 OK, the attractiveness of the subject, which I have to present, I just want to show what makes it interesting, is the Schaele curvature. And most specifically, it is very much concerned with Schaele curvature bounded from below by something. And these kind of objects remain in many folds, which I say about there in a second. So what makes it interesting is that it displays features of two kinds. One, characteristic of geometry, something which we can see when it conveys the main individual space, which we presume we understand. They are very well shaped and we have good description of them. And we may think we understand them unless we go deeper than because many things we don't understand, but basically we know what they are. And then there is topology, which we also understand for opposite reasons. And when objects are kind of not like that, but maybe like that, but the point that we are studying in such a way that we kind of all this complexity disappears and we come with simple invariance, kind of linear algebraic essentially, homotopy theory or homology theory. And there are few. And actually I know only three examples in geometry, which medias between the two. So we have on one hand we have like convexity, on the other hand we have topology. And there are only three domains in geometry, which medias between two. And the most developed for today probably is simple-active geometry. And the second one is gauge theory in dimension three. So you don't understand theory, specifically kind of. And then this K. Likovician, specifically bounded from below, because all these objects you cannot classify them and to say where they are. So interestingly enough, that the specificity already seen in the algebra going behind them. And in all three cases, I would say, this algebra was first kind of used and kind of manifested in physics rather than mathematics. Right, say, simple-active geometry associated with classical mechanics. Gauge theory is physics and physics is first realized. There are some special about three four dimensional connections. And the same about K. Likovic, which appeared in general relativity in the Einstein equation, at least in the Hilbert derivation or Einstein equation. And mathematically, still K. Likovic is well behind all the three until we don't quite understand what it is. And the kind of interesting point is that we understand it so poorly, right? And so in just my kind of interest recently was the following reason that some new development happened in the last couple of decades. And so more we learned about the K. Likovician and more its relation to simple objects like this, like convex ones. And so, but instead of showing that we are giving better perspective of K. Likovic, it is showing how poorly we understand this one. So we can see something new from the point of K. Likovic concerning simple objects and you'll see how badly we understand them. And so let me give an example, which I find quite amusing, exactly of that kind. So say I just formulate some simple kind of question and some answers related to K. Likovician and the method involved to study this. And I must say that concerning comparing other domain of geometry when you speak about curvature, the level of sophistication of techniques here is much higher, because one of the main tools here is the index theorem for Dirac operator, which is not, doesn't appear anyway in Riemannian geometry, kind of of this kind. And so let me give an example. Where is the razor here, which we shall eventually converge. So I want to formulate very, very simple question, but first I have to remind you kind of some terminology. So when I have a, say, sub-many-fold of hypersurface, particularly in the Euclidean space, say Y in the Euclidean space. And this may be first dimension, n minus one in a second. I want to have any dimension. What is a, what is a curvature? When I say a curvature of this, a curvature of this Y. So here we curve, you know, what the curvature is. If you have a surface, you can see the old geodesic line with them and look at their curvatures, in, externally. And supremum of them, it takes supremum of the curvature. And of course, if it's hypersurface, that's particularly easy, we just take this hyperplane, then it's locally become a graph of a function, which is have vanishing first derivative, second derivative of quadratic form, and the principle, I know, principle values or principle axis of the corresponding ellipsoid is curvature. So how much it curved? And so the question which we raise is as follows. I have a sub-many-fold of dimension m sitting in the unit ball in the Euclidean space. And for example, it's a simplest example of boundary sphere, but it may be something more complicated. And the standard examples we might have in mind, you can consider, for example, product of two spheres, or n spheres, if you have, say, circles, right? If you take them m times, this naturally sits in the Euclidean in the ball of radius square root of m. I'm sorry. It's a dimension to m and radius square root of m in the Euclidean space with dimension to m, right? This is an example. And if you want to go back to the unit, to the unit ball, you have to divide this, scale it by square root of m, and then here we have one. And then, by the way, the curvatures become, so these curvatures, the unit circle, have curvature one, when you scale it, it will be there with curvature, right? So when you put this product of circles inside, curvature behaves like this way. And so this square root of m must be kind of very respectful of that, because this, of course, Pythagorean theorem. And Pythagorean theorem, of course, everywhere will be there. And so, yes, people can say, oh, use Pythagorean theorem, and they smile. But I think if in 2000 or 10,000 years, you write a textbook on mathematics, and then it will be written, this Pythagorean theorem, and little applications like Hilbert spaces, you know, quantum field theory, everything, the kind of trivial kind of, in a way, manifestation of Pythagorean theorem. It's everywhere. Some of us go back to something. It's amazingly how well it works and how often it being forgotten. So my experience, I'm going to take proof, say, in six thousand years, I had the experience in the time you have a 10 pages argument. Behind, of course, this Pythagorean theorem. You throw all this argument, use exactly theorem, and the argument disappears. It's an extremely remarkable thing, yeah? It's used everywhere. This quadratic linearity for squares, not for lines. So it should be respect to this square root of m. And here, of course, this square root of m has many other kind of, I feel, appears everywhere, and probability is everywhere. So this Pythagorean theorem. So it will be everywhere behind what we are doing, up to a point, because this can be the basic rule. It has quadratic rule, and all that computational mechanism of reminding in geometry depends on that. But once we understand this rule, of course, we want to break them. And this, of course, purpose of my talk, lectures will be to indicate possibility of breaking it, which I don't know how to realize. And this, of course, common, not only for mathematics, everywhere in science, first you learn the rule, you learn them very well, so you know how to use them, and then you learn them even better, and then you know how to break them. Which was well articulated in the lecture, not the lecture by Fleming, when he was describing how he was discovering the discovery of penicillin. As he said, there are rules how to operate with bacteria, but the main point is how to break the rules, and then make discoveries. But this is not easy. So first I will, so far it will be describing rules, breaking them up to the audience. I don't know how to do this. So, but this is easier, so we know what the curvature is, and so the question we raise, so I give you this manifold y, our dimension, m, and I want to put it into this ball, and I want this maximal curvature of y to be small, so it's minimized. So this is, this maximal curvature, this maximum taken over point y, and this after, with respect to all this embedding, so what is the optimal kind of way to do it, to put it with minimal curvature. And so amazingly, how little you can say about that, and all you can say only when scaly curvature is concerned. So what you can show, rather elementary, that this kind of exercise, if you don't do it yourself at home, I will come to that, is that if this y, say, just for simplicity sake, let it be n minus 1, just to invoke. So if y, as topological manifold, and this may look admissible metric with points of scaly curvature, actually it's not like that. No, I think it's better to say that, otherwise it's actually quite essential difference, quite respectful. Then by elementary argument, you can show that this maximum of the curvature of y, n minus 1, in the ball, might be greater or equal to square root of n. We're missing some constant here, maybe 1 1⁄2 just to be comfortable, I don't remember. So it must grow at least a square root of n. This by elementary argument. However, for this, you have to know that some manifolds do and some don't have this property. And it seems this example, to write down at least, is n dimensional torus. You know, and this was conjectured by thesis, actually by Gerrach, in early 70s it had been proven in some cases when Shonyahu and then by using minimal surfaces, and then with the use of dirac operator. So torus is like that. So for torus, this is true. So if you put this very product of the circles, as in this example, it doubled dimension. And we don't know actually what happens for double dimension variable. So we cannot prove that this is optional embedding. At least I can't prove. And I don't think it has been studied. However, if you go to a dimension one, you know you cannot do better. But if you go deeper and see not only the result, but look in the proof, you see this estimate. And which is not optimal either, I guess. But then already you have to, so the proof of this elementary statement, that torus, when it goes to the ball of unit radius, might be curved that strongly. And then if you look at the example, when you actually embed it, it curved even more. So probably a right exponent will be something like, I guess, maybe 3 over 2 or something. Possibly, I don't know. Certainly, it's pointing something else. But what is the assumption on the co-dimension? This is co-dimension one. This is true for co-dimension one. And this is also known for co-dimension two. And then it becomes co-dimension three. You can't say anything. It's amazing. Nothing you can say co-dimension two. I mean, except square root of n still works. Square root of n works for any small co-dimension. But n works only for the proof, which I know. Only for co-dimension one and two. And moreover, with certain degree of sophistication, if you go to the next level, there are other examples of that kind. And this I will describe, which don't admit this metric. And among them, there are exotic spheres. So there are spheres actually under of dimension of this dimension plus two. So there is this exotic sphere. So many false homeomorphic but not deterministic spheres in these dimensions, which are known also not to admit the metric of Poisson-Kirikovitch. And in this case, it's truly kind of application of the index theorem, actually a subtle index theorem, which was proven 10 years