A visit to the Finsler world­

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A visit to the Finsler world­
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the think other 2 and we a half and it's a tremendous
honor to be here to to speak at the the occasion of violation peers retirement In talk from my congratulations have when I agreed to speak I didn't realize I was going to also have to follow Sir Michael at the creatures as well as you know always a daunting task
but that's a real pleasure to hear up to here your thoughts on spinner a geometry and your and your sense that that our geometry is is the inspiration that we should be at which be headed for in in
understanding these modern constructions and a of a window into the physics and and higher mathematics of and in fact that's that's what I'm hoping to give give some flavor of today but for someone say sale little bit
about about a mile experience with patients here I think we met in 1979 at the Institute for Advanced Study on this special year in differential geometry was fantastic year for me and not the least of which reason
was that I got jump here and there and learn about his insights in geometry and spent geometry Riemannian geometry of the and also the other significance the INS in beyond just the just mathematical experience over the years our is that having just finished 6 years at a mere 6 years of being our director of of the Mathematical Sciences Research Institute and in Berkeley I think I'm in a somewhat rare position to to appreciate the jump is amazing accomplishments over there over the last 18 years both of us but as a mathematician and illustrator and a leader of the of the international community and if I if I did wear you will be off to you today definitely on no John as contributions to mathematics however have been I'm very diverse and and far reaching we just heard about the spin geometry which I jump wrote has has made done many contributions to our and our end of course I believe that we can survey everything but I wanted to pick out a single but I hope you'll agree very important thread in jump years work that of the on the role of calculus of variations in differential geometry I was very pleased to see that and somewhat amazed at say all to see their job here recently wrote a book on account of variations in differently geometry while being director and which is part of the reason for the amazing not that does nothing would spend time thinking about status variations which of which I think we all agree is very important
what I want to talk about is something of that sort of on the commerce of this estate is the role of the role of differential geometry and calculus of variations and in in in a sense to try to understand by geometric means what that you and approach to the status of variations and I infer that I wanted to start with a few remarks about this geometry and if there's time talk about all the hard shown on generalizations that in extensions that that are now being approached using differential geometry so the young defensive story starts on banking started further back than and then Sir Michael did in 1854 it was reminder this famous gave his famous lecture on the on how the basic notions of geometry that is let distance metric and so forth all all all I can be thought of as arising from the calculus of variations that use of that is to specify a 2 to understand what GDS 6 hours straight lines and so forth .period should think 1 should think in terms of minimizing functional and that functional being and so here this the and I'll slightly more modern language that To start with the functions on the tangent bundle all that years of that year's wants on the idea that that if you have occurred you associate elects to that depends on this function by immigrating from 0 to 1 the function applied to the velocity vector the curve now of course in order for this to make sense but that is not to do it to be a geometric thing not have not been prompt physicians that impose some conditions on and the typical conditions are that you want to be the want the function of a homogeneous a degree 1 and for my purposes I'm only going to take our are among them to take that in the in the positive case because I want to allow for the possibility that the debt of the length of occurred in 1 direction may not be the same as the land and the other Asian true reverse in the other direction and for reasons that have to do with the regularity of the status variations applied this problem you wonder required that you will require that Esquerre should be or should be strictly convex on each tangent space on and on the exam for all its and the picture you should have in mind is that if you look at the level set in the so-called indicator if you Sigma X is conservatively until exam such that definitely is 1 of these is usually called the retention index trips they should be a convex curve are enclosing the origin of typical you here in sync explicitly space end under these assumptions the the idea of finding a minimizing curve joining 2 points is on behaves well and you get a nice so a nice family of planes that are infinitesimally anyway the straight lines the notion of shortest curves joining any 2 points which become the G-8 6 now the other the natural question is how do you distinguish would it not so you have a notion straight line joining you need to sufficiently nearby .period highly distinguished to such geometry from each other and that of course you can sing should looking only at 1 1 1 because all lines will in turn look the same but but if you consider how raises separated on you begin get a way of distinguishing the geometries so the race separation by then that is starting to To raise starting in on