 It's a tremendous honor to be here, to speak at the occasion of Jean-Pierre's retirement. And to offer my congratulations, when I agreed to speak I didn't realize I was going to also have to follow Sir Michael Atia, which is, as you know, always a daunting task. It's a real pleasure to hear your thoughts on Spinner Geometry and your sense that Geometry is the inspiration that we should be headed for in understanding these modern constructions and a window into physics and higher mathematics. And in fact, that's what I'm hoping to give some flavor of today. But first I want to say a little bit about my own experience with Jean-Pierre. I think we met in 1979 at the Institute for Advanced Study. This special year in differential geometry was a fantastic year for me and not the least of which reason was that I got to meet Jean-Pierre. And learn about his insights in geometry and Spinner Geometry, Rumanian Geometry. And also the other significance beyond just the mathematical experience over the years is that having just finished six years, a mere six years of being director of the Mathematical Sciences Research Institute in Berkeley, I think I'm in a somewhat rare position to appreciate Jean-Pierre's amazing accomplishments over the last 18 years, both as a mathematician and administrator and a leader of the international community. And if I did wear a hat, it would be off to you today, definitely. Now Jean-Pierre's contributions to mathematics have been very diverse and far reaching. We just heard about Spinner Geometry, which Jean-Pierre wrote, has made many contributions to. And of course, we can't survey everything, but I wanted to pick out a single, but I hope you'll agree, very important thread in Jean-Pierre's work, that of the role of calculus of variations in differential geometry. I was very pleased to see that, and somewhat amazed, I have to say, to see that Jean-Pierre recently wrote a book on the calculus of variations in differential geometry while being director, which is part of the reason for the amazement, not that he would spend time thinking about calculus of variations, which I think we all agree is very important. What I want to talk about is sort of the converse of this, that is, the role of differential geometry in the calculus of variations, and in a sense, to try to understand by geometric means what an approach to the calculus of variations, and for that, I wanted to start with a few remarks about Finsler geometry, and if there's time, talk about the higher dimensional generalizations and extensions that are now being approached using differential geometry. So the Finsler story starts, I can even start further back than Sir Michael did, in 1854 there was the, Riemann wrote this famous, gave this famous lecture on how the basic notions of geometry that is length, distance, metric, and so forth, all can be thought of as arising from the calculus of variations, that is, to specify, to understand what geodesics are, straight lines, and so forth, one should think in terms of minimizing a functional, and that functional being, and I'll start here, this is, and I'll use slightly more modern language, that to start with a function on the tangent bundle that gives lengths, the idea that if you have a curve, you want to associate a length to the curve that depends on this function by integrating from 0 to 1 the function applied to the velocity vector of the curve. Now of course, in order for this to make sense, that is not to depend, to be a geometric thing, not to depend on the parametrization, you have to impose some conditions on F, and the typical conditions are that you want the function to be homogeneous to the degree 1, and for my purposes I'm only going to take that in the positive case, because I want to allow for the possibility that the length of a curve in one direction may not be the same as the length in the other, as you traverse it in the other direction, and for reasons that have to do with the regularity of the calculus of variations applied to this problem, you want to require that F squared should be strictly convex on each tangent space, on txm, for all x. And the picture you should have in mind is that if you look at the level set, the so-called indicitrix, if you sigma x is the set of v and txm, such that F of v is 1, this is usually called the tangent indicitrix, that it should be a convex curve in closing the origin, typical u here in sigma x, this is the tangent space. And under these assumptions, the idea of finding a minimizing curve joining two points behaves well, and you get a nice family of things that are infinitesimally anyway, the straight lines, the notion of shortest curves joining any two points, which become the geodesics. Now, the natural question is how do you distinguish with it, so you have a notion of straight line joining any two sufficiently nearby points, how do you