after the first one by C.A. Singer. And so such spheres exist for them. But I say it's true. However, this estimate one over n, I cannot claim it. Because I use for this, you have to use some theorem, which is published, but which proof I don't understand it, who knows. I'm really certain it's true. But the proof of this particular theorem, it's beaten by a lot of log camp. And it is not really written in detail. And so who knows the proof may be incorrect. But the theorem, I'm pretty certain it's correct. So this was how I'm using. So the proof eventually goes by using index theorem, diracoperated, but you prove absolutely elementary. The only thing about this, what you use on the ambient space, only what is the scalar curvature of the sphere? You only lose local geometry of the sphere. You don't use the scalar curvature of this as something. You don't use geometry at all. And actually prove much more stronger statement. You don't prove not only about this eigenvalues, whatever, but you prove a much more profound statement about this geometry. And there is absolutely, at least I tried, no elementary way to see it. So the point conclusion is the geometry of the sphere, you found out, I thought I understood it fully. I realized I don't understand it at all. The simplest question you can't answer. And exactly because there is some progress of, and this is because of how science works, the more you understand, the less you understand. That's very satisfactory. And this is what I want to explain. This was kind of preamble, as to say, but also kind of quite amusing. And so you may have questions, which you may point out. They are so absolutely elementary. And if you take very simple manifold, like product of circles, you can say something that is in local dimension. If you put here two spheres, it will be the same picture except if you hit now four, right? And then you don't know what happens. So if, when you put the product of spheres in the Euclidean space, the curvature must grow like square root of m in high co-dimension and like linear, its co-dimension is getting smaller. Absolutely no, no guess how to prove this. And because the scale of the curvature here, of course, you do have point of scale of the curvature on spheres. So it is a rather amusing situation. And I will give you another example later on, even more elementary, in a way more elementary statements about Euclidean geometry, even about convex sets, which are posteriori follow from relatively sophisticated techniques used for the study of scale of the curvature. Now definitions, yeah. So what is scale of the curvature? And that's tricky, of course. So what kind of definition you want? And specifically, I want to know what it means scale of the curvature is greater than something, this point is your negative constant. Once you know that, of course, by limit, by using dedicating definition of real numbers, you can say what the number is. But for me, it's most crucial understanding that. So first, it is assigned to Riemannian manifold. So I have Riemannian manifold, so I smooth manifold, and Riemannian metric, which is, of course, for pieces, I look at this symbolic thing. And it's unclear what it should be, correct definition. And for the moment, what you have to know about that is give you a metric, this metric, and the volume on the manifold, how it is defined. It's just defined by simple, simple, simple functionalities and the volume out of the metric. But then, of course, in a second, I will be using, of course, some calculus to make sense in computers. So first point is, so a scalar curvature of this manifold is a function. So it's a function manifold defined in terms of that, and it must be defined in invariant terms. So manifold isometric. And then fundamental property, kind of the most fundamental property, which is, of course, it's curvature. It has some feature of the curvature, which I say in a second. But the most fundamental is additivity. That if you take a scalar curvature of x times x1 times x2 with the metric g1 plus g2, when this sum is Pythagorean sum, of course, which meaning that when you say distance, it means this. So the distance in the numeric is understood like that. So I put here this, right? Then it adds up here. It's equal to the sum of the two, the scale of x1 plus scale of 2. So sometimes I'm writing it's like that, or sometimes I'm writing this scale of g. So when it's add-tomatic and Pythagorean sum, it's addit, this one. And this already shows that dimension plays a certain role. Yeah, you just cannot limit dimension, which is very different from other curvatures. So if you understand it one dimension and send another dimension, because of this additivity, it's property number one. The second property characteristic of any kind of curvature, how it scales. If I multiply this metric by constant, then it's OK. So it's a fixed notation. Did you just fall down? OK, I can do it. So this means I multiply this. But of course, when it's addit-tomatic, you have to remember that it's actually right scaling is quadratic. So think about this metric, not as a metric, as a quadratic form. But for the moment, I keep it to this way. And then the rule is the scalar curvature of lambda x equals lambda minus 2, the scalar of x. Sorry for you. In large sphere, in a second, I say that. So what is the normalization of this? Maybe you're going to anywhere further. It's important to remember. For a normalization, and this is the key point for computations, the scalar curvature s2 equals 2 everywhere. It's not 1, but 2. And that is a good reason it might be 2, not something. And then it adds up for that. And now you have to relate it to geometry. So what you can say after that, now we have to specify it. There are several ways to say it. Algebraically is, beside of course, being invariant on the disometers, which is implicit. You have to give some a little bit of substance to that. And one of them, if you work in terms of Riemannian metric and remember that Riemannian metric, jj, is a family of quadratic form. So it is quadratic form jj, depending on the point. So you can say, if your manifold locally is Euclidean space, you have these functions, jj. That's it, sir. And then they may pose a definite quadratic form. So it's a field of quadratic forms. So it's a bunch of n squared over two roughly functions. And curvature is expressed by differentiating this function. So any kind of curvature. Involve g, ij, a priori, the first and second derivative when you compute it. And scalar curvature linear in the second derivative. And then that's it. Then it's uniquely defined with what I said. So the only fact, it's being linear in the second derivative. The only expression we can write down, which will be, which gives you a kind of invariant expression, involved in the first second derivative, linear in the second derivative. And this is how it appears, kind of the reason it so kind of appears in formulas and computation in physics, because linearity in the second derivative. Which is certainly not quite satisfactory geometrically. And this, of course, linearity has something well-agreed with this linearity, with this additivity. But additivity is more kind of fundamental thing. Another way to say it, it's kind of quasi-geometric. Now I give some description, which is kind of geometric but extremely deceptive, which is often appears as justification with which completely useless, completely wrong. I think there's good reason why it's wrong. However, it's very easy to say. And then I will say. And then this give you illusion, you know the definition. But it will be illusion. So what I do, this can be done in several ways. And one of them is as follows. So if I have too many formulas of the same dimension, x1, in some point here, on the other hand, x2, g2. I want to say what does it mean, this inequality. Scalia of x1 at point x1 is less than scalia of x2 at x2. So I give kind of geometric definition of this inequality. Again, once you know this inequality, you can, scali curvature become kind of linear ordered quantity and also is addition because we can multiply many false and so define the scali curvature. And it's very simple. It's said that if I take both in x at the point x1 of radius epsilon and compare it with the both of at x2 here also of epsilon and take the volume, the both n-dimensional manifold, n-dimensional volume, then this will be greater than that for all sufficiently small epsilon volume. So if curvature becomes smaller, both become bigger. You see, it's important to have strict inequality otherwise not true. For all sufficiently small. And this small depends on the point, the pole, depends on many, many things. And that's what makes definition kind of useless because you cannot integrate it. And I tell you nothing about actual ball. You take small epsilon, but it doesn't go to zero, and this inequality immediately breaks down. That's the problem. And usually, in the description of different curvatures, this kind of inequality integrates. You can say it not for volume, but for some other invariance and then you can integrate it. This doesn't integrate. And then, of course, it's uniquely defined curvature. For this condition, it defines curvature. You can check, by the way, that this agrees with this identity. But again, using Pythagorean theorem, this property of both is kind of stable under the multiplication of manifolds, well behaved. There is another way to say it, slightly more informative, but still not very good. But on the other hand, this, what enters in some moments, it's more serious study of scaling curvature, is that instead of this little ball, you can see the tiny little spheres. And instead of their balls, you look at the integral mean curvature. In a second, I will play with this. And then there is similar inequality. So the more negative curvature, the bigger this integral. And when you turn the world inside out on this point put to infinity, this mean curvature is called a kind of thesis, a kind of muscle of some kind, which I don't understand, but it has some reinterpretation in general religion. So that's a little bit serious. As it stands, you can choose. It looks kind of nice, but it doesn't integrate. And so I don't know a really good definition of that kind. However, well, something, something will be there. You can make this kind of definition, but this will take some effort. And to go first, we need to have a means to actually compute or evaluate scaling curvature in specific examples. OK, so let me take a little pause because this again still was rather, so you have to prove something that is really well defined definition, which is give you the scaling curvature. And as an example, OK, kind of exercises, if you try to do it directly without knowing some way how you manipulate with formulas, it will be quite difficult. And the example you have to have in mind, the scaling curvature of unit sphere equals nn minus 1. And you see it may look strange normalization. Why don't you say it 1? But you see because I have this additivity, you just force if you accept for two-dimensional sphere a scaling curvature is 2, then it's like that. Or if you, in other corollary of that, either a hyperbolic plane, if you look at a hyperbolic plane, it is metric on R2, say coordinates x, y with the metric dx squared plus e to the 2x device squared. So this is a good representation of a hyperbolic plane. So I have