OK, the attractiveness of the subject, which I have to present, I just want to show what makes it interesting, is scale curvature, and more specifically, it is very much concerned

with scale curvature bound from below by something, and these are objects, Riemannian manifolds, which I'll 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 at convex domain in degree and space, which we presume we understand. They are very well shaped and we have a good description of them, and we may think we understand them unless we go deeper, and of course, 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, at 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, homotopic theory or homology theory. And there are few, and actually I know only three examples in geometry which mediate between the two, yeah? So you have on one hand, you have like convexity,

on the other hand you have topology, and there are only three domains in geometry which mediate between two, and the most developed for today probably is simple axial geometry, and the second one is gauge theory in dimension three, say Adonelson theory,

specifically kind of, and then this scale equation, specifically bounded from below by constant. So all these objects, you cannot classify them in the same way 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 axial geometry associated

with classical mechanics, gauge theory with physics, and physics is first realized. There is something special about three-dimensional, four-dimensional connections, and the same which appeared in general relativity in Einstein equation, at least in the Hilbert derivation.

And mathematically still, scale equations were behind all the three, and so we don't quite understand what it is, and the kind of interesting point is that we understand it so poorly, right? And so, 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 scale equation, and more its relation to simple objects like this, like convex ones, and so, but instead of showing that we are giving better perspective with scale equations, showing how poorly we understand

this one, right? So we can see something new from the point of scale equations concerning simple objects, and we realize how badly we understand them, right? And so let me give an example which I find quite amusing, exactly of that kind. So say I just formulate some simple

question and some answers related to scale equation and the method involved to study this, and I must say that comparing other domains 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 doesn't appear anywhere in Riemannian geometry of this kind, and so let me give an example. Where is the razor? It's here,

which we shall eventually converge, and 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-manifold or hypersurface in particular 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 a dimension,

what is the curvature? When I say curvature of this, curvature of this y, so we have a curve, you know what the curvature is. If you have a surface, you can see the all z-axis line with them and look at their curvatures in extended, and supremum of them with x-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 just have vanishing first derivative, second derivative of quadratic form, and the principal, I know principal values, or principal axis of the corresponding ellipsoid is curvature. So how much it curved? And so

the question which we erase is as follows. I have a smooth sub-manifold of dimension m sitting in the unit ball of m in the Euclidean space, and for example, the simplest example

is boundary sphere, but it may be something more complicated than 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 Euclideans

in the ball of radius square root of m, the dimension to m, and the 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 this curvature unit circle have curvature one, when you scale it, it will be with curvature, right?

So when you put this product of circles inside, curvature behaves this way. And so this square root of m must be kind of very respectful of that, because this, of course, Pythagorean theorem.

In Pythagorean theorem, of course, everywhere will be there. And so, yes, people can say, oh, I use Pythagorean theorem this mile, but I think in 2000 or 10,000 years to write textbook on mathematics, and then it will be written this Pythagorean theorem, and little applications like Hilbert spaces, you know, quantum field theory, everything. They're kind of trivial kind of,

in a way, manifestation of Pythagorean theorem. It's everywhere, some of square is equal to something. It's amazing how well it works, and how often it's being forgotten. So in my experience, sometimes you take proof, since I had the experience, in the time you have 10 pages argument, behind, of course, the Pythagorean theorem, you throw all this argument, use exactly theorem, and the argument disappears.

It's extremely remarkable thing, yeah. It's used everywhere, this quadratic linearity for squares, not for lines. So, it should be respectful of this square root of m. And here, of course, this square root of m has many other kind of, if you, 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, we have quadratic rule, and all computational mechanism of Riemannian geometry depends on that. But once we understand this rule, of course, we want to break them. And this, of course, is the 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, yeah. Which was well articulated in the lectures, Nobel lecture by Fleming, when he was describing how he was, his discovery of penicillin.

Yes, he said there are rules, how it interacts with bacteria, but the main point, how to break the rules, and then make discoveries. But this is not easy, yeah. So, at first I will, so far I will be describing rules, breaking them up to the audience, yeah. I don't know how to do this.

So, so, but this is the issue, yeah, 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 bowl, and I want this maximum curvature of y to be small, so it minimize it.

So this is, this maximum curvature is maximum taken over point y, and this our, 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 scalar curvature is concerned. So, what you can show, rather elementary, is that by 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 one, just to invoke. So if y,

is the topological manifold, and this may look, admits no metric with Poiseu-Skate curvature, actually it's not like that, no, I think it's better to say that, yeah, otherwise, actually it's quite essential, different, quite, quite respectful, yeah.

Then by elementary argument, you can show that this maximum of the curvature of y, n minus one in the bowl, must be greater or equal,

I guess, square root of n. We're missing some constant here, maybe one half just to be comfortable, yeah, 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 is a, and this was a conjecture by physicists, actually by Gerach in early 70s, and then proven in some cases by Shen Yao and then by using minimal surfaces,

and then 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, a double dimension. And we don't know actually what happens for double dimension very well. So we cannot prove that this is optimal embedding.

At least I don't, I can't prove, and I don't think it has been studied. However, if you go to co-dimension one, you know you cannot do better. But if you go deeper, and you see not only, 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. It certainly explains something else. But this is co-dimension one. This is true for co-dimension one, and this is also known for co-dimension two,

and then become co-dimension three. You cannot say anything. It's amazing. I think you can say co-dimension three. I mean, except, of course, 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 only of dimension, of this dimension, plus two.

So there is this exotic spheres, so many forms, homeomorphic, but not diffeomorphic spheres in these dimensions, which are known also, not to admit metric of Poisson's Karacharyya. And this, in this case, is truly kind of application of the index theorem, actually, of the subtle index theorem, which was proven ten years after the first one.

And so the spheres exist. For them, what I say is 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, who knows.

I'm really certain it's true. But the proof of this particular theorem, you know, it's written by Lohkamp, and it is not really written in detail, and so who knows, the proof may be incorrect, but the theorem I'm certain is correct. So this was, so I'm using, so the proof eventually goes via using kind of,

in the theorem, but you prove absolutely elementary. The only thing about this, what you use, kind of on the ambient space, only what is the scalar curvature of the sphere. You only use local geometry of the sphere.

You don't use the scalar curvature of this as something. You don't use geometry at all, right? And actually prove much more stronger statement. You don't prove exactly, 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, I found out, I thought I understood fully. I realized it, I don't understand at all. The simplest question you can't answer, and exactly because there is some progress of, and this is of course how science works, the more you understand, the less you understand.

