 So welcome all to the beginning of this course. So my name is Claudio Arezzo. And I'm your lecturer in a topic which is called differential geometry. I hope you all know something about it, but we will start from the very beginning. So you are not supposed to know anything about this before. So what is differential? Actually, a few instructions before starting putting our hands on the subject. So you are absolutely invited to come every time you want to my office. I'm always here. I don't usually fix an office hour if you prefer to have a specific hour to come. We can fix it. But please come whenever you want, OK? I will not give you notes for this course, but because there are plenty of books and you are free to choose the book you prefer to follow similar lectures, the best book that I consider on the thing we will do is by Manfredo do Carmo. And it's called The Geometry of Curves and Surfaces. But as I said, there are plenty of very nice books that you can choose on this topic. Every two weeks, I will give you a homework with a list of exercises, five or six exercises. And so there will be one week the sheet with the exercises. And the week after, I expect you to give me your homework written. This will not be part of the evaluation of the exam, OK? So you are strongly invited to do it, to write down the solutions. You are free to do them working together, discussing in little groups all together as you want. It doesn't matter. It's not part of the evaluation, OK? But it's important you try to write down things. Mathematics, 50% of our job is having ideas. The other 50% is being able to communicate the ideas, OK? So try to make an effort also to learn how to write down things, OK? The first homework will come in 10 days or so, because now, of course, it's too early, OK? So what is differential geometry about? Well, you have seen certainly some geometry in Euclidean space and analysis in Euclidean space. You are able to study functions defined on open sets of RN or even infinite dimensional spaces, but in some sense always vector spaces. All the spaces that you know probably up to now are spaces where you can make translations and so on. In geometric terms, you would say these are flat spaces, OK? So differential geometry is the part of geometry which studies how curvature influence our ability to study, for example, differential equations or to characterize spaces in terms of their curvature. So in fact, the main problem is what is curvature, OK? Giving a sensible definition of what is something curved compared to something flat, it's almost a big, I mean, it's almost 50% of the problem, OK? Once you have a good definition of curvature. And in order to have a feeling of what is a good definition of curvature, the idea is to start from the lowest dimension of spaces and then going up, OK? So of course, zero dimension doesn't make sense because it's a point. So what is a flat point or a curved point doesn't make any sense. So we start with curves. So one dimensional objects, we will be quick about this because I assume probably you all have seen already something about this. And then we will move quickly to surfaces, so two dimensional objects. And already two dimensional objects have already all the features that make things complicated and nice to go higher dimensions, OK? So let's start with one dimensional objects, curves, OK? And this will be the topic probably for the first three or four lectures of our course, OK? The first thing to note, this is the first definition. So you would like to say that a curve is a set, OK? It's a specific set with some property, but this is not what we are going to do, OK? And we will have to think about why we are not doing that. But the most common definition of a curve is in fact of a map, OK? So we are going to confuse in our mind in some sense, but we have to be careful about this confusion, of course. It's a plain confusion. The image of the map and the map itself, OK? So a differentiable curve, by differentiable, I will always mean c infinity unless I specifically say that I want lower regularity. Sometimes it will be useful to speak about c1, c2, c3. But if I don't say anything, if I just say differentiable, it means smooth, OK? A differentiable curve is a map, let's say alpha, from an interval 2r3, where i is an interval, let's say, with extreme a, b, a subset of r, OK? And that's it. Of course, this is a smooth map, OK? It's a c infinity map, OK, from an interval 2r3. Of course, what does it mean smooth? Well, you know, this is a map from a subset of r to a subset of r3. So it's a usual definition, but it's useful to underline. That means that if we call t the parameter on the real line r, that means that alpha of t will be a triple x of t, y of t, z of t. So it will have three components being in r3. And smooth means, of course, that x, y, and z are smooth functions. So few notations, words, and other definitions. So given a differentiable curve, we will say we will call the vector given by the first derivatives of these components. We will indicate it by alpha prime of t. This is just the vector whose components are the first derivatives of the original functions. This is called the tangent vector of alpha at alpha of t. And just to add the last definition in this first series of definition, we will say that a differentiable curve is a planar curve. So alpha is called planar. If there exists a plane, so a two-dimensional plane, is dimension 2, a plane p in r3, such that the image of this map, alpha, is contained in this plane p. Now, since by a rigid motion in r3, so imagine you have your curve. So this is r3, x, y, and z. And your curve lies in some plane p. This is the image alpha of i. But by a rigid motion, I can change coordinates of r3 in such a way, by a rigid motion, p, I