the I'm sorry hinted at an initial point and you look at that a new look if you go the distance guests along 1 of the wounded that U.S. 6 and you ask how fast how rapidly due the separately this notion this is by the way is the usual are the main theme of the lack of courage joining 2 points then are there the but the interesting quantity just to remind on every 1 of this picture of course there is the 1st known in the 1st quarter to 1st order this depends on what you might think of the angle but you don't necessarily know what angles or in in Riemannian geometry and so on 1 of the EU's actually just put out some constant growing when nearly all of the depends on the 2 raised and then the and then to the next order what happens is you get something like this and I put 16 because that's the classic normalization up to the order of something to the 4th powering or the square but all the 2nd power you know in this in the separation this lowest order separation costs the costs as as Myanmar approaches Road see gamma goes zeros as you expect but this is the this is the expansion you would expect this quantity on which are which sort of measures the deviation from the way straight lines would seperate is
is the is the flag curvature His ,comma flanker Richard because it but it depends not just on the other nite just on the 2 plane in which this year infinitesimal Essex but it depends on which ordered you do this evening and the independence of wit on the actual specific rolling down that is it take is not necessarily Carol again in and that and it depends on you the ideas you fix rowing in you see how see how damning varies and this is in some sense the which might think of as the 1st geometric art piece of information that comes out of of given up Romani structure all this is this is what's known as the flag curvature sorry something although owners probably and I making mission or maybe they can adjust the volume or something which I keep and and it always is in some sense the fundamental invariant enough in In this all this notion of Junction this metric notion geometry that reliance on remand established the 1 thing you can see for example example always like to give media in pictures like this is a fiscal example the the river all the example you imagine you have a imagine you have for simplicity let's make it a major this make it a straight River and you have current flowing faster in the middle and slower at that hour slower at the at begins at the banks of and I and the set of displacements that you can reach unit times is not of of course centered on the centered on point where you are but shifted over by by the current it's my drawings of great that but you can see you have to have something strictly convex Towards the Towards the Twas the origin and each point it's not centered on the origin if you look at the gymnastics are leading up .period saying center as they go downstream what happens is that there is of the Virginia 6 separated faster than faster than you would expect that some of the negative but if you go upstream the Essex tend to do something like this and that the GDS excel U.S. extending come together so even in even in the two-dimensional case that you can see the flanker Richard depends on the depends on the on the GDS it's not just responded to point that set off in the direction you're going in all in on of course remind look at this sort of situation he figured out what the what the case if the if efforts actually invest squares is actually quadratic which is the case we now call Riemannian geometry it appears quite the Gentry then I then take constant by exists what is classified we don't know what those answers are that so-called space In reminding backed himself gave the gave formula for what the metric looks like after change of wouldn't say if the intifada if the if the flag curvature is identically concept but the in money injuries but in case that's still that's still very much an open problem exactly what the what the cake was constant constant things look like in fact it's is not obvious from the definition that I gave a case actually differential in the area that it can be computed by by some kind of doing differentiation of al-Salem about how that stand in the middle of the street and so on and so there is a natural question is what Watanabe equation is K constant in the general case Hilbert gave examples that turned out also called the Hill were projected metrics on date examples of convex domains that are using cross racial formulas that I will write down that gave examples of the use of these kinds of on metric structures with the property that plays England minus 1 on what they're not money that is their right there on the the spirit each point is not is not a is not a that is not an ellipse centered at the origin also in 1919 coffins Paul fistula made area started a kind of geometric studies of all of these kinds of structures and that's what we call this geometry and this is 1 of a string of a string of other people involved in 1934 church cotton wrote The road area of Wilkinson Sword geometry the 1943