That's very satisfactory. And this is what I want to explain. Okay, this was kind of preamble, just to say, well, it's kind of quite amusing. And so you may have questions, which you may ponder now. They're so absolutely elementary. And if you take very simple manifold, like product of circles, you can see something that is in local dimension. If you put here two spheres, it will be the same picture, except it will be here now four.

And then you don't know what happens. So if, when you put the product of spheres in Euclidean space, the curvature must grow, like square root of m in Heiko dimension,

and like linear, it's got dimension getting smaller. Absolutely not a guess how to prove this. And because scale curvature here, of course you do have Poise-Kelley curvature in spheres. So this is a rather amusing situation. And I will give other example later on,

even more elementary, in a way, in more elementary statements, about Euclidean geometry, even about convex sets, which a posteriori follow from relatively sophisticated techniques used for the study of scale curvature. Now definitions. So what is scale curvature?

And that's tricky, of course. So what kind of definition you want? And specifically, I want to know what it means scale curvature is greater than something. It's Poise-Kelley whenever you're constant. Once you know that, of course, by limit, by using the definition of real numbers, you can say what the number is.

But for me, it's most crucial understanding that. So first, it's assigned to Riemannian manifold. So I have Riemannian manifold, so it's smooth manifold, and Riemannian metric, which is, of course, for this, it's a symbolic thing. And it's unclear what it should be, correct definition. And for the moment, what you have to know about that is it gives you a metric,

this metric, right? So it's the end, the volume on the manifold, right? And how is it defined? It's just defined by kind of simple, simple, simple functionality, 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, the first point is the 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 varying 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. But if you take scalar curvature of x times x1 times x2, with the metric g1 plus g2,

when this sum is Pythagorean sum, of course, yeah? Which meaning that when you say distance, it means this, yeah? So the distance in the new metric is understood like that. So I put here this side, right? Then it adds up here.

It's equal to the sum of the two, scale of x1 plus scale of x2. So sometimes I'm writing this like that, or sometimes I'm writing the scale of g. So when it's add, thematic in Pythagorean sense, it's additive, this one.

And this will already show that dimension plays a certain role. Yeah, you just cannot limit dimension, which is very different from other curvatures. So if you understand in one dimension and another dimension, because of this additivity, it's property number one. Second property, characteristic of any kind of curvature, how it scales. If I multiply this metric by constant, then, yeah, so it's actually a fixed notation.

So this means I multiply this.

But of course, when it's a manual metric, 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 this way. And then the rule is the scalar curvature of lambda x equals lambda minus two,

scalar of x. Sorry for you. In large sphere, so the second I say that, what is the normalization of this? Maybe you're going to anywhere, anywhere further, it's important to remember.

What is normalization? And this will get a key point for computations. The scalar curvature S2 equals two everywhere. It's not one, but two. And that is a good reason it might be two, not something, yeah.

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 algebraically is,

beside of course being invariant under isometries, which is implicit. You have to say some, 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 J, J is family of quadratic form.

So it is quadratic form J, J, depending on the point. So you can say that your manifold locally is Euclidean space. You have these functions, J, J, that's it. And then they make positive definite quadratic form.

So it's a field of quadratic forms, right? 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 and J are priori, the first and second derivatives, when you compute it.

And scalar curvature linear in the second derivatives. And then it's, that's it. Then it's uniquely defined by what I said. The only fact is being linear in second derivatives. And the only expression we can write down, which will be, which gives you a kind of invariant expression involved in 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 subsequently well agrees 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 of it, which is completely useless, completely wrong. I think there is good reason why it's wrong. However, it's very easy to say. And then it will say, right?

And 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 two manifolds with the same dimension,

x1, g1, and some point here, on that hand, x2, g2. I want to say what does it mean, this inequality. Scalar of x1 at point x1 is less than scalar of x2 at x2.

So I give kind of geometric definition of this inequality. Again, once you know this inequality, you can, scalar coefficient become kind of linear order quantity, and also is addition, because we can multiply manifolds, and so, you know, define the scalar coefficient.

And it is very simple. It says 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, right? And so it's n-dimensional volume. Then this will be, so I wrote this inequality, greater than that, for all sufficiently small epsilon.

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 ball, small epsilon, but it doesn't go to zero, and this inequality immediately breaks down, right? That's the problem. And usually, in other, 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 scale curvature. You can check by the way that this agrees with this additivity, again, using Pythagorean theorem. This is property of balls.

It's kind of stable under 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 is what enters, in some moments, it's more serious scale 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 the second day, it's clear what it is. And then there is similar inequality. So the more negative curvature, the bigger this integral. And we should turn the world and set out on this point put to infinity. This minicovitch is called kind of a thesis,

called kind of a master of some kind. You cannot understand, but it has some reinterpretation in general relativity. So that's a little, but still, as it stands, you cannot use it. It looks kind of nice, but it doesn't integrate. And so I don't know 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 further, we need to have a means to actually compute

or evaluate scale curvature in specific examples. Okay, so let me take a little pause, because this again still was rather, so just you have to prove something. That this really well-defined definition, which is give you the scale curvature.

And as an example, kind of exercise, which is if you try to do it without, directly, without knowing some way how you manipulate with formulas, will be quite difficult. And the example you have to have in mind, the scale curvature of union sphere equals n n minus one.

And you see it may stroke strange normalization, why not say it one? But you see, because I have this additivity, you just force, if you accept for two-dimensional sphere scale curvature is two, then it's like that. Or if you, in other corollary of that, in the hyperbolic plane, if you look at the hyperbolic plane, it is metric on R2, say coordinates x, y with the metric

g x squared plus e to the two x g y squared. So this is a good representation of the hyperbolic plane. So I have one coordinate x squared, and we go another exponential expand with this coefficient. It has scale curvature minus two.

And it follows from the previous definition, because you take this and you multiply it, for example, with s two, then the scale curvature must be one, and you have to check it. Indeed, this volume property will be there. So you see 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 the big question is, so what happens? It's very unclear what happens in the infinite dimensional setting, which is, I'm not saying it's meaningless, but it's a big mystery.

It begs to be treated simultaneously in all dimensions, including infinity. But now how to compute it? Before starting proving anything, you have to compute it. For example, I want to explain maybe today that y, what I said before, that if manifold y admits no metric,

so if scalar curvature cannot be positive, then y, when it sits, say n minus one in dimension n, then scalar curvature must be greater than square root of n, approximately. I'm not certain what is the constant.

This needs some computational means, and this will be crucial for what we do. So all proofs depend on that. So maybe, just historically, which is, well, first I must say I'm writing something and I put on my web, so kind of it will be

adelated to my course here. It will 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, with the IP Atchisinges theorem for elliptic operators. And then the next year, Lihnirovich

proven that there are manifolds, actually very simple manifolds, I'll bring them first in a second, which do not admit metric of Wyspoytius-Kelikovich. And the basic example, which seems rather impossible to prove without inductive theorem.