can always think of p as the plane z equal to 0. I can always put a plane by a translation and a rotation. I can always think of this plane p as to be z equal to 0. So a planar curve, in some sense, after a linear change of coordinate, will always be of the form, alpha of t will be of the form, x of t, y of t, 0. So being 0, the third coordinate, I drop it. So a planar curve is just given by two functions, x and y. Be careful, up to a change of coordinates. OK. So this is definition one. Now, comments. Is this when you give a definition, you have always to check if the definition is capturing exactly what you wanted before? Because in some sense, you have, in your mind, some kind of idea of what a curve should be. And now the problem is, is this a good definition? I mean, does this correspond to what we want? So one. So what kind of objects can we study with this? Well, first, alpha is not required, note that in definition, alpha is not required to be injective. What does this mean? This means that if you think of the image, I mean, the image of alpha can be something which has self-intersections. For example, take as a planar curve now, alpha. E g is a shortcut to say, for example, in case you want. For example, alpha of t equal, well, this is t cubed minus 4t, t squared minus 4. If you just check, you can quickly check that if you take the two values, t1, 2, to be equal, plus or minus 2, you get alpha of t1 is equal, alpha of t2 is equal to 0, 0, 0. And actually, you can have a feeling of what the graph of this curve is, what is the image geometrically in the plane x, y. This is a planar curve. You see, I've already written just two coordinates. And this is a curve which looks like this. So when t comes from minus infinity, you see that x comes from minus infinity and y comes from plus infinity. So it comes from here as t moves going from minus infinity onwards, then you reach to the first value minus 2. So you are moving in this direction on this curve. At t1 is equal to minus 2, you are in the origin. Then you make this loop, and at t2 is equal to plus 2, you are again here at the origin. And then as t goes to plus infinity, this becomes plus infinity, and plus infinity means you are going in this way. So first, first comment, a curve can have self-intersections. That's good, because so in principle, we could have been studying only objects without self-intersection, but the definition is sufficiently flexible to allow also this situation. A bit more surprising is that comment number 2 is that the graph of alpha, in some sense the image of alpha, alpha of i, could have corners. So you see somehow you're saying alpha is made of smooth functions. So you might suspect that the image of alpha, in some sense, it's a smooth thing without corners. That's the idea behind being smooth. Well, that's not true. Alpha could have corners, even if all the components are smooth. For example, again, alpha of t is equal. This is a famous example. You see, this is made of polynomials. So these are even better than smooth functions. But what is the image of this function here? Well, you can think of it in many ways, because it's also a graph of a single function, if you want. So this is also the function, is the graph. So alpha of i, in this case, i, of course, is the whole real line. There is no problem, is the graph of the function 2 over 3. You see, if you think this is y, you divide by 3, you get x. And then you see why corners can arise. Because this is not a smooth function. This is made of smooth functions. But all together is the graph of a function which is not smooth. And in fact, if you draw it, you get something like this. It's the cusp. It's the prototype of what we call a cusp. This is the prototype of what we call a node, a self-intersection like this is what we usually call a node. And this is what we call a cusp. How do you write it? Sorry, can you repeat? Yes, to work with the cusp. Ah, cusp. Sorry, to which one are you referring? Oh, yeah. You are. Take your test. Yeah, yeah, yeah. OK, OK. Well, it's enough to call this x and this is y. Now, the third comment is even a bit more delicate and requires a little bit of topological insight. Comment three, even if alpha is injective, so even in the good case, alpha could be node and homeomorphism onto his image. This requires a few seconds of thought. So what does it mean? Well, you might be led to think, well, after all, alpha is what? Alpha is a particle moving in space. So of course, if it has self-intersection, somehow the topology of the image is different from the topology of the domain. So you are thinking of a map going from an interval, so topologically trivial, to something if it has a node, clearly here there is some topology. And so the two things are not homeomorphic. That's OK. That's clear. But then you might be led to think, well, this is the only possibility to introduce some topology in the image. So now the comment I'm making is no. There is something more subtle than this. How do you see that? For example, let me explain what I mean with an example again. Suppose you take alpha, now it's defined from minus 1 plus infinity into R2. Again, it's enough to take planar curves. And you define it to be alpha t is equal 3t over 1 plus t cube, 3t squared 1 plus t cube. How does this curve look like? So here we are in the xy plane again. So if t goes to plus infinity, well, let's go first to minus 1 from the right. So if we approach the left extreme of the interval, what's going on here? Well, on top, on the numerator, there is nothing strange going on. But below, it's going to 0. And it's going to 0 from the positive side. So this is going to plus infinity. So this curve from t equal to minus 1 starts here. Plus infinity, plus infinity in some sense. Then it