Varieties in topology & differential geometry, structures in geometry of SR & GR, Spinors & Twistors

0:00 There is no structure of the frontier. Once we have a topological variety, once we have a range of the frontier, we can continue.

2:30 For example, we can put connections, or we can put Riemannian structures, or a symplectic structure, and then we can define several things at the same time which will have relations with each other. But, at the moment, we are at the second level of structure, the differential structure, and already, it allows us to define a lot of things, namely vectors, forms, tensorial products of vectors, that's how we will arrive at the tensor, We will see that there are natural structures of fibrils, a fibril, I don't know if you know this, but as soon as we have a differential structure, we have vectors, we have vector fields, and we have, in a canonical way, fibrils that appear, namely the flammable fibril, the flammable fibril, all the tensile fibrils, but also, we have glyphosate agents that also appear in a canonical way, something that is very powerful. So already, if we have a differential variety, We have a lot of things on our variety, so that's what we're going to study a little bit to see how different notions related to the dimension structure and especially the tensors are introduced. Then, if we want to go further, especially if we want to define the speakers and registers, we will have to define the additional levels of structure on our variety. Because from that, we will be able to define the tensors, for example. Many other things too, but we don't know how to define spinners or twistors, so we'll have to introduce... We are going to define additional levels, and we are going to define two things that are different, which are on the one hand the connections. In a differential variety, we can define the connections, we can define the different infinities, and we are going to define a metric.

5:00 Likewise, if we give ourselves a differential variety, we can define the metric tables on it. And, in general, we are going to use one, one that will have the property of being, for example, Lorentzian if we are interested at the moment. But we can arbitrarily define the state of different collections, we can arbitrarily define the state of different metrics, but we will see that there are certain relationships, that is to say that when we have a metric, for example, we will see that there are certain collections, among an infinity of collections that can exist, which are linked to this metric. And from this matrix structure, Lorentzian, we are going to design what we have here, the twisters. We call this a chamber, it's the same thing, we usually call it a key.

10:00 This is a Witten space structure, since if we have a functional derivation, we can add them. We can multiply it by a scalar, since we will simply have ... Derivation A times the derivation applied to the function f will give A times the derivation applied to f and we realize that when we have two derivations that I call f and v, that I call v and w, I can apply two times the derivation to the function f, so it is not a derivation, we realize that it does not apply to the rule of 10, however if I call f w minus w v, We can see that this is a derivation, so when we have two derivations B and W, their commutator, their next unit is also a function.

12:30 So we have a sort of multiplication of functions, an antisymmetric equation, and it gives our function an algebraic structure, which is an algebraic unit. So we know that algebra in general, that is the theory of algebra, is associated with a certain group of transformations. Here, this algebra is associated with a group of transformations, indeed, and as this algebra is huge, it is huge, the group will be huge too, this is what we call the group of dichromorphisms, and on the contrary, the group of dichromorphisms on a variety, these are all transformations that keep the differential structure. There are a lot of them, that is to say that each time that the mathematical physics group defines transformations, we do not always say it. There are a lot of particular dichromorphisms, such as isometry, conformal transformations, etc., but dichromorphism is an absolutely enormous gas. And if there are two differential varieties of dichromorphisms, then we can consider that, as differential varieties, these varieties are equivalent. The notion of equivalence and identity is quite subtle, and it is quite subtle in general relativity, because what relativity says is that space-time is a differential variety on which we have built a certain structure which is a non-linear metric. That said, if you have, for example, the Minkowski space, the Minkowski space, we can consider it as a central space and a solution of the rationality of the center. What is the Minkowski space? It is a certain differential variety in which we have defined a method. Now you can look at another solution, which is the De Sitter's central space, which is a different solution from the Minkowski solution. As a differential variety, Minkowski and Minkowski are diffeomorphs, so as a differential variety, they are the same variety. However, diffeomorphism does not transform Minkowski's memory into Minkowski's memory, so as a non-differential variety, but a Minkowski-Rinamian variety, they are more than equal.