In some cases, you can avoid direct use of inductive theorem, but only a kind of fragments of it has elementary fragments of this. But here you need the following one, the basic example is, it's a Kummel surface, so it's given by this equation,

and this is sitting in the complex projective space, right? So it's real dimension four, right? So complex co-dimension two, or one, so it's a real dimension four. And this is the simplest instance of many forms where Lihnirovich theorem applies. And so this admits the metric of Wyspoytius-Kelikovich.

Nowadays, due to our solution of calabic conjecture, we know it admits metric of flat, which has flat zero Kelikovich, but it cannot be positive. And what he used some formula relating Dirac operator in kind of,

this is whatever is Dirac operator to which index theorem applies, which is, he used the formula, this plus one quarter of Kelikovich. 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, it's written in German, I forgot 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, immediately have contradiction with Wyspoytius-Kelikovich. And this is a very simple, innovative proof,

if you understand what is a Dirac operator, which I don't. You are not, because it's spinous, as I keep saying, was explained to me, my achi are not supposed to understand them. Nobody understands them. You live with them without understanding, that's fine. Unlike the way you understand, say, differential forms.

But anyway, this is a kind of mathematics involved. And then, about 10 years later, in about 1972, maybe I'm not certain, that appears in the sequence where you force paper by A.G. Singer with the proven, more sophisticated, index 30 mod 2. And then it was pointed out by Hitchens that this applies

to the zoonotic spheres of dimension n, k plus one or plus two. So there are spheres, manifold, which are homeomorphic to spheres, but not diffeomorphic to them, and which carry a nomadic of Wyspoytius-Kelikovich.

And the invariant, actually spin kind of invariant, this is also, everything was known, immediately have in the theorem. And so the logic is like that, in this kind of proof. In the theorem tells you, in certain manifold, granted sufficient topological complexity.

They carry harmonic spinners, whatever they are. So the solution of some partial differential equations and harmonic functions, but it's a 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 the theorem came up, you go, what to do with this? It was very unclear. And then this was kind of development. And then, well, there was different approach developed, five, six, seven years later, by Sean Yao. I think following something with the suggestion of a thesis,

with 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 hyper surfaces, where singularities cause problems. And so one of the main issues in my view,

that Skelikovitch indicates that there is a relation between, on one hand, zero cooperator, on the other hand, minimal hyper surfaces. And they're both elliptic equations involved by very different nature. And they talk to each other in this instance, but you don't know what they say. And that's one of the big problems.

And Skelikovitch is just the 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 mixture of the two methods today. They do interact in applications, but they don't interact internally, right? So you can imagine the picture.

You can have in mind, so here is a big thing, something, which you don't understand, that goes here on the ground of Skelikovitch, and sometimes they meet here, but you don't know how they meet there. In their fundamental nature. And you make some conjectures. Okay, so, but now I have to describe basic formulas in relation to curvature.

How do you find curvature in Riemannian manifold? So one thing about Riemannian manifold you have to know, it will be smooth-emetic, and again, smooth-emetic will be essential,

actually, for the purposes of that C2 is sufficient for curvature. So Riemannian metric. And the reverse point is that you have this Riemannian metric and you can locally share this 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 O epsilon, so it approximated with an epsilon error, by a flat one, but in fact, this is better than that. Because you can choose coordinates because the group of diffeomorphism, or linear group x, operate there, and allows you to cancel x return.

And that's quite remarkable. It's very hard to say what was motivating Riemann in his definition, how much he thought about that. But they're very special among all other metrics, Riemannian metric.

Closer to Euclidean than 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, say, hypersurface inside Riemannian manifold,

its local geometry, infinitesimal geometry, we speak about this curvature, is exactly the same as for Euclidean space. So the curvatures are defined to just replace Riemannian metric to be Euclidean metric, define what is curvature, principal 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 this is the main, I call it second main form of Riemannian geometry, because the first one is a Gauss theorem, but this is as follows.

So I have my submanifold, and you can see that this family of this parallel of them. So you move them by epsilon. So you have this family of manifold, y sub t, inside of x, t close to zero, maybe, or maybe not. And each of them carries its own metric, called h sub t.

How this metric developed, right? So first, what 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, 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 here, disposed by Pythagorean theorem. So already, all you say is behind it is Pythagorean theorem. I want to say it's Riemannian theorem.

Highly non-trivial for the fundamental phenomenon. This allows you to make all computations. And what is the derivative? Now this is supposed to be the first line and, of course, in differential geometry, even in the Euclidean space, right? So you'll fail, yeah?

Amazingly enough, in textbooks, people don't know it. Most authors textbooks aren't aware 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 this function. Function has zero, first derivative, so there is second derivatives.

It's a quadratic form. The second fundamental form, right? And it depends, of course, on the sign. Depends which direction you go. Reverse direction of time, sign changes. And that's, again, this plane with sign crucial, right? So it's a form depending on direction, plus or minus. So it actually has extra structure. How are you defining the formation again? Yeah, you take parallel, family of parallel hypersurfaces.

Just by? Yeah, parallel, yeah. By distance, it's displayed. The simplest possible thing. And it's first fundamental form. Curvature of the hypersurface, right? This is, I would say, that's minus one formula in the Riemannian geometry, right? And now we come to the second one, first one I will explain later on.

And you see just, this is kind of tricky point. When you, myself, when I was learning Riemannian geometry, when you look at textbooks, you just can't find it. People write, and you just, when you start writing simple formulas, they take pages. When it takes usually no line, yeah? Even no line, there's 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 correctly define it, you don't need it. You can define properly second fundamental form even in the fine space for getting the metric. In a slightly different definition, right? Because Hessian is defined finally.

And interestingly, if you, the difference is, you can see it already in the Newton kind formulation of the dividing law of physics and the first and the second law. If you take the first law of inertia, you have a fine 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 you will not fool your mechanics. It defines structure in the space. And secondly, it is another one. It's kind of really also fundamental.

And the second may be defined if you have measurements of time also, a lot of space. But you have to measure something, right? But if you only have concepts of a quality, so what is the straight line, yeah? So I go straight because you move my hands in the same way. So same use of my legs define a fine structure in the space. Not for me also, for any bug in this mechanism of your brain.

This allows you to understand the world. Same use of my motions give you a fine structure in the world. But energy or measurement of time is a higher level structure. And this is different. But 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.

