Abstract (Manifold-Free) Approach to Differential Geometry

0:00 set or even when there is another point of view where we have much more singular parts of the space, much more, many singular points. And we have also the same . And don't forget, of course, the other type that is considered recently with these incidents, and where these are, of course, are very peculiar from the point of view of the standard theory. But instead of it, we do have these theoretical techniques. And, but what is the, let me say still something here. What is the, say, the instrumental intervene of the manifold in that size, in the case of differential geometry, is that this provides us with these algebra of differential geometry. And, well, another point that it is of importance is that we, if we, I can say something about the use of Schiff theory and, of course, the notions of Schiff-chothomology that we can apply here. So by looking at the Sifco homology, et cetera, we realize that, for instance, the complex is an important notion, which means . So if the space is not good enough, then we cannot expect that we would have the dilemma We cannot substitute it with any, say, because it's very deep and analytic and of course, geometrical. We need the contractability of the space. So we have the Poincare-L-M. In that sense, we cannot, because we do have, as we shall see, be a space, but we don't assume anything from that space. So in that case, we accept that .

2:30 And the strange or the unexpected thing here is that there are particular cases where this is true. For instance, that happens with the and that happens also with the incidence. And this is due because we start with some initial space, although that space is not And another point that would be also probably obvious is some interpretation of physical notions that we can have in terms of these notions here. Many, for instance, say that an electromagnetic field is a line bundle which has also a connection. A line bundle with a connection. And, of course, instead of line bundle one can use the sections of the line bundle, which means the SIF, which is produced by Lundberg. And another question is why a SIF? We can say that we prefer functions than the space on which the functions are defined or the functions are or to where to where or we prefer the space to where the functions are arrived, or the space on which the functions are defined instead of the space. We don't know the space, but probably it's easier to consider the functions which live on the space or arrive on the space. So sections, this is important, not the space. So you say that it's more important the range of a function than the domain? Inrespectively, either the range of the function or the domain of definition of the function,

5:00 we prefer the function and not the space because we don't have many information about the space. And the space also creates anomalies or inconveniences. On the other hand, we don't need the space in itself to work with the functions. So in that sense, we can say that we prefer the functions. And of course, the sections are more important here than the space. For instance, in differential geometry, we prefer and actually we work with the sections, which are the vector fields, not the bunches. we don't work with tangent of course in a sense also for a physics a physics wants to have the equations and the equations are in terms of the vector fields and of course the connections so in that sense sections are preferred again the space we can say that the electromagnetic field is a line shift instead of a line bundle. Line shift is the shift that we take from a line bundle by considering each section. And, of course, if we consider the C-infinity function on the manifold or the shift of C-infinity function on a manifold, then, of course, a line bundle So if a line battle is, say, L on X, then the sif of sections of this line battle, say script L, is a module with respect to this other. So this is a line sif. And formally speaking, a line sif in the module with respect to this sif of altruals, which locally free. This means that locally the L is of the form some power of this algebra. C infinity x and since we have lines if dimension 1 then this equation is valid on a neighborhood

7:30 of any point on the space X. Say, every point X has an able on which one has this behavior of the line C. So it's locally free. Locally, the C is a power of the C for coefficient. And, of course, we have a connection. So we have a pair. L from a D. Our connection. Of course, what substitutes here the notion of an equation of the classical derivative, but here we speak of a connection which acts between shifts, vector shifts. Vector shifts mean that instead of having here power 1, we have power n, a natural number So locally, a vector shift gives us the same relation, c infinity sub x, on u, u is a neighborhood of a point of a space, but here we have a power, n, n belongs to natural So this is the notion of a vector sieve. So bosons, in a sense, can be associated with line sieves, while fermions with vector sieves. Of course, line sieves, because we consider the symmetry of the functions of the sections, while in the case of fermions, we have anti-symmetry, so exterior for them. So we have an exterior power. So T is the dimension, say, is 1. And as I said, the notion of the connection is a function between the sieves, which means morphism between the sieves. So, D, the function defined on the shift L, and then the values is from the shift L tensor with what we can name one-force.