15:00 So, indifference, you see, is something very, very, very, very low and, of course, if we want to distinguish the varieties that we want to distinguish, we have to define the minimum. So, you see, the notion of vector field, we defined it before the notion of vector. What is a vector? We will define a vector in a point M of a variety, even in each point M of a variety. This is simply the reduction of the vector field. In this point, that is, the vector field is something that has a function f corresponding to its derivative, which we will write vf, the vector v to the point n corresponding to the vector field, which we will call v of n, is something that has the value of the function f in n, f of n, which will correspond to the value of the derivative. So we define, from the vector fields, we define the vectors in each point. And this is interesting because it means that in each point we have a certain set of all the vectors. We call it the tangent space and it has, in a rather obvious way, a vector space structure. So the set of vectors of point M is called the tangent space in M. And we note that tangent from point n to the variety n has a vector space structure, that is to say that in the end, since all vector spaces of finite dimension are somewhat equivalent in terms of vector spaces, well, it is almost R to the power of n. For the moment, we do not have a metric line, so all varieties of the same dimension have the same vector space in each point. And we know that in Rn, as a vector space, it is true of any vector space of dimension n, there is a completely natural action of a certain group, which is the binary group of n. And this binary group of n, on a vector space of Rn, is simply the group of transformations... The idea is therefore to preserve the structure of the vector space. So this is something absolutely universal. So you see that already we have all this only from the differential structure. That is to say that in each point we have a certain vector space which is well defined with a group acting on it. So this is going to be true for any variety. No need to talk about theories, whatever they are. On the other hand, we have elementary algebra. Each time we have a vector space, we define its dual. What is the dual? It is the set of what we call uniform values. It is, in the end, the set of linear applications in the vector space.

17:30 Each time we have a vectorial space E, we have its dual, we have E star, and we know that it is also a vectorial space, and that it has exactly the same structure as E, they are isomorphic, except that there is no isomorphism one by one to define them in a canonical way, but we know that they are isomorphic. So here, in each point we have a tangent space which is a vectorial space, so in each point we have its dual, which is also a vectorial space, and we have it like this, and we call it the cotangent space in a unit. So we already have these two things, and it's interesting because every time we have an element of tau and tau, that is to say what we will call uniform, uniform is something that we can apply to a vector and it will give us a number. So naturally, the tangent space acts on the tangent space, and vice versa, the tangent space acts on the tangent space because in fact the tangent is the dual of the tangent. The dual of the dual is the space of the two. So we have that and we can do it in a very natural way. The tensorial product of these things, that is to say that our tensorial product, I think that you know what it is, we will put two angles, simply, we have two vectorial spaces between E and the prime, we will consider the couples of vectors B and the primes, and then the sum of the couples like that, and it will define a tensorial space, so we will write that, so we have two spaces. We have a vector space, a tensorial product like this, and an element of this tensorial product, probably, like this, knowing that it may not necessarily be... Not all elements are written like this, but all elements are written as a sum of tensorial products, of elemental vectors. So this is something that is tangent. Well, here we will be able to make tensorial products as much as we want, of the tangent space and of the tangent space. And the tensioner is none other than these constant products. So, if we have the vector vector V, which belongs to...

20:00 What we also need to know is that when we have a vector space, we always have a base in it. That is, if the vector space is of dimension n, there is a set of n vectors that we can decompose, any vector, in relation to these base vectors. And we have written that with a 10. The vector is written as a sum of the base vectors with here the components, which are the components of the vector in this base. But you have to be careful because the vector itself is something that is really defined geometrically, it is a geometric being that is defined in the tangent space, whereas its components must have the effect of a base, and there are many bases, so we see that this is something, the component is something that exists only relative to a base, so it is not really a well-defined geometric object, while the vector itself is a geometric object of an effect. So, I told you that we can make tensorial products. Every time we make a tensorial product, we obtain what we call a tensor. So, if we make a tensorial product of the tangent space, we eat it at the same time. By itself, like this. And then, we make products of the cotangent space, also by itself. And then, we have q times this and p times this. We are going to say that we have built a P-type dancer, so in particular, if we take the tangential space itself, there is no tangential space, u equals 0, p equals 1, so the vectors are 0-type dancers, and the forms are inverse-type dancers. And remember what I told you, this is important for the next part, we have the linear group acting on it. What is a linear group? As I told you, it is simply a group of a linear transformation. When we have a vector space, we write the vectors, the columns with the component, in a base. The linear group is simply the matrices, since a linear application on a vector space is simply a matrix. So, the linear group is simply a matrix.