It's quadratic form depending on t 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 Weyl, you know, and probably in Twenties, whatever. And probably it was known before to Cartan or to probably other people. And this is the main formula. 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? Right? 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 it to operator, right? If you have Euclidean space, Riemannian tangent space over Riemannian manifold,

it's Euclidean space and you have quadratic form. Always there is operator associated to it, symmetric operator or vice versa, right? You have operator, then you have this quadratic form with respect to metric. Scale the product. Actually, I don't want, it's enough for me to have x.

Just like a, it's called quadratic form rather than bilinear form. And vice versa, any quadratic form uniquely defined with this operator. If it is symmetric, operator is symmetric. So 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 handier 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's defined with the curvature. Which encodes curvature of your operator.

So it's quadratic form, but it depends on which direction you go. So it's three tensors. There are these two directions and then normal directions. So it's a number assigned to these three vectors, it's 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 it was third vector, 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 agrees with, doesn't depend, you change your hyper surface, et cetera, et cetera. But that is the main formula.

So what exactly it is? So if I have this, this normal vector, and I take this. So can you say A is a operator corresponding to second fundamental form? A is times zero, times zero? No, no, okay.

Of course, at any moment, yeah. But you do it, apply it, if you define curvature, you do it at time zero. But you do it for all t. You can specify it for t zero, you're absolutely right. But this is for all t. So this is the so-called Gauss-Kadasi equation. No, Kadasi is not here. Kadasi is the next derivative.

Kadasi was the third derivative. That's the Gauss-Kadasi equation. No, I don't think so. I don't think it's Gauss-Kadasi. I think it's Weyl equation. This formula, you see, it is Weyl cube formula. It's called Weyl cube formula. Kadasi is not there. Kadasi is the third derivative. Gauss-Kadasi, no, no, no. Gauss-Kadasi is the relation with external internal curvature.

It's really different, much more fundamental. Gauss-Kadasi, everything is correlated with it. There is square of that and this is sectional curvature involved. So again, if you have this vector, this is in y and this normal vector, then this is just sectional curvature on this plate.

And now, and just to check, even in the Euclidean space, you just look at this circle and see how it develops. The curvature is one over r and this length has r squared.

You differentiate exactly, have its identity, things cancel off very nicely. So my understanding is from this example, just circle in the plane, this formula must follow. But I don't know exactly how to say. It's algebraic question. All formulas in differential geometry, you look at one example and the realities must follow.

So usually it's being proven in pages and pages of short computations. But you don't have to make computations. 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 figure is to prove this statement. Anytime you can verify it, always works.

And the point is that people who work with these formulas, they 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 you're impossible to look at the net, look at textbooks, you never can find any explanation and you can see a reasonable proof of any formula. It becomes a huge computation.

People use your moving frame, exterior form, God knows what. It just follows from one simple example and from the reality. But I don't know where it's written. And that's one of the big technical problems. Because otherwise, so you have to check all formulas by yourself, it's such a pain. To write them correctly. So that's, and that's the formula which we have to know.

And this allows, and then you have to combine it now, of course, with the Gauss formula. So what is the Gauss formula says in this case? Specifically for the scale equation. So first look at the Euclidean space. So what Gauss formula says and what the usual definition of scale equation is.

But that's kind of the formula to remember. But I don't think it is Gauss-Kadasi, or Gauss comes next. Gauss before that. Yeah, Gauss-Kadasi is not Gauss. Gauss-Kadasi is this, it's something separate from Gauss-Kadasi.

No, Gauss-Kadasi, it's Euclidean space. Third derivative involved in the gravitation of the curvature. And the Gauss, no Gauss, Gauss is just, for surfaces, basic formula. Gauss, the curvature, sectional curvature of the surface. And this is the principal eigenvalue of this a, or principal ox is this. Product is the curvature.

That's Gauss formula. But it's not about that. It doesn't apply there. It's different. This is called, in all textbooks, Gauss formula. This one? No, which one? Physics textbook, not mathematical. Physics textbook, yeah, it's standard in Einstein equations. Yeah, sure, sure, sure. Of course, they always use that. Let me say it, though people who use it, they know that, yeah. But in differential geometry textbooks, it's not there, right?

But this wild tube formula, because it appears in the wild computation of volume of tubes, there are also many forms. Right? If you take trace of that, when you trace it, it becomes richer, it becomes volume, and it becomes this wild tube formula. This is a precursor of the wild tube formula. By the way, cadets, I don't understand because cadets are third derivatives.

Cadets always involve third derivatives when you look at surface in the space, and you write this Gauss equation when there are curves which determine higher dimensions, and there are integrability conditions. And this integrability condition, third derivative of the metric, it's not here. It's only, Gauss-Cadets need third derivative, Cadets. And Gauss, only second. And this is similar to Gauss, but different.

But now what is Gauss in our case? We have a hyper surface in the equation space. And at every point you have quadratic form. You have this principle, these eigenvalues of A operator called principal curvatures. Because, as I said, it's quadratic form, so it's diagonalizable.

It has these values in this principal direction. These are principal 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 remaining manifold, is that it's something assigned to the tangent space. You take two-dimensional planes. It's numbers assigned to two-dimensional planes, the 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. It diagonalizes in this plane.

And then sectional curvature of some manifold is equal to the sectional curvature. So kappa on some dimension of my y equals this kappa on x plus this product. This is Gauss formula. On every tangent plane, I take these principal eigenvalues.

When I restrict quadratic form, I 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 original Gauss formula. And so what's remarkable in this formula, what excited Gauss, of course, that you have this surface and you have this. So two eigenvalues, you multiply them. What you get? Invariant on the bending of the surface.

You deform it into the space without tearing it. This number doesn't change. And this is 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 cancellation somewhere or something, yeah? Because it's a wrong derivative.

A priori has formulas involved in each derivative, but they cancel off. This may be saccharides, but they cancel off. But there are, of course, so derivatives. 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 for 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 orthonormal frame of vectors, take all this plane, which you say, and

add together all the scalar curvatures. I'm sorry, all this sectional curvature. Then you get scalar curvature. So which, again, we can think as some kind of trace. Or you can integrate with the right coefficient. It's, again, by Pythagorean theorem, integral of the sphere, the same as sum 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 of manipulating with this kind of computation, they're routine. Yeah, they don't have to think. Yes, you do it by naturality.

However, it's easy to break this naturality. And, you know, even great people making similar computation in Hilbert were making some rather bad mistakes, yeah, for thesis, know that. Not in here, but in mechanics, actually. Einstein equation was OK, but Hilbert was also writing some derivation of hydrodynamic

equation from Boltzmann equation. He was exactly not properly using naturality here, might be. Well, 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 featuring the examples.