10:00 Of course, since both of these objects here, L and omega 1, are modules with respect to this algebra of coefficients, let us, for convenience, write it, call it A. Then this tensor product is with respect to A. And this D is a map. In fact, it's a shift between these two shifts, which is a complex linear. Of course, this C infinity of X contains the complex number, considered as a constant shift. So this map is, by assumption, complex linear, and of course, it satisfies Leibniz's relation with its behavior, say, with respect to a product. A product in the sense that because L is a sense of vector, it's a vector space, so we don't have any product in L, but we have a product with a coefficients from, because L is a line shift, so locally is a vector here. So we can consider a differential from, say, map alpha at a section S on line C. So we have to check the behavior of D on such a product, of a function at a section of L. So, and this is, of course, homogeneous with respect to alpha in the first step, D S. this is the obstruction here we have the section so we apply D on the section S but here we tensorize with the differential of alpha what is the differential of alpha alpha is a local section the shift of coefficients. So it appears an object here. What is this d? Of course, this d is the basic differential we know.

12:30 This is the dx, if x is dx, or df, if f is a differential function. So since we don't have anything, nothing apriot, So we accept that we have such a map D, which plays the role of a basic differential. So this map D is a gain between the shift of coefficients, A, and this shift, we consider here, and explain what is omega 1. Omega 1, as we say, is, again, another module with respect to A. It's an A module. And so D is a map between these two shifts, which is, again, a type of derivative. So it is, again, complex linear and satisfies the analogous relation here, but A is an algebra. like two sections away. And this behaves like the ordinary derivative on the product of function. So we have here S tt plus tds. So we accept that we have such a function, such a secret of morphemes. And omega 1 is an A molecule, which also is given. and it plays the role of one form, differential over function. We need to remind us always that these are not forms in the user sense. In the case of the algebra of Tobrongeier, we construct such forms. And of course, in the case of the incidence algebra, we also construct such forms. We construct the machinery, the tools of the machinery. The machinery works. We have to construct, in special cases, the tools. So this is the connection. And of course, the connection is a fundamental notion.

15:00 I would say in our physics, since in a sense, it represents causality or the field itself and of course in the case of the electromagnetic field this is the of the field while we can say that D is the field action and of course we can another information of this generalization is that connection does not always exist but if we have a connection then we do have curvature which means that exists say it is not always given we can say we realize the connection through the strength of the connection which means through this is the field strength so we can have we can formulate a an equation with respect to the connection and if and if we have a solution of this of this equation then we realize the field we have to realize the connection Up to this point, I said that the curvature always exists, which means that we can consider if we have these two operators, say, or functions, or the connection and this d, and then we have another derivative exterior derivative d1 from omega 1 into omega 2. Omega 2 means the second exterior power of omega 1 with itself. And of course we require again classical properties for the first exterior derivative. So if we have a connection, L and omega 1, D, we can put it here, D0 or D, and here we have D1 which is from L and so

17:30 omega 2, then the composition of the two functions is D2, the operator, D2. So in terms of this, this is actually the capital of the connection D. So, of course, we assume that these two operators have the relation, that satisfying relation D1, the convolution with D is 0. And this, if we come here, we realize that this D1 convolution with D is actually the the curvature of the connection, because this is also a type of connection, of the initial connection. So the initial connection, D, which is the basic differential, is flat, as we say, because its curvature is zero. This is the curvature of D. So this is the curvature of the connection. which always exists if we have this. Of course, in the classical case, the connections always exist because the shift of differential functions is a nice shift, is a fine, as we say. The underlying space is paracompact. So we have connections for any vector button or for any shift. So if you consider the higher order extension, say, of your D, you have, you define, I tell that, you know, D to the D to the D to the D. Yes, and you build what we know, the complex. Then the presence of curvature would mean departure from exactness.

20:00 Yes, because of the curvature, we don't have actually a complex. Because a complex means, say, a sequence of differentials between the potential max, of morphing between modules, et cetera, so that we have the relation dn plus one composition with dn equals zero. This gives us a complex of these marks, dI. But in the case of this d, capital D, then we don't have this. My question is if one naively say wanted to write a change down but with D capital instead of D and with D capital I take it to represent a non-flat connection, right? Yes what would that mean, that I cannot build the complex it would mean that would be, why do you need to have a curvature zero. Anyway, we don't have this complex, exactly, because the curvature is here, if we consider, say, section, then the curvature is here. So curvature means departure from the exactness. Anyway, and well, departure of exactness, since we have the right type of differential so that to detect it, because we have exactness if the differentials are the ordinary exterior derivative. Say d, d, i, proposition with d, i minus 1 is 0. This is something that we prove. In the first two degrees, we accept it, 0, 1, 1, 2. But after that, we have this. actually, well, these are the prolongations, so to say, of the basic differentials. And these prolongations do give an exact complex because the initial differential, D0, B, is