distinguish two such geometries from each other? And of course you can't distinguish it looking only at one line, because all lines will internally look the same, but if you consider how rays are separated, you begin to get a way of distinguishing the geometries. So if you look at ray separation, that is starting at two rays, starting at an initial point, and you look at, if you go a distance s along one of the geodesics, and you ask how rapidly do they separate, this notion of distance, by the way, is the infimum of the length of curves joining two points, then the interesting quantity, just to remind everyone, this picture of course, there's the first order, the first order, to first order, this depends on what you might think of as the angle, but you don't necessarily know what angles are in Riemannian geometry, and so what I'll do is actually just put out, put some constant growing linearly that depends on the two rays, and then to the next order what happens is you get something like this, and I put the 6 in because that's the classic normalization, up to the order of something to the fourth power in S, or the square of the second power in this separation, this lowest order separation constant. Of course, as gamma approaches rho, C gamma rho goes to zero as you would expect, but this is the expansion you would expect. This quantity, which sort of measures the deviation from the way straight lines would separate, is the flag curvature. It's called the flag curvature because it depends not just on the two plane in which the infinitesimally the two geodesics lie, but it depends on which order you do this, and it depends on the actual specific rho and gamma, that is K gamma rho is not necessarily K rho gamma even, and it depends on the idea is you fix a row and then you see how gamma varies. This is in some sense what you might think of as the first geometric piece of information that comes out of a given Riemannian structure. This is what's known as the flag curvature. Sorry, something, oh yeah, it's probably am I making too much noise, maybe they can adjust the volume or something, I'll try to keep. It is in some sense the fundamental invariant in this metric notion of geometry that Riemann established. One thing you can see, for example, an example I always like to give in pictures like this is a physical example, the river example. You imagine you have for simplicity, let's make it a, let's make it a, let's make it a straight river, and you have current flowing faster in the middle and slower at the ends, at the banks, and the set of displacements that you can reach in unit time is not of course centered on the point where you are, but it's shifted over by the current. My drawing is not so great, but you can see you have something strictly convex towards the origin at each point, but it's not centered on the origin. If you look at the geodesics leaving a point say in the center, as they go downstream what happens is that the geodesics separate faster than, faster than you would expect. That's K negative, but if you go upstream the geodesics tend to do something like this, and the geodesics tend to come together. So even in the two dimensional case, you can see the flag curvature depends on the geodesics, not just on the two plane that it's, and the direction you're going in. Now in, of course Riemann looked at this situation and he figured out what the case, if the, if F is actually, if F squared is actually quadratic, which is the case we now call Riemannian geometry, is quadratic form Riemannian geometry, then K constant exists, is classified. We all know what those answers are, the so-called space forms. And Riemann in fact himself gave the, gave a formula for what the metric looks like after change of coordinates, if the, if the, if the flag curvature is identically constant, but in the Riemannian case, but in the non-Riemannian case that's still, that's still very much an open problem, exactly what the, what the K equals constant, constant things look like. In fact, it's not obvious from the definition that I gave that K is actually a differential invariant, that it can be computed by some kind of, by some kind of doing differentiation. I'll say a little bit about how that's done in a minute, but it is true. And, and so there's a natural question is what, what kind of equation is K constant in the general case? Hilbert gave examples that turned out so-called the Hilbert projective metrics. Gave examples on convex domains that using cross ratio formulas that I won't write down. The, gave examples of these, these kinds of metric structures with the property that they have K is identically minus one that are not Riemannian. That is their, their, the, the unit sphere at each point is not, is not a, is not a quadratic, is not an ellipse centered at the origin. In 1919, Fensler, Paul Fensler made a, started a kind of geometric