You'll be right. But the simplest is tricky. It's not simplest as analysts understand, yeah, but simplest in the true sense. OK, so that's about scale curvature. And that's enough, for example, to compute it for the sphere. If we have sphere in Euclidean space, it's sectional curvature one.

So it's n dimensional sphere. So there are n by n minus one pairs of normal vectors or different normal vectors. So scaling curvature of the sphere is what I said.

So, yes, let's well look at some examples how this thing being computed. Now, I'm not certain. I remember the formulas I put them on the web. So I go next step because someone who understands computation numbers are kind of crucial.

But still I can explain now maybe what I said about embedding of submanifold, say of dimension n minus one into the ball.

Why they should have big curvature somewhere if they admit normative of Poisson scale curvature. Namely, this estimate that this maximum of this lambda y, now I introduce them, must be greater.

Well, maybe here I put here one half, I don't remember. If this y admits itself nomadic of Poisson scale curvature, which by the way again is very tricky because there is no single instance of such manifold beyond dimension two when you can show it by simple means. You need to use some techniques rather sophisticated, well, a different degree of

sophistication, but it's not at all from this naive geometric definition you can prove. Probably for fundamental reason. However, this is elementary. How to prove that?

And so you have to see in 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 manifold, it's like spheres. Any manifold can be embedded in any hyper surface sitting there, maybe just anything.

But for example, the torus, which torus certainly, n minus one sits on n plus one. And you know that torus does admit a posteriori, which we shall prove at some moment, it's not a simple theorem, it was a big conjecture,

why the torus admits a nomadic of Poisson scale curvature, but then from there it follows, it sits in Euclidean space, but you cannot compress it into a small ball. If you start bringing it into a small ball, and necessarily, regardless of what metric is induced here, this metric doesn't have to be a Poisson scale curvature. It must, it must, curvature might blow up with this rate.

But in fact, I said it must break with the rate much faster, like at least one over n, which is more difficult. So why it is so? And this is just elementary, elementary computation, because circle, in bigger space, circle around it, circle around it, circle around it.

And then of course, in this example, you see it blows, I think like n squared, in this kind of simple embeddings.

So how to show that? So we know the scaling curvature has this formula in the Euclidean space. It will be product of lambda i, which you can, it's 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 in that. And this is a norm, squared norm of the second fundamental form of this operator A.

Right, it's trace over square. But nothing happens here in Euclidean space. Yeah, yeah, purposefully I made it in this form. And there is no contradiction. However, here a little geometry enters.

There is little, from this perspective, difference between this ball and sphere of dimension n. And this ball of dimension n. They're kind of almost the same, yeah? You just can move the ball into there, but actually the projected transformation is very small distortion. Right? The ball in Euclidean space and the ball in this sphere, they're projectively equivalent.

And this unit, because it's a unit ball, distortion is very small by a constant of two or something. This way I'm all saying there is a constant of two. You just take radial projection. So my sphere, here is the ball, projected radial from here.

And so this ball goes to some piece in this sphere. Okay? 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 the sphere. And here I introduce a distortion of a factor of two.

And from this moment on, all... No, in this moment on, I'm sorry, I'm using the full geometry of the sphere. In a more sophisticated theorem, I don't have to do that. And so now the formula becomes, the scalar curvature of my y becomes this sum of lambda i squared minus...

Plus, this is the curvature of the sphere. The scalar curvature of the sphere. And now you can see how well, if this, if all lambda i are 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.

If you have all these numbers less, I'm sorry, than square root of n. If they're less than square root of n by two or something, then this quantity will be positive. And therefore, you cannot do it.

However, again, this is an amazing thing. You believe, you understand torus so well. Of course, even something like exotic sphere, how we can say? And only not for all exotic spheres, but only for particular ones. Where this invariant is not certain topological invariant on zero. So for those who knows, they don't bound spin manifolds.

They're not bound, they're not spin boundaries. But if they are spin bound, the majority, many of them are spin boundaries, there's 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 immerse. And this is certainly super elementary kind of corollary of this theorem.

Therefore, 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 manifold with small curvature may be very, very special. But there is absolutely no, in my head, I had at least any idea how to prove that.

So, but this I think you're convinced by this computation. You 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, you have the number of terms here, that, as it is. Hope it's correct. I mean, just amazing computation, a couple of times, it seems so good.

My algebra is up to this level. But I'm certainly more convinced. You're right, absolutely. Some, ah, yeah, absolutely.

Of course, we see they're all square homogeneous. So again, the assumption on why it's embedded in S n? This y, a priori it was in the Euclidean space, now it's in the sphere. If it is in the n-dimensional sphere, then this formula for the scalar curvature. Yeah, y is the dimension n minus one. The a priori was in a ball, but from this ball we went to the sphere.

But again, from geometric point of view, it's kind of the same. Ball, sphere, y should be there. But again, when you, you know that in truth, it will be one over n. And for this, I mean, there is no, there is no kind of, you have to, you have to prove it.

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 with the proofs how you prove this. Non-existent with 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 minutes break? You can relax and you can hold, hold tight, you 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? Pierre, you decide, yeah? It's supposed to be two hours, yeah.

But now I can run away. So, before going to mean curvature, let me say extra justification. Because I was emphasizing many times all the time that this is condition which I'm concerned.

Very much, but not that. That is geometrically significant and this is not, yeah? And so let me give you two theorems which justify that. And this is as follows. We can see that the space of Riemannian metric of C2 smooth, say, Riemannian metric on a manifold X.

And there there are, you 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, you can see the C2 smooth matrix, yeah? But now I want to equip, give 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 if you approximate your manifold with Poiseuille scalar curvature, it still remembers the geometry, right?

So, if you have manifold of Poiseuille scalar curvature. As I'm general, manifold has whatever geometry you cannot approximate Poiseuille scalar curvature because geometry gives you abstraction for that. And here it says no, no abstraction. Any Riemannian metric can be uniformly approximated by metric with negative scalar curvature or any bound to scalar curvature from another side.

And this was proven earlier about this old theorem, about 20 years old at least, by Lockham. And this proof is just kind of some geometric construction. Just do this, this, this and just do it. And this I proved some time ago, and actually my proof using the rock operator.

And then there was another proof soon thereafter by Bamler using Ricci flow, which I must say I don't understand how it works. At some moment you use something there which I don't understand, but I think it's okay. And, but you can do this also, but nowadays using again some recent results by Shun Yao, some regularity results in minimal surfaces.

But both proofs are not quite simple, whichever, in both directions, they're not very transparent. And this by the way, this very much for me was motivated and related to the fact that the same is true about simplex structures.

