 You know that we will record the session, so if you remain or assume you have given permission to record the session, so that it will be available for people afterwards. So thank you, Walter, you can record the session, yes. Yes, please, record the talk. Okay, so it is really great pleasure for me to introduce Kadim Wal. So Kadim Wal is a, really is a, the, is a, is a star of the system of the Abdul Salam International Center for Physical Physics. Kadim was an ICTP diploma program, which is our starting program, let's say pre-PhD program at ICTP, and he then stayed here to do his PhD under the ICTP CISA program here in Trieste. And he then did a postdoc for two years at Bochum University in Germany. And he was a postdoc for one year in Chicago, and he's currently a ten-attack, he has a ten-attack position at the INPA Institute of Peer and Applied Mathematics in Rio de Janeiro, Brazil. So I'm very, very pleased to have him here. This mini-course will consist of four sessions, two this week and two next week. The first two are particularly introductory on geodesic flows. And then next week you will get into a little bit more advanced, and on the last session he will talk about some two recent preprints of his. So please feel free to ask questions either through the chat, or if you realize it might be difficult to see the chat, you can just unmute yourself and ask questions live. If you have any questions, then it's likely that other people have the same questions, so please do not feel shy and do not hesitate to ask questions. Okay, so thank you very much. So, Hadin, the floor is yours. Thank you, Stefano, for the introduction. So thank you for this introduction, and I'm very happy to give this series of lectures. Yeah, Stefano said, so I'm going to give these four lectures, which are going to build up towards our two preprints that I have. One is already accepted, and another one you can find it in the archive, which is mainly a joint work with Gerhard Knipper, which is in Bochum and Von Klimanaga in Houston. But for today's lecture, today's lecture will be more about the geometry, like background geometry that we need to understand these results. And so today's lecture, so I will try to lecture one, so geometry of Von Klimanaga. Today might be very, I mean, elementary for some of you who are very familiar with the topic, but as I know that there are many students here, so maybe this is not something that people are used to, like Von Klimanaga point. So what I will do today is just to try to define it and see what are the properties that we derive from this condition of no Von Klimanaga points. So the first part today will be a recall about Riemann's metric. So I'm supposing that I'm given M is a Riemann's manifold. So I would just say it's an n dimensional manifold. As I was thought, maybe not everybody, I'm sorry if this is Gerhard Knipper, but maybe not everybody is used to thinking about manifold. But this, I want us to focus in three examples of manifolds that we will be looking all over today. One is Rn, which we know, so Rn. And then we have also this sphere Sn, which is a subset of Rn plus one. So that, like the unit here, okay, and it goes one n plus one. And there is another example, which is more interesting to us, which is a hyperbolic model X1, Xn plus one. That is that Xn plus one is positive, and that Xn plus one square equals one. So I will not give like this explicit definition of a manifold, but think of these three examples. They are very interesting in what I'm going to define later about the curvature and, yeah. So, the definition of a Riemannian metric, P in M, is a choice of, is a family of scalar products. So GP, which is defined from GP M times GP M, and that it will soon be the same that if I take, if X, Y are two microphones, function f of t minus GP X, P, Y, P is differentiable, okay? So, yeah, this is the definition of the metric in Rn. In this case, we know the usually euclidean metric that is given by DW is just a usual unit product. We know the IWI and in Xn also, you can see Xn as a subset of In plus one, and you also have the induced metric, okay? The induced unit product, the induced metric. And in the hyperbolic plan, in the hyperbolic model, sorry, it's not a plan. So, if you can consider this metric, which is given by HX, so H would be Gp of Vw is just the sum of the IWI for I plus one and minus Vn plus one, Wn plus one, okay? Because you see this as a subset of In plus one, okay? So, the vector V has n plus one components. So, those are three metrics that we'll be looking at. And next is, I want to talk about covariant derivatives, covariant derivatives. So, usually in, if I have a, I mean, before I go, maybe I will just say few notations before. So, this guy, gamma m, gamma gm, is the space of vector field on m, okay? So, denotes the space of vector field. My vector field, I mean, is the small vector field, okay? The space of, can you see it here? Yes. The space of the whole vector field. The vector field is just X, which takes from m to the tangent bound. So, this is the tangent bound. So, it's just like a section where there's a choice, a smooth choice of tangent vectors, okay? So, given a vector field in Rn, so given Y for vector in Rn, a vector field and a vector V in Rn, you might ask the change of Y along V, like the derivative of Y along V. So, that's, so the derivative of Y along V is, so this is how we denoted this. So, V of Y, point P, this is a limit when T goes to zero of Y, P plus Tv minus Y, P over T. So, this is just the usual