 I'm thinking maybe the zoom, is it okay if I turn off my camera? That might be the problem because I remember when I was teaching I would always turn off the camera because the computer would overheat. You can do whatever you're okay. So I will turn off the camera hoping that that problem doesn't happen again. All right, okay. So we were talking about parallel transport, right? Okay, so we have a Riemannian manifold, which means a manifold with a metric, which is a positive definite quadratic form on every tangent space of m and it varies in a sinfinity way with the point of m, right? Okay, so on our Riemannian manifold then we give ourselves a sinfinity vector bundle and a connection on it. And then how do we do parallel transport? We also need a curve. So we give ourselves a smooth curve. Okay, now the apple pencil isn't working. I need to reconnect. Okay, so on the smooth curve, gamma, which goes from the interval zero one into our manifold r sinfinity manifold m, right? Okay, so how do we do parallel transport? We can pull back. So the pullback gamma upper star of e is a sinfinity vector bundle on zero one with fiber e sub gamma of t at any t belonging to zero one. Okay, and the connection nabla defines a connection on this pullback vector bundle, which I will denote by gamma upper star of nabla. This is a slight abuse of notation. So what am I doing? I want to go from gamma upper star of e to gamma upper star of e tensored with a tangent bundle of the interval zero one. That's a cotangent bundle of the interval zero one. That's by definition a connection on gamma upper star of e. But that's not what I'm going to get, right? So normally when I just pull back nabla, I'm not going to go to the tangent bundle of zero one. I'm going to go to the pull back of the cotangent bundle of m, right? But then this guy has got a projection onto the cotangent bundle of zero one. And this composition here is gamma upper star of nabla. And in local coordinates, you can write down what it is. So you can write gamma of t as x1 of t xn of t. And then you have the velocity vector gamma dot of t, which is the derivatives x1 dot of t up to xn dot of t, right? And then for all sections e of e, actually, you just need to do local sections again, you can define the value. So nabla gamma dot t of e. Well, what is this going to be? This is going to be nabla, the sum of x dot i of t d dx i sum over i applied to the vector e, which is also the sum for i going from one to n of x dot i of t nabla sub d dx i of e. Okay, so you can differentiate. So what are we doing here? We're differentiating this section, little e of big e, in the direction of the curve. So in the direction of the velocity vector of the tangent vector of the curve, right? Okay, so now we have, this is now parallel transport, right? What do we mean by parallel transport? This is a definition and proposition. If little x is gamma of zero, so that's the starting point of our past gamma, little y is gamma of one, right? Then for all sections e, not sections, sorry, just elements of the fiber. So little e is not just in the fiber of e at the point x. This is also, remember, this is the pullback of e fiber at the point zero, right? There exists a smooth, sorry, a unique smooth section little s of gamma upper star of e, such that s of zero is little e. And if I look at the connection gamma upper star nabla of s, then that is actually zero. What does that mean? It means that when I take the derivative in the direction, in the tangent direction to my curve of my section little s, that is zero. So in some sense, so this is the best substitute for having a constant function, right? Here we don't have functions. What we have are sections of a vector bundle, right? And so we want to talk about what it means for a section of a vector bundle to be constant. And now here we're only talking about being constant in a given direction. So this is the direction of our curve. We have a path from one point of our manifold to another point of our manifold. That's the path gamma going from little x to little y, right? And we want to say what it means for a section of e to be constant on that path. And what it means is that this derivative, the connection applied to that section is zero, right? Okay. So what this is saying is the proposition part, the proposition part says that there exists a section which is actually constant, right? And it is unique. Now the definition part, the parallel transport of little e along gamma to little y is what we call p gamma of e, which is by definition the value of our constant section s at one, which belongs to the fiber of our vector bundle at y, which is also equal to the fiber of gamma upper star of e at the point one, okay? And then the, and again, the proposition part, the map, this map defines, this operation defines a map, right? p sub gamma, which goes from the fiber of little e, big e at x to the fiber of e at y. And the claim is that this is a linear isomorphism. Okay. So this is a parallel transport. So this is, this is very nice. If you have a connection, then you can, you can transport one, one fiber of your vector bundle to another fiber of your vector bundle. Okay. Now we use parallel transport to define holonomy, right? So what is