22:30 flat. So since we start with a flat differential, then we are led to an exact complex of propagation. We start with a differential, which is flat. So it has zero color. And so we are led to an exact complex of differential. But if we start with a general differential, as it is a connection. It is a disconnection So this D has, of course, if we have D, as we say, we always have R of D, if it's strength, which is not zero. Then we don't get an exact sequence or an exact complex, because in the calculations we have So we need the 0, we need the, in that sense, we need the cover to be 0. So the general rule is that one has to start with a flat differential in order to have the exact sequence of progress and well and here we have some classical theory which can be generalized and also applied for instance we know that if we have a closed form then this closed form gives us a cohomology class and also with the theorem and also with application for instance in notion related with that theorem of Weyl concerning the integrity of the power field. So the theorem of Andre Weyl, you have a line shift L, which is equipped with a connection D. Then we have the character R of D. This is a two-form. So this is an element of omega 2.

25:00 And so it's a closed form, actually. So we have a cohomology class, and this cohomology class should be integral in the sense that the coefficients should be taken from the . So the curvature of the connection of a line shift is integral in the sense that the respective cohomology has integral coefficients, this is necessary. Actually, why prove the converge of this? So any time we have an integral, a two-dimensional integral of homology class, then this should come from a bosom set, because this should be the curvature of a connection of a line shift. Well, probably from a boson, where we have the phenomenon that a boson can be represented or maybe represented as a pair consisting of a line-siever connection, but it does not this, does not mean that any time we have this, it's a possibility to have a boson there any time we have a cohomology class which is integral of dimension two. Some bosons might be around. And another indication, still indication which is connected here, which is connected with this is the interconnection of a line sheet and say we have a vector sheet and then if we tensor this vector sheet with a line sheet then this tensoring is locally undetected. Say we have a vector sheet or we can say a power of a So we tensor it with a line C, L, or always the tensors are over L.

27:30 So locally, this means that we have omega N over U, tensor L over U, but L over U is A. So at the end, you have only omega N over U. So this tension and this interaction between the fermion, a fermion, say, which can be represented by, in a sense, by a vector shape, and the interaction of a fermion with a The boson here is not detectable. And the result is, the profit is that in terms of this line shift, we have a quadrizing line shift, as we say. So we endow the fermion with a quadrizing line shift by tensoring it with a boson, which is but locally we don't detect we have such an interpretation it is absorbed in some sense the whatever you call the sections of the line sheath which represents it is absorbed by our by our measurements yes yes yes well it's a it's a it's an interpretation because the, say, the boson, the sections of the boson are differentiable, say, in potential marks or which. The sections of the boson are such functions. So such functions exactly are absorbed by the analog function from omega n. Say, typically you have omega n times of L over U and this is omega N over U tens of L over U but this is A, A over U so omega N is an A module so in that sense these functions are absorbed in omega N so finally you have But in terms of red, which is a positive, so, and we accept a connection, so we do have

30:00 curvature, and this curvature, so we have this line part, and this line part can be pulled back on omega n. In that sense, we can say that since L is, has a line, a quadrising line c, then also omega n has such a line c, a quantizing line c. Well, these are some indications of the whole story. Probably the interesting information was at the beginning concerning these relations. I think I have. Thank you. Thank you for your time. I, well, I prefer to give such an exposition and not to stick on the formalities and the technical results I wanted to talk about your remarks about these, say, these phenomena of differential geometry and the possible applications in problems of physics. I didn't quite understand, well, I didn't understand at all, what you were saying about quantization, did you mean literally quantization in the sense the quantum field theorists would this is actually I literally quantized in the family field I just didn't understand what you meant why did you call it quantization? a very silly question no no I didn't get all the questions that's it in fast English you're talking about quantization a contribution I thought you kept using you're talking about quantization quantization in which sense? well that's what I was asking you that was the question did you mean to be related at all quantization that's a definition a definition of quantization it's a definition say we we accept a like what is in lines here what it is we know that the curvature of a line sieve is an integral so this is the say the basic aspect on geometric what they