derivative we know, but this is the directional derivatives, okay? So, see that this has a function that takes two vector fields and gives another vector field. This is a vector field. So, this is a vector. So, given two vector fields, so you can see that this operator, Tm times gamma Tm to gamma Tm, which takes X, Y gives X, Y. This is a well-defined operator. Okay, so if you want to, it's a one-two tensor, like, so, yeah. I mean, you can see that for any F smooth function on M, you have the gradient, the directional derivative on F plus Y is F times the directional derivative of Y and that it is also satisfied of X, F of Y. You can take this, I mean, for the usual directional derivative in R, N, and like a tensor or an application like this that satisfies these properties is called a governing derivative. So, this guy is a governing derivative, it satisfies these properties. So, this is X, F, Y plus F, X, Y. So, this is a two, a one-two tensor that satisfies these two properties is called a governing derivative. If you give a governing derivative, you can define the usual, I mean, curvature tensor. Yeah, so, for those of you in the tensor, so, that satisfies these two properties, plane star and n-star. So, another definition that I want to play is the curvature tensor. Curvature tensor is, so, is a one-two tensor, so, defined from gamma Tm to Tm. So, this takes, you take two vectors, R, X, Y, Z. This is a new bracket between two vectors. So, you can see this setting that we're talking about, it's just given by Y minus Y. And from this curvature tensor, now you define the curvature of your model. To define a curvature in high dimension, what we define basically is like section of curvature, to define curvature of planes. In surfaces, it's just one, you don't do sectional, it's just one section, one plane. So, given sigma as a plane, this is the time and space of M and P. And Vw, Vw, two vectors that generates sigma, so, basis, so, this is the basis of sigma. So, now you define the curvature of this plane to be g bar. So, this is mainly called for, mainly called for Vw, Vw, Vw, Gw, Vw, minus Gw, Vw. So, this is the definition of the, of the curvature, of the sectional curvature using the curvature tensor. So, an exercise that you could check is that the three models that I was writing, an exercise, an exercise that. Excuse me, what is G here? Yes. In the definition of curvature, what is G here? G is a metric. Okay. G is a metric. So, G is a metric, G is a metric, yes. And this guy gives you, this is a vector, this is a vector. You calculate the inner product between these two vectors. I should have, I mean, P of P, okay. You can say this is a metric of P. I'm sorry. You missed the W in the definition of, definition of curvature. So, you can take that. So, this is a usual inner product that I have before, the curvature is zero, for any sectional curvature is zero. You can kind of see it. I mean, to do this curvature calculation, you just do it in coordinates. You can see that the two coordinates so that this guy will vanish and also these vectors they commute like the two. Covenant derivative they come in so you can, yeah. And also you can see that, I mean, this exercise before, you have curvature one. And that is before is a metric G, that is G, curvature is minus one. So, these are three important models that kind of gives you the different curvatures, you have zero curvature, curvature one and curvature minus one. So, you can try to do it because for this space which is defined, you can, you can see the charts that define this manifold and then you have to charge it and do this computation. All right. So, another section that I want to move is about geodevics and dark materials now. So, this was just about defining a metric, what is a metric and define the curvature. So now I want to talk about geodevics and jacobic fields. This notation so given. I mean, again, this is something that many people know that I will just because this is a lecture and complete and so even out from differentiable curve. And so I did not buy them at sea. The space of back to the long, long, okay. You can define the, you know, the, the direction. See if you have a vector field that you define along the curve. So you can change the change of this by the field along this curve. So that's the derivative of the vector field. So you can define this one here. DB of the DT, which is this alpha dot. So this is a program derivative along alpha and you see that B is parallel transport along alpha if this one is zero. Okay, so this is parallel to the position D is parallel along alpha. This one is, it's just either the vector that will change. This parallel transport is long. And now geodevics is just now a curve moves. The tangent vector is parallel along the definition. So when you write D like this, in D, there's alpha in it, like here you specify that this D is the directional derivative for fixed problem. So this understand this as alpha dot, alpha dot. So this is like the user definition of two deserts. And again, again, so you can check existence. So on this page that I wrote, and here the geodevics are the straight lines. And these are the geodes are given by the great source. And in this space, the geodesics are obtained by taking a plan in RL plus one and you intersect these action. I have a question if possible. So, is it true that if the curve is a satisfy this equation that it's geodesic or only the opposite is true? I mean, I think not I mean it can be possible that something is a singular curve for in