churned out of applied the method of equivalents to office geometry of Andorra and in some sense we you are applied the method of equivalents all end of an incense is derived the thereunder the funneling variants and I'm not going to go through that derivation we don't have anything like the time like that but I I I will say that I will have to call the cell colonies thanks for some for some of the ideas before some the results and and it's not known are just give another example beyond beyond this is now known for example that that even on the 2 spheres you don't the space of about officer metrics Clifford J. identically 1 is international the I'm commodity more efficient all the even even if you already spirit even if you fix in the require that the Virginia Essex would be great circles all that is unique in that is even if he is saying that the Kyrgyz 1 and is projected Lifland this still attend parameter family of of of of the state it's symmetric for the property it's even plus predictably flat 10 perimeter family Oh no no this is something I should say I this of the examples the temporary orders are actually find it turns out that especially complex mental spaces the space of solutions turns out to be a complex medical and that in itself and it's the 1 of the things until about his is how complex geometry comes into this picture because it's not at all obvious that is any complex geometry at no but now yes so such law look locally locally an international family of solutions as far as I even of projected with flatulence that's right yeah right still environmental predictably flat cables 1 right here luckily is that there is definitely a global theorem new media deformations use the fact that you you find rational curves somewhere and you look at it and what the deformations of rational curtain home manifold somewhere and find its defamation space from his bundles yes on the other hand up in 1988 up was ordered on showed that thought into his Compaq and Keyes identical minus 1 then then all of them in better Atherstone ETA's Romani all although I mean if you think about it what you what's happening here is that is a little bit just give you a sense of what's what's odd is that even before the infamous surface case if you look at she is the defined is being defined basically in terms of this sort hyper surfacing in the media in the in the tangent bundle so it's that so actually the variable is of 3 manifold monitored manifold and I N K is a function on that on that object it depends on for derivatives that hyper surface sitting in sitting in the sitting in tangible and it's not political equation isn't more hyperbolic it's quite complicated and you use the 9 use the use the gym did generously on it to prove notes certain sort of certain conservation laws show up to prove this statement in the cables months 1 case but nothing like that works and it was possible you some sense of how hell you ordered is from the point of view he might think will the 4th order because in the remaining cases 2nd order and the reason is it turns out that that if you impose the remaining condition the higher during the coefficients of a hard to reduce dropout and so you actually only if it reduces to a 2nd order equations whereas it's naturally 4th order .period prices it is a form of discrimination in the job used was to remind you of I don't know how to answer that up but it is important New York you just raise a really important point that there is a Riemannian structure on all in all in fact on the other on the unit tangent bundle and that's what I that's what I wanted to do for now it said the other 2 are in more detail Bob we started out with Sigma sitting inside the tangent bundle him and let him be impossible one-dimensional for them and not back to the general economic recession is oriented on to 2 4 5 simplicity there's a natural inclusion of signaled that they the Legendre transformation in that segment of the cutting edge of the point being that if you have a point of hears that his unit sphere bundle peers you point you in in Sigma there is a there is a 1 form Italian In the coating to bundle that satisfies the town you applied to use 1 and Tower is less than or equal to the no wonder :colon sitting that is is that it's the 1 form that guy that whose level set equals 1 is exactly the tension pointed to the end that hurts at that point that naturally maps towel into the coat and a bundle and it's invading and so that the net the canonical 1 content Munhall pulls back the same it's a signal as it has a canonical 1 Sigma has has a contract structure and of course the Classic Car picture is that Is that that contact structure gives rise to the to the Jurassic flooded the radar vector field of Alpha is a flaws safety is Jessica Lange now as I as John mentioned there's a of it follows in particular from the work of Chernobyl although it's kind of implicit in cartons analysis that that although there is no natural Riemannian metric induced on em there is a natural Romani Mexican distancing squared on signal and I think he has no 1 with respect to that I certainly is the union of all landed the that is each similar acts as a hybrid services instead .period you take the union of those high reserves in the coat intent upon that's right exactly it's level so that the Eagles 1 on and it turns out there is a natural Romani magic on this up the art the dynamical properties of this year Essex flow a very interesting I'm a synthesis mood that's right yeah if you don't send smoothness then then all kinds of bad things can happen up on and I should be here for the for the for us a very nice interpretation of not only this metric but the flow and geometrically in terms of dynamical systems you should look at the of the patent for long work on on the dynamics all he was an interpretation of this of this geometry but what I want to focus on is is here is a consequence of that identically one-tenth of a solid is 1 case today's minus 1 case has a similar development but that but ,comma focus on cables 1 case today this is following proposition that it stays identically 1 there then then of the metric is invariant under the under the the on off genetic flaws this canonical metric is invariant genetic flaws In fact that's almost as almost equal to the 1 on and so have following picture which is about which will turn out to be important if you look at Sigma cost mapping out