holonomy? Again, this is another definition and proposition. If gamma is a loop, which means of course that gamma of zero is equal to gamma of one, which is equal to little x equal to little y, then, then this p gamma, right? Well, we already know that this is a, this is in an element of gl of e x, right? The holonomy group, the holonomy is the image, sorry, the holonomy h o whole x of nabla at x is the image of p gamma. Well, of, of p, really. So, so what, what, what does that mean? It means that your holonomy, so whole, let me write it down here, whole x of nabla is by definition, the set of p gammas, such that gamma is a loop based at little x. Okay. All right. So the holonomy group has got the following properties, right? Number one, this is a lisa group. Whole x of nabla is a lisa group of gl of e x. Okay. Well, what, how do you do that? Well, there are two parts to this. You have to show that it's a group. And then you have to show that it has a manifold structure. Okay. So I'm not really going to get into too much into the manifold structure, right? But I will, I will explain to you how it's a group. So if I, well, if I want to compose, right, composition, which the usual thing, so you, if you have two parts, gamma and delta, right, you can define their composition in the following way. What's it going to be, you're going to take from zero to one half, you're going to take delta. And then from one half to one, you're going to take gamma. Okay. So you concatenate. This is the concatenation. And what's the inverse? The inverse is just gamma of one minus thing. You're just going in the opposite direction, right? And what, what do you have? You can prove easily that P gamma delta is equal to P gamma composed with P delta. And the inverse P gamma inverse is equal to P sub gamma inverse. Okay. All right. And the second, another nice property of this holonomy group is the following if gamma is a path, now from, from between possibly distinct points, right, the points don't necessarily have to be the same from X to Y, then we can, we can describe, we can relate the holonomy at little x to the holonomy at little y, right? So what you get is that the holonomy at little y is the conjugate of the holonomy at little x. So you just conjugate by P gamma. So what do you get here? So you get that up to conjugation. You get that the, that the holonomy depends only on the connected component of, of M. containing X. And here we're thinking of it as a subgroup gln, which is gl of ex gl, let me say glm if ex is isomorphic to RM. Okay. So if M is the dimension of the fiber of your vector bundle, then you can identify ex with RM and gl of ex with gl of RM. And, you know, then the holonomy group is well defined up to conjugation regardless of which point you choose in a given connected component of M, right? So this is something that's actually associated to the manifold and the vector bundle with its connection. Property three is that if M is simply connected, then the holonomy group is actually connected. And the way that works is that any, you know, just roughly speaking, what's the idea of that proof is that any loop can be shrunk to a point, right? So then if you take a homotopy from your loop to the constant loop, which is just a constant function mapping to a point, then you get these homotopies will give you a path in, in the holonomy group. So the as I said, I don't want to get into the details of this, we get a path in whole ex of nabla from, from one, from one holonomy to another holonomy. Okay. To another, sorry, from one parallel transport to another parallel transport. Yeah. So, so, so first of all, the holonomy group depends only on the on the connected component. And then if your manifold is simply connected, your whole homotomy group is actually a connected list of group of GLM, GLM, right? And then we have some, another very nice property of the holonomy, it, it puts a restriction, I mean, on the curvature. Okay. So how does that work? So that's property number four. It, this is the relation with the curvature, curvature R of our connection nabla, right? So, okay, so what did we have? We have that, you have the holonomy group, right, whole ex of nabla, which is contained in GL of EX. You can look at the Lie algebra. This is a Lie subgroup, right? You can look at its Lie algebra. It's Lie algebra. I'm going to denote it by little whole of ex of nabla. This is contained in the Lie algebra of GL, which is just the, and the morphism ring of EX, right? And the morphism ring of EX. And remember that so we had the curvature R of nabla, right? Where, where was this? This belongs to one way of looking at it was C infinity of the endomorphisms of E, tensored with which to TM dual, right? So then at a point, at a point of M, the point little x of M, you can look at the fiber of the curvature R of nabla sub X, when this belongs to C infinity of end EX, tensored with wedge two of TXM dual, right? So that's the fiber of the curvature. Then what's the claim? And this is the relation between the holonomy and the curvature. This R of nabla at X, a priority, it belongs to C infinity of the endomorphism ring. Sorry, sorry. There's no C infinity anymore. Okay, because I'm at a point, right? I'm just, I'm just evaluating