one coordinate x squared and we go another exponent, you exponentially expand with this coefficient, it has a scaling curvature minus 2. And it follows from the previous definition because you take this and you multiply it, for example, with S2, then the scaling curvature must be 1 and you have to check it. But indeed, this volume property will be there. So you see once this definition allows you even to understand what is negative curvature. Only looking at example is positive curvature because it's additive. And this, of course, has some meaning. I'm pretty certain quite significant meaning what makes it so nice. Which, of course, makes you ask what happens when dimension goes to infinity, but then it's a big question, so what happens? It's very unclear what happens in the infinite dimension setting. Which is, I'm not saying it's meaningless, but it's a big, big mystery. Because it begs to be treated simultaneously in all dimensions, including infinity. But this, but now how to compute? Yes, before starting proving anything, you have to compute. For example, I want to explain, maybe today, that y, what I said before, that if manifold y admits no metric, so if its scalar curvature cannot be positive, then y, when it sits, say n minus one in dimension n, then curvature must be greater than square root of n. About approximately, I'm not certain what is the constant. At least it needs some kind of computational means. And this will be crucial for what we do. So all proves dependent on that. So maybe historically, which is, well, first I must say I'm writing something and I put on my web, so it will be related to my course. See, I feel not exactly what I'm saying, but more or less what I'm saying in a different order. So historically, by the way, the story was like that. That it was the same, I think in 1962. But I appeared at the single theorem for elliptic operators. And then the next year, Lichny-Rovich proven that there are many faults, actually very simple many faults. I bring them fast in a second, which do not admit metric of this positive scalar curvature. And the basic example, which seems rather impossible to prove it out in the theorem, in some cases you can avoid direct use in the theorem but only kind of fragments of it, how the elementary fragments of this. But here you need to find one of the basic example is, it's a coulomb surface. So it's given by four with this equation. And this sitting in the complex projective space, right, so it's real dimension four. Right, so complex co-dimension two, a one, so it's a real dimension four. And this is a kind of simplest instance of many faults where Lichny-Rovich theorem applies. And so this admits the metric of positive scalar curvature. Nowadays, due to our solution of collaborative conjecture, we know it admits metric of flat, which has flat zero scalar curvature, but it cannot be positive. And what he used some formula relating Dirac operator in kind of pieces, whatever is Dirac operator, which in the theorem applies, which is, he used the formula, this plus one quarter of scalar curvature. So, but the point is, this is something positive. And interestingly enough, this formula was known prior to Schrodinger, who wrote in even more general context. But I don't think, I don't know, I sit in Germany, I forgotten what purpose he had in mind to look at this article briefly. But the point is, we just take this theorem, plug in this formula, you immediately have contradiction, this point is scalar curvature. And this is a very simple innovative proof, if you understand what is a Dirac operator, which I don't, which I'm not, because spinus, as I keep saying, was explained to me, my idea, you're not supposed to understand them. Nobody understand them. You leave with them without understanding, that's fine. Unlike the way you understand the differential forms. But, but anyway, this is a kind of mathematics involved. And then, about 10 years later, about 72, maybe I'm not certain. The idea in the sequence where you force paper by Archie Singer with the proven more sophisticated index 30 mod two. And then it was pointed out by Hitchin that this applies to this exotic sphere of dimension N, K plus one or plus two. So there are spheres, manifold, which are homeomorphic to spheres, but not defiomorphic to them, and which carry a normatic of point of scalar curvature. And the invariant, actually, spin kind of invariant, this is a, also everything was known, immediately have an index theorem. And so the logic is like that, in this kind of proofs. Index theorem tells you that in certain manifold granted sufficient topological complexity, they carry harmonic spinners, whatever they are. So they're solution of some partial differential equations. And harmonic function, but it's different operator. On the other hand, when you have this kind of formula and this is positive and this is positive, you cannot have them. So that's the proof. It's extremely kind of simple proof. On one hand, on the other hand, completely telling you nothing about the geometry of this manifold. So when a theorem came up, it was, hmm, what to do with this? It was very unclear. And then this was kind of development. And then, well, there was a different approach developed about five, six, seven years later by Sean Yau. I think following some of the suggestions with thesis, whether it was some conjectures and ideas how to do that, and they were implemented first in a very, very heavy way. Now they can be done much shorter. Using completely different mathematics, namely, minimal hypersurfaces, where singularities cause problems. And so one of the main issues, in my view, that Scaly Kovic indicates that there is a relation between, on one hand, Dirac operator, on the other hand, minimal hypersurfaces. And they're both elliptic equations involved by very different nature. And they talk to each other in this instance. But we don't know what they say. And that's one of the big problems. And Scaly Kovic is just their meeting ground. And this is how you can think about that. So what makes it so tantalizing? And there are examples which I explain, when you need a mixture of the two methods today. They do interact in applications, but they don't interact internally, right? So we can imagine a picture. So you can have a mind. So here is a big thing, something that you don't understand that goes here on the ground of Scaly Kovic. And sometimes they meet here, but we don't know how they meet there in their fundamental nature. And you make some conjecture there. OK, so, but now I have to describe basic formulas relating to curvature. How do you find curvature in Riemannian manifold? So one thing about Riemannian manifold you have to know. They will be smoothenetic. And again, smoothness of a metric will be essential, actually, for the purposes of that C2 is sufficient for curvature. So Riemannian metric. And the first point is that you will have this Riemannian metric. And you can locally chair the Euclidean space. And then for every Riemannian metric G, there is, at every point, say, x0, there is Euclidean metric G0, such that the difference G minus G0 is O epsilon on the ball epsilon. So O epsilon squared. Not epsilon, but epsilon squared, which is quite remarkable that Riemannian metric a priori defined as with O epsilon, so it approximated with epsilon error by a flat one. But in fact, this is better than that. Because you can choose coordinate because the group of d-phiomorphism or linear group x operates there and allows you to cancel extra term. And that's kind of quite remarkable. Certainly, it's very hard to say what was motivating Riemannian's definition, right? How much he thought about that. But they're very special among all other metrics, Riemannian metric, the closer Euclidean, then you may expect. However, this is what you eventually want to break. Because when you look at the logic of the proofs, it should go beyond this phenomenon. And then because of that, when you have a hyposophist inside Riemannian manifold, its local geometry, infinitesimal geometry, when we speak about this curvature, is exactly the same as for Euclidean space. So the curvatures are defined to just replace Riemannian metric with Euclidean metric, define all this curvature, principle curvature, whatever I said, applies. And then back to Riemannian at every point makes sense. That's simple. However, what happens on the next level? And there is a second main formula in Riemannian geometry. Because the first one is Gauss theorem. But this is as follows. So I have my submanifold. And you can see that there's a family of this parallel of them. So we move them by epsilon. So you have this family of manifold, y sub t, inside of x, t close to 0, maybe, or maybe not. And each of them carries its own metric called h sub t. How this metric developed. So first one is the first derivative of this. So when you differentiate it, you have, again, some quadratic form. So this is a quadratic form. Now you think about the metric as quadratic form. From this moment on, the same, like Euclidean metric, you don't speak about the length of the vector, but the square of the vector. And then you have linear algebra to be disposed with Pythagorean theorem. So already all you say is behind it is Pythagorean theorem. I want to say it's really highly non-trivial and fundamental phenomenon that allows you to make all computations. And at what is the derivative? Now this is supposed to be kind of the first line on the Euclidean differential geometry, even in the Euclidean space, right? So we all fail, yeah? Amazingly enough, in textbooks, people don't know it. Most textbooks are on the way of that. They write some nonsense, but they don't say it. This will be second fundamental form. It's exactly curvature at every point. It is a quadratic form, right? Because I said you take this plane, you think about it as a function. Function has zero first derivative, so there is second derivatives. It's a quadratic form, the second fundamental form. And it depends, of course, on the sign. Depends on which direction you go. Reverse direction of time, sign changes. And that's, again, this plane is sign crucial, right? So it's a form depending on direction, plus or minus. Actually, it has an extra second. How are you defining the deformation again? Yeah, you take family of parallel hypersurfaces. Just parallel. Yeah, parallel, right? By by distance, it's displayed. There is simply a sports function. And it's first fundamental form, curvature of the hypersurface, right? This is, I would say, that's minus one formula in the Riemannian terms, right? And now I come to the second one. First one, I will explain later on. And you see, this is kind of tricky point. Myself, when I was learning Riemannian geometry, and when you look at textbooks, you can find that people write. And you start writing simple formulas that take pages, when it takes usually no line. Even no line, just more or less definitions. However, there are two definitions, of course. Second fundamental form is this way defined using Euclidean structure, or Riemannian structure. However, all you need is a fine structure. You're correctly defined, and you don't need it. You can define properly second fundamental form, even in the defined space by getting symmetric. And it's slightly different definition, right? Because Hessian is defined, and finally. And interestingly, the difference is you can see it