study of this, of, of these kinds of structures. And that's why we call them Fensler geometry today. And there's been a, a, a string of, a string of, of people involved in, in 1934, Churn, Carton wrote a, wrote a, a book on Fensler geometry. In 1943, Churn applied the method of equivalence to Fensler geometry. And, and, and, and in some sense, okay, applied method of equivalence. And, and in some sense derived, derived the fundamental invariance. And I'm not going to go through that derivation. We don't have anything like the time, but, but I, I, I will say, I will have to call on these, call on these things for some, for some of the ideas. I mean, for some of the results. And, and it's now known, just to give another example beyond, beyond this, it's now known for example that, that even on the two sphere, even on the space of, of Fensler metrics with K identically one is infinite dimensional, mod diffie morphisms. Even, even if you, on the two sphere, even if you fix, even if you require that the geodesics be the great circles, that is, that is, even if you assume that the curvature is one and it's projectively flat, there's still a ten-parameter family of, of, of distinct Fensler metrics with that property. So, even plus projectively flat. Ten-parameter family. Oh, no, no, this is something I should say I did this. Family of, of examples. The ten-parameters are actually five, it turns out. It's actually a complex manifold. The space of, the space of solutions turns out to be a complex manifold. And, and it's, and it's the, one of the things I'm going to tell you about is, is how complex geometry comes into this picture. Because it's not at all obvious that there's any complex geometry involved. So, yeah. Now, now, yeah, so. So, locally, locally there's an infinite dimensional family of solutions as far as, even, projectively flat ones, that's right, yeah, right. Still infinite dimensional, projectively flat. K equals one. Right. Yeah. Locally. It's definitely a global theorem. I mean, the deformations use the fact that you, you find a rational curve somewhere and you look at it's, the deformations of, rational curve in a holomorphic manifold somewhere and you find it's a deformation space. Compute its normal bundle. Yes. On the other hand, in 1988, Ackbarzade showed that if M2 is connected to a complex compact and K is identically minus one, then, then F is, F is Riemannian. I mean, if you think about it, what, you know, what's happening here is a, is a little bit, you know, just to give you a sense of what's, what's odd, is that even for the, even for the surface case, if you look at K, K is defined, is being defined basically in terms of this hyper surface in the, in the, in the tangent bundle. So it's, so actually the variable is a three manifold, not a two manifold. And, and K is a function on that, on that object. It depends on four derivatives of that hyper surface sitting in, sitting in the, sitting in the tangent bundle. And it's not an elliptic equation. It's neither elliptic nor hyperbolic. It's quite complicated. And you use the non, you use the, use the degeneracy of it to prove, you know, it's a sort, sort of certain conservation laws that show up to prove this statement in the K equals minus one case. But nothing like that works in the K equals plus one case. Gives you some sense of how, how, you know, odd it is from the point of view of PDE. You might think, well, how could it be fourth order because in the Riemannian case it's second order. And the reason is, it turns out that, that if you impose the Riemannian condition, the higher derivative, the coefficients of the higher derivatives drop out. And so you actually only, it, it reduces to a second order equation. Whereas it's naturally a fourth order equation, surprisingly. But it's the fact that the fourth order is connected to the fact that the fissile jump is used wise to a Riemannian junction of tangent bundles. I don't know how to answer that. But it is important that you're, you just raised a really important point that there is a Riemannian structure on, on sigma, in fact, on the, on the unit tangent bundle. And that's what I, that's what I wanted to, to point out. Yes. So the, to, in more detail. We started out with sigma sitting inside the tangent bundle of m. And I'm going to let m be m plus one dimensional for, I mean, now go back to the general. And I'm going to assume m is oriented to, to, for simplicity. There's a natural inclusion of sigma though. The, the Legendre transformation that maps sigma into the cotangent bundle. The point being that if you have a point, here's the, here's the unit sphere bundle. Here's u, here's a point u in, in sigma. There is a, there is a, a one form tau u in the cotangent bundle that satisfies that tau of u applied to u as one and tau is less than or equal to one on sigma x. That is, it's the, it's the one form