terms for this equation but still it's not geodesic. I mean a point to like a curve for which this operator D over DT on alpha prime is equal to zero is not like it can be not this is like it's not necessarily geodesic like in the sense that it minimizes the distance between two points. So, no, I think if you are in this, in my understanding, this is equivalent to this. Oh, this is equivalent. Yeah, yeah, this is equivalent to it. Okay. Okay. Yeah, you can, I mean you can rewrite this using coordinates you can rewrite this and see the equation that you usually have from Christoffel symbols. Those details because that's not what I want to say. So there is this remark, which I think is important is that given. So there is a unique geodesic. So there exists a unique geodesic. See, if I'm smaller than small m is see, so I will see a CB, CB5 at zero. So there's a unique. I mean, this just follows from the uniqueness of solutions or code. Okay. And this allows to define what we know as like the exponential map. Okay. So, that says that even being to be an exponential. The time from the team to be an M by this exponential. So, what you do is you have your page and space here, and you have your manifold that looks like this is an M is the end. You give me a vector tpm. I can find this guy. This is CB because this is zero. And the exponential map just take the vector one. Yeah, the points are. So, so till now this is just generally called from, you know, the management space and stuff. So, yeah. So now we want to be fine. What the job of it feels given. So she's fine. So I will just say GCN, you know, gamma C at the end. So this is just saying that it's a vector field along. Along C. It's a job of it. If it satisfies this. So these were over these two. Again, this derivative is this type to see is all see J. Class. So, see that J. This is a simple equation that the vector should satisfy to be a job of it. You can rewrite it in simple setting like in dimension. If you're looking for job review that are transverse to your design, if not in this direction, you can see this. You can write it again. That's K. Okay. Okay. So this is a simple, take an order. And, you know, I was taking the time to define curvature dancer and exponential map to just show you a nice way to derive from the exponential. So, exercise. So, let. And you can consider this. As T. That's defined as exponential P. So it takes a surface to check that this back to the team. Yes. He said, yeah, I like very much this. I mean, actually, whenever you have a geodesic variation, not only the exponential map, if you have alpha, so therefore each S alpha is alpha s is a geodesic. So this is the way to say that. So, see that. So, yeah. Yeah. This is that you get. Yeah. The exponential of the exponential map. So, this is the way to say that. So, see that this is satisfied zero equals zero and. Yeah. Zero. Because that. So this is a, this gives a job that's that means this. I mean, if you have the second order equation, you just need boundary condition to have units like we need to know you are zero and he prime zero. So this specific job of appeals, it goes with this foundation. I'm going to finish. Two points. So, see, is a geodesic. See, at the one and see to get a conjugate. They exist. See, in my C. Yeah. Yeah. Yeah. It was zero, but James. This is what you have two points. And along this. Zero here and start going and not start the game. So, if you have this, these two points are going to get you can see that from this property here from this example, that's why I have this example. Because it, it kind of tells you that. No point you get points. It's kind of related to, you know, the fact that G of this month is not single. Like the differential of the exponential month is not singular. So let me write here. So even question. The definition of country points does not actually depend on the geodesic. The geodesic is fixed once you fix two points on the one. Yes. Yes. So it does not need to have a. Yeah, yeah, that's true. Yes. Yes, but even two points. Yes. The condition is not there. Yes. Yes. So two points are going to go to the existing geodesic. That means that property. Yeah, actually, yeah. Yeah, so we'll talk about that there. So, so you can define this step. Yeah. But there's a nice lemma that lets you get into the exponential map. That's it. So given v in GPM and Cv, so Cv is a geodesic that starts at v. The m is a geodesic. You have the dimension of the terminal of this p at v is equal to the dimension of this dimension of J, 0, 1, Cv. The proof of this is that it's very difficult. You consider this map, the set that goes to the set of j-4b-fields is back at 0, that's J at 0 equals 0 to Tc0m. What does it do? It just takes the j-4b-fields and map to Vj over Vg at 0. J to the power 0. And you can see that this is a linear isomorphism. And you see that if you calculate this, the differential at v over Vg at 0, what you get is J1. And this implies that in particular that p of this space, J, 0, 1, Cv equals the kernel of the exponential. So in particular, it gives this being an isomorphism to this guy and this guy. They have the same dimension. So no conjugate points, it's kind of related to this exponential map, not singular. So this in particular, it ruled out the sphere that I got before. So no point to get points, no point to get points implies that it's not singular. So the point I guess I was trying to make is that the existence of conjugate points depends purely on the method. Depends on the method. Yes, the condition of