 it's so meeting it's being recorded. Very good. All right, so can I share screen? Yes. All right. Screen. All right. Hi, everyone. This is a series of lectures on hypercalor manifolds. I'm going to start with fairly elementary things, but we will relatively quickly get into more advanced stuff. So don't be afraid to ask questions. And here we go. So something is wrong with my apple pencil. So the first thing that I'm going to start to do is mainly to fix some notation. So in this first lecture, we'll be working with C infinity manifolds. Okay. So that's our first part. So if we have a C infinity manifold, we're going to denote it by M, right? And we denote by Tm, the tangent bundle of M and Tm star, the cotangent bundle. So this is the real tangent bundle, right? My C infinity manifold is a real manifold. So when I write star, I mean the real dual, right? Okay. And we also will talk about tensors. So for any pair of non-negative integers, k and l, the sections, and by sections, in this part, we always mean C infinity sections, the sections of the bundle. Now I'm going to take Tm tensor power to the k, and then I'm going to tensor that with the cotangent bundle Tm star, tensor power l. These are called kl tensors. Sorry, something is wrong again with my apple pens. Oh, there we go. Are called kl tensors. Okay. And of course sections of wedge k of Tm star are called differential one forms, right? C infinity differential one forms, I'm sorry, k forms, not one forms, k forms. And also the the sections of the tangent bundle, we also call them vector fields, right? So alternatively, vector fields, which are the sections of Tm, right, can be defined as first-order differential operators on C infinity functions on your manifold, right? So most of the time we think of a vector field in terms of how it acts on a C infinity function, right? And this action is basically taking derivatives, right, of C infinity functions, and it has the usual properties of the derivative, you know, it's linear with respect to constants, and it satisfies the Leibniz rule, right? Okay. And let me also set down some notation about the coordinates. So in a coordinate chart, so in a local coordinate chart, okay, sorry, my computer is heating up a little bit. Okay. So in a local coordinate chart with coordinates x1, xn, the local vector fields, right? ddx1 and then ddxn, they form a basis of the tangent bundle, right? This is, of course, a local basis. It will only work on your coordinate chart. And then you have local one-forms, right? So the local one-forms, dx1, etc. dxn form a basis of differential one-forms. And then you can write a local local KL tensor. That would be, you could write it as a sum, a big sum, of some C infinity functions, which I will call ti1ik and then j1jl. And then I will have my basis of tangent vectors, so ddxi1, ddxik, and then I will have my differential one-forms. Okay. So this is our notation. And let me now remind you what the Leib bracket is. This is something that is the basis of a lot of what people do with vector fields, right? So given two vector fields, I'm going to write them in local coordinates. So some of the vi ddxi, i going from 1 to n, and w will be the sum of the wi ddxi, again, i going from 1 to n. Then you can locally write down the Leib bracket. Of course, this is something that's defined globally, right? And we're just writing here the, we're just writing the local expression of it, right? So the Leib bracket vw can be locally expressed as, so you write it like this. So it's the sum for j going from 1 to n of the sum for i going from 1 to n. And now what we're going to do, well, it's the thing that you probably have seen before. You want to think of this actually in terms of acting on functions, right? So for c infinity, let me write down how it acts for c infinity functions, right? You're going to say that, so you wanted to take the commutator of v and w, right? So this is going to be v acting on the c infinity function w of f. So w is the first differential operator that acts on f, right? And then v is going to act on the output of w, right? And then minus, as I said, it's the commutator. So you're going to do minus w of v of f, right? So if you think of what this means in terms of local coordinates, right? So in local coordinates, v is the sum of the wi ddxi, v of f is the sum of the wi df dxi, right? So if you think about what this means, then you will see the local expression of the Liebrakhet up here. It's going to be vi. Remember, I have to act on f by w first. And then I act on it by v, right? So I'm going to have the ddxj here, right? Which acts on f. And then the coordinate will be the coordinate of w, right? And this is v acting on w of f, right? It's a simple exercise that you can do at home if you have trouble following me right here because I'm going a little bit fast. All right. So this is the Liebrakhet. And as I said, it's a very foundational thing. You can show that it's independent of the choice of local coordinates. And so it puts a, it puts also a structure of the algebra on the, on the space of sections of the tangent bundle, right? On the, on the space of all vector, all synchrony vector fields. All right. So the next thing that we're going to need to introduce is connections. So what's the connection? That's at the basis of the study that we will do of manifolds, right? So, and they're, they can be defined for any vector bundles. So for any, for a C infinity vector bundle E on M, a connection is a linear map, which goes from where to where. So usually people write connections with the letter nabla. And it goes from the space of C infinity sections of the, of your vector bundle E to the space of C infinity sections of the vector bundle E, tensored with the cotangent bundle. Okay. So now if you, if you think of the fact that, you know, the cotangent is the dual, giving yourself a map like this is equivalent to giving yourself a map like this. So you can say nabla goes from C infinity of E tensored the tangent bundle into C infinity of E. So giving yourself a map of this shape or a map of the shape that I had before are the