that, that whose level set equals one is exactly the tangent plane to the, to the endicatrix at that point. That naturally maps tau into the cotangent bundle and it's an embedding. And so the, the net, the canonical one form on the cotangent bundle pulls back to sigma. So, so sigma has a, has a canonical one form on it. Sigma has a, has a contact structure. Alpha. And of course the classic picture is that, is that that contact structure gives rise to the, to the geodesic flow. The rave vector field of alpha is the geodesic flow. Say e is geodesic flow. Now, as I, as Jean-Pierre mentioned, there's a, it follows in particular from the work of Cherne, although it's kind of implicit in Carton's analysis that, that although there's no natural Romanian metric induced on m, there is a natural Romanian metric induced on sigma. There is ds squared on sigma. And, and e has unit length with respect to that. Sigma is the union of all the endicatracies. That is, each sigma x is a hypersurface in each tangent plane. You take the union of those. It's a hypersurface in the cotangent, in a tangent bundle. That's right. Exactly. It's the level set of f equals one. And it turns out there is a natural Romanian metric on this. The, the dynamical properties of this geodesic flow are very interesting. Mm-hmm. I'm assuming offensive smooth. That's right. Yeah. If you don't assume smoothness, then, then all kinds of bad things can happen. Right. And I should, you know, for the, for a very nice interpretation of not only this metric, but the flow and geometrically in terms of dynamical systems, you should look at Patrick Flan work on, on the dynamics of e and interpretation of this, of this geometry. But what I want to focus on is, is, here's a consequence of k is identically one. And I'm going to focus on the k is one case. The k is minus one case has a similar development. But, but I'm going to focus on the k equals one case today. There's the following proposition that if k is identically one, then, then, then the metric is invariant under the, under the, under the geodesic flow. This canonical metric is invariant under the geodesic flow. In fact, that's almost a, that's almost equivalent to k equals one. And so you get the following picture, which is, which will turn out to be important. If you look at sigma, of course, mapping down to m by, it's normal projection. The, if this, if this, if the flow is ge, is geodesically simple, that is, if there's a quotient by the flow of e, call that q, the space of geodesics, call that lambda mapping down to the space of geodesics. This is n plus one dimensional, two n plus one dimensional, two n dimensional. That says that in the k equals one case, what happens is that there's a well-defined quotient metric. Yes, bar squared. It's, well, you can ask it locally and globally. If the, if you want, if you just want to look at the local solutions, you can just take a, take a local convex patch. And they're always in this, in the geodesic flow, will be geodesically simple on the convex patch. But this can actually happen for the n sphere over the n plus one sphere. In fact, there are this two-dimensional case. This fact generalizes to the, generalizes to the, to all dimensions that the, that the space of projectively flat k is identically one metric on the, on the n sphere or the n plus one sphere turns out to be a complex manifold in a natural way. In fact, it's identifiable with the space of quadrics in Cp n plus one without real points. Interestingly enough. But so there's, there's both local and global questions. Right now, I mean, we can focus on the local. What it says is that there's a natural metric on the space of geodesics. Now, just from, just from classic, classic calculus of variations, we know that because this guy has, because this guy is the quotient by the rave vector field of, you know, in this contact structure, there's actually also a symplectic structure where omega has the property that when you pull back omega to, to this guy, you get the differential of, of the contact form alpha. So this guy comes equipped with both a metric and a, and a, and a symplectic structure. And a priori, you know, it's not obvious that they have much to do with one another, but it turns out, and here's the, the beautiful fact that in the case, identically one case, when case identically one q ds squared omega is a scalar manifold. In fact, in fact, my geodesics are always oriented because I took, yes, but I only divided out by the geodesic flow, right? Oh, oh, yeah. Oh, sorry. Okay. My definition of geodesic is oriented geodesic. My curves are all oriented. Right. Right. No, I mean, put in oriented if you, everywhere if you want. Okay. Right. So it turns out that this is in fact a scalar manifold. Very, very, well, it was a big surprise to me. And in particular, it shows that the, that the natural, so of course there's