comes down. I mean, it does something. When you go to 0, when you reach t times 0, what's going on? Well, below now it's 1, 1, and above it's 0, 0. So you are passing through the origin. You see, y is always positive. So you are always in the upper half of the graph, of the half plane. And then when t goes to plus infinity, what's going on? Well, now, guess, 0, 0 again. So it goes back to the origin. But it never reaches the origin, of course. It's just the limit as t goes to infinity. So that means the curve must do something like this. Well, this is a famous curve, so let's give it a name. Again, you can also characterize it in terms of polynomial, because it's the 0 set. This is just for your curiosity. But I mean, it's 3xy is equal x cubed plus y cubed, OK? So which again shows you that this phenomenon can occur even as a 0 set of a beautiful function, because it's a polynomial function equal to 0, like in the case of the cusp. So let me go back to the comment. Why this is an interesting example in our theory now, because you see the image of this curve, of course, the initial topology was the standard topology of the real line. The final topology on the image as a topological subspace of the plane is different. You see it? Of course, everything boils down to what's going on here, around here. See the topology induced by the plane on this set, how do you get it? You get it by taking open set of the ambient, intersect with the subset. That's the induced topology. That means that open set, open neighbors of this point are of the form interval around this, which is OK, but plus something like this. So you see, neighbor of this point are always made like this. And this topology is not homeomorphic to the standard topology of the interval, OK? So this is kind of curious, but shows you how delicate things are. So the image of a curve, even though everything is defined on intervals, the image can be topologically more complicated than the interval. And not just because you can make loops. OK, loops are OK. That's a way to change the topology. But there is something else that we have to be careful about. OK, so these are three comments about the definition there will be others coming later. But now let's make other examples. We need to have a good set of examples to go on. And in some sense, up to now, I've shown you pathologies, which is typical of mathematicians. You give a definition, and you start by going to the most delicate things. But now let's stay to the most standard situations. So what are the simplest curve? The simplest curve you can think of, of course, is the straight line. How do you realize a straight line in this language? In some sense, what we are doing now in this definition of the differentiable curve is what you are used to call a parametrized curve, probably. You have already seen something. So how do you parametrize a straight line? Well, you need a starting point and a vector, which gives the direction. So you just write it something like this. Given two fixed vectors of R3, the straight line passing through the point v0 with velocity or tangent vector or directional vector or whatever you call it. v is this one. And here, of course, you see that t, the parameter is free to move in the whole real line. So this is, of course, the simplest. I mean, if our theory does not cover lines and circles, then our theory is to be thrown away. So circles, how do we parametrize circles? What do we need? Well, now I'm parametrizing it as a planar curve, because, of course, a circle is free to be on the plane that you want. So again, I pick the standard plane to be z equal to 0, and I give you the parametrization in x, y. And if you want, so you need a center and you need a radius. So if you give me a center and the radius, I just construct a curve like this. Now the c will be the center of my circle. So it's just any point in the plane. R is just any positive number. It's the radius. And then t, again, it will be free to be any real number. If I parametrize the circle with the t varying on the whole real line, of course, this means I'm going around and around and around infinitely many times. Sometimes it will be smart to restrict yourself to the piece of the interval which covers the circle only once. But it's not necessary. Depends on the situation. So the question is why I'm dividing t by r? Just to have this question. Just to stimulate a question. In this moment, it's totally irrelevant. If I'm letting t to be free, if I call it t or t over r, it's the same thing. It's just a scaling. It's usually convenient once we will fix a range, an interval which will cover the circle only once. It's usually convenient to have this range to be radius independent. You will see it. It's just a choice of, after all, if you call it s, you can do it whatever you want. The third type of curves that I want to introduce right away is what are called helices. And these are what? Let me give you first the parametrization. Then we will comment on the graph on the image of these curves. So alpha of t, I will call something to be an helix if there are two numbers, a and b, such that these are space curves now. I really need three coordinates. So these are not planar curves. Cos t over square root of a squared plus b squared, a sine t over square root a squared plus b squared, and then bt over square root a squared plus b squared. You can make the same objection. So now what are a and b? In fact, it doesn't matter the sine of a and b. It doesn't matter the sine. So a and b are just real numbers. Of course, you avoid both to be equal to 0. Otherwise, this is nonsense. So let's say a and b are both different from 0. Otherwise, they are degenerate. These are just any two real numbers. You can make the similar objection that your colleague was