things at the point. So this is just, this is just and EX tensored with wedge two TX dual of M, right? So that's all I've got. So then, you know, if you look at the holonomy, it's at least sub algebra of end EX. So what happens is in fact that this curvature belongs to the holonomy. So this tells you, this is good news because this tells you that looking at the holonomy is a natural thing, right? So we define this holonomy as, you know, these linear operators that take a vector to its parallel transports along different loops, right? You take a vector to its parallel transports along different loops, different loops. And that defines an operation on the fiber of your vector bundle at a point, at a given point, right? And what's nice is that the le algebra of this least subgroup, the holonomy group contains the curvature operator, right? So the value of the curvature. Okay. Okay. So now we're going to look at look a little bit more at at this holonomy and curvature. So it's useful to talk about connections on tensor powers. Okay, so the connection nabla induces connections on all tensor powers. E, you know, tensor to the K, tensor with the dual tensor L, right? And all exterior and symmetric powers as well. When we will denote all of these again by the letter lab nabla. So it's an abuse of notation, but, you know, it would do the notation will be will become very heavy if we started to use indices for these things. So we will denote these also by nabla. And so again, just as I said, to further, you know, relate the curvature and the holonomy, we're going to make a definition, a tensor S by a tensor. Now what I mean is just some section of some tensor powers of power, you know, of E and E dual or symmetric powers or exterior powers. So some section of one of these bundles, right? It's called constant or sometimes people say covariantly constant, if nabla of S is zero, right? So I have a tensor, I can apply the connection to it. And I'm going to say that it's constant when, when, you know, if I apply the connection to it, I just get zero. Okay. So now we have a nice theorem for a tensor S nabla of S is zero if and only if S is fixed by the holonomy group. Okay. So the holonomy group, remember, a priority it acts on the fiber of E. But, you know, if you have a linear action on a fiber of E, you can also define a linear action on the fiber of E, E dual on the power of power on all the tensor powers of E or the tensor powers of E dual, et cetera. Right. So you have an action of the holonomy group also on all of these tensor powers. And so now you're saying that being constant is the same as being fixed by the action of the holonomy. Right. And that is also what the same are saying that P gamma of S of X is equal to S of Y for all X and Y in M. Okay. So what you're saying then is that your, your tensor is constant if and only if you can, you know, it's parallel transport. It's itself, which makes sense. Right. We were talking about the parallel transport as being, you know, some kind of a constant transport. So to say that, to say that the holonomy, being fixed by the holonomy group is the same as being constant, I mean, that makes sense. Right. And of course, you know, technically speaking, you would have to write it down and verify that if this all works, you know, prove this theorem rigorously, but intuitively it kind of makes sense. That's how we define things. Right. Okay. All right. So we're done with our basic definitions. Now, let's get closer to our classification. So what we want to do here, we want to do some kind of classification of the monion manifolds. And then our hypercaline manifolds were naturally dropped out as one of the big classes of manifolds in that classification. All right. Okay. So and we do the classification by using the holonomy. All right. So now the holonomy that we're going to use is the holonomy that's intrinsically related to our Riemannian manifold. Okay. So Riemannian manifold has got a canonical connection on its tangent bundle, which is called the levy-chivita connection. And that's the connection that we're going to use. And, you know, we're going to do its holonomy. And then we're going to classify the holonomy groups of these levy-chivita connections, and then say that this these hypercaline manifolds, you know, form a big class of manifolds with a given holonomy group. Okay. All right. So let me now talk about what the levy-chivita connection is. So suppose then M is a Riemannian manifold. And I will usually denote it by a pair. So M is the c and c manifold, and then little g is our tensor, which is the Riemannian metric. Okay. So is there Riemannian? So g is the metric, right? Is there Riemannian manifold? So we have the fundamental theorem of Riemannian geometry. And what is that theorem? It says that there exists a unique torsion-free connection, nabla, on the tangent bundle Tm, such that nabla g is zero. So g itself is a zero-two tensor, right? And we remember we can apply the connection to any tensor, and we want this nabla g to be zero. This unique connection is called the levy-chivita or Riemannian connection, M together with g, right? So the metric is very important here. The metric defines the levy-chivita connection. Okay. And it's not too difficult