the M bye it's normal projection of the if this if this summer if the floor is yours is geographically sample that is if there's a quotient by the flow of E. coli in the space of genetics Of polar lander mapping down to the space to do their successors one-dimensional to impress one-dimensional to invention up that's as in the cables 1 case what happens is that there's a well-defined quotient metric His more square is well the gasket locally and globally it that but if you want to use will look at the local solutions you can just take a take a local comics patch and they're always in this in Jesse flow will be genetically simple comics pact but this did not happen for the instigator of the impose once-feared invite there this side two-dimensional case this fact generalizes to the generalizes to the 2 organizations that the that the space of projected reflected resonantly 1 metric tons of food in spheres for the feared turns out to be a competent people away in fact it's identifiable with space of Quadra CP plus 1 without real points interestingly enough but but so this both local and global questions right now and we can focus on local all up but it says is that there is a natural metric on the space injured now just from my just from our classic classic calculus of variations we know that because this guy has because this guy's the quotient by the raid vector field of you in this context structure is actually also a simplistic structure where a major has a property that when you pull back when they get to To this day I you get the differential of all the contact form so this guy comes equipped with both a metric and a and a R and invited structure and all priori yeah it's not obvious that they have much to do with 1 another but it turns out and fears that the beautiful fact that in the case and in 1 case when pleasantly 1 2 the best where America is a Cayman in fact are in fact Essex are always oriented because it took off 1 yes but not only divided up by the geodesic flows right but you also all always outside of my definition of geodesic is oriented Utusan my curves are already in effect but doubt that he put in oriented if you everywhere if you want to right so it turns out that this is in fact a killer all very out of various worlds big surprise me and in particular the shows that the that the natural so courses natural Lady to connection at all MIT says is that the whole army of the Home Army of Nabila gears Ewing In general how best the largest it could be that if the mechanism is Taylor this is a special case In the special case where I where where you actually where you start out with that impulse 1 sphere with standard metric the whole actually dropped special case you imports 1 a half years as in plus-one 1 canonical then then to actually turns out to be isometric to SO IN plus 2 new modified S 2 crosses so and so the whole family is actually incident process set which is a proper subgroup of Western Union but all and fall in the general case it does not reduce something of this survive in the sense that there is it turns out unnatural circle action that dog and up while there's not quite a natural SON structure there is that there is a baby there is a sort of a relief not Huntington describe that the point is this that over it was a couple of hours a point of Q is of course a curvilinear oriented occurred him and if you then appointed him on appointing him when you look at why corresponds to over and cute Is you lifted up and push it down it's a Lagrangian manifold it's an individual quite Rongen said manifold the scoreless Escobar's C X which is land of high verse politics for exon in here and so there's is a inventors a 1 parameter family of love a 1 parameter family of Lagrangian submanifolds sitting in here and the next later turns out to be true to very very beautiful geometry is that if you look at the if you look at potential spaces all of these on these guys if you look at the changes space at this guy which I failed to name chief if you look at the Tengiz basic unit 2 to these Of these Lagrangian submanifolds they all turned out to be eaten the FAA times of detainees face of 1 of them In other words what happens as you as it as you move along the as you move along father is you have this 1 from a family of special public Rongen submanifolds they're all all the all the all intersecting Q may insect uniquely to and they just stop and they just write it but that so actually it turns out that the that there's a natural there's a reduction of the structural here from New England to a subgroup which is which is basically S 1 times so far his union I'm sitting in there there's S 1 time zone in there is a natural reduction of structure from from From this step down to this and all unfortunately it happens that when that this this reduced structure is not portion free remember Taylor is the same as a UN structure distortion free the Nabila reduces reduced to this is not toward him for it well but that of course you is sitting here in GAO and City this has 1 times and has another enlargement tests 1000 times GL that is not Jordan free except in the flight sorry it said In this is very likely in this symmetric space Romanian so there is a reduction to this which is not toward him free but then because of course this grouping contains this 1 which is sitting inside in the miracle is that that this this enlarged structure Nabil actually turns out to be taught free member is tossing preview it's compatible with "quotation mark I should say so on that's right now but it is compatible with this structure actually note I have to be careful when you reduce you can Split sent radio into but that connection splits in 2 sir she said of the current the connection splits into a reduced connection