doing before for the circle. So after all, if t, again, here t is free to be any real number, so you might say, well, if t is any real number, it's enough to give a different name to t over square root of a squared plus b squared, and this becomes a cos s, a sine s, and bs. You will see a similar convenience later for having this notation. So geometrically, what are these? Well, how do I get a feeling of what is this curve in space? Well, there is no rule for that, geometric intuition. So one way is to make projections and see what are the projections on the coordinate planes. So here we are, x, y, and z. So how do I understand what is the projection on the z equal to 0 plane? Well, I just put this equal to 0. So this is not there. It's like if this is not there, and I just want to see what is this curve here. Well, but this curve is exactly something like this. It's a circle with center d origin because there is not a translation here. And with radius a, mod a, I mean, sometimes I will automatically assume that a and b are positive, because otherwise you can switch things. So the trace on this plane will be a circle. So what is the effect of the z coordinate? You see, in the z coordinate, this is a linear function. OK? So what's going on? Well, after all, I already told you the name. So you know what's going to be. So what's going to be? This is drawing a circle, but not on a fixed plane. It's drawing a circle while you are going up or down. I mean, depending on the sign of b, but suppose b is positive, you are going up and you are moving on a circle at fixed speed. So you should imagine something like this. My drawing is terrible, but more or less gives you the idea. So the project, you should imagine a cylinder. A cylinder. On this cylinder here, you draw this kind of helix at fixed speed. OK, so this is a little series of examples. Can I erase the blackboard? Because now what is the first geometric property of a curve that we want to learn? Well, the first geometric thing associated to a curve is its length. So let's learn how to measure length of curves. The problem is, again, giving a good definition. So giving a definition that corresponds to what we know, because somehow we feel that we know, but we have to mathematically formalize it. You see, this game, the reason why we are studying curves is to practice in this logical thing. Because I can tell you, when we will study two or three or four or six-dimensional objects, it will not be even clear what we have in our mind. So let's play this game in dimension one when everything is easy. It's also a little bit boring. But we understand the method to do it in dimension 26, where it will not be clear at all what is the area or the volume or something of an object. Because after all, what we are doing now was known at least 2,000 years ago. Not mathematically formalized as we do it now, but they knew it. So given a differentiable curve, alpha in space, and suppose we take a compact subset of the interval where it is defined alpha, because after all, we can measure lengths only of finite pieces of a curve. Otherwise, we should expect the length to be infinite. So the question is, what is a good definition of, so in this way, I introduce the symbol that I want to use, the length of the curve, of the piece of the curve between A and B. Well, the idea is to break the curve into pieces. So suppose that this is your curve, and suppose this is alpha of i. And let's say this is alpha of A, and this is alpha of B. So the idea to solve this problem is take a partition of the interval AB. So what is a partition? A partition is just, i.e., a choice, so P of numbers, which starts from A and gets to the final one has to be B. So remember, if this is alpha, here you have AB on the interval. A partition means just, let's call this t0. Let's call this tn. And let's put some numbers in between, ti. This is just a partition. So then you go and look what are the corresponding points here, alpha t, so t0, t1, alpha t2, alpha t3, and so on. You go and see the corresponding point. And then you define the length, so this is a definition, the length between A and B of the curve alpha subject to the partition P. I'm not going to write it down. You read these symbols like the length between A and B of the curve alpha subject to the partition P. To be what? To be the sum from i equal to 1 to n of the length of the segments, alpha ti minus alpha ti minus 1. Think for a moment. We are in Euclidean space. So given two points, I can draw the segment. And I know how to measure its length, the length of a segment. And this is the symbol here. So I'm saying the length of the curve subject to P is essentially the length of this series of segments. Of course, this can be a good approximation of what I think should be the length of the curve or a bad approximation. So how do I get closer and closer to what I think is the right notion of length? Well, let me call it now the blackboard is getting small. Another notation, which is useful. I call the norm of the partition to be the maximum over i. So what does it mean, this norm? This norm is big, of course, if there are two points on the interval in this partition which are very far away. And this norm becomes very small if all the points in this partition are very close to each other. So the theorem, which will justify our next definition, so theorem one is the following. Under these notations, so given alpha, given a complex subset, complex subinterval of the domain of alpha. We can say that for any epsilon there exists delta such that if the norm of the partition is less than delta, then the length between a and b of alpha relative to the partition p minus this number, minus the integral between a and b of the norm of the tangent vector in dt, this is less