to show the existence and uniqueness of the levy-chivita connection. You will find it in a standard Buchan-Riemannian geometry, and I'm happy also to provide references if you wish. So now let's look at the, so now the fact that this is a very special connection means a little bit more about the curvature, right? So the curvature, so now we have this connection nabla, which is the levy-chivita. So nabla goes from Tm into Tm tensor Tm dual, right? That's be the levy-chivita connection. And then you have the curvature R of nabla. What does that do? It goes from Tm to Tm tensor wedge two of Tm dual, right? And we can also use, because we have the metric, right? You have g, the metric. We can think of the metric as an isomorphism between Tm and Tm dual, right? And we can combine the curvature with the metric to define a different, a new tensor. So define the zero four tensor. So the curvature tensor, this is a zero three tensor, right? Oh, sorry, a one three, a one three tensor. The metric is a zero two tensor, right? And so we're going to define a zero four tensor r tilde as the composition. So how does that go? You go Tm goes to Tm tensored with wedge two of Tm dual here. I'm going to put r and then here I'm going to put g tensored with the identity and I'm going to go to Tm dual tensored with wedge two of Tm dual. So this composition here, this is my r tilde, okay? So why do we do this? We do this because we can actually exhibit some symmetry properties of of our Levy-Chivita connection, okay? So what are the symmetry properties? You can show, for instance, something nice, you can show that r tilde. So a priori, let's look at r tilde again, right? So a priori r tilde goes from Tm to Tm dual tensored with wedge two of Tm dual. Or you can think of it as an element of Tm dual tensored twice, tensored with wedge two, right? But what happens is that it actually belongs to sim two of wedge two of Tm dual. If you think of this inside wedge two of Tm dual tensored with itself, which is also inside Tm dual, Tm dual, and then wedge two. So the bottom guy is what I had, a priori, right? The priori, I just know that r tilde belongs to this bottom guy, right? But inside the bottom guy, I have wedge two of Tm dual tensored with itself. And then inside that, I have sim two. And what happens is that you can actually show that r tilde belongs to the subspace, all right? And in fact, there are also some identities that people call Bianchi identities. Maybe I won't really go into those. Again, you can find them in references or in the notes that will be published soon. So again, these are again more symmetry properties, right? Of this r tilde. So it's much easier to see the symmetries, you know, when you use r tilde, when you use the g, the metric g to define something from the curvature, right? And another thing that you also see is that, you know, so we remember the Holognomy group again, right? Where was the Holognomy, right? The Holognomy group is a subgroup of gl of Tm, Txm right now, right? Because our vector bundle is now the tangent bundle. So this is gl of the tangent space of m at x. And then you have the Holognomy, the Lie algebra of the Holognomy, which is in the endomorphism ring, right? And we had that for an arbitrary vector bundle, right? We had this thing right here, which is that the curvature belongs to the Lie algebra, the Holognomy tensored with wedge two of Tm dual, right? But what we have in fact for when our vector bundle is the tangent bundle, we have something even better. We have that r two, this r tilde is actually, if I look at it at a point little x, right, it's actually belongs to seam two of the Holognomy group at x, which is contained in seam two of wedge two of the tangent bundle. And it's also contained in whole x tensored with wedge two of the tangent bundle. Okay, so this, so you can show that this new curvature that we just defined by just putting g into the equation of the curvature is actually belong, it's much more restricted. So again, these are symmetry properties of the curvature, all right? But I'm not going to get too much into that. These are the things that people use, you know, when they, as I said, what we want to, what we're going to, what people wanted to do was classify all Holognomy groups. So you have this, you know, you have your Levy-Civita connection, right? The Levy-Civita connection is going to have a Holognomy group. And then people classified all of these Holognomy groups using all of these properties of the Levy-Civita connection, and then the curvature tensor of the Levy-Civita connection, which are these symmetry properties and all that. So then they can, as I said, then they can classify these Holognomy groups, and then that allows them to classify the Riemannian monophones themselves. So I don't know, should I, should I stop now? Am I out of time? Or can I, is this the end of my time? Or do I have another five minutes? You have 10 minutes. But there is a question. You also have a question in the chat, yes. Yes. Yeah, I just saw that, right? In the last expression, you have the Holognomy of the metric. Oh, yes, yes, that's right. Well, I should really, okay, I'm going to change that. It is the Holognomy of the connection, of course. But because, you