same thing, right? And let me put this one in parenthesis because I'm not using it right now. And so you have this, you have this linear map, right, from the space of C infinity sections of E to the space of C infinity sections of E tensored the cotangent bundle. And what you want is you want it, of course, to again satisfy the lightness. The connection is a substitute for taking derivatives again. So, you know, if you have, if you have local coordinates, you know how to take derivatives, but but that's only for functions, right? So what you want to do here, you want to take derivatives of sections of a vector bundle. And that's not always going to work. So you need to actually say how you're taking derivatives. So because it's a way of taking derivatives, it has to satisfy the lightness rule. And which would say what, which will say nabla of f times E is equal to f times nabla of E plus E tensored with the differential of the function f for all c infinity sections, little e of E and c infinity functions f on m. Okay, the connection actually doesn't just act on global sections, it also it also will act on local sections on any open set, right? So this is what we're not actually saying here explicitly is that this is actually well defined on a open set. So you can take c infinity sections on any open set, and you have the same thing. All right. And whenever you have a connection, it defines a useful linear map. And this is the map, you know, this is what I was talking about for taking derivatives, right? So for any vector field, little v, nabla defines a linear map nabla v, which goes from the space of c infinity sections of e to the space of c infinity sections of e. So this is the taking derivatives. And what do you do here? You send the section e to nablas, what we call nabla sub, sorry, nabla sub v of e, which is by definition, nabla of e acting on v, right? So remember nabla is something that goes from the c infinity sections of e to the c infinity sections of e tensor t and dual, right? So this nabla of e is a section of e tensor that cotangent bundle, and you're acting with it on this tangent vector. So you're pairing the cotangent bundle with the tangent bundle. And then so they kind of cancel each other and you get a section of e, right? So that's what you get. Or if you want to think of nabla as going, sorry, my handwriting is getting messed up. Or if we think of nabla as going from c infinity of e tensor tm into c infinity of e, then this nabla v of e will be nabla of e tensor v, right? So that's, I mean, that's another way of thinking about it. That's all. All right. So this is a connection. Now, most of the time, we will be interested in connections, not just on any vector bundle, but on the tangent bundle itself, right? So for that, for that particular case, there's another notion, right? So when e, when e is equal to the tangent bundle, we can define the torsion of a connection, where here I will also again use this, the other point of view. So here I have e tensor tm, I bet e is tm. So I get tm tensor tm going to c infinity of tm is defined as t of, now I'm going to take v tensor w and this guy is nabla sub v of w minus nabla sub w of v minus the bracket of v and w, all right? So all of these are vector fields. And we say that nabla is torsion free or some, in some, in some of the literature, people would also use the word symmetric if the torsion is zero. Okay? All right. So this is a useful notion for Riemannian manifolds and we will see shortly why. Could someone remind me when I'm supposed to stop for the break, please? Yes, you should stop at 450 and we will have five minutes break. Yes. So that's in 25 minutes? Yes, right. Yes. Sorry. No, no, that's fine. That's fine. Yes, I forgot you're in another time zone. Sorry. I have to translate. All right. So okay, so we're doing good time, time wise. Are there any questions so far, anyone? Did anybody want to ask me about anything? Okay. All right. So the next, the next thing about a connection is the curvature. So let me explain about that. So what's a curvature? You probably heard of curvature. Euclidean space is what we call flats. It has no curvature. It's not curved, but a lot of the time when you have a Riemannian manifold, which I haven't told you yet what it is, it's not flat. It's curved. Okay. So and what does the curvature tell you mainly? I mean, you know that if you take partial derivatives, usually when you take mixed partials, you know, that the mixed second partials commute, right? So if I do d2f, dxi, dxj, that's equal to d2f, dxj, dxi, right? I can switch the order of partial differentiation for functions normally, right? That's on Euclidean space. And it's also true in local coordinate charts for any manifold, right? But it's not going to be true in general for an arbitrary connection. As I said, now we think of the connection as our way of taking derivatives, right? So if you take a derivative with respect to one vector field, then you take the derivative with respect to the other vector field, it's not going to be the same in general as switching the order. So switching the order is going to change the final output of your second derivative, all right? So that's, so the curvature kind of measure that measures that error, you know, and what and how much, how different are, is it, how different are those two differentiations from each other, right? So what's the difference between taking one derivative first and taking the other one and then switching the order, right? So, but let me give you the formal definition. So the, the curvature of a connection, nabla, is a linear map, which people usually denote by big R. And it goes from C infinity of E2, C infinity of E tensor wedge two of the cotangent bundle. Or you can think of it as just like before, we can think of it as a linear map going from C infinity of E tensor with wedge two of the tangent bundle into C infinity of E. Or we can think of it as a third way. Now you see, because I have a wedge two of TM here, I can take one of the tensors, one of the TM factors and put it on the other side, right? So I can think of it as an