than epsilon. Before proving the theorem, why this is an important theorem? Because you see, this is telling me that this number here, you see this number is constructed only using alpha. So p has disappeared. And what this is saying is that if p becomes, I mean, denser and denser in the interval, because the norm of p getting smaller and smaller means that the number of points is increasing and they are becoming very well distributed in the interval, then the length of this segment, the sum of the length of these segments is converging to this number here. So this is a good definition of length. At least this is what I was suspecting as a good definition of length before starting. I mean, you would say if you pick three trillions points here and you construct all the segments and you sum, this should be a good definition of the length of this curve. So in fact, let me give it directly. Now we will prove the theorem. But this justifies the following definition. We will call the length between a and b of the curve alpha. Now it's not subject to any partition. This is the length of the curve alpha to be exactly the integral between a and b of the norm of the tangent vector in dt. So before proving the theorem, are there any questions now? Is it clear the logic that we are following? Well, the proof of the theorem is actually simple. It's essentially the mean value theorem. It's an application of the mean value theorem. This is the statement that's right here, the proof. So let's define the function f in this f from 3 times i. So i cross i cross i into r. So this takes three numbers, t1, t2, t3 into r and gives me the square root of x prime t1 squared plus y prime t2 squared plus z prime t3 squared. Now this is not completely crazy. What is the norm of the tangent vector? So the tangent vector, so this is a remark just to explain because now the proof will work and it seems magic. No, it's nothing magic in mathematics. So alpha prime is the vector x prime. Alpha prime of t is the vector x prime t, y prime t, z prime t. So what is the norm of the vector? Well, the norm of alpha prime is just square root of x prime of t squared plus y prime of t squared plus z prime of t squared. So here the only trick is to split. For each coordinate, I use a different parameter. See, when t1 is equal to t2 is equal to t3, this function is exactly giving me the norm of the tangent vector. So this is just a trick to decouple the variables. Now, f is clearly a continuous function. Maybe not more than continuous because it's a square root of something. So in principle, it could be just continuous. If the thing inside goes to 0, maybe it's not differentiable. But it's certainly continuous. So hence it is uniformly continuous on compact subsets. On, you see, we are restricting ourselves to a compact sub-piece of i. So AB, AB, AB, OK? In fact, let me call it AB cross AB cross AB with some abuse of notation. This is by definition AB cubed, OK? Just a shortcut, OK? So what does it mean? It's uniformly continuous. So for any epsilon, there exists delta such that if I pick two points, any two points in this range, so such that if t1, t2, t3, and s1, s2, s3, both lie in AB cubed, which differ from each component with t1 minus s1 less than delta, t2 minus s2 less than delta, and t3 minus s3 less than delta, then uniformly continuous means if I take two points sufficiently close, I mean, there exists a measurement of closeness which guarantees that the image is not too far away. So that f of t1, t2, t3 minus f of s1, s2, s3 is less. I should say less than epsilon. Here, for the proof, it's convenient to say epsilon over b minus a, which is just another epsilon, OK? So this is just the statement that this function is uniformly continuous. How do I use it? I mix this information with the mean value theorem in the following form, OK? By the mean value theorem, I know that the length of this vector ti minus alpha ti minus 1, this is equal actually to f of beta i gamma i delta i times ti minus ti minus 1, OK? For some, beta i gamma i and delta i in this interval ti minus 1 ti, OK? And this is simply the observation I was doing before, OK? The relationship between f and the norm of the tangent vector, OK? Just convince yourself that, I mean, it's immediate, OK? But then it's just a matter of putting things together. So once you have these two bits of information, and now, so now the length between a and b of the curve alpha subject to the partition p is what? It's the sum of these by definition, no? I take a partition and I take the sum of the length of these segments. So using this property, I can write it as the sum from i to 1 to n of f of beta i gamma i delta i ti minus 1 ti minus t1 minus ti minus 1, OK? So this is one side. Remember, the statement of the theorem was that for any epsilon there exists delta such that if the norm of p is less than delta, this number is essentially, it's close to the integral of the norm of the tangent vector. So what do we have on the other side? We are comparing this with what? With the integral of the length of the tangent vector. So how can we express it in this language? Well, of course, we can split it. The integral are additives in the extremes, OK? So if I have a partition, this becomes just the sum of the integrals between ti minus 1 and ti of the length, OK? This is just a partition. But then again, by the mean value theorem, each of them, so now apply the mean value theorem to this function here. And this is equal, has to be equal, so this is the sum of the value itself at some point, xi i, OK? ti minus ti minus 1. But what is alpha prime of xi i by the initial observation in the definition of f? This is f of xi i, xi i, xi i. So this is equal to the sum from 1 to n of f xi i, xi i, xi i, ti minus ti minus 1, OK? For