already in the Newton kind of formulation of the dividing law of physics in the first and the second law. If you take the first law of inertia, you have defined world. And everything is defined, but you don't normalize squares. And when you have energy, you have squares, then you have the second law. And this is why logically, you have to separate first and the second law. You can make mechanics with first law, but we will not fool you with mechanics. It defines structure in the space. And secondly, it is another one. And it's kind of really also fundamental. And the second may be defined if you have measurements of time also, not of space. But you have to measure something, right? But if you only have concepts of a quality, so what is the straight line? So I go straight because I move my hands in the same way. So same with my legs, define a fine structure in the space. Not for me, also, for any buck. And this is a major mechanism in your brain. This allows you to understand the world. Same with my motions, give you a fine structure in the world. But energy or measurement of time is a high level structure. And this is different. Here, I use this because I need to write the second formula. And now, what will be the second formula? So I have my second fundamental form. I want to define it like that, because I'm really possessed by this quadratic form depending on c on this y sub t. Now, I want to know the next derivative, and I want to differentiate that. What is that? It's, again, a quadratic form. What is this quadratic form? Now, if you, and then, this is a remarkable formula, and it's kind of written by Herman Weill, and probably in 20s, whatever, and probably was known before to Cartan, or probably other people. And this is the main formula that you might enjoy. If you know this, you can compute everything. You don't have to know anything else, if you properly. But what is this formula? What is this quadratic form? And this now depends on the curvature of the manifold, but it has the following shape. This equals, first, it is quadratic form. It's convenient to turn into operator, right? If you have a Euclidean space, a Riemannian tangent space of a Riemannian manifold, it's Euclidean space, and you have quadratic form, always there is operator associated to it, symmetric operator, or y sub t, right? If you have operator, then you have this quadratic form with respect to a metric. Scaly product. Actually, I don't want, it's enough for me to have an x. Just like a quadratic form rather than a linear form. And vice versa, any quadratic form uniquely defined by this operator. If it is symmetric operator, it is symmetric. So I'm operating slightly more convenient here just to write it, because the formula is the following. It is minus a squared, just square of the operator. So it's more handy to take square of the operator than the quadratic form. And here, plus some other operator, and this is defined with curvature. In a second, I explain how it defines with the curvature. It encodes curvature of the operator, so it's quadratic form, but it depends on which direction you go, so it is three tensor. There are these two directions and then normal direction. So it's a number assigned to these three vectors, actually operator. In order to have a number, I have to also turn it into quadratic form. So as a quadratic form, it depends on these two vectors, but there were three vectors, so it's kind of a three tensor. It essentially, I describe it as a curvature of the manifold. And of course, you have to show that this has enough linearity and algorithm doesn't depend on your chain. You have a surface, et cetera, et cetera. But that is the main formula. So what exactly it is? So if I have this normal vector, and I take this, A is an operator corresponding to second fundamental form. A is a zero, a times zero? No, no, it's OK. Of course, at every moment. But you do it applied, if you define curvature, you do it at times zero, but you do it for all t. You can specify it for t zero, you absolutely right. But this is for all t. So this is the so-called Gauss-Cadazzi equation? No, Cadazzi is not here. Cadazzi is the next derivative. Cadazzi is the third derivative. That's a Gauss-Cadazzi equation. No, I don't think so. I don't think Gauss-Cadazzi, no. At least it's a wild equation. This formula, you see, it is a wild-tube formula. It's called a wild-tube formula. Cadazzi is not that. Gauss-Cadazzi is the third derivative. Gauss-Cadazzi, no, no, no. Gauss-Cadazzi is relation with external internal curvature. It's really different, much more fundamental. Gauss-Cadazzi, everything is corollary over there. That is square of that. And this is sectional curvature involved. So again, so if you have this vector, this is in y and this normal vector, then this is just sectional curvature on display. And now, and just to check, even in the Euclidean space, it's just quite, you can just look at this circle and see how it develops. The curvature is 1 over r and this length is r squared. You differentiate exactly how this identity of things cancels off very nicely. So my understanding is from this example, just circling the plane, this formula must follow. But I don't know exactly rigorously how to say it. So it's a break question. All formulas in differential geometry, you look at one example and they, by far, from the realities must follow. So usually, it's being proven in pages and pages of certain computations. But you don't have to make computation. You know there's only one natural way to write it. And this must be the right one, which agrees. It's natural, simplest one, agrees with this example. It must be true. But I don't know where this rigorous to prove this statement. Any time you can verify it, all this works. And the point is that people who work with these formulas that don't write these formulas, they just know them. And those who actually write them, don't understand them. And that's the problem. So if you impossible look at the net, look at textbooks, you never, you can find any explanation and you can reasonably prove any formula. It becomes huge computation. People use moving frame, exterior form, God knows what. It just follows from one simple example in functoriality. But I don't know where it's written. And that's one of the big technical problems. Because otherwise, that's all you have to check all formulas by yourself. It's such a pain to write them correctly. So that's the formula which we have to know. And this allows you to, and then you have to combine it now, of course, with the Gauss formula. So what is the Gauss formula that says in this case, specifically for the Scaly curvature? So first, look at the Euclidean space. So what Gauss formula says and what is the usual definition of Scaly curvature is. But that's kind of the formula to remember. But I don't think it is the Gauss catastrophe at all. Gauss comes next. Gauss before that. Gauss-Caudati is not Gauss. Gauss-Caudati is this. It's something separate from it. No, no, Gauss-Caudati is an Euclidean space. Third derivative involved in the gravity of the curvature. And the Gauss, no, Gauss is just for surfaces, basic formula for Gauss. The curvature, sectional curvature of the surface. And this is the principal eigenvalue of this A, or principal axis of this. The product is the curvature. That's Gauss formula. But it's not a real that. It doesn't apply that. It's different. This is called in all textbooks, Gauss. This one? No, which one? Made in physics textbooks, not mathematical. It's never. It's not a textbook anymore. It's standard in Einstein equation. Yeah, sure, sure, sure. Of course, they always use that. Let me say it. Those people who use it, they know that. But in differential geometry textbooks, it's not there. But this while tube formula, because it appears in the while, computation of volume of tubes are out of many forms. If you take trace of that, when you trace it, become rich, and become volume, and become this while tube formula. This is a precursor of the while tube formula. My wife, Kadasa, don't understand because Kadasa is a third derivative. Kadasa always involves a third derivative when you look at surface in the space. And you write this Gauss equation when there are curvature determined in high dimensions. And there are integrability conditions. And this integrability condition, third derivative of the metric, is not here. It's only said, Gauss-Kadasa needs third derivative, Kadasa. And Gauss, only second. And this is similar to Gauss, but different. But now what is Gauss in our case? We have a hypersurface in the collision space. And at every point, you have quadratic form. You have this principle, this eigenvalues of the A operator called principle curvatures. Because, as I said, it's quadratic form. So it's diagonalizable, it has these values in the principle direction. These are principle curvatures. And so one of the formulas says that if you speak about sectional curvature, which I didn't define, of a manifold, of a remainder manifold, is, so it's something assigned to have tension space. You take two-dimensional planes. It's numbers assigned to two-dimensional planes with sectional curvatures. And then it says that if I have, I restrict everything to some two-dimensional plane. In this two-dimensional plane, I have different eigenvalues. I diagonalize on this plane. And then sectional curvature of sub-manifold equal to the sectional curvature. So kappa on some dimensional of my y equal this kappa on x, largely this product. There's Gauss formula. On every tangent plane, I take this principle eigenvalues. When I restrict quadratic form, we see they're different. They're not describable in terms of the original eigenvalues. But you have them there. You multiply them and add to the sectional curvature of the ambient space. And you get curvature of your inside space. For example, if it's Euclidean space, this term absent of x disappears. So in the case of surfaces, you have a regional Gauss formula. And so what's remarkable in this formula, what excited Gauss, of course, that you have the surface and you have this. So two eigenvalues, you multiply them. What you get, invariant and the bending of the surface. You deform it into the space. Without tearing it, this number doesn't change. And this is a rather amazing why it works. Because you see, if you look preliminary, you should think it shouldn't be true. If you kind of understand what you're doing, you say, oh, it can't be true. But then there's some secret cancellations somewhere. Or some change. Because there's wrong derivatives. A priori, the first formula is derivative, but they cancel off. This may be second answer, but they cancel off. But the rest, of course, are derivative. So this is the formula. But now coming to scalar curvature, you need all this full lambda i. And the formula for scalar curvature of y in the Euclidean space, it will be sum. And where you take all ordered sum. Here, essentially, you could remember this. You don't take symmetric sum, but you take all sums. So the surface sphere you have two. And this agrees with another definition of the scalar curvature as a kind of trace. And trace meaning you have this, your sectional curvatures in the Riemannian manifold of each plane. And you take an orthonormal frame of vectors, take all this plane, we should say, and add together all the scalar curvatures. I'm sorry, all this sectional curvature. Then