endomorphism. So, or you can think of this as a section, a global section of the endomorphism bundle of E tensored with wedge two of TM dual. Sorry, sorry, I didn't mean to take one of the TM factors, I meant take the E factor. If you put, if over here I take my E factor on the first line, I take the E and I put it on the other side, it becomes an E dual, right? So this endomorphism of E, right? This guy here is also C infinity of E dual tensor E tensor wedge two of TM dual. Okay, all right. So these are, you know, three different ways of writing down the curvature. So, and how do we define it, right? So it can be defined via its action on sections little E of big E and vector fields V and W as follows. So I'm going to take the point of view, the second point of view, right? So I'm going to think of the curvature as something going from E tensor wedge two of M into E, right? So I'm going to take a section little E of big E and then tensor it with V wedge W and what is this going to be? This is going to be, now I'm going to take nabla V of nabla W of E. So this is what I was talking about, like I'm taking the first derivative of E with respect to W in the direction of W, then I'm taking the derivative of that in the direction of V, right? So this is a second derivative here, and I'm going to subtract nabla W of nabla V of E, right? But now actually, because we have arbitrary vector fields, right? And arbitrary vector fields themselves do not commute, right? The commutator of arbitrary vector fields is actually the lee bracket, right? So what I want to do here, really, I do want to subtract the lee bracket, okay? So normally, in an ideal world, right, if you had a flat connection, if you had a flat world, right, if your manifold were something that you can think of as flat, it's not curved, then when you switch the order of differentiation for two arbitrary vector fields, right? It should give you the lee bracket. So then what the curvature actually does, it measures the difference between switching the order and the lee bracket, right? And so in terms of local coordinates, right? And now you'll see that in local coordinates, it's exactly what we wanted before, right? X1, Xn, then what do you get? You get that R of, I can do the R of ddxi wedge ddxj, right? And what's this going to be? This is actually going to be nabler ddxi of nabler ddxj of E minus nabler ddxj nabler ddxi of E. And that's it, because what you know is that the bracket of ddxi and ddxj is zero, right? This is a local thing that we're doing here. And you know that the bracket of partial differentiation is zero. So in local coordinates, this is really what you get. So now, let me give you a definition. So before we had the definition of torsion, right, you will see how it's a little bit similar, right? The torsion that we have here was, well, it was only defined on a pair of vector fields, right? So you acted with the connection on one vector field, right? And then minus the action on the other vector field minus the lee bracket of the two vector fields, right? That was the torsion, but not that the curvature is a little bit different, although it's reminiscent of the torsion, right? So you're doing the difference between the differentiation with respect to the two vector fields and then the lee bracket. Now we have the fact that it was torsion free when the torsion was zero. So now we talk about when the curvature is zero, right? So the definition here is, we say in the nabler is flat, if its curvature is zero, okay? And you will note if you go back to this description of the curvature as I see infinity section of the endomorphism of E tensor wedge two of T and dual, well, okay. Now maybe I will get back to that later. Sorry. No, let me not say that right now. Okay. So this is our definition of a flat connection. It's flat if the curvature is zero. All right. Any questions? Now, because we're going to pass to a different section now. All right. No questions. So the next thing we're going to do, we're going to talk about Riemannian manifolds, right? Those are the most important objects for this series of lectures, right? So what are Riemannian manifolds? So as see infinity manifold is called Riemannian if it has a Riemannian metric. Now I will explain what that is. Okay. What is that? That is what we call in the language of tensors. This is a two-zero tensor. And what's a two-zero tensor? If you don't remember, a two-zero tensor is as the global section of the second tensor power of the cotangent bundle. Okay. And well, it's not just a section of the second tensor power, but it's also symmetric. What does that mean? It means that it actually belongs to see infinity of seam two of tm dual. And it needs to have one more property. It has to be positive definite. Okay. So and defines a positive definite quadratic form on the tangent space txm for all x in M. Right. So your form is a section of seam two of tm dual. So it naturally acts on tm, right? It's a bilinear form on tm. And you can, you know, when you have a bilinear form, you can just plug in the same vector twice and you get a quadratic form. So you're asking that that quadratic form be positive definite, meaning that so what you want is if I take the value at the point x, I want gx of vv to be strictly positive for all v in txm, not zero of course. Okay. All right. So this is a Riemannian manifold. That's it. It has to have a Riemannian metric. And this is what a Riemannian metric is. So now when you have a Riemannian metric, and if you have a vector bundle with a connection, you can do something that we call a parallel transport, right? Okay. So what is that? Let's explain what that is. So now suppose that we are given a vector bundle, a C infinity vector bundle E on M with a connection nabla, which goes from, again, I'm thinking of this nabla as going from C infinity of E to C infinity of E tensor tm do, right? And now, as I said, we wanted to try parallel transport. Sorry, Elam. I think that you are muted. You did yourself. We lost signal from the microphone of the iPad, I think. Yes. Elam, could you turn on your microphone on your computer?