some, of course, here, what I mean is for some xi i in the interval ti minus 1 ti, OK? And now you are done, because what does it mean? Now, these are the two objects we are comparing. And we are comparing under the assumption that the norm of the partition is less than delta, OK? So now if we take a partition with norm less than delta, that means that all these ti, the length of these intervals are all less than delta, by definition, OK? But then what happens? If ti, so this implies ti minus ti minus 1 is less than delta for any i, OK? But then beta and xi i, beta i and xi i are both in the same interval. So they must be delta close. They are inside an interval of amplitude delta, OK? So this implies beta i minus xi i less than delta. And the same for gamma and the same for delta i. Of course, don't be confused about delta i and delta, OK? I just had a few. So gamma i minus xi i is less than delta, and delta i minus delta is less than delta, sorry, minus xi i is less than delta, OK? Because they all fall in the same interval, and this interval is small by assumption, OK? So now that's it. Put things together, and you get to the conclusion, OK? There is nothing left. What do I mean by put things together? Remember, we were evaluating the difference between this and this in absolute value. So now you just take this formula and this formula. So you are taking this minus this in absolute value, knowing this. How do you use it? But there is written here how you use it. So you apply the uniformly continuity to, instead of t and s, to b times xi to the gamma and so on. So basically, the t will be xi i, xi i, xi i, xi i, and the s will be beta gamma delta, OK? And here there is written exactly what you want, OK? So this is the end of the proof. It's formally not very beautiful. I have to admit, but it's simple, OK? It's just the mean value theorem applied with a certain ingenuity, OK? As I said, so the basic comment was after the statement that this gives us a good definition, at least gives us a good candidate for a definition of length. But now we have to be careful what is this definition taking us to. So remember, this theorem is justifying and giving us also a way to explicitly compute for hundreds, even thousands of years. This was the way to compute length of curves. So by really putting denser and denser partitions, OK? For example, all the, I mean now we know the value of pi up to few trillions digits, OK? But the way Greeks, for example, were estimating pi was because of course they knew that the length of a circle was 2 pi, I mean radius 1, OK? And then how you compute the length of a circle, well, you put 100 points, 200 points, 300 points, OK? And that was the way you do it, OK? So in practice it could have been even useful. Now, of course, with our computational skills, this is getting a bit, but I mean still. So in any case, let me remind you, we got to this point, OK? So this was justifying the length, the definition of length of a curve between two points as the integral between A and B of the length of the norm of the tangent vector, OK? Now, in order to be a good definition, so this seems a plausible definition, but in order to be good, we have to be careful. We have to check one creep property, which if it turns out to be false, it means it's not a good definition, every time in geometry, OK? I mean, let me, this is a key point. So what does it mean? It has to be invariant under isometrics. We have defined this length only using our ability of measuring length of segments, OK? Now, what are isometries of R3? So we have our curve alpha in R3. And the point is, if we have an isometry of R3, then of course, our curve will change under this map. Now, the point is, the length of a curve before and after applying an isometry, has it changed or not? OK? Now, this is a quick question, because if it has changed, it means that this object is not really a geometrical object. A geometrical quantity is something which has to be invariant by the group of transformation, which leaves the things that we are using invariance. We are using the measurement of length of segments. Isometry is the right thing, OK? So this is a good geometric object if it's invariant by isometries. Otherwise, it's not a good geometric object, OK? Now, the point is, so to answer the question, we need to understand which are the isometries of R3. Do you know it? Do you know them? Translations and rotations. Perfect. So, well, even though I must say, when I dig in my experience of teacher, sometimes this is the definition. So the point is, an isometry is by definition something which respect distances, OK? And then you prove that for R3, it's a composition of translations and rotations, OK? So, well, is it clear to all of you? Because so let me speculate a moment on this, OK? So an isometry, an isometry of R3, of course here we mean R3 with the usual way of measuring distances, OK? So R3 has so many structures because it's a group, it's a vector space, it's everything, OK? So when you speak about R3, you need to say exactly what kind of structure you are looking at. And now, since we are doing kind of metric geometries, the key property of R3 that we are interested in is that we are able to measure distances between points, OK? In some sense in a canonical way, OK? An isometry of R3 is a map, is a transformation, is a transformation of R3 such that the distance between two points is the transformation, let's say phi over 3 such that the distance between two points, p and q, is equal to the distance between phi of p and phi of q. OK, so it respects distances. Of course, this is for any choice of p and q. Now, it's a nice exercise, it's not easy of standard classical geometry to prove that out of this condition, you get that this implies that phi is up to