32:30 So if we have a quadrizing line shift, in that sense, so... That's what I'm talking about, then it comes to my general question. This type of structure, with several of these sorts of things floating around, work very well for linear. Linear. Yeah, basically things are linear, because the Ds are... More. ...in ordinary, if these were ordinary modules and so on, the maps D would be linear. D, yes, D is linear, yes, of course. That's right. And as a result, these things, these types of structures, tend to work very well for linear field equations. This was like the work of Siebel and all sorts of people over the years who've sort of played with this type of thing. The difficult things are for non-linear. Yeah, directly from the non-linear problems, then the whole thing becomes, typically, and I'm not saying the case here, but typically becomes much less than the form. Of course, D is linear. D is linear by definition. So what would happen if you were presenting a non-linear fermion equation, for example, a fermi-field equation? Well, we have hints here. By using the type of one of the graphs I apply. These are transferred to the coefficients. Because the equations are in terms of the... Well, we don't need to stop at the... The differential equation is D. The D does not need that the differential equation is linear. The type of functions we can say, how can we apply this machinery in differential equations? Non-linear function. So what is the contribution of Rosiger's algebra here? The contribution is that we have the machinery of differential geometry by using these non-smooth functions. Although this does not seem to answer the question, the difficulty is not with the, of course, if we don't consider linear but non-linear, the difficulty still arises because of the singularities of the function which are involved.

35:00 And in that point, we don't have problems. We can consider functions which are non-smooth. And again, consider the same type of equations that we have at the classical theorem, because we have the differentials. We can consider dn, higher-order differentials. There is no problem. The profit is from the type of functions we can apply, So these functions did not be differentiable. This is the contribution. Yes, but Chris seems to, maybe I'm wrong, but seems to imply that the difficulty lies with the nonlinearity of the problem. The nonlinearity of the problem. In some sense, this is developed in a linear. In which sense linear? Well, the nonlinearity of the problem, not of the machinery. The classical differential theorem. Yes, and the equation? The equation is the same equation as before. We change the items which are involved in the equation. We don't change the equation. We have the same Yang-Mills equation, or we have the same Einstein equation. We just change the machinery. In which sense? in the sense of the coefficients that we may use. The coefficients are the algebra A. Classically, A is the algebra of differentiable function. But now we have UR. Or Orsinger's algebra. This is not an ordinary algebra. It's a very straight algebra. We didn't expect that such an algebra would be applied. But then, Joey Marks, for quantization, suggested rather than definitive your remarks would be sorry i'm more suggestive rather than suggestive they are more suggestive rather than definitive well nothing is definitive

37:30 Well, what I mean is that I'm expressing a slight question. Can we say that nothing is definitive? Well, the question is whether there's techniques that work at all for a quantized non-linear system. Well, quantized is the definition. Well, quantized means quantizing line-siff means a line-siff which by itself is quantizable. A lineship by itself is quantizable in which sense? That the curvature is a form has a cohomology class with it is a coefficient. What is this? Well, it's saying in the classical, well, these gives us sections integrable functions on a human space and now the human space is not standard, say, but it consists of the section of a line but geometric quantization yes but that only works for linear systems normally that's the trouble Siegel produced a very elaborate quantization scheme in the 70s which looked very geometrical and very beautiful but unfortunately it was completely wrong when it came to a non-linear system it just didn't give you the right answer because you just can't you get a very functorial way of doing quantization don't know why it looks so attractive it looked very natural and it looked very powerful and very geometric it simply was wrong it wasn't just a question of not being well defined it actually was wrong in non-linear systems in non-linear systems? yeah, non-linear field systems it's wrong to consider geometric quantization for non-linear systems? no, it's wrong to do it the way that Siegel did it and there's something about the way you talk that reminds me a little bit of his ideas I was just wondering how seriously do you take this and laughs about quantizing, which is not normally an easy thing to do. Linear quantum field theories can be done in many, many different ways. Yes, yes, yes. But most of them, almost all of them break down when you come to a non-linear system. And very often people think about linear systems, they use those. But then the problem is with the theory of who we apply, say geometric quantization.

40:00 Yes, or some . Yes, yes, yes. So I just wonder, in this case, how confident you were at this work to a non-linear system? That's a good question. So geometrical digestion does not have difficulties in non-linear systems. In field theory. Well, here there is difficulties come from differential geometry then we can try to apply these quantum theory but yes we have to analyze the problems we have to analyze the problems which are the problem the problems are physical or geometrical geometrical in a sense because we have again to see what do we mean by say the problems are geometrical and then you get this deep general question what extent quantum systems can be discussed in geometrical language there's no question that classical physics fits very well into what classical geometrical language is debatable this is really true geometrical geometrical language for the classical theory yeah is the language concerning differential geometry is the classical differential geometry applied to classical right okay then the question Okay, then, can we apply again differential geometry in that area or not? Well, that's what I'm saying. It's problematic. It's problematic. Why it's problematic? Because we find a singular infinities. So he said the basic notions of space are not applicable because of the infinities. so we have to distinguish what is the problem the problem is physical and in which sense it's physical or it's mathematical so here it is not the physics which well of course the physics has problems but we increase the problems through the language we apply the mathematical language we apply well even many are saying For example, have you actually done, has anyone done, shown you can analyze the ordinary quantized scalar field