22:30 So, we say that a group has a representation when we specify how it acts on a vectorial mesfab. So, the mesfab of the representation of the linear group GL would have to be N, when we say that it is a matrix N by N. The most natural representation space, which we will call fundamental, is simply the space of vectors, a vector space of the dimension n, for example, Rn. But since the tangent space, we saw that it was the same thing as Rn, it means that the tangent space in each point, the tangent space constitutes a representation of the group. This is a representation of space and vectors, and it also acts in the same way on forms, so the space of a form, i.e. the cotangent space, is another representation of the linear group. And in fact, we realize that if we have tensors like that, the linear group acts linearly on it and linearly on it, if we have others it's the same, so we also have any space of tensors. It is also a more complicated representation of the same linear group, that is to say that when we have spaces that are representations of a group, all the tensorial products of these spaces are also representations of a group. This is very important because we have to play the role of the linear model. So this notion of group representation is really important. We have an object that is made of elements. Each element g acts on a vector, this belongs to the space of representation, by giving something which is simply the product of the action of the element of the group g on the vector v and we can express this as the product of a vector, the product of a square matrix by a vector, it is indeed a vector. So this notion of group representation is very important, and of course, if for example we make the product of the vector space by itself, the tangent space, it will be dimension n, the product of the tangent space by itself will be dimension 2n,

25:00 So, when we're going to have the representation of a group on a tensorial product which is a vector of a dimension 2n, we'll have a matrix of a dimension n, a matrix of a dimension v, etc. And if we have a tensorial product with many factors, we'll have something here which is going to be a square matrix of dimensions, not n by n, but p plus q times n by p plus q times n. Is it the representation of the same group? It's always the same group. But here, it's always the finite group. GL , which has its natural representation in the N. We call it the fundamental representation. For example, the group of rotations, the group with three dimensions, GL , which obviously goes to G3. But if I do the tensorial product of R3, and this will be the matrix 3-3, and if I do the tensorial product of R3 by itself, I obtain a new vectorial space of dimension 6, GL3 will act on it, I will simply turn each of the two, it is not a representation, it is a general, and so I will have a representation of this group in a vectorial space of dimension 6. I have taken all my games. A priori, I can not go down. I can, I can't, I can't, I can't, I can't, I can't, I can't, I can't, I can't, I can't, I can't, I can't, I can't, I can't, I can't. All of this, by the way, when we go to the webinar, this question will come to mind. So, yes. So, we defined a tangent space in each point. So, since at the top of each point of the variety, we have a variety with points everywhere, and then we just saw that at the top of each point there was associated a thing that is a tangent space.

27:30 If we have a surface, for example, we see well what happens on the surface. So if we have a surface, each time we have a white tangent. If we have a space above each point, we have a tangent space in each dimension. If we have a space-time above each point, we have a time-time space in each dimension, which is flat by definition, which is the time-time space of the universe. So when we have a general relativity, that is to say that above each point of the space-time, that is to say below each element, if you will, associated with each element, we have a vector space of the universe. So we can see that we have a kind of structure here where we associate a vectorial space at each point. This is what we call a fibrous structure. And as here, the vectorial space that we associate at each point is a tangent space, this fibrous is called a tangent fibrous. So, from the point of view, it is a variety. It is a variety that is much larger since it has the same dimension. The dimension of the space of the starting variety plus the dimension of this which is equal to n. So it's something that will have a dimension of n, a variety of dimension of n, and which has a particular structure since it's not any kind of n-dimensional variety. So we immediately see a first thing, it's that when we are going to define... Here we have our starting variation, with here, above, above, when we have points in it, we see a thread that projects itself here, on this thing there. So, when we have a future of microfiber space like this, we have something we call the base, on which all the points project themselves. In fact, all the points that are in a tangent space, So we have a projection in this sense, but we don't have a projection in this sense.