a natural Levy-Chavita connection on it. And what it says is that the holonomia, the holonomia of Nabla is u n in general. That's the largest it could be if the, if the metric is a, is scalar. There's the special case, in the special case where, where, where you actually, where you start out with the n plus one sphere with the standard metric, the holonomia actually drops. Special case, n plus one f is s n plus one canonical. Then q actually turns out to be isometric to s o n plus two mod s o two cross s o n. So the holonomia is actually s o two cross s o n. So, which is a proper subgroup of course of u n. But, and while in the general case it does not reduce, something of this survives. In the sense that there is, it turns out a natural circle action that, and while there's not quite a natural s o n structure, there is a, there is a, there is a sort of a remnant on it. And I need to describe that. The point is this, that over an m, oh yes, good. A point of q is of course a curve in m, an oriented curve in m. And if you, and a point in m though, a point in m when you look at what it corresponds to over in q is you lift it up and push it down, it's a Lagrangian sub manifold. It's an n dimensional Lagrangian sub manifold. Let's call this, let's call this c x, which is lambda of pi inverse of x for x in here. So, there's a, in fact there's a one parameter family of, a one parameter family of Lagrangian sub manifolds sitting in here. And the, the next thing that turns out to be true, it's a very, very beautiful geometry, is that if you look at the, if you look at the tangent spaces of these, of these guys, if you look at the tangent space at this guy, which I failed to name as q, if you look at the tangent space at q to, to these, these Lagrangian sub manifolds, they all turn out to be e to the i theta times the tangent space of one of them. In other words, what happens as you, as, as you move along the, as you move along the fiber is you have this one parameter family of special, of Lagrangian sub manifolds that are all, that are all, they're all intersecting in q, they intersect uniquely at q and they just, and they just rotate like that. So actually it turns out that the, that there's a natural, there's a reduction of the structure group here from u n to a sub group, which is, which is basically s one times s o n. So here's u n and sitting in there, there's s one times o n. There's a natural reduction of structure from, from, from this guy down to this guy and unfortunately it happens that when that this, this reduced structure is not torsion free. Remember, Kailer is the same as a u n structure that's torsion free. The Nabla reduces, reduced to this is not torsion free. Well, but of course u n is sitting here in glnc. This s one times o n has another enlargement, s one times glnr. It is not torsion free except in the flat case, sorry, except in this, this very flat case, in this symmetric space case, Riemannian. So there's a reduction to this which is not torsion free, but then because of course this grouping contains this one which is sitting inside glnc. The miracle is that, that this, this enlarged structure Nabla actually turns out to be torsion free. Nabla is torsion free here, is compatible with this. Oh, sorry, I shouldn't say that. So, no, that's right. Nabla is compatible with this structure. Actually, no, I have to be careful. When you reduce, you can split this into a, into, the, the connection splits into, sorry I should say it, the connection splits into a reduced connection and a, and a torsion tensor which only vanishes in that case. Well, Nabla not is compatible with this structure. And in fact, its holonomy is generally equal to this, equal to s1 times glnr in general. When I first noticed this, and it's kind of the result of a calculation, I was astonished because there'd been a classification of the possible torsion free holonomies in, in, in affine geometry that was completed by, that was started by Berger with up to a finite number of exceptions and, and then carried, carried further in the 1980s and 1990s. And the published list still did not have this one on there. And, but turns out that this actually gives, gives examples. And if you go back and look at the analysis that was done, you can see where the gap occurred. This, so these actually turn up to exist. And the beautiful thing is that there's a nice converse, which is that if you start with a two-in dimensional manifold, it's a complex manifold that has a, has a connection on it, whose torsion is, is this, you know, complex multiples of the identity times this glnr. If you start, start with such a guy, there's a natural, the curvature form associated to the S1, that's a form of type 1, 1 you can prove. And if it's positive, then you can retrace the steps and locally anyway recover a fensinometric with K is 1. So the, so the fensinometrics with K