42:30 in four dimensions this way? Do you get the usual results back? I would say, well, this is, as you said also, this is an aspect of quantization. But if this aspect has some problems, then, again, we have to look of the type of the problems. here we conflict the the difficulty of mathematics with the difficulty of physics and because we have difficulty of mathematics we say that we have difficulty in physics in a sense, or not. So would you put the problem like, for instance, would you put the problem, so you are telling us to relax, I mean, and not care about, very much about the mathematical logic? Well, up to a certain point. Up to a certain point, because it seems to me, what, I mean, Feynman did not need, you know, functional integration. There is actually the notion of space which is not a rich space, rich space in that tip. So it's the way we look to solve the problem, the physical problem. Is there any space in the quantum tip and in which sense? And so together with the notion of space we apply comes a type of geometry. So a remark here is that we can have the same type of geometry irrespective of the space, the same type of geometry we use to apply. This is something. Well, of course, a concern in problems, quantization problems. Again, we have to clear up the dimensions, what is the knowing and why we choose these type of mathematics and not another one. Can you take a quick question, just a technical question. Supposing you wanted to talk about a C omega space

45:00 rather than C infinity. Suppose you want to talk about C omega rather than C infinity, so you're a real analogy rather than C infinity. How does that fit into this ? Just as a motivation, just as a . C infinity, well, this is a good C for one, by means of which we can do differential geometry in the classical sense. But c-omega spaces do appear sometimes, so how does that fit in the region? C-omega space. C-omega. C-omega. Yeah. Yeah. Real line. Ah, real line. Well, we can, the machinery is the same. How then can you tell whether c-omega or c-omega, and how do you distinguish it? from the algebra we start with. It depends on the algebra. This machine depends on the algebra one starts with. So we can start with another algebra. And try to apply again the machine. Differentials is a lot. What I actually meant was, is there some, just as you present here an abstract analogue for C infinity, is there a corresponding abstract analogue of C omega, is what I was asking? Yes, I choose it as a motivation, as a standard motivation. Is there an analogue of C-omega, of real analytics? You choose here C-infinity. I didn't try, but the imaginary is quite general. So do you have, of course, this is a general question. If we have a differential, then we can look whether this differential works or not. That's right, if you do infinitely many times, That dimension, well, say axiomatically, well, we don't have to, say, we don't evolve questions which have to do with the space in the classical sense, with the space from which we start in the classical case but the of course from in the classical case the dimension and all other properties of the space give to the algebra we consider particular properties but we start from those properties here so we accept that we have such a good algebra so that to have the result so in your case so we can

47:30 And then we have to look, what can we do with this? Same. Okay. Then the algebra, in some sense, then . Okay, okay. But this does not affect the whole story, because then the algebra will have properties, maybe we do not need. So properties of the space, we don't accept anything for the space, but only for the other one. For instance, we don't accept that to formulate Einstein equation, we need a metric. But I don't say that the space has a metric. The space is a topological space. So we assume that the metric is on the other one. Yes, because the space does not intervene. And of course, we may have, I didn't look at it, but probably we may have a representation theory. Say, under which conditions for the enterprise used, one is led again to the classical case. but then we have to think of the properties of the algebras that one has to put on them except for them so that to go backwards to go backwards from the algebra to the space yes yes yes of course even in this case we have a such type of algebras whereby consider maximal ideas in the case or other type of ideas in the case so we have the space The important thing is the algebra, which we start with, and the differentials. So this is differential geometry, the machinery. The machinery does not have anything to do with the space. And if we have analytic problems, analytic questions, then differential, cohomology shift is involved. I mentioned the Poincarélema, how can we prove Poincarélema in it? But we accept it for the abstract formulation. And in particular cases, important particular cases. You accept it as an axion? We accept it that we have exact sequence,

