh Uh so B intersect lambda in a small chart. Okay. So in this blue line here but then since it is in a small chart you can find a diffomorphism of lambda. So a defomorphism of the zero section isotopic to the identity that send the blue line. So the intersection of B with the zero section with uh inside the purple line. Okay. inside B prime the intersection of B prime and the zero section and then you can lift this. So you have a defomorphism isotopic to the identity of the zero section. You can lift this defomorphism canonically to the one jet bundle. So there is a yeah there's a way to lift theomorphism of the zero section to uh the one to contacttomorphism of the one jet bundle and then this new uh contacttomorphism of the one jet bundle. It will send this B the red line inside the dotted red line above the dotted blue line here.

49:03 And then at last you just need to use the contacttomorphism of the title of the slide. So in the title of the slide you have a contacttomorphism that shrink everything for t small enough this contacttomorphism shrink everything to the zero section. So by com composing the contacttomorphism we constructed with this contacttomorphism of the title uh you can send B inside uh B prime and you can make everything compactly supported because everything was contact isotopies and B was bounded. Okay, so this was uh the two proposition that were uh the most important to prove the b and then yes this is just a reminding of this uh notation. So I I have a leandrean. I take a neighborhood of this leandreen. I take a contacttomorphism that send it to the whole one jet bundle. I take a ball b which intersect the zero section in a small ball and then I take curly b to be the corresponding set inside the original

50:22 contact manifold. So curly b intersect the original lambda into a small ball and the corery of these two uh previous proposition is the following. If you take a distance which is invariant on the space of leandre and isotopy then for any contacttomorphism which is supported in curly b then the distance between lambda and phi of lambda is less than two times the distance between lambda and lambda prime for any lambda prime which is not equal to lambda. Okay. So let's prove uh this corary. So you take lambda which is different from lambda prime. So there exist an open set P prime inside your initial neighborhood of lambda which does not intersect lambda prime but that intersect lambda.

51:24 And then by the first step by combining the two proposition of the first step you can find a contacttomorphism that sends B to B prime and that fix lambda. And so the support of the conjugate of 5 minus one by this contacttomorphism will be inside B prime because the support of the conjugate is just inside the image of the support but the support of five was inside B and S of B is B prime. So the support of conjugate of 5 minus one by pi is inside b prime and then is just some algebraic manipulation. So the distance between lambda and ph of lambda is the same as the distance between s lambda and si lambda because the distance was invariant but s preserves lambda. So it's just the distance between lambda and si lambda and then you use the fact that you are a distance. So you have triangle inequality. So it is less than distance between lambda lambda prime plus distance lambda prime si lambda.

52:48 So the first term remain the same on the next line. But then the second term you use again that the distance is invariant and you compose both sides by s 5 - 1 s - one and then you reach the last line because 5 minus one s minus one has its support in b prime but b prime it does not intersect lambda prime. So this contacttomorphism fixes lambda prime and and s also fixes lambda. So the distance uh the last term is just the distance between lambda prime and lambda and then you use yes of course symmetry of the distance between lambda prime and lambda is the same as okay.

53:42 Um do you have questions? Because I I I wanted to point at some time what I'm talking about at what inequality or equality here I am but I was not able to point it. So okay if no questions I I go yeah how to deduce from this cor theorem B then it's not hard. So what we want to show actually it's enough to show that you fix lambda and you suppose by contradiction that there's there does not exist discontent C such that distance between lambda and lambda prime is greater than c. So by contradiction we assume that there exist a sequence lambda n such that each lambda n is different from lambda and the the distance between lambda and lambda n goes to zero as n goes to infinity and then you take this contactorphism you take so [clears throat] kibb just recall kibb is a small ball that intersect uh lambda in a small chart.

55:01 But then you can construct a contacttomorphism supported in curly B such that five lambda is different from lambda. to just push a little bit lambda in this uh in this small chart and u and then [clears throat] so since the five lambda is different from lambda the distance between lambda and f lambda is strictly positive but at the same time the corerary said to us that the distance between lambda and f lambda is less than two times the distance between lambda and lambda n for all n and So you have a contradiction because this sequence goes to zero but at the same time it is bounded away from zero by a constant. So this is the contradiction and it gives the proof of theorem B and um okay so in the last minute maybe uh if you have no question I I I put some open questions uh that will be maybe interesting to yeah to think about. So [clears throat] the first one is uh in some sense if we if

56:18 we forget about the contact structure if we just look at the smooth setting I would like to show that this kind of results cannot be expected. So it's really the contact distribution that gives enough rigidity to expect existence of unbounded distance. Okay. So more precisely the question is the following or I put it conjecture here. So if you take a closed submanifold inside M and you take its isotopic class so the space fi lambda for fi uh defomorphism isotopic to the identity and if you assume that the dimension of your sub manifold is small enough.