and a and torsion intensive which only vanishes in that case well that will not is compatible with the structure In fact this whole anomie is generally equal to that please 2 that's 1 times children .period when I 1st noticed this but in this kind of result in the calculation of I was astonished because there have been a classification of the possible portion Freehold only in in in all at geometry that was completed by that was started by a bear J but with up to a funny number exceptions are and and then are carried carried further in the 1980's and 1990's on the published list still did not have this 1 and but also turns out that this actually gives us his examples of it you go back and look at the analysis that was done you can see where the gap heard on the Street so these actually turn out to exist and all the beautiful thing is that there is a nice commerce which is that if you start with the 2 Mitchell manifold are it's a complex manifold are and that has a has a connection to his portion is is this year complex multiples of the identity times those children are the talks start with such a guy there is a natural the curvature form associated to the S 1 that's 0 form of tied 1 1 you can prove and if it's positive then you can retrace the steps and locally anyway recover offense symmetric with phase 1 so the source so that there's a mattress with Keyes 1 actually turn out to be intimately tied to hold on the problem in in our financial and of I want to give this example because it shows that there's a there's actually a lot of geometry in the cat in countless variations problem that that you if you pursue the the the general structure and look at what it tells you there is this it turns out a lot of the tools of differential geometry ,comma come into play to help us understand things like not just this but calibrations on when things are actually minimizing as opposed to merely locally minimizing and so on and I think that we are really all while we made some progress on this is still a lot to do with the soul of lot more to going a lot more to discover in the then the differential geometric aspects of Japan's variations now I'm almost out of time and I was supposed to allow more time for questions so I just wanna say that the that the 2nd part of the talk of was to have been about what happens if you look in higher dimensions that is instead of instead of specifying he in each direction size suppose you do as cotton didn't know in a 1933 wrote little book called on metric space is based on the notion area where what you do is start with in his particular case on 3 manifold he started with a geometry that specifies the volume of every 2 points and wanted to know have liked what kind of geometry can you get out of that it turns out is a very beautiful geometric story that you can know that you can tell up something that filled riffs and I have been thinking about for 4 of the last few years is generalizations of this to higher dimensions where it turns out that unlike impotence cases where there was a unique canonical form that you can attach the pump record-tying form there is now enough when you're going higher coding services and 4 spaces surfaces and 6 basic things that are worth 4 folds 6 places it turns out that the notion of canonical forms much more subtle In the end depends on hired reviews all but there's still a geometry there that I think is not well understood and and there's a lot a lot more forceful about his relationship a lot more forceful by analyzing using the tools of differential geometry and so that's where I was stopped so that people who should look for I think or all of this he was that tickles 1 says that this is true but from what you might as well in this actually there's actually you have to replace this natural Romani matches with the suitor Riemannian metric and in the ceremony case what happens is that what happens is that the the thing that pushes down is itself is somewhat different structure but you can still pursue it but there's still a differential geometry and it leads to a kind of a different color on me that's not what this the you have to the wall that modified still 0 that's correct right great total the National School this is she added I mean in some sense the fact that a complex structure shows up here Our whereas is no this nightmare national Romani instructor here but that the actual killer structure shows here says something about that says something about you know killer geometry is Saskatchewan squaring of of Riemannian geometry were geometry in general this is a vindication that on yes is the company's 1st appeared here in the history of the Parliament not in there not in the sense that I that I understand twisted theories normally it's actually although there is actually a connection turns out with on something I haven't said is for example and enables 1 case it turns out that these things when you look at the trends from over here they they turned out to actually be Zoll matches and and the sole metrics that are you know all liberally Mason had shown that you can that you can understands all metrics on the 2 sphere in terms of the account of a twister construction looking at that at looking at home Norfolk desks whose boundary line certain real of manifold all complex rejected Tuesday looking the module a war against that line boundary and I so vote they have a very nice improvement understanding of Zoll metrics and construction all matches that of the color twister they call a twister constructions all metrics but that's not what it's not you know it's it's twisted more in the sense of using complex geometry to solve some of a global Riemannian geometry thing than it is in terms of the the classic twisted construction of the Balkans mm