30:00 That's very important. That's precisely why it's not a space product, not a Cartesian space product. If we had a Cartesian space product, we could project... In both directions, one and the other, they are very similar, but here we cannot define a projection in one direction. If we want to do that, we have to introduce additional structures. So at the moment, we can only project vertically. And what is interesting is that, in our case, we have this famous linear group that acts in the same way. In each vector space, the linear group acts, and in fact, we can consider that the linear group acts on the entire retorne film like this, but there is a particular action, which is that when it takes a point here, in the film, we call it the film, above the theorem, when it takes a point in the film, it will make it move, but in the film. That is to say, it will never take a point and be here to put it in the film. So it's a transformation we call vertical because it does not exist in the structure of physics. And in fact, this notion of the group's action is very important because as soon as we have a free space, the free space of convergence in the case of a particular free space, we always have a group action that is vertical. We call this the main group associated with the final. Our fibrin is something where we have the variety that plays the role of the space at the bottom of the fibrin. Above each point of the variety, we have a fiber, which in this case is the tangent space at the point M. And what is interesting is that we can define what we call a section of the fibrin. What is a section of the fibrin? We will say that at each point M, I associate a point of the fiber. This is a kind of function. I have a section called S. At each point M, it corresponds to something called S of M which inhabits in the fiber and the fiber inhabits the tangent space. What is a tangent space point? It is a vector. So it means that a section of the tangent fiber is something which, at each point of the variety of the base M, associates a vector with the point M.

32:30 So a section of the tangent fiber is a vector field. So what we had called the set of vector fields, which we had noted earlier, and the set of derivations, if you will, the set of vector fields, it's the same thing, it's also the set of sections of the tangent fiber. Well, it's a bit of a special language. In this case, it's quite simple, but this notion of tangent fiber is extremely important because it is fundamental, of course, in relativity and in the structure of human varieties as we see here. But if we do the quantum cycle, we realize that it is exactly the same structure that is going to evolve, except that it will no longer be tangent fiber, we will have other fibers. These are, for example, vector spaces on which groups act, and these groups will be, for example, group 1, group S2, group S3, and so on. And the entire quantum theory of the field, which we call geological theory, is defined on fibrous structures. And this is interesting because it means that the general volatility, But once we have introduced the matrix, these structures, the tangential fibers associated with their general activity, will come very close to the structure of the theory of Joule, which are quantum physics, quantum theory of the fields. So we see that through this formalism, we will have a very close relationship between General relativity and the theory of Jones, that is to say, it allows almost to see the general relativity as the theory of Jones, analog and string theory. So the section, it's not a section, it's an application that each time corresponds to an element of the space. Exactly, it's a section. So it's not a rule, it's a continuum. And we know that when there is topology and geometry, we know what is topology and geometry.

35:00 So it turns out that we can't always do that in any field. But we can do it in geometry. We can always do it in geometry. We can always take open spaces and, in these open spaces, create a continuous section. We can't always do it in a global way. For example, on a sphere, we can't do it in a global way. We say that the sphere is not parallel to itself. It's another way of saying that on a sphere, we can't define a continuous vector field, but it doesn't exist. We can't paint a sphere. We can't paint a sphere. We can't paint a sphere in a way... There is always a place where there is a singularity. So, we defined the tangent fiber, we know what the sections of the tangent fiber are, it's the same thing as the change of colors, and we know that this group G acts on the tangent space by preserving the fibers that we have. So, if we make the tensorial products, we will define, obviously in the same way we can define the tangent fiber. In other words, instead of taking the reunion of all tangent spaces, we take the reunion of all cotangent spaces. It has the same structure, so it's almost the same thing, and we're going to call it the cotangent fiber. So it means that it's the fiber that, above each point M, provides the cotangent space. This notation means that we have a fiber that projects itself in M. M is the variety of the parts, it's the base of the fiber. And a section of the fibrous is N. It is no longer a vector, it is obviously the duality of a vector, it is something that acts linearly between vectors, so it is a differential form. So you see that all this is canonical, that is, at the beginning I have defined my vector fields, I have my set of vector fields, all this is well defined, and I could have, instead of taking vector fields at the beginning, taken differential forms and build everything from vector fields from differential forms. For some reason, it is often easier to work with differential forms than with vectors, so it is quite often that we see the differential structure presented from differential forms rather than from vectors, but it does not change anything, it is completely different.