Right? This is a simplexic forms, omega, the space of simplexic forms omega is closed under uniform limits. And this was all resolved by Eli Ashberg, which is explained, or rather suggests that there must be non-trivial simplexic geometry.

Otherwise there would be none. If you could approximate everything, there would be no kind of a, no substance to the subject matter. Of course, proving this density is also quite interesting, but it shifts the subject to a completely different field. It goes out of geometry and topology. And then the geometry can more or less forget about it.

But it went close, it says, well, it only starts, it starts, it opens its 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 Riemannian manifold, but for our purposes, that's fine. You have this principle curvatures, right?

And their sum is a mean curvature. Also functionally. And the point is that the shape of these manifolds, of these hyperservices,

very much goes along with the shapes of manifolds of positive scalar curvature. First you can say, huh, they're quite simple. For dimension one, it's just convex. We understand it 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 it, 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, as you see, it looks very convex here, very concave here. But because this radius is very small, compensate for that.

Actually, it's not just one formula. Actually, no simple formula. It's really a little bit of an argument. You have to do this approximation. It's an infinite kind of geometric progression. Its conversions play a certain role. But anyway, they may look rather complicated.

So what you can prove about them, about this mean curvature. 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, 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 scalar curvature.

So firstly, I state one very simple theorem which justifies this parallel. I just give you one relation and this is quite simple. So first, when I say positive 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, dramatically. You may twist it in any way you want. Interesting one which I embedded. And again, my emphasis will be in this situation.

When the mean curvature is positive or moreover greater than some positive constant. So how they look like. Now, the major link with scalar curvature is as follows.

So I have this domain with positive scalar curvature. For example, the one which I described here. So it may be highly oscillatory. Of course, in the plane, convex cannot oscillate in high dimension, it 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, so this is sitting in the Rn and this Rn is sitting at n plus one. So imagine this plane figure in the simplest example here. This kind of figure in the plane in this sitting in three space. Now in this three space, so call this y, call this x one half in a second to understand y.

So I have this x one half sitting now in this n minus one dimensional in the space of dimension n plus one. And what I do, I take epsilon neighborhood with this.

And take boundary with this. So here is a model picture of my interval in the plane. I take epsilon neighborhood is like that and here is my boundary. And since this boundary is straight here, just to have a picture more transparent. Straight here, straight here, and here will be two semicircle.

Observe that this is not smooth fully. Right? Because here is C2 jump, derivative jump. We can smooth them. It's nice to smooth them. And the point is when you smooth them in this example, even has positive curvature. Then scalar curvature here will be positive. And if you slightly perturb the metric, it becomes positive everywhere including here.

After you smoothed it, by small C infinity perturbation, it becomes really positive curvature. So every shape you have here has representative scalar curvature. Just the same, absolutely the same shape. Yes, epsilon, the smaller epsilon the better. Epsilon must go over very, very, very, very small.

And so, now. And now we just may say, aha. And what can be the surfaces with mean curvature? When you think of course, yeah, of course these are very special manifolds. They came out of Euclidean and so you must understand them anyway. But amazingly sometimes, the only way to understand them, go to the scalar curvature,

and then look the Dirac operator there, and then make some conclusion and come back, and then absolutely mystery what actually happens. So these of course some limit. So because usually this epsilon will go to zero, and this Dirac operator which you there degenerate to something else, but I don't know what the something else is. This corresponding object, I don't know what it is.

So I do 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 is as follows.

If I take inside of the Euclidean space Rn, some piecewise smooth set called Y0 of dimension n minus 2. And if I take now a small neighborhood of this, because we have some care,

then the boundary of this will have the smaller it comes, the bigger it becomes the Minkowski, and it actually blows up to infinity. And moreover, if I already have some part of that, which was this Poisson Minkowski, add here something like that, you can add this a little bit.

At this moment slightly smaller, right? So if it was flat, I cannot do it. Although the Poisson still can do it, and the whole thing will have the same scary curvature as before, except here it becomes smaller. And this is by the way inevitable, right? So there is this little theorem, and again I don't know any proof of this yet.

I haven't mind one, but you cannot do it. So if you take plane, hyperplane n minus 1 Rn, and you say, aha, can it be turbid with Poisson Minkowski, and keeping it at infinity, at infinity like that?

You cannot, but there is no quite simple proof here. And then the proof immediately comes. The moment you use this trick, you go to Poisson scalar curvature, there you know it, and you say, because already 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 problems. You have hyperplane in the Euclidean space. You cannot modify it with compact support, keeping here curvature non-negative, 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 convex, join them by this in two, and as I said, you can thicken it and have curvature a tiny little bit

because it was strictly convex, so it remains Poisson Minkowski. But if it was flat here, it would be impossible to do that. So it's rather kind of delicate. If it's Poisson, you can add this thing, this arbitrary small perturbation of Minkowski here, but if it was flat, then flat. And this is in the context of scale curvature,

this was a similar statement. It was apparently conjectured by Guerra, and it says, if you have Euclidean space, you cannot make perturbation of the magnified geometry in the compact region

without making scale curvature somewhere smaller, somewhere negative. So if you make such perturbation, necessarily scale curvature becomes negative somewhere. I want the mean curvature to be positive in the perturbation. Where in that statement is the mean curvature positive? No, I'm saying, you cannot make it.

Yes, I know, but where to be positive? Everywhere, outside, non-negative here. Compact support? Compact support, yeah. Why can't you just put a hemisphere and smooth it out? No, you cannot. You produce negative curvature. You cannot do it. That's, of course, a simple but non-trivial statement.

Of course, you cannot. By perturbation? The moment you start smoothing it in this corner, it produces negative sectional curvature, negative mean curvature. But where is the compact support? If it's not compact support, you can't 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. You can do it. But you can't exactly say in the reasons which perturbation you can make, probably the same as allowed by the positive mass theorem. This, by the way, because it's a positive mass theorem, I don't think it was formulated here. But you can immediately guess what it should be

now from knowing from scary curvature. So it must, roughly speaking, when you go to infinity, if you have positive curvature, it might go slower a little bit than Euclidean case. If it goes faster, you cannot do it. Or when it's equal, you cannot do it. But let me say, indicate, yeah, maybe I can educate how we can prove it.

Which I never checked it carefully, but by the end of the lecture, I say, indicate a proof which reduces to elementary computation. But you see, also there might be some subtleties here. So the three-dimensional case, high-dimensional case, slightly different because of kata noise, of different shape of kata noise. So there is some distinction between two and three-dimensional case.