the method. Condition of the method, yes. So the exponential map is not singular from this property, because if this has a, this has dimension, so this one will have dimension, and then this is, this will be singular, okay? I just got this equation. And so in particular, so another remark is that SN doesn't have a romantic result because in SN, you can, you can see that result, yes, result. You can see that in SN, you have these great circles that are so, that are two deserts. So you go this way, you go that way, the exponential map is not, it's singular, right? The exponential map is singular, so this sphere is not, it's not romantic that we are looking at, but there is this nice lemma that tells you that now if there is no positive coverage, it implies that it involves no conjugate points, has no conjugate points, and also the hyperbolic model that I said, and so it has no conjugate points. And SN has, so I'm looking for money for that, more like RN and SN. So how to prove this is actually quite simple. So if you have non-positive coverage here, so you take any jacobic field, let j be a jacobic field. So along gamma c, dm be a jacobic field, be a jacobic field. So let's suppose that you have j at t1 equals j at t2 equals 0. So I want to prove that j is identically 0 under its assumption, non-positive coverage. So you consider this function at t to be the norm, which is just in the case of jt. So this function is 0 at t1 and t2. But see that f prime t, f prime t is twice d over dt, dt, dt. And you can now do f double prime, f double prime t, which is twice d square dt2 dt over dt plus, so I should just open here bracket. And this is now g over dt, dt square. So I'm just trying to, I mean, yeah. So this time I was just saying that no point get points is a generalization of non-positive coverage. So and then you see here, if you use the, if you use the fact that j is a jacobic field, what we'll have is f double prime t is 2 times, so this guy, this guy is a curvature tensor. If you remember the definition of jacobic field, so this is minus r of d prime j, d prime, j. Yeah, because, okay, because j is a jacobic field. So j double prime is minus this quantity here. And this quantity is, if you use your, is not, this quantity is not negative because of this condition. So this quantity is positive, and this guy is positive, so f double prime is positive. So you have this, this function is a convex function, and it is 0 at 2 points of its identity. So which implies that j is 0, so non-positive coverage implies no point get points. So yeah, so there is another, so I want to end up with this, resist, I will stop here, which is due to, I don't know, which tells you that if m has no point get points, so no point get points, I mean, it's part of the theorem, the theorem is no point get points, imply that the exponential map gpm to m is a covering map, covering map, I mean, so this is a topological property that says that every point has a neighborhood, which put back is a union of sets where each is homomorphic to this one copy, okay, yeah, covering map. And yeah, so this is a very nice theorem that, that you derive just from no point get points. So in particular, it is again a way to rule out these fields. So the corollary for, there are any assumptions on the manifold and just no point get points, means many things are going to get points. No compactness. No, no, no, no, just a many things are going to get points. This is a covering map, so in particular, the universal cover is a copy of rm in particular. So this is the way you see that simply connected domains like, simply connected manifold like in here, they're going to have the metric without any get points. So the universal cover to rm because this is a copy of gpm from this guy and so in particular from this you can also derive that, you know, you're looking for manifold like torus or hydrogen surfaces or, you know, this manifold, the universal cover should be, should be a copy of rm. I have a question, please. So in general, if we have rm and manifold, then on a point, there is always a positive radius where the restriction of the exponential map becomes diffeomorphism on the ball of this radius, the tangent space, right? And so saying that the exponential map is a covering map that makes it whole diffeomorphism, right, onto its image. It's roughly diffeomorphism, see that? I mean, no. Give an example. Yeah, I mean, you can just think of something on the torus, which is here, if you have the torus, you do the tangent space with r2. Yes, yes, yes, yes. Okay, let's see. So this is a very nice thing that we derive from point to get point, no point to get point. And yeah, and let me just say a few words about how to do this, because to study what we do is to study the geodetic flow. And to study the geodetic flow, it's more practical to work in the universe of cover, to look at the geodets in the universe of cover. And then we need to know what the universe of cover is and in this case, it is a copy of rm. And yeah, so proof of that theorem. So for the proof, so there is this I mean, lemma, maybe I write it as a lemma, that if you have nm2 completely minding the manifold, and you have that p is a local isomagnum, and p is a column m. So yeah, this part, this had a material that follows easily from this, let me just maybe say how to derive from this lemma. So you know that this guy is locally a different organism, it's not single. So you can put a metric here in this space, so that it is a metric, like there's no metric here. So define a