we get scalar curvature. So which, again, we can think as some kind of trace. Or we can integrate with the right coefficient. It's, again, the Pythagorean theorem is going to go over the sphere. The same as some of the quadratic frame. And so that's traditional definition. And it takes some time to realize that exactly agrees with the way I described it. It's really kind of, again, the point is that the way manipulating with this kind of computation, they are routine. They don't have to think. If you, yes, you do it by naturality. However, it's easy to break this naturality. And even great people making similar computation, and Hilbert were making some rather bad mistakes, for thesis, know that. Not in here, but in the mechanics, actually. Einstein equation was OK. But Hilbert was also writing some derivation of the dynamic equation from Boltzmann equation. He was exactly not properly using the trials here. But again, this net, not formulated. I know how to formulate it in just once and forever. Eliminate all computation which you make. You don't make to have computation. You have to look at examples and write only the simplest formula fitting of the examples. It will be right. But the simplest is tricky. It's not simplest as analysts understand. But simplest in the true sense. OK, so that's about scalar curvature. And that's enough, for example, to compute it for the sphere. If you have sphere in the Euclidean space, it's sectional curvature 1. So it's n dimensional sphere. So there are n by n minus 1 pairs of normal vectors of different dimensions. So a scalar curvature of the sphere is what I said. So yes, it's worthwhile to look at some examples. How this thing being computed. Now I'm not certain. I remember this formula, say, I put them on the web. So I go next step. Because some one-infinitiveness computation numbers are kind of crucial. But still I can explain now maybe what I said about embedding of sub-manifold, say, of dimension n minus 1 into the ball, why they should have big curvature somewhere if they admit normative of scalar curvature. Namely, this estimate that this maximum of this lambda y. Now I introduce them. Must be greater. Well, maybe I put here 1 half. I don't remember. If this y admits itself normative of scalar curvature, which, by the way, again, is very tricky. Because there is no single instance of such a manifold beyond dimension 2. When you can show it by simple means. You need to use some techniques rather sophisticated. Well, different degree of sophistication. But it's not a talk from this naive geometric definition you can't prove. Probably for fundamental reason. However, this is elementary. How to prove that? And so you have to see in the Euclidean space, of course, there is no problem, right? If you don't have bound on the sides, you can embed everything. You have any manifolds, like sphere seed. Any manifold can be embedded. Any hypersock is sitting there, and maybe there's anything. But for example, with the torus, which torus certainly n minus 1 seeds are n plus 1. And you know that torus doesn't admit episteriori, which we shall prove at some moment. It's not a talk. Simple theorem. It was a big conjecture. Why the torus admits normative of positive scalar curvature, not? But then from there it follows. It sits in the Euclidean space, but you cannot compress it into a small ball. If you start bringing to a small ball, and necessarily regardless of what metric induced here, this metric doesn't have to be a positive curvature. It must, curvature must blow up with this rate. But in fact, I said it must break with the rate much faster, like at least 1 over n, which is more difficult. So why it is so? And this is just elementary computation. Because we didn't get two n maybe torus. They circle in bigger space, circle around it, circle around it, circle around it, and they have this. And then, of course, curvature, in this example, you see it blows, I think, like n squared or something. In this kind of simple embeddings. And so how to show that? So the scaling curvature had this formula in the Euclidean space. It will be product of lambda i, which is nice to write in the following form. It will be sum of lambda i squared minus sum of lambda i squared. Because this is nice quantity. It's a mean curvature. It's also a kind of thing. In a second, we shall put some weights on that. And this is a norm, square norm of the second fundamental form of this operator 8. It's stress or it's square. But nothing happened here in the Euclidean space. Personally, I made it in this form. And there is no contradiction, a priori. However, in here, little geometry enters that there is little, from this perspective, difference with this ball and sphere of dimension n. And this ball of dimension n. They kind of are more the same. You just can move the ball into there. But actually, by projective transformation, it's very small distortion. With the projective with the ball in Euclidean space and ball in the sphere, they are projectively equivalent. And this unit, because it's a unit ball, distortion is very small by constant of 2 or something. This way, I'm all saying this is a constant of 2. You just take radial projection. So my sphere, it is ball, I project the radial from here. And so this ball goes to some piece in this sphere. So from this moment on, I can say, aha, I don't have to look at what happened in the Euclidean space. And this is a little piece of geometry. But I have to speak about this sphere. And here, I introduce a distortion of factor of 2. And from this moment on, all, no, in this moment on, I'm sorry, I'm using the full geometry of this sphere. In a most sophisticated theorem, I don't have to do that. And so now, the formula become the scalar curvature of my y becomes this sum of lambda i squared minus plus, and this is the curvature of the sphere, right? Scalar curvature of the sphere. And now you can see, yeah, well, if this, if all lambda i kind of small, then this quantity will be still positive. This elementary algebra, I think it's correct. I checked it once, and I'm unable, of course, to do it here. And I think you can check. But if you have all these numbers, less, I'm sorry, then square root of n, say, if they are less, then square root of n by 2 or something. Then this quantity will be positive. And therefore, you cannot do it. However, again, this amazing thing you believe, you understand torus so well. Of course, even something like exotic sphere, yeah, how we can say. And only, you know, for all exotic spheres, but only for particular ones, where this invariant is not certain topological invariant non-zero. So for those who knows, they don't bound spin manifolds. Yeah, they're not bound, they're not spin boundaries. But if they are spin bound, in the majority, many of them are spin bound, this is not an absolutely no idea. Of course, you cannot embed them in this dimension, but you can immerse them. The manifold doesn't have to be embedded, maybe immersed. And this is certainly super elementary kind of corollary of this theorem. Say, for the case of the sphere, when you use this index theorem and have this conclusion, it's completely mystifying. Because it seems that all no complicated, but there's many small curves that are very, very special. But there is absolutely no, in my case, had at least any idea how to prove that. So, but this, I think, you're convinced by the computation, you can probably can see if you can compute it. And you can see that if all these numbers are small, then this sum will be positive. Of course, see the number of terms here, that, as it is. Hope it's correct, yeah, I mean, just a major computation a couple of times, it seems to, my algebra is up to that level. But hey, I'm sorry, the more convict, hmm? Yes, you're right, absolutely. Side, side, side. Some, ah, yeah, yeah, yeah, yeah, absolutely. Because you see, they're all square homogeneous. So again, the assumption, why is it embedded in S n? This y, a priori, it was in the Euclidean space. Now it's in this sphere. If it's in the n-dimensional sphere, then this formula for the Schuyler coefficient. Yeah, y is dimension n minus 1. If a priori was in a ball, it was from this ball, we went to the sphere. But again, from a geometric point of view, it's kind of the same ball, sphere-wise, would be there. But again, when you, you know that in truth, it will be 1 over n. And for this, I mean, just there is no, there is no kind of, you have to prove it. It says, which I shall prove at some stage, but it's much more complicated. It's not terribly complicated, but you use, you have to go inside of the proofs how you prove this, in an existence of this metric. And just carry it over here. So that's one thing. So what we want to know next, now I was speaking about mean curvature. And so I want to bring it forth again, because it will give you a very good frame to understanding scalar curvature. Because one can say that scalar curvature is just Riemannian incarnate. By the way, you can make a break now. What about five-inch break? So you can relax, and you can hold those so the tide can run away, so being polite at the same time. How long is it supposed to be? It's another hour. Two hours. Two hours? No, I thought it was two hours, no? Yeah, it was good. Pierre, you decide, yeah? It's supposed to be two hours, yeah. But now it can run away, it's a good time. So before going to mean curvature, let me say extra justification, because I was emphasizing it many times, all the time that this is conditioned, which I'm concerned very much, but not that. That is a geometrically significant, and this is not. And so let me give you two thirds which justify that. And this is the follows. We can see that the space of Riemannian metric of C2 smooth, say, Riemannian metric on a manifold X. And there, we can consider subspace where scalar curvature is like that, or where scalar curvature is like that. And what I want to say, they have fundamentally different properties, opposite kind of properties. So this itself, I can see the C2 smooth matrix, yeah? But now I want to equip, give to the spaces uniform topology, so conversions when you forget derivatives. And then with respect to this uniform topology, this space is closed, and this is dense, everywhere dense. So the meaning is that if you approximate your manifold with positive scalar curvature, it still remembers the geometry, right? So if you have manifold of positive scalar curvature, some of the general manifold has whatever geometry you cannot approximate with positive scalar curvature, because geometry will give you obstruction for that. And here it says no obstruction, given any Riemannian metric can be uniformly approximated by metric with negative scalar curvature, or with any boundary scalar curvature from another side. And this was proven earlier about this old theorem, about 20 years old, at least, by Locke-Hamp. And it's proven with just kind of some geometric construction. Yes, do this, this, this, and just do it. And this, I proved some time ago, and actually, my proof using Dirac operator. And then there was another proof, soon thereafter, by Bambler using Ricci flow, which I must say, I don't understand how it works. Yeah, at some moment, you use something there, which I don't understand, but I think it's OK. And but you can do this also, but nowadays, using again some recent results by Schoen-Yau, some regularity results in