Three-dimensional, high-dimensional case here, but not at this level. There are slightly more questions here. So, but the point is, I was explaining this example here, how you can make this kind of highly wiggle thing. You take this very narrow but still convex thing,

add this interval and glue them together, and you can do it with positive curvature. And this is not obvious, I mean, it's easy, but you have to do it. And so, well, maybe again, at some point I will explain what the proof is. But once it's being said, of course, it's exercise. Once you know that, it means it's a one-dimensional problem

because everything is symmetric. You have to solve some simple differential inequality and it's easily solvable, but the fact it is solvable, you have to know that. But now, so what Zafu, when you have any kind of thing with positive mean curvature or sudden mean curvature,

you can add these huge kind of things. A long co-dimension, one subset there, so in three space, certainly around points, yeah? But apparently you can't multiply 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 you have some sort of positive mean curvature, and more specifically, just in real space, there is Rn, and so it's a good measurement, this mean curvature is greater or equal to n minus one.

Because n minus one, you compare to the unit sphere. Unit sphere has mean curvature n minus one, right? Because there are n minus one principal curvature, they're all one. So that inside, you cannot put ball greater than the unit one because if it were a bigger ball, you move it,

it touches somewhere, and here, of course, there is kind of maximum principal 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 a usual kind of perturbation, idea of stability, completely breaks down. We can say, oh, this is not bad. No, let's see if we have fun stats, yeah? When you usually say something, when you prove some theorem and some inequality, and then how it's slightly perturbed, it's a small error. And we're very happy about that. Up to your point, when you say it's a ball, I mean, it'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 you make 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 fact that positive mean curvature makes the bulk of this small. But of course, you can have another one, another one, you can slightly perturb the curvature, but you cannot spread it in high dimensions,

only spread in the dimension below a certain level. And the second theorem here is, again, quite elementary. This is just maximum principle. Maybe in the past you can say something else about that.

I'll show you some aspect of that in a second, I tell you, which I understand very poorly, but quite amusing, but before that, look at the following thing. So I have this convex thing, I'll say mean curvature, greater or equal then, n minus one. And inside I take this hyper surface, inside, yeah?

But this also cannot be too big. And this I want to formulate in the following way. Now I consider the hyper surface. Now it's in there, so it's n minus two dimensional. So I'll call this z in Euclidean space, the dimension n minus two. And it serves a boundary of some y zero

of dimension minus one. Imagine you have this picture. By the way, you see, I put this here. And my reason is, of course, when you make it in tech, it's difficult, because when you write something like that, this means x times x times xn times. Right, so it's very, we'll say Euclidean space, okay, right?

But sn is certainly not okay. And I don't know what the right notation is, yeah? So to avoid this kind of confusion. But anyway, and assume that this one. But he just said that you put sn in 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 I have to, this hyper surface must be coordinated. So it's a Möbius band, it makes no sense. But it's coordinated, 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, y1. Also this boundary, as we had before,

such that this new surface sits in the bowl of the integrals. So more specifically, if you take any point from this y,

so distance, Euclidean distance of all points, y1 to the z, less than one. So the hole cannot be too big.

If you can feel it, by surface with large mean curvature, the hole cannot be too big. I cannot because I have a big hole from which you can go. So this is elementary. There are slightly variations of that which are later on which are not. So how do we prove that? This is an exercise also.

We can give an exam 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 feel it, which will be more efficient in a way. And of course, the extreme thing is just the sphere of coordination too,

where it is, where it works. And you see, it's a really different feeling for sphere. It feels by hemisphere with curvature greater than one, but this optimal feeling will be flat.

How do you get that? And again, you only use what you use for its maximum principle. You don't use anything sophistication except understanding. Understanding means somebody's boundary of something.

The claim is that if I have some co-dimensional two things, filled by submanifold curvature such one, there is another one, another feeling, y1. No, nothing about curvature, but it's sitting 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. Why is there a boundary distance from z? Not only it will be less than one. This new one will be just within distance one, all points,

all points y1 will be close, I'm sorry, all points from y1 to z will be less than one. So again, with this example, I have a circle, it's filled by hemisphere of curvature, mean curvature here less than one, and then there is a disk filling it, and so everything is in distance one.

Because here it's not, yes, this is too far. But you can do that. So what's the proof? Again, this is an exercise. You have to have some assumption on the y1? Huh? You have to have some assumption on y1?

No, but it has the same boundary. But it lies within distance one from z. For example, if you have a big circle, right, it's bigger than one, you can fill it by a small thing, yeah? Always there will be somebody far away. But this whole cannot be too big. The fact it's filled without control of the size,

only with control on the mean curvature, implies there is another filling which is small. Because, you see, this original filling could be huge, yeah? Let me give another example. It is my circle, and I fill 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 filling close within distance one to the boundary. This is model example and exactly what you want eventually to prove for scale curvature, which is still far from that. But this is elementary.

And one is wondering if there is a similar elementary argument for scale curvature. For scale curvature, there is no single elementary argument. Nothing can prove that. Either you prove minimal surfaces when you're stuck with high dimensions. And there are extremely technical papers by Shon Yao and by Lokham. Shon Yao is probably more readable.

But certainly, I have to say, it's one of the most technical papers in the field. And Lokham 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 a nice one. I can prove something in one line. The proof is as follows. What does it mean? You cannot fill it within distance one.

It's a negative statement, but you want to turn it into a positive statement. And so what do we use? Of course, duality, right? That's the power of topology. This essentially is Poincare duality, which has a different name. Poincare duality says, if there is no cycle here, there will be a cycle there.

And this is extremely powerful statement. Poincare duality tells you that two opposite dimensions talk to each other. And here it's kind of, I think it's exactly a kind of talk, which must eventually become essential for scale. The curvature says, you have this thing, and you cannot fill it by surface close to it.

Then there is a curve going from this side, going here and here, into infinity, which everywhere lies between distance more than one, and which has non-zero link number with this. And this is, well, Poincare duality is called, I think, Alexander duality.

So if there is a cycle in Euclidean space, it's not filling. It 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, now take a ball and move it in this way.

And I'm claiming that at some moment it will touch it like that. And if it touches like that, okay, it's impossible. Because this curvature is bigger than one. So you have to be sure, when it touches the first time, it may touch it from the wrong side. Because this feeling could have been, you know, like that.

And when you go from here, it attaches from the wrong side, not from this side. So I have to show you some moment to attach from the right side. 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 Morse theory here, then attach with some moment from the right side.

How do we do it technically? So what you do, you can see that this kind of tube, which goes here, or this sphere times interval going here, take pull back of this thing, because you don't touch the boundary,

you become closed manifold, it has so many components, so in the cylinder, you get some cylinder. And then it is hypersurface here, when you meet your hypersurface. And because index was non-zero, there are many components of that.