a translation, a linear map whose, if you want, once you know it's a linear map, it's usually nice, I mean, quick to associate a matrix by using a standard basis. A translation is a linear map whose associated matrix, A, satisfies a A transpose equal the identity, OK? So it's what it's called an orthogonal transformation, OK? So in fact, strictly speaking, it's not a composition of a translation and a rotation, because that could be also a reflection, OK? Out of this equation, you cannot decide that the determinant of A is equal to 1. The determinant of A is equal to plus 1 or minus 1, OK? If it's plus 1, it's called a direct isometry. If it's minus 1, it's called an inverse isometry sometimes. But it doesn't really matter, OK? This is just names, OK? OK, this is enough for us. This observation is enough for us to conclude that our definition of length of a curve is invariant, because of course, length of segments is preserved by isometries, certainly preserved by translations. I mean, if you move a segment by a translation, the length doesn't change. And if you act on a segment by an orthogonal matrix, its length doesn't change, OK? And so being our definition of curve of length, essentially, the limit of lengths of segments, then it's OK, OK? So this is a good geometric definition of a quantity. So length is a geometric quantity, OK? This is the philosophy of a statement like this. Now, another nice curious remark, which gives also the first exercise I want to leave you, is the following. I leave the definition of length there, OK? So now that you know what is the length of a curve, it's quite natural to ask which are the curves of minimal length, OK? And of course, again, remember, I know it might sound a bit boring at the beginning, but we are building, again, I'm saying, again, we are building a method for situations where the solution will be highly non-trivial, OK? So we want to check that we are building a theory, even when we already know the answer, OK? Because now we know the answer to the problem. What are the curves of minimal length between two points? I mean, we would be amazed if the answer was not this one, OK? But there will be a situation where we have absolutely no clue of what will be the answer, OK? So let's spend two minutes on this, even if it has to be simple, OK? So we have a curve. We take a compact subinterval of its domain. So what do we want to prove? We have two points. So we have our curve, which we are looking already at the piece of it, so between alpha of A and alpha of B, OK? So we want to argue that the length of the segment, which is just alpha of B minus alpha of A, the norm of this vector, is less than or equal to the length between A and B of the curve alpha. Well, is this true? Yes, 30 seconds proof, OK? What is alpha of B minus alpha of A? This is just the fundamental theorem of calculus, OK? This is, I can say that this is the absolute value of the integral of alpha prime without the norm, OK? But then, by the simple properties of the integral, I can put the norm inside, and then I get less than or equal, OK? And this is exactly the length, OK? So very simple. It is true. Segments minimize length in Euclidean space, OK? So exercise. If you get equality, does that mean that alpha is the segment? Actually, this will force you to think. It's one of those problems where when you understand the question, the solution is immediate. But understanding the question is not because there is an ambiguity here. What is a segment? The segment means the geometric object or the parametrized object. I mean, you can take, we said, a line is something of the form alpha of t is equal to tv plus v0, no? So I guess a segment could be, by definition, just this between with t in some fixed interval, OK? So this could be one way to say what is a segment, but then this will not be true, OK? Think about this. You see, out of the length, the point that you have to convince yourself is, out of the length, you are not able to reconstruct the way the particle is moving on the curve. So it is true. This will be true, has to be true, but you are not able, you cannot decide if the point going from alpha of a to alpha of b will go in the uniform motion way or it will go, for example, like a pendulum. The length of this curve is the same. I mean, if the particle, for some reason, is stupid or has some forces and wants to go back from time to time and go on, at the end, the length of the curve is always the same thing, OK? So here you have to adjust. I mean, is the segment up to parametrization? I mean, if you parametrize it well, this is a geometrically meaningful question, OK? And in fact, this is exactly the problem we are facing in the last 10 minutes, OK? What is the effect? After all, we want to do geometry, OK? We are using analytic tools to do some geometry. Parametrizing object is very convenient, but it introduces a problem. Passing from, I mean, you can parametrize the same geometric thing in an infinite number of ways. So how do I go back to the geometric properties of the original object, OK? Which quantities are independent of the way the particle moves, OK? You understand this is a key point. So in some sense, up to now, it was very nice to give this definition of curve using smooth functions. But there is a price that we are paying. It's that many things, so many geometric things, can be described in completely different analytical ways, OK? And since I'm really interested in the geometry of this object, I need to have some way to kill this freedom of the possible parametrizations, OK? That's what I want to try to do now. So what is edithiomorphism? Now, I'm assuming i and j are both open intervals, OK? I never really said if the interval for me meant closed, semi-closed, doesn't really matter. Up to