57:06 So here strictly less than half the dimension of your initial manifold which is the case of course for leandrean. Then I would like to to show that an invariant distance on the universal cover of the isotopic class of lambda is bounded. So it's really the fact that we added the contact structure and uh that make uh the existence of unbounded distance possible. If we forget about contact structure then it's not possible anymore. And actually there is similar results that was shown by Burago Ivanov and Pterovich and also boy not on the not in the relative case let's say but in on the group of defomorphism itself actually they shown that on the group of defomorphism isotopic to the identity you cannot expect um existence of unbounded uh invariant B invariant and distance and here it will be in some sense uh yeah an analog of this result but for for yeah the relative case okay so this was uh one question another one uh it's

58:29 also um based on actually for the contact sphere we know uh a lot of uh flexibility results For example, Fraser Pirovich Rosen shown that for the standard contact sphere, if you look at the group of contacttomorphism, then you cannot find any interesting bean variant distance on this group. And again a bit in the same vein I would like to show that it should be the case also for Leandrean but this I I don't know but it would be interesting to to see if uh for any leandrean of the on the sphere you take the leandrean isotopic class of this leandrean then you cannot find any invariant distance that is unbounded on this space and uh and maybe a last question and uh actually all uh all the thing I was talking about was for no not all the things but all the unboundedness results for Leandrean was not on the leandrean isotopic class itself but for the universal cover of the leandrean isotopic class and actually there is no

59:43 unboundedness result I think for closed landrean for yeah and for the leandrean isotopic class itself not going to the universal cover. Actually, it's easier to find a unboundedness or rigid detail rigidity results on the universal cover because you remember more about the leandrean. You don't just remember the time one, you remember all the isotopy or at least maybe not all the isotopy but the homotopy class of the isotopy. So as you remember more it's easier to construct to have rigidity results but if we forget about this all isotopy if we just think about the one leandrean then I think we there's no answer to the to this question about unboundedness for leandrean isotopic class itself and uh so I I think this was the yeah pre question I wanted to talk about. So, thank you very much for your attention.

1:00:55 Thank you very much, dear Alexandra, for your beautiful talk. Would somebody like to ask already a question?

1:01:05 Maybe I have a little comment about this uh matrix on demorphisms, right? So all is okay except dimension two because in dimension two when you have surface of genus one or higher so there is a result by both enhanc web very non-trivial highly non-trivial result which shows that there are non unbounded distances ah

1:01:35 and so I mean god knows so maybe in small dimensions like in in dimension three something weapons.

1:01:43 So that's I mean in legend case. So that's I don't know of course. Mhm. Okay. But I mean that's Yeah.

1:01:52 I didn't know this. Yes. And there is also some delicacy in dimension four which I don't remember. Probably in dimension four it's still unknown but otherwise indeed it's unbounded. Yeah.

1:02:02 Okay. So that's uh yeah and and I have another question. So uh I mean when you prove that I mean you have this unbounded metric. So next natural question is what is the course geometry of this metric and so for instance kind of is so true or not true that there exists a embedding of a line.

1:02:26 Mhm. Let's say quasi isometric such that all the group I mean all the space of leandrians is in a neighborhood of 100 around this line. So can you prove or disprove this?

1:02:40 Uh so can you repeat the question? Yes. So you have unbounded matrix space. So in principle this matrix space could look like this that you have like quasi isometric embedding of R

1:02:54 and all the group lies in a neighborhood of radius 100 allows this line. And in simple case for instance for demorphisms of spheres there were several groups of people who kind of proved this with very hard work. Yeah. So I mean that this this is not the case. So it's not in principle untrivial non-trivial result kind of. Yeah.

1:03:18 But for leandreans I have no clue what happens and that's very very natural question. So that's

1:03:24 Yeah. Yeah. [clears throat] Uhhuh. No I I I don't know. I don't know. Uh, I don't know. Yeah, I I I would think that it's not only around the line.

1:03:43 I would think bigger than that, but uh I'm not sure. Yeah, but then I mean it would be nice to prove this maybe to find other flats or I mean to to work on asytoic cons. So there is kind of this g ofic con which captures behavior of the space like when you look at it from very far away. Yeah. So so maybe you can do some calculations there. So in simple case I mean there there is at least some examples of such calculations.