is 1 actually turn out to be intimately tied to the holinomi problem in, in affine geometry. And I wanted to give this example because it shows that there's a, there's actually a lot of geometry in the, in calculus variations problems. That, that, you know, if you pursue the, the, the general structure and look at what it tells you, there's, there's, it turns out a lot of the tools of differential geometry come, come into play to help us understand things like not just this, but calibrations, when things are actually minimizing as opposed to merely locally minimizing and so on. And I think that we are really, while we've made some progress on this, there's still a lot to do. There's still a lot, a lot more to go and a lot more to discover in the, the, the differential geometric aspects of the calculus of variations. Now I'm almost out of time and I was supposed to have allowed more time for questions. I just want to say that the, that the second part of the talk was to have been about what happens if you look in higher dimensions. That is, instead of, instead of specifying in each direction a size, suppose you do as Carton did in, in 1933, he wrote a little book called On Metric Spaces Based on the Notion of Area, where what you do is start with, in, in his particular case on a three-manifold, he started with a geometry that specifies the volume of every two-plane and wanted to know what, what kind of geometry can you get out of that? And it turns out there's a very beautiful geometric story that you can, that you can tell. Something that Philip Griffiths and I have been thinking about for, for the last few years is generalizations of this to higher dimensions, where it turns out that unlike in Carton's case, where there was a unique canonical form that you could attach, the Poincare Carton form, there's now in, when you go to higher co-dimension, that is you look at surfaces in four-space or surfaces in six-space or things like that, or five, you know, four folds in six-space, it turns out that the notion of canonical form is much more subtle and, and depends on higher derivatives, but there's still a geometry there that I think is not well understood and, and there's a lot, a lot more for us to learn about its relationship, a lot more for us to learn by analyzing it, using the tools of differential geometry. So that's where I want to stop. So thank you for. Questions? There's a question for, so you do statement here, it's a local statement, the k equal one, the k equals one says that this is true, that's right. For k equal minus one. Well, in, there's actually, yeah, there's actually, you have to replace this natural Riemannian metric with a pseudo-Riemannian metric. And in the pseudo-Riemannian case, what happens is that the, what happens is that the, the thing that pushes down is a, is a, is a somewhat different structure, but you can still pursue it. There's still a differential geometry and it leads to a kind of a, a different holonomie that's not. So then the statement that the derivative of the, of the, of that modified thing is still zero. That's correct, right? That's correct. To follow on, so Michael, is there a natural square root of insta geometry? Gee, I don't know. I mean, in some sense, the fact that a complex structure shows up here, whereas there's no, there's not even a natural Riemannian structure here, but an actual Kehler structure shows up here, says something about, says something about, you know, if Kehler geometry is some kind of square root of, of Riemannian geometry or geometry in general, this is a vindication of that point of view. Yeah? Is there any complex structure that appears here in any relationship with Twister, Twister theory? Um, not in the, not in the sense that I, that I understand Twister theory normally. It's actually, although there is actually a connection, it turns out with, something I haven't said is, for example, in the n equals 1 case, it turns out that these things, when you look at the transform over here, they, they turn out to actually be Zoll metrics. And, and the Zoll metrics that, you know, LeBron and Mason have shown that you can, that you can understand Zoll metrics on the two sphere in terms of a kind of a Twister construction looking at the, at looking at holomorphic disks whose boundary lie on a certain real sub manifold of complex projective two space. You look at the moduli of holomorphic disks that, that lie in that boundary. And, and so they've, they have a very nice improvement of our understanding of Zoll metrics and constructions of Zoll metrics through that kind of, they call it Twister, they call it a Twister construction of Zoll metrics. But it's not, it's not, you know, it's, it's Twister more in the sense of using complex geometry to solve a, solve a global Ramonian geometry thing than it is in terms of the, the classic Twister construction of, of Penrose.