37:30 So, what I also recall is very important. The tangent space, well, in general, a vector space and its dual have the same structure, so a tangent space in one point and its dual have the same structure, they are isomorphic as a vector space, it is the same for the fibrous, they are isomorphic but there is no canonical isomorphism, that is to say, if I give you an element I don't know how to make a tangential space correspond to a particular differential form. If I want to do that, I'll need a supplementary structure. For example, I'll need a home. So, we have the tangential fiber. We have the tangential co-fiber. We can make tensorial products in the same way. In the same way, we can make tensorial products of as many copies as we want. After the discussions we had, from the tangent space to the tangent space, I can do the same thing with defibrils and I will have tensorial defibrils, which will be defined like this, by tensorial products of tangent space, of tangent fiber and tangent fiber, so a section will be what? A section of a fiber like this, it will be, of course, the tensorial product of the entropy of the tangent space itself. and Q-copy of the cotangent space itself. What is this? This is obviously what we call a tensorial field. So tensorial fields, there are types as high as we want. I said type Q, it goes as low as we want. Knowing that 1, 0 are vector fields, 0, 1 are uniform fields. And there is a particular field. So when we will be able to define a metric a little later, here, we will define a particular field. A vector space is a vector space which is a vector space which is a vector space which is a vector space which is a vector space which is a vector space which is a vector space which is a vector space which

40:00 What we call a moving frame is something that corresponds to a base of the tangent space at each point m. That is to say that E is by definition a vector base of the tangent space at the point m. And E is the application that corresponds to this point m at each point m. So, it can be done in a continuous way, and when in general we cannot do it in a continuous way on all the variety, we do it locally, open to the public, and we work in general in an open space where this is well defined. So, in the same way that we had defined the silver fiber, we can define the silver fiber. It's not that difficult. What is it? Above each point of my variety, above each point N, I associate here the set of all possible landmarks in my vector space. What is a landmark? It is simply a set of four linearly independent vectors. So, to verify that the vectors are independent, it looks like the product of the tangent space four times by itself. So, there is no problem in defining that. What is interesting is that, once again, This is the set of all the landmarks of all the possible bases in the vector space. Remember, it's basically Rn. How do we pass from a vector base of Rn to another? By a base change. What is a base change? It's actually a matrix always of the same group, Gn. So that means that finally, provided that we chose one at the beginning,

42:30 we can identify the set of all the bases here, of all the bases of the tangent space, In the end, each group is transformed into another, and so on. So, this is the same thing as this. Which means that this fiber of all the landmarks we just built is the same thing as the fiber that exists as a base above each point M, the group GF. And as... This fibrin has to feed the group itself that acts on it, we call it a main fibrin, and it is very important that the main fibrins are at the same time interregional and interregional. So, what is a section change? What is a section of this benchmark? As you have understood, a section is something that each point M associates with an element of the film, that is to say, a benchmark for a section. This is exactly what I called a model benchmark for a film. A section change is a model benchmark change, that is to say, it is a base change at each point. In a base change, it is an element of M, that is to say, it is an element of the group linear at each point. That is to say, a section, or an object, is the same thing, and we can also interpret it as being given by an element of a linear group in each one. According to mathematical expression, by saying that this fiber of the object which has a base M is a main fiber in the group, is the linear group. All this, I remind you, is merely the differential structure. We do not yet have a geometric definition, we do not yet have a collective definition. So, the connections, I'm going to go through them very quickly. Note that in the differential structure that we have just studied, which has already been superimposed on the topological structure, we can associate other levels of structure, such as connections or batteries, all of this in a completely independent way. On a given differential variety, we can build an infinity of different connections. We can also construct an infinite number of different metrics, which are defined, I repeat, independently.

45:00 We can also define many other things, if we want, we can define simple structures that you will find. At the moment, we are going to talk about connections and metrics. So, I don't want to go into details about connections, but simply know that we call them linear connections because, once again, it is linked to the natural level. In short, remember what I told you earlier. On the subject of tangent fibers, we are going to focus on the main fibers, the main fibers, as I said, that is to say that we have a variety that serves as a base with points and on top we have fibers which, in the case of the main fibers, each fiber is the set of all the main fibers in its own way. Remember that by definition of a fiber there is always a projection. They project the fiber on itself. But, I told you, there was no projection like that. Well, we can define one. We can say, by definition, I will say that this point, this project, I choose here... So, I define a projection. I do not define it between any two fibers. I define it between two fibers that are infinitely adjacent, i.e. I define it in an infinitesimal, differential way, and this correspondence that we define is called a connection. It will allow me to define in there, when I go in tangent space to that, to distinguish a vector that will be... Vertical is a vector that is going to be horizontal. That is to say, the vector is horizontal if, basically, it connects the elements to the corresponding elements. But this horizontality, you see, did not exist before. I had to define it. Verticality, it existed because projection already existed in the library. Horizontality, it does not exist. I have to define it. To define it, I introduce an additional level of structure called connection. Thank you for your attention and see you in the next lecture.