now, now I require i and j to be open just to avoid nonsense, OK? So what is edithiomorphism? Edithiomorphism is a map between two open intervals, smooth, which is smooth invertible with smooth inverse, OK? So every time we have edithiomorphism phi, and we have a curve, so given now a parametrized curve, a smooth curve alpha from i to r3, we can construct a new curve that I call beta, simply composing with this map phi. And this will go from j, because of course, I wrote the intervals in the wrong way. OK, I'll have to composite with phi inverse. So let me correct this, OK? Otherwise, I have to put phi minus 1, which is, OK? Let me go from j to i. And so beta will go from j to r3 as, so beta will be just alpha composed phi. So you see, phi takes a value from j to i, and then alpha will take it to r3. So you see, this is a new curve, because it's a different map. It's a completely different map. So there is no reason why we should say that this is the same curve. But on the other hand, as geometers, we would like to say, yes, it is the same curve, because the image is the same. We are just going around this object in a different way. But the object is the same, because after all, the image of beta and the image of alpha are the same. So beta is called a reparameterization of alpha. First observation, so proposition if you want. It's really up to you to decide what has the dignity of a theorem. Somehow there is a proposition. It's a bit less important, OK? So phi from j to i, defiomorphism, alpha, a smooth curve, and a b, our closed subinterval of j. Let's say that phi maps this closed sum interval into another closed interval that we call cd. Then the length between a and b of alpha composed phi is equal to the length between c and d. So reparameterizing, so changing parameters, if you know these properties, OK, this is crucial. You need to have these properties. Doesn't change the length of the curve, which is expected, of course. I mean, again, it's something you have to check, but it would be if this was not true, you have to go back from the beginning, erase everything we did up to now, and start from scratch. Because we are trying to measure something which is a geometric property of the image, and these two objects have the same image. So if this was not true, it means we are on the wrong track. So these are just indications that we are doing the right thing. Let's prove it. So what do we have to compare? We have to compare the norm of the tangent vector of this with the norm of the tangent vector of this, of course. The length are given by integral. So what is the norm of alpha composed phi prime at t? Well, I computed this by chain rule. It's a composition. So this is the norm of alpha prime at the point phi of t norm times, it's the same symbol, but now it's the absolute value, because phi is a map from r to r. So one is the norm of a vector, and the other is the absolute value of a number. But we use it the same symbol. But now, so this is in general. Now phi is a diffeomorphism. What does it mean? It's a diffeomorphism. It's invertible with smooth inverse. In particular, that means that phi prime can never be 0. So it's either always positive or it's always negative. So call s is equal phi of t. So let's see, what are we comparing now? We are comparing the integral between a and b. What is the length on the left? It's the integral between a and b of the derivative of this with respect to t in dt. So this length alpha compose phi prime t norm in dt. We did the first computation above. This is equal to the integral between a and b of alpha prime at phi of t times the absolute value of phi prime of t in dt. And now there is the only delicacy here. Because if phi prime was always positive, this absolute value is equal to the phi prime itself. And suppose for a moment, if phi prime is positive, what can I say? That this object here is actually ds. And this object here is d in ds. So this is exactly the integral between. So by the change of variables in the integral, this is the integral between c and d of ds alpha of s in ds. OK? Now, if phi prime was negative, this formula was also true. And you have to convince yourself in the sense that, of course, you have to also switch d. So this formula holds, if you always assume that c is less than d. So you have to switch sign also. OK? Excellent. That's it for today. So what do we do? So summary of the lecture of today. Definition, start having a feeling of which kind of objects are we covering. Self-intersection, casps, smoothness. In some sense, how smoothness is covered by our definition. Few examples. And length. Length, and how does it depend on isometries? Meaning, does not depend on isometries of the ambient space. And how does it depend on reparameterization of the domain? And again, does not depend on the reparameterization of the domain. So this leaves us the fundamental question, which is where we will start from on Thursday. So since it does not depend on the parametrization, is there a best parametrization? That's maybe there is a canonical parametrization. Remember, we want to study the geometry of the object. We don't really care about how the particle moves on it. So if there is a canonical one, it would be great. We always will fix that one. And this is kind of the accident in dimension one. The answer in dimension one you will see in two days is yes. For curves, there is always a canonical way to move. There is uniform motion, velocity one. But this exists only in dimension one. So we will use it, we will be happy, the theory of curves will be almost over because of this accident, almost immediately. But then when we go to dimension two, we will face the problem that this theorem fails. There is no canonical parametrization of a surface. That's for Thursday.