50:00 which means we accept Poincarélema action. How can we prove it? But we do have it, in particular case, in Rosiger, in Euron. So without recourse to constructability of the... No, no, no. In a nutshell, yes. You consider the shocking space. So somewhere there was, but not the final form of the space. It was hidden somewhere. It appears to me, Chris, that this such a scheme shows that space is like really a scaffolding. And what actually you can do on space is what really inhabits on space, say this function, whatever you have. For the construction of this abstract theorem, we don't use the space. But if we want to apply this machinery in some particular case, then we, of course, we have to realize that we do have this machinery. So you start with another, we start with another. Do you have the same machinery? It depends. Do you have the differential? And do they work in the same way? And even because there are some categorical, say, information, you don't have to stay at this space, shortly. Again, you can consider projectively. It's not the use of space. But the machinery works. This is a general kind of physics question. I was always confused about the connection, the differential geometrical connection, general relativity and gauge theories, where you see the connection as, well, the U1 connection as, for instance, the bosonic gauge field, or the Christoffel connection, in my language,

52:30 as, in a sense, you can identify with the metric, with the symmetric spaces. So my question is, the difference is that in one case, you see this principle, the equivalence principle arising in the case of gravity. So in a sense, well, anyway, so let me just leave it at that and ask the question, is there, therefore, any obvious connection or way to think of an equivalence principle with gauge, the gauge connection, the way that you can identify the Christoffel with the metric? I would say, in general, we have all these notions even here. So, for instance, we realize that from a metric implies a connection. well we of course we can have a connection with automatic so the again the question the the question is the same s in the classical case the equation that does not enter does not the the possibility of answering because you have other machinery other functions this is the profit you have the same einstein equation as before but this equation is valued the former did not. And which is the meaning for you, for a physicist? That's it. We have metrics. Again, we cannot say that gravity is due to metric. This is a misnomer, again. Misterminology. Why gravity is due to metric? No. Metric is a property of the space. what does reality have to do with space well some people say that I've been corrected actually no I asked some question to another physicist and he said who's a relative he said the metric is dynamical I say even if these sounds peculiar or strange we say because we accept that the field is is the object which determines other things If, say, can we, well, let me formulate it.

55:00 Space is the set of the object which constitutes it. So these objects are the fields. And so gravity is a field. And so we want to know its relation with other fields. It's OK. and as a field means we have a connection connection the field is the connection connection is the field itself it's not a metric metric is our interpretation of that particular field but here let me say it again we have another possibility of extending the in more space than at the beginning, than former. This is something. And Janis made a nice interpretation of that phenomenon with Finkelstein, where we have some infinity in the systems. system they use is the, well, William calls it the Edicton-Finkelstein coordinate system or et cetera. So there we have a singularity. And then Finkelstein changed the algebra essentially. And so he extended his equation. And what did that mean? Well, you mean singularity? Yes, yes. And Finkelstein said that probably there we have an antipartic. So this is the information, by extending the domain of definition of the equation, by removing the singularity. We don't have to remove the singularity of Ullembeck, again, this is Ullembeck, by removing the singularity of the north pole, of the pole. So we can apply the enanthema of Rosier, where the singularity is inside of the fine to try to remove the singularity it's well of course it's a problem of explaining or in a reflection of this possibility

57:30 I didn't understand your reliance on the Leibniz rule Yeah, the behavior of the connection of a derivative on a product. Yes, d alpha times s equals alpha ds plus s tensor dn. That's a relation which we accept for. Let's say the derivative of an F with a vector field. Yeah, it's a derivation. And is this crucial to what you're doing or could you lose it? Sure. I didn't understand where it was crucial to what you were doing. Crucial as a notion, the notion of a connection. Of course, this means that. So no, subsequently it was not crucial, but it would be nice if it sounds like a line. We don't have, we don't stop here, then connection would be a tensor. Because it would behave good with respect to the inquiries. A tensor is something that respects the algebraic. So in that sense, connection is not a tensor. Connection is not a geometric notion. Connection is an LED notion or algebraic notion. But curvature it is, because for curvature we do have R alpha times S equals alpha R of S. So this is a tension, because it respects the equation. In that sense, we can say that the field, if we identify field with a connection, which means we identify the field with the equation that it generates, from the solution of the which is inside, then if we say that the field is a connection then we realize that the field is not a geometric notion and that's a this also comes together

1:00:00 with the interpretation of gravity because we mixed the result of the gravity with the gravity. The result of the gravity is the curvature of the connection which represents the gravity. Every field is represented by a connection and by the carrier. The magnetic field the connection is the field and the carrier is the photo L the line C so in the case of gravity again we have a field and this field should have a domain of definition say a carrier this is the gravity there must be a if it is a field then this field should act well should have a carrier. The field is the essence, the starting point. We cannot say that we have the result and in that sense we say that... In that oblique sense, you are calling the field to design the way. It's a blind part of the talk. It probably is the most interesting part of this discussion, question. That is why I wanted to run again. Thank you.