minimal surfaces. But both proof are not quite simple, whichever, in both directions, they're not very transparent. And this, by the way, this way very much, for me, was motivated and related to the fact that the same is true about symplectic structures. The symplectic forms, omega, the space of symplectic forms, omega, is closed under uniform limits. And this was all resolved by Eliasberg, which is explained, or rather suggests, that there must be no three-view symplectic geometry. Otherwise, there would be none. If you can approximate everything, there would be no kind of a substance to the subject matter. Of course, proving this density also quite interesting, but shifts the subject to completely different field. It goes out of geometry topology. And then you can, as a geometry, you can more or less forget about it. But when closed, it says, well, it only starts, it starts, it kind of opens these possibilities. Again, I will prove that if I have time in my following lectures. But now, what's about mean curvature? How is it related to mean curvature? So what's about mean curvature? So what it tells you? So mean curvature is in euclidean space, or in the remaining manifold. But for our purposes, that's fine. You have these principle curvatures, right? And their sum is mean curvature, plus the function of y. And the point is that shape of this manifold of the hyperservices very much goes along with the shapes of manifolds of positive scalar curvature. First, you can say, they're quite simple. For dimension one, just convex. You understand them very well. However, already in dimension three, you may have the following kind of thing. You take this kind of curve. It doesn't look convex at all. And take a revolution. So this highly oscillating thing as much as you want. And then you can approximate it by somebody when this mean curvature will be positive. And of course, you see it's looked very convex. It's here, very concave here. But because this radius is very small, compensate for that. And actually, it's not just one formula. Actually, no simple formula. It's really some little bit of argument. You have to do this approximation. Infinite kind of geometric progression somewhere enters and its conversions play a certain role. But anyway, so they may look rather complicated. So still, so what you can prove about this mean curvature. So if you look at it, so what you know, what you don't know about the mean curvature. And again, you can prove something new about them using this scalar curvature results on one hand. On the other hand, you start realizing how poorly you understand them. This was, again, for me kind of a surprise. You take a question about positive scalar curvature, reduce it. Look at the parallel question here. When you think immediately you can prove it, you cannot. So what you can prove, however. And then you may ask what would correspond to the corresponding in scalar curvature. So first, maybe I state one very simple theorem, which kind of justify this parallel. We just give you one relation. And this is quite simple. So first, when I say positive mean curvature, it's fundamental that they're not immersed but embedded. If they're immersed, it may be everything. You may have any kind of pathology kind of geometry. You may twist it in any way you want. Interesting one, which I embedded. And so again, some kind of my emphasis will be in this situation, when mean curvature is positive or moreover, greater than some positive constant. So how do they look like? Now, the major link with scalar curvature is follows. So I have this domain, this positive scalar curvature. For example, the one which I described here. So it may be highly oscillatory. Of course, in the plane, convex can not oscillate. In high-dimension, may be rather oscillatory. And I shall show examples how much it can oscillate in a little while. But it bounds something here. And so what you do is this sitting in the Rn. And this Rn sitting at n plus 1. So imagine this plane figure in the simplest example here. This kind of figure in the plane in this sitting in 3-space. And now in this 3-space, so call this y. Call this x 1 half in a second to understand y. So I have this x 1 half sitting now in this n minus 1 dimensional in this space of dimension n plus 1. And what I do, I take epsilon neighborhood with this and take boundary with this. So here is a model picture in my interval in the plane. I take epsilon neighborhood is like that. And here is my boundary. This boundary is straight here. I just have a picture of my transport. It's straight here, straight here. And here will be two semicircles. Observe that this is not smooth fully, because here is C2 jump, derivative jump. You can smooth them. It's nice to smooth them. And the point is when you smooth them in this example, even if it has positive and curvature, then scalar curvature here will be positive. And if it has slightly perturb the metric, it becomes positive everywhere, including here. But after you smooth it, by small c infinity perturbation, it becomes really positive curvature. So every shape you have here has representative is scalar curvature. The same, absolutely the same shape. Yes, epsilon, the smaller epsilon, the better. Epsilon is got very, very, very, very small. And so now, and now I guess we might say, ah, and what can be the surfaces between curvature? But when you think of course, yeah, of course, these are very special manifolds. Yeah, they came out of Euclidean, so you must understand them anyway. But amazingly, sometimes, the only way to understand them go to the scalar curvature, and then look at your operator there, and then make some conclusions and come back, and then absolutely mystery what actually happens. So these are of course some limits. So because we usually, this epsilon will go to zero, and this your operator which you did generate to something else, but I don't know what the something else is. Corresponding object, I don't know what it is. So I do it with my argument very, very stupid, I guess. No use what is known, and just apply it here instead of properly defining intermediate objects. But now, how it may look like, what are the basic, basic examples, and how you make them, and this as follows. If I take inside of the equation space, Rn, some piecewise smooth set, called y zero of dimension n minus two. And if I take it now small neighborhood of this, because we have some care, then the boundary of this will have the smaller it comes, the bigger it will become the mean curvature, and that she blows up to infinity. And moreover, if I already have some part of that, which was this positive mean curvature, add here something like that, you can add this a little bit. I make it at this moment slightly smaller, right? So if it was flat, I cannot do it. But if it was already positive, I still can do it, and the whole thing will be, have the same scary curvature as before, except here it becomes smaller. And this, by the way, inevitable, right? So there is this little theorem. And again, I don't know elementary proof of this yet, might be, I haven't mined one, but you cannot do it. So if you take plane, hyperplane, n minus one now, right, and you say ha, can it be turbid, this positive mean curvature, and keeping it at infinity, at infinity, like that? You cannot. But there is no quite simple proof here, yeah? And then the proof immediately come, once you use the streak, you go to spoil positive scalar curvature, there you know it, and you say, yeah. Because it's already up, of course, it might be here, actually I have some idea how to prove it elementary, but it requires some effort. And you must be careful, if you're a little bit kind of wiggle with that, you may have some, you may have some problems. Which I, you have hyperplane in the Euclidean space. You cannot modify it with compact support, keeping here curvature non-negative, scalar, mean curvature non-negative. On the other hand, as I say, you may have this kind of, how now again this very pathological example, you make this one, they convex, join them by this in two. And as I said, you can thicken it and have curvature tiny little bit, because it was thickly convex, so it remains positive mean curvature. But if it was flat here, it would be impossible to do that. So it's rather kind of delicate, if it's positive, you can add this thing, this obituary small perturbation of mean curvature here. But if it was flat, then flat. And this is in the context of scalar curvature, this was a similar statement, was apparently conjectured by Gerr, and it says, if you have Euclidean space, you cannot make perturbation or modify geometry in the compact region without making scalar curvature somewhere smaller, somewhere negative. So if you make such perturbation, necessarily scalar curvature become negative somewhere. What do mean curvature to be positive in perturbation? Where in that statement is the mean curvature positive? No, I'm saying, you cannot make it. Yes, I know, can be, where to be positive? Every way, outside, non-negative, yeah. Come back support? Come back support, yeah. Why can't you just put a hemisphere and smooth it out? No, you cannot, you produce negative curvature. There's, you cannot do it, yeah. That's of course, simple but non-trivial statement. Of course, you cannot. By perturbation? The moment it starts smoothing at the disk corner, it produces negative scalar, negative sectional curvature, negative mean curvature. But where is the compact support? If not compact support, you can make perturbation. You can slightly bend it in all directions, yeah. You see, you can make it like that. And then it will be non-compact perturbation. And you can do it. But you can exactly say in the reasons how each perturbation you can make, probably the same as allowed by the positive mass theorem in relative, when you're looking at, this by the way, corresponding point of mass theorem, I don't think was formulated here, but you can immediately guess what it should be now from knowing from scalar curvature. So it must, roughly speaking, when it goes to infinity, if you have points of curvature, it might go slower a little bit than Euclidean case. If it goes faster, you cannot do it. Or when sequel, you cannot do it. But let me say indicate, yeah, maybe I can indicate how I can prove it. Which I never checked carefully, but then with the lecture, I say, indicate the proof which reduces to elementary computation. But you see also there might be some subtleties here. Say the three-dimensional case, high-dimensional case, slightly different because of the, no, of the, of the, of Catanoia's, of the differential of Catanoia's. So there is some distinction between two and three-dimensional case. Three-dimensional, high-dimensional case here, but not at this level. And there are more slightly, slightly more questions here when it involves. But so, but the point is, I was explaining this example here, how you can make this kind of highly-vigorous thing, you take this very narrow, but still convex thing, add this interval and glue them together, and you can do it, this point you've been coaching. And this is not, not obvious, I mean, it's easy, but you have to do it. And so how, it's so well. Maybe again, at some point, I will explain what the proof is. But this once is being said, of course, exercise. Once you know that, I mean just you. It's one-dimensional problem because everything is symmetric. Become