There will be some component which separates to ends. And because it separates to end, when you go, you 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 the 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-1, sitting in the Euclidean space, closed hypersurface, mean curvature greater or equal than n-1. 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, right?

It's just strictly distance decreasing. So if it's smooth, you can say the differential is less than one, then the map is contractible. Actually, it may be easily subjective, but here, topology again is fundamental. So this feature of scalar curvature and this mean curvature, all their properties, even if it's not apparent, are linked to topology,

namely to their fundamental cycle. And when you start proving various theorems, you see just how it kind of interferes everywhere. And for example, when you see the proof, which goes with the YDRAC 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 a number which assigns to some differential operator, but it gives you a kind of fundamental homology class of your manifold, because it gives you an index of your operator twist within your vector bundle, so it becomes dual to the k-th theory.

So it's really a kind of fundamental homology class, all the time enters the game, right? And here it is, especially the sphere, not the same, you 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, obvious, elementary, right? So I'm saying that if I have a map, which decreases distances as understood, so here is my manifold,

if distance is understood like that, and accordingly distance in the sphere, where a map also is understood like that, it's again the sharp inequality where equality holds for the sphere, but the proof is elementary, which I'm going to present. However, if you take intrinsic distance, because extrinsic distance and intrinsic are very different, right?

Here is extrinsic as one and here is huge, so it's very big, so you can imagine, you can spread it, but you cannot. So everything says that it's essentially 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, in the literature related to physics, there is a so-called Penrose Conjection, which was proven, which says what happens for certain manifold dimensions of positive scale curvature. Indeed, this picture is partly justified in some examples, but not at all everywhere.

But again, there are two different kinds 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 theorem about Dirac operators of the style which I described already.

So, which I didn't say about Dirac operators, but they are rather sophisticated objects compared to this. So how do you prove it's elementary? So indeed, so 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 y n minus one, we bound some x, and here is my sphere s 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 was non-contractable, this relative map, so non-contractable means, of course, you have to know in this case, it means it has non-zero degree.

Contractable, I say, is just to avoid appeal to homology, which is more kind of, but on the other hand it's impossible to do it without it. Then you know this map to the ball will also have non-zero degree, and in particular it will be onto, right? But essentially, there will be extension and necessarily onto, because the map was of non-zero degree.

If it were contractable, even onto, this doesn'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 the fundamental theorem, very elementary, again very simple, and kind of quite, quite, quite, quite, quite remarkable in my view.

Here's the bound theorem, it says, I have a hidden space of any dimensions, maybe an infinite dimension, if I want. And there is any subset there, y. 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 a specific property of Euclidean spaces, it's not true in any metric space. It's rather, rather, rather amazing property. It's rather amazing property. Again, if you know it, even if you know it, the proof is not so simple, I mean, you have to make some guesses here.

I don't want to make a pronoun, I know, I always forget what computation I have to make. But it's not obvious. What is the hypothesis on y? Any subset, any, any, any subset, any, no assumptions.

I have myself bad experience with this theorem, because I heard about this 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, there was some kind of situation I want to generalize and I couldn't. Because if you, the original proof, which you find in textbook by Federer, but of course it's a 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, I forgot, Shirodi with somebody. And it's an independent proof by Petruin, from whom I tell on first. And it says the following, that, I say slightly, slightly.

So x was distance decreasing? Yes. With Rn euclidean? Yes, euclidean, of course. And then it extends the whole euclidean space also in euclidean. So, and the theorem is as follows, that this, this property is true more general.

If you have x and y, such that sectional curvature of x is greater than some constant, kappa. And here, sectional 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 to make little, to avoid some obvious count examples. But if you, locally it's always true. Globally you must be slightly careful.

Any is less than n, or any is? No, no, any, any, any, any. You might put Hilbert spaces if you want, yeah. Dimension completely immaterial. Here is zero curvature, so here is no dimension at all. Maybe in three dimensional spaces. But if this curvature is greater than kappa, and is 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. But the more sophisticated proof, more elementary proof, I don't know, different in my mind, and I just couldn't prove it to my shame.

But this is true, and this, by the way, gives you a definition of curvature. This way you can define what curvature is greater than something or less than something. So you compare it with Euclidean spaces, right? If you use some of them flat, then this is necessary in sufficient condition. So if this property defines the sign of the curvature, or the sign bound on the curvature from either side.

So this is kind of the best definition in my view. It's really a functorial kind of property of the curvature. And there is no such definition of scalar curvature. Hopefully, you want to find such a definition of scalar curvature. And this is, you see, because on the other hand, you think of sectional curvature, you understand very well, compared to scalar curvature.

However, if you want to prove that some exotic sphere has no metric with positive sectional curvature, you know nothing except what comes from scalar curvature. So in a way, this results with scalar curvature. On one hand, they can say, aha, they tell you some new story about sectional curvature.

But the major thing they tell 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's maybe a wrong question to ask what exotic spheres have metric with positive sectional curvature. Maybe they are wrong questions. Because there is a reason for such thing with 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 the theorem which I said. This is Kirchbaum's theorem. And, well, this is maybe an exercise for you to prove.

And so, let me give one form of the theorem which will be relevant for some moment. It says the following. You have two simplices in any dimension.

And they kind of correspond for faces to face, etc. 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 all the dimension three. Dimension two is obvious because sum of angles is given. Dimension three is this. First, you invite you to derive it from that theorem. And secondly, try to prove it independently. For simplicity, this is easy, relatively easy.

And this is unknown for general polyhedra. Convex polyhedra. And it is also related to the scalar curvature. So you can prove it in some cases using Dirac operator. But this case, you can prove it 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. It says it follows from some kind of particular statements in the Hodge theory for toric variety. Something you can prove in this respect.

But for general polyhedra, 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 and remembering this is kind of positive mean coefficient, this is called scalar coefficient, and then you have this Dirac operator. And then again, of course, when Dirac operator becomes kind of very degenerate operator,

it probably reduces to something elementary. And then Hodge theory of toric varieties may enter. But this actually, I must say, I don't know. But this is at least some exercise like this.

For this one, in Euclidean space, in my view, this kind of number one theorem after Pythagorean theorem is number two. First Pythagorean theorem, then there is Kirchbaum theorem. But it's not so well known. I mean, you have the textbook full by some nonsense, but that remarkable simple theorem. And it's really the proof is kind of, rather kind of, you know, you have to guess it.

I mean, just it's not, doesn't 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're invited. So I put something that I wrote on my web today. And then there are the form which contains some of what I said and something I'm planning to say later on.

Okay. So you have questions. Now you have one minute. You have questions.