1:04:18 Yeah. No no of course it's an interesting question. Yeah. Yeah. Thank you. Yeah. Do you have kind of a natural uh conjecture to characterize Leandre instead admit an invariant unbounded metric? Is there or do we know not yet enough to have a intuition for that? I would say orderable is a good uh is a good start. I think at least orderable I think without orderable for the moment. Uh even for metric on group of contacttomorphism also every time orderability is assumed and uh boundedness. So this is yeah for unboundedness and for boundedness actually there is a beautiful paper of uh Patrick Maso and Silylvan so about boundedness also on the group of contacttomorphism where they I don't remember exactly I think they they show that uh for contact manifolds that admit open book decomposition such that the page of the open book is sufficiently flexible. I don't exactly remember what is the sufficient flexible condition

1:05:50 uh then the the group has to be bounded. So here I guess similar statement can be done for but maybe a good start will be yes to try to start with the sphere try to start with the sphere to show that for the sphere it's bounded and then uh then see uh what we use for boundedness maybe also it allows to understand more what is important for unboundedness but I would say that a good start for unboundedness that orderability is a important condition maybe not sufficient

1:06:36 but uh maybe not necessary neither but um I think it's a good uh yeah good candidate to be yeah very important condition thank you are There are the questions.

1:06:56 So Dylan wrote something in in the chat. So Oh yeah, thank you Dylan. There is the shukin oscillation distance. Yes. Distance to rep flow. Oscillation distance which measures distance to rep flow. Ah that's true. Okay. And then this distance you say that it is unbounded sometimes and then it It's on the cover right or

1:07:27 it's not invariant right so it depends right it's not invariant yes but the question is whether you can you can build I mean if you take some super or something whether you can build something invariant that's I don't know so you go can you comment about this

1:07:47 um not sure I mean uh taking suprema usually does not really work because we don't know if it's bounded or not. But uh on the other hand, I'm pretty sure that um so so there are some methods that are used to prove that uh this kind of distance to rev flow this kind of not by invariant distance to ref flow is unbounded. And these methods might be possible to apply to prove uh something like question uh special cases of question three. So sorry can you can you repeat uh sorry I I um so it was a very long thing I said I'm not sure. Yeah, sorry.

1:08:42 No, what what you said about dimension free what you can use to prove so so there there's this distance to rep flow and in certain cases uh it was proven that this distance is not bounded and I think that if one looks into the methods that are used there then um one might be able to show something about bvariant distances as well. Yeah. Yeah. Yeah. Yeah. Yeah. I think so too. Yeah. Okay. Thank you, Ego.

1:09:47 So in the Hamiltonian setting when one takes fins matrix they are all comparable to Her right now here everything is discrete. Is there a way still to normalize and maybe to see that all unbounded metrics are somehow comparable or did people see such a thing?

1:10:08 No, no. I I think there are really unbounded distance that are not equivalent. I I don't know if it answers the question but uh I I have examples of uh unbounded distance not not on space of leandreans but on contacttomorphism such that uh yeah one of them is uh uh unbounded when you look at iteration of the red flow but the other one is unbounded for complete different reason when you look at it on the red flow it's just uh it's say constant for example.

1:10:47 So yes, I don't know. So there is a much bigger zoo somehow. Yeah. Yeah. Yeah. There are interesting invariant distance that are not equivalent and they Yeah. I saw you would like to ask a question. No,

1:11:32 not much. But thank you very much for the talk, Don.

Questions about this episode

What is discussed in "Invariant Distances on Legendrian Spaces - Pierre-Alexandre Arlove"?

In order: 0:00 contact distribution, contacttomorphism, contact geometry, Daru theorem; 10:00 contacttomorphism, contact distribution, Vansula and Shila Sandon, Elashburg and Poltervic; 20:00 contacttomorphism, contact geometry, discriminant distance, one jet bundle; 30:00 contacttomorphism, leandrean isotopy, contact distribution, Elashburg and Poltervic; 40:00 contacttomorphism, one jet bundle, B and B prime; 50:00 contacttomorphism, contact distribution, one jet bundle; 1:00:00 contacttomorphism, open book decomposition, Patrick Maso and Silylvan; 1:10:00 contacttomorphism.

What is mentioned in this Institute for Advanced Study episode?

18 people, works and ideas are mentioned, including contact geometry, contact distribution, Daru theorem, Vansula and Shila Sandon, contacttomorphism, leandrean isotopy. Each one on this page links to its own page with every episode that mentions it.

How long is "Invariant Distances on Legendrian Spaces - Pierre-Alexandre Arlove"?

The episode runs 1 h 12 min; the transcript is about 9,155 words in 80 timestamped sections.

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