someone, or you have to solve some simple differential inequality, and it's easily solvable. But the fact that it is solvable, you have to know that. But now, so what, so what, therefore when you know, when you have any kind of thing with point of, I mean, culture, or certain culture, you can add this huge kind of things. A long core dimension, one subset there. So in three space, sort of around points, yeah? But apparently you can't find it more. So it is unknown, really, how they look like. So what you do know, what is elementary kind of statement. The most elementary one is that if I have some sort of positive mean curvature, I am more specifically, yes, we in the previous space are n, and so the good measurement, this mean curvature is greater or equal to n minus one. Because n minus one, it is compared to the unit sphere. Unit sphere has mean curvature n minus one, right? Because there are n minus one principle curvature, they're all one. So that inside, you cannot put ball greater than the unit one. Because if it will be a bigger ball, you move it, it touches somewhere, and here, of course, there is kind of maximum principle involved. And moreover, it is sharp. If the ball sits there of unit radius, the whole thing must be the unit ball. But you see, it's a little bit perturbed, right? And then you may have any kind of thing attached to this. So it's highly unstable. So usually, a kind of perturbation, idea of stability completely breaks down. We can say, oh, it's not bad, no. That's really what fun starts, yeah? When you usually say something, when you prove some theorem, and some, it is some inequality, and then how it's slightly perturbed to a small error, and we're very happy about that. Up to your point, when you say it's a ball, I mean. There's really something completely different happens. It's much harder to explain in what sense it's stable. And intuitively, it's stable in a way that you can cut away this whole thing, and then make a correct cut, and cut will be very small, as this example suggests. And then the remaining thing is stable. And this is unknown, but something is known, something you can prove. So this is one theorem, another elementary theorem. So all of them, what I'm saying, to the effect that positive mean curvature makes think the bulk of this small. But of course, you can have another one, another one, if you slide a bit of the curvature, but you cannot kind of spread it in high dimensions, only spread in the dimension below kind of certain level. But in the second theorem here, it's kind of against quite elementary. This is just maximum principle. We've been passing and say something else about that. Some aspect of that in the second I tell, which I understood very poorly, but amusing. But before that, look at the following thing. So I have this convex thing, I say mean curvature, greater or equal than n minus one. And inside, I take this hypersurface, inside, yeah? But this also cannot be too big. And this I want to formulate in the following way. Now I consider the hypersurface. Now it's in there, so it's n minus two dimensional. So I'll call this say z in the Euclidean space, this dimension n minus two. And it serves a boundary of some y zero of dimension minus one. I'm going to have this picture. But the way you see it, I put this here. And my reason is because of course, when you make it in the tech, it's difficult. Because when you write something like that, this means x times x times xn times, right? So it's very, say Euclidean space is okay, right? But Sn is certainly not okay. And I don't know what the right notation is, yeah? So it avoids this kind of confusion. But anyway, and assume that this one. What did you say that you put Sn the index below? Yeah, but it's not also not true. Dimension is special. It's not, it's not it's kind of counting. You don't count them, you just, right? That's the whole point. Yeah, it's unclear, unclear, unclear. It's a big problem, I think you said. So, and imagine this mean curvature of this guy is greater than n minus one. Of course, mean curvature, you see, depends on the sign. So you have to, this hypersurface must be co-oriented. So it's a merbuous band, it makes no sense. But it's co-oriented, so it looks in one direction, it's like that. In one of the two directions, greater. Then what is true? That there is another surface, y one. Also this boundary, as we had before, such that this new surface sits in the ball of the inter-radius. So, more specifically, if you take any point from some, from this y, y, so okay. So distance, Euclidean distance of all points, y one to the z, there's one. So it's, the hole cannot be too big. If you can feel it by surface with large mean curvature, the hole cannot be too big. Yeah, it cannot be kind of a big hole from which you can go. So, this elementary, there are slightly, some variation of that which we should, later on, which are not so elementary. So how we prove that, yeah? This guy, exercise also. This again, yes. We can do, give it an exam, yeah, on differential geometry. So I repeat, we have a manifold with boundary, which have mean curvature greater than something, look at this boundary. And there is another way to fill it in, which will be more efficient in a way. And of course, the extremal thing is just sphere of coordination two, yeah? Where it just, where it is, how it works. And you see, it's really, really different feeling for sphere, it's filled by hemisphere, with curvature greater than one, but this optimum feeling will be flat. How you get that? And again, you only use what you use because maximum principle. You don't use any kind of, anything sophistication, except understanding. Understanding was it means somebody's body boundary of something, yeah? So. Excuse me, can we say again, the claim is that if I have some co-dimension to think, filled by some manifold curvature such one, there is another one, another feeling. Why one? No, nothing about curvature, but it's sitting inside, inside of the ball. Possibly, if you take this minimal surface with this property, it may do, I don't know. But the argument which I have in mind is just quite elementary. Once they have boundary distance from zeta. No, not only, it would be less than one. This new one will be just a little, within distance one, all points, all points y one will be close. Ah, I'm sorry. All points from y one to z will be less than one. And so again, with this example, I have circle. It's filled by hemisphere of curvature, mean curvature here less than one. And then there is a disc feeling it. And so everything is in distance one. Because here it's not, yeah, this is too far. But you can do that. So what's the proof? I can. This is again exercise, yeah. You have to have some ascension on the y one. Huh? You have to have some ascension on y one, or always in between. No, but the boundary has the same boundary, but it lies within distance one from zeta. For example, if you have big circle, right, it's bigger than one, you can feel it by small thing, yeah? Always, there will be somebody far away. But this side of this hole cannot be too big. The fact it's filled without control of the size, only is controlled on the mean curvature, imply there is another feeling which is small, right? Because you see this original feeling could be huge, yeah? Let me give another example. Here is my circle, and I feel it like that. And then up to some time, I can add here, God knows everything, yeah? Goes anywhere far away. But I know I can forget all this, and there is a nice feeling close within distance one to the boundary. This is a model example, exactly what we want eventually to prove for Skelikovich, and we're still far from that. So how to prove, but this is elementary. And one wondering is there is a similar elementary argument for Skelikovich. For Skelikovich, there is no single elementary argument, nothing can prove that. Either you prove minimal surfaces, when you're stuck with high dimensions, and there is extremely technical papers by Sean Yau and Willow Humph. Sean Yau probably more readable, but certainly I have to say it's one of the most technical papers in the field. And Willow Humph is, you know, it's about 200 pages, which is unreadable. But here it is just one line. So what is this line, yeah? This is nice, but I can prove something online. The proof is as follows. What does it mean? You cannot feel it within distance one. It's negative statement, but you want to turn it to positive statement. And so what do we use? Of course, duality, right? That's the power of topology. There is, essentially it's Poincare duality, which has different name, right? Poincare duality says, if there is no cycle here, there will be cycle there, right? And this is extremely powerful statement. Yeah, Poincare duality tells you that two opposite dimensions talk to each other. And here it's kind of, I think it's exactly kind of talk, which must eventually become essential for Skelikovich. It says, you have this thing, and you cannot feel it by, you cannot feel it by surface. Close it. Then there is a curve going from this side, or going here, maybe here, into infinity, which everywhere lies between distance more than one, and which has non-zero link number with this, right? And this is, well, form Poincare duality is called, I think, Alexander duality, right? So if there is a cycle in Euclidean space, it's not feeling, cannot be filled in this one neighborhood, it's not homologous to zero, then there is a hole which we can go around in this way. So, yes. And then I'm saying, aha, and I will take a unit ball, and move it in this way. And I'm claiming that some moment it will touch it like that. And it will touch it like that, okay? Okay, it's impossible. Because this mean curvature, kind of, this ball is bigger than the one. So we have to be on the shore. When it touches it for the first time, it may touch it from the wrong side, right? Because this feeling could have been, you know, like that, right? And when you go from here to attach from the wrong side, not from this side. So I have to be sure you have some moment to attach from the right side, right? But this is topology. So if you think a little bit what topology tells you, it says, well, if you look at the properly organized more sphere here, so then attach with some moment from the right side. How we do it technically? So what you do, you can see that you will, you can see this kind of tube which goes here, or the sphere times interval going here, take pull back of the thing, because you don't touch the boundary, become close manifold. It has some many components, so in the cylinder, you get some cylinder. And then there is hypersurface here, when you meet your hypersurface. And because index was non-zero, there are many components with that. There is some component separates to ends. And because separates to end, when you go, we take the minimum point, minimum to the complement, it will be the right point. So this will appear the wrong ones, but they will be in the wrong component. They will disappear, they will be right component. So elementary topology. But still this is true. So this is one thing. However, let's ask something more. There is another statement now, which has no such simple proof, which is in a way even look maybe simpler. And it says as follows, again I have my x, my y, n minus one, sitting in the euclidean space. The closed hypersurface mean