 Okay, so let's start yesterday's lecture was about still about on technical thing We need to I mean we need to be able to speak about differentiable function We need to be able to compute differential We need to be able to speak about chain rule and so on but these are kind of Technical things that are required but are not the heart of our of our study So let's go back in business So now we really want to understand how to attack the study of the geometry of a surface in round three So let me start with another but another definition and Just for convenience as usual s will be a regular surface in us in our three and Now since we learn what is a differentiable function. It is very convenient to define what is a vector field. So a vector field V on s is a A differentiable Function and now we know what it means V from s to our three Okay, so basically to each point of the surface you associate a vector But you need to do it in a differentiable way and now you know how what what this means, okay? Now in fact just to complete the definition. Let me say because Often this will come up. So a vector field V V is called Normal and unitary If V of P for any point for any P in s V of P is orthogonal to the tangent space okay, this is normal Unitary if the norm of V of P is equal to 1 for any piece Okay, so so today we are going to start studying the behavior of Vector fields and normal vector fields on a surface before doing that since there is always an ambiguity You will see it and that for my experience. This is a source of confusion for students. So let me Freeze for a moment this definition and make just a comment when you have a surface in our three And you want to write down a vector field along it What you usually do and you will see it now in practice is that's around around some point Since you know that it's a regular surface you have the usual chart the usual picture Okay, so you have this open set you and the map X Okay So usually we will confuse so now the vector field is really here Okay, it's on the surface. It's a map which assigns to every point of the surface In fact, not even inside the local chart. I mean this local chart It might be here here here a differentiable map to our three Okay, we will often confuse make a confusion. You have to be clear about this It's an irrelevant confusion, but it's a confusion. So we have to emphasize it The map V so the vector field on S with its composition with X inverse So if you have at every point of the surface some vector field here V of P Okay, you can also think of it. So this P will come from a point here Okay, of course X is one to one Okay, so in some sense you can think of a vector of R3 pointed at this point of view Okay, and we will make this confusion You will see it. So you have to be Always is a bit careful about where is the domain of the of the vector field? So for example, but this is just a simple example To warn you about something that's going on later Let's take this is also an excuse to make Little exercise. So suppose we have a surface which is a graph for example, for example the graph In fact, it's given by 2z is equal x squared plus y squared Okay in R3 now S In our notation is the graph of a function f because you see for example just dividing by 2 I have z equal to f to something okay f if I look at f as a function from R2 to R taking x y Into one half of x squared plus y squared You immediately see that this is exactly the z to be put here So s is equal the graph of this function. Okay, in particular, we have seen in general So why this is a regular surface? Well, the function is certainly differentiable. It's a polynomial Okay, and how did we prove it that the graph is a regular surface? Actually graphs have the special properties of being covered by a single chart There is there is one single chart, which is good for every point of the surface Okay, and which is this chart is exactly in some sense the map f suitably Changed. Okay, so x for every point of the surface I can think of x now my domain will be the whole of R2 because actually I will should take in general the domain of f Here f is defined over the whole of R2. So I take R2 here and I go to R3 Taking remember our notation is to call the coordinates on R2 Umv not to be confused with x y and z. Okay, so umv and this goes to Some some in some sense the graph map. So u v one half u squared plus v squared Okay And now I mean we know it in general so in this particular case you can double check that this is actually This verifies all the all