metric, you just pull back this metric by the exponential. Define a metric gpm by pulling back, pulling back a metric by pulling back g, which is an m under the exponential m, because it's not singular. So you can pull it back. This is the point where you know when to get points, because this guy is not singular. You can do this, you can pull back a metric in here and define it in a metric. So define a metric, let's call it g tilde on this tangent space. So in this way, you just define it so that this guy, it means g is an isomagnum. It's local isomagnum. And if you have local isomagnum, there's this guy that tells you that. Yeah, so you define a metric here, so that is an isomagnum, because you already know that it's local dpo, and to be able to pull back the metric, you need the exponential to be non-singular. That's what you get from no conjugate points. Okay, and yeah, so now let's just sketch the proof of this before we start. The proof of the level. So this guy being an isomagnum, you can see that it commutes with the exponential map. So the first step is that it commutes with the exponential map. So this is what I mean is that if I have q in the set in n, and p, which is p, q, so you have exponential m, p, p to p, p at q is equal to p in four, please, exponential. So I just put m here to say that exponential, so I have p from n to n. So this is the exponential map at q in n. I mean, this is just, you can just see that if this is the case n, and this is the case m, you have this geodesic here, q is here. This is cd. We mark geodesic with geodesic because it's an isomagnum, so and if you also mark this with geodesic. So the vector, this is v, this is dp to v, and if you exponentiate this vector, you get this curve. It's the same as taking the exponential and then multiplying. So this is p. That's true for, and then because the isomagnum is geodesic, geodesic, geodesic, and then there is another part, is that p, so I mean, this we know that p from v q epsilon to v p epsilon, this guy is imaginative, and it's not difficult to see this other property that if I look at p minus one, p epsilon, this is the region of union of q in p minus one, q at d, q at z. The way to see this, you can just see it in the picture here later. So think of having here the tangent space at n, q n, and think of taking away the balls, so this is your independent space, and this is epsilon, and here I can exponentiate the distance, and yeah, so I exponentiate this ball, I get the ball in the minus one, so this is q, and this is again epsilon, this is geodesic, so this is dp epsilon, so here you do exponential, and q, and here you can see that if you do p, p is an isomagnum, so it marks the optimum ball to an optimum ball, so this is your p, and your p in the tangent space, okay, so this is epsilon, and you can exponentiate here, and what you have is again another, what you have is another epsilon ball here at p, and this is d, and this is the exponential mark, and p. You can see that this diagram comes, and yeah, so you can actually easily from this diagram you can write the proof of this of the second step, and these are the two steps that you need to do that they say the covering map, these are the two steps that I need to do that this is the covering map, and yeah, so basically what you have to get from today's lecture is that just without going to get points, like many points without going to get points that you cover is our end, I think that's where we're going to go today, I will think more, but what we will do in the next lecture, we will push that by, like, starting the two days is not in the universal cover, and from there we will define a measure that is invariant, and we will put that this measure realising, but we will see how it goes, but next lecture we will, we will study more properties, more geometric properties now of these derivatives in the universal cover, because now you know that the universal cover is like our end, that helps a lot, and what we are going to do basically is to try to reproduce this, if you're familiar with this negative curvature, this point carry model, like I have my universal cover is our end, I can do a compactification, and stuff like that, and to do that you need to define an ideal boundary, and the mission of the ideal boundary in this no point you get points is not so trivial, it requires something which we call a divergence property that I will define here, so I will stop here for today, I'm going on with that, thank you. Thank you very much. Are there any questions? I would say there's any question in the chat, so there's a question about where to submit the access sign for us. So it's due tomorrow. Are you willing to look at the access sign for us? Yeah, I mean, if you think about it, they can say it's three years, so yeah. Can you write your email? Yes, I'll write my email here, and you can turn the egg back on. Is my name at IMPA? Yeah, maybe you can write it in the chat. Okay, so are there any other questions? Any other questions? So do you have an example of a local commitment which is not a covering map? At least the way this, so of an example, but yes, so at least, I mean, this group doesn't work if it's not an resume, because you would not have that this invitation is an exponential map, but I don't know for an example, it would be good if one time that was an example, but I don't know if it's an example. Okay, so thank you very much everybody, so the next lecture would be on Thursday at the same time as this time. Okay, thank you. Thank you. Bye bye.