curvature, gray to recall then n minus one. So what I am now claiming, that if I have any map F, just continuous map F from this hypersurface to the sphere of unit radius, the size of this map is distance decreasing. It just says strictly distance decreasing. So if it smooths you can say the differential less than one, then the map is contractable. Actually it may be easily subjective with the way. But here topology again is fundamental. So that's feature of scalar curvature and this mean curvature. All their properties, even it's not apparent, are linked to topology and namely to their fundamental cycle, right? And when you start proving various theorems, you see just how it kind of interferes everywhere, right? And for example, when you see the proofs which goes via Dirac operator, you have to also all the time just to make it work properly, you have to keep in mind that index theorem is not just number which assigns to some differential operator, but it give you a kind of fundamental homology class of your manifold because it give you index of your operator twist with any vector bundle so it become dual to the k theory. So it's really kind of fundamental homology class. All the time enters the game, right? And here there's a special sphere. Not the same, we can say slightly more than that. But that's again I'm saying, so again it can be big and when I say distance decreasing, it's essential that I mean intrinsic metric of the sphere. If I use extrinsic, it's obvious. So let me prove it again, obviously it's elementary, right? So I saying that if I have a map which is decreases distances as understood, so here is my manifold, if distance understood like that in accordingly distance in the sphere where a map also understood like that, it's again the sharp inequality where equality holds for the sphere but the proof is elementary, which I am going to present. However, if you take intrinsic distance because extrinsic distance and intrinsic may be very different, right? Here is extrinsic as one and here is huge, yeah? So it's very big so you can imagine you can spread it but you cannot. So everything says that this essentially is such a surface. It consists of bulk of this essential part like sphere and then there are narrow stuff attached to it by narrow bridges. By the way, literally to fix the so-called Penrose conjecture which was proven, which says what happens for certain many for the dimension three of positive of positive scale of which I did. Indeed, this process, this picture is partly justified in some examples but not at all everywhere. But again, there are two different kind of statements. One for extrinsic distance, which is elementary and for intrinsic, which the only proof I know relies again on Dirac operator and some things about Dirac operators of the style which I described already. So which I didn't say with Dirac operators, ah, but they kind of rather sophisticated objects compared to this, yeah? So how you prove it elementary? So indeed, so I imagine, so you have this thing and you map it to this sphere and so there is a domain and then there is the following theorem that I'm claiming that the boundary was mapped to the boundary, so this is my yn minus one, we bound some x and here is my sphere as n minus one, we bound the ball. So I have a map from here to here. I'm saying this extends to the ball with the same distortion of the distance and degree if the map has, kind of, was non-contractable, this relative map, so non-contractable means, of course, you have to know in this case, means it has no zero degree. Contractable, I say, to avoid appeal to homology, which is kind of, but on the other hand it's impossible to do it without it. Then you know this map to the ball, you'll also have no zero degree and in particular it'll be onto, right? What is essential, that there will be extension and it's necessarily onto because the map was of non-zero degree. If it were contractable, even onto, this don't have to be onto. And when you know, take the pullback of this point, take it here and then the distance will be bigger than one, which you know is impossible. Now, the only point here is extension and this is fundamental theorem, very elementary, again very simple and kind of quite, quite remarkable in my view. Here's a bomb theorem, it says, I have Euclidean space of any dimensions, maybe infinite dimensional, so on. And there is any subsets there, why? And if there is here a map, F, which is distance decreasing, say strictly or non-strictly, it extends to such a map between spaces and it's specific property of Euclidean spaces. It's not true in any particular case. It's rather, rather, rather imagine property. It's rather imagine property. Again, if you know it, even if you know it, the proof is not so simple. I mean, you have to make them guess it, yeah. I don't want to make a problem. I know, I always forget some computation you have to make. But it's not, no, not obvious, yeah. What is the hypothesis of why? Any subset, any, any subset, any, no assumptions. I have myself better experience with this theorem because I thought about it and I tried to prove it, I found some proof and I was quite happy. But then, you want to generalize it and I couldn't just throw some kind of situation, I want to generalize it and I couldn't. Because if the original proof, which you find in the textbook by Federer, but of course some readable book, but some people read it. This proof, you have remarkable generalization and say, I guess, you have an occasion to say it, found by, by, shh, I forgot, shriyadu with somebody. And they also, and they were independently proven by Petruyne from whom I had learned first. And it says the following, that I say slightly, slightly. S was distance decreasing? Yes. Or are N euclidean? Yes, euclidean, of course. And then the sense of how euclidean space also is in euclidean. So, and the theorem is as follows, that this property is true more generally. If you have x and y, such that section curvature of x is greater than some constant, kappa. And here section curvature is less or equal of this y, is less than the same constant, kappa. And here you get any subset, and the same is true. Any map, distance decreasing extends here. For positive curvature, you must be slightly careful. You need some, to make little, to avoid some obvious count examples. But if you, yeah, locally it's always true. Globally, you must be slightly careful. And is this the M or N is? No, no, N, N, N, N, N. You might put here Hilbert spaces if you want, yeah. Dimension completely material. Here's zero curvature, so here is not dimension level. I've been through in dimensional spaces. Complete maybe, if you only have to say. But if this curvature is greater than kappa and this less than kappa, then again any subset, map from any subset, distance decreasing extends. And this is again, if you know the definition of curvature and the original proof is more or less instantaneous. And but the most of its decay proven, or more elementary proof, I don't know, different moment and I just couldn't prove it. To my shame, and but this is true. And this, by the way, gives you definition of curvature. This way you can define what will be curvature greater than something or less than something. So you compare it with Euclidean spaces, right? If you sum them flat and then this is necessary and sufficient condition. So if this property defines sign of the curvature or a side of a bound on the curvature from either side. So this is kind of the best definition in my view. It's really functorial kind of property of the curvature. And there is no such definition of a scalar curvature. Hopefully you want to find such a definition of a scalar curvature. And this is, you see, because on the other hand, with this, you think the scalar section of curvature understand very well compared to scalar curvature. However, if you want to prove that some exotic sphere has no metric with points of section of curvature, you know nothing except what's come from scalar curvature. So in a way, this results with scalar curvature. On one hand, they can say, they tell you some new story about sectional curvature. But the major thing they tells you, you understand nothing about it in a way. Maybe you don't have to understand it. Maybe it's a wrong question. And it may be a wrong question to ask what exotic spheres have metric with points of scalar. Or a six-sectional curvature. Maybe a wrong question. Because there is no such thing as topology. And here I tell more about my time is a little bit out. So I don't tell more. Today, about scalar curvature. And then, so this is a theorem which I said. Yeah, this is Kierzbaum theorem. And well, this is when we exercise for it to prove. And so let me give one form of the theorem which will be relevant for the, for the, it's a moment. It says the following. If you have two simplices in any dimension, in other syntax, and they kind of respond for faces to face, et cetera. And here we have some dihedral angles, six of them, right? And here is called the beta-i. And if you have this inequality, it follows in fact there is equality. So you cannot make, already dimension three. Dimension two is obvious because some of angles are given. Dimension three is this. And yes, you first you invite you to derive it from that theorem. And secondly, try to prove it independently for simplices. Easy, relatively easy. And this is unknown for general polyhedral, convex polyhedral. And just, and it is also related to the scalar curvature. So you can prove it in some cases using kind of Dirac operator. But this case kind of can prove independently. This kind of phenomenon. And there is some justification explained to me recently by somebody why it should be true. And apparently this is the infinitesimal deformation as Paul said, adiprastata. It says it follows from some kind of particular statements in the Hodge theory for photoric variety, something you can prove in this respect. But it's simple, of course, elementary. But for general polyhedral, it's probably even not true. But under certain situations it may be true. So it's unclear when it's true and it's not. But interesting, again, when you start developing remembering this is kind of positive mean coverage and this is point scale coverage and then there is some Dirac operator. And then again, of course the relevant Dirac operator become kind of very degenerate operator, probably reduces to some sort of elementary. And where then Hodge theory of photoric varieties may enter. But you know, this actually I must say I don't know. But this is at least some exercise like this theorem. And just what I want to, for this one in the Euclidean space, in my view this kind of number one theorem after Pythagorean theorem, it's number two. First Pythagorean theorem, then there is Kirzbaum theorem. But it's not so well known. I mean, you have the textbook full of some nonsense but that remarkable simple theorem. And it's really the proof is rather kind of, you know. And you have to guess it, it's not, then come to you. It's very easy, I mean just one line if you know it. But you have to guess this line. So you invite it, to invite the two of you. So I put something that I wrote on my web today and then I read it in a uniform which contains some of what I said in something I'm playing to say later on. Okay, so you have questions. Now you have one minute. You have questions. Thank you.