see, because the connection is, yeah, it's the Holognomy of the connection. Because the connection is determined by the metric, people actually write the Holognomy of the metric. You see, so if you look at the theorem, it says that if you have a Riemannian manifold, right, yeah, if you have a Riemannian manifold, it has a unique torsion-free connection such that the metric is constant with respect to that connection, right? So, so then what you get is that the Holognomy group of this connection depends on the metric, right? Because the connection is determined by the metric. So then, yeah, so that's, that's what we get. So because, so let me actually introduce that notation as well. So since NABLA is uniquely determined by divide the metric, we actually write whole x of g equals whole x of NABLA and the same for the Lie algebra. Okay, so this is notation. Okay, so now before, before I can actually get to the classification of the Holognomy groups, I just need to talk about symmetric spaces and then being a product and stuff like that because we need to exclude those. Okay, so we need to get, you know, to not have something that's symmetric and we need to have something that is not a product. Okay, so I just need to briefly say something about what it means to be symmetric and what it means to a product so that then I can get rid of them. Okay, so let me start by, so first of all, I think everyone can do a product, right? So, you know, if you have two manifolds, two Riemannian manifolds, you can take the product of the two differential manifolds and then you can put a product metric on that product, right? So that's a product. And so now let me talk about symmetric and locally symmetric spaces. So what's our definition? A Riemannian manifold is called actually, before I get to symmetric, I have to talk about reducibility, sorry. Okay, so Riemannian manifold is called locally or not locally reducible if every point has a neighborhood isometric to a product. It is called irreducible if it is not locally reducible. Now you have a proposition about the holonomy of a product. So suppose a neighborhood of little x in M is isometric to the product, some product M1G1 let's say times M2G2, then the holonomy of G1G2 is equal to the holonomy of G1 times the holonomy of G2. So the holonomy group shows you the product structure, okay? And then you have a nice theorem. If M is irreducible at x, then Rn which is TXM is an irreducible representation, the holonomy group, okay? So it's kind of nice, you know, if it's reducible, you have a product. And if it's irreducible, then you have an irreducible representation, all right? Okay, so this is reducibility. And as I said, we're going to just break down, break Riemannian manifolds into their factors, right? Into products. And we're going to look at the products one by one. And the next thing, as I said, I want to exclude is being symmetric. So being symmetric means that basically you have an action of a big group on your manifold. So, and here's the rigorous definition. So a Riemannian manifold is called symmetric. If for every point of the manifold, there exists an isometry. By an isometry, I mean something that will preserve the metric. So an isometry SP from M to M, such that the square of your isometry is the identity. So basically, it's an evolution, right? And P is an isolated fixed point, SP. All right? So it's allowed to have other fixed points, but you don't want any fixed points in some neighborhood of your point. And now this allows us to define what locally symmetric is. So a Riemannian manifold is called locally symmetric if every point has an open neighborhood, isometric to an open subset of a symmetric space. It is called non-symmetric if it is not locally symmetric. Okay. And then let me just give you the theorem here, and I will stop. MG is locally symmetric if and only if NABLA R is zero. So NABLA is again the Levy-Civita connection here and R is the curvature of the Levy-Civita connection. So you're saying that your Riemannian manifold is locally symmetric if and only if the curvature of the Levy-Civita connection is constant for the Levy-Civita connection. Okay? All right. So this is a nice characterization of locally symmetric Riemannian manifolds, all right? And so what I will do next time, then I will talk about the Durand decomposition theorem and the Berger decomposition, no, not the Berger, the Berger classification of polynomial groups, and then how we get hypercalamant manifolds out of that. And I think I am out of time. Is that right? Perfect. Thank you. Okay. Well, thank you. All right. Any questions, anyone? I have one. Yes? Go ahead, please. So what are the examples of symmetric Riemannian manifolds besides the spheres and Riemannian? Oh, okay. Yeah. I was going to say that next time. Well, you can show that symmetric guys are basically just portions of leagroups. Yeah. That's all of them. So you can, so each time you just, each time you have a connected leagroup, you take a close connected leagroup, you take the portion, that's a symmetric space. And they're all obtained in that way. Yeah, that's all of them. Thank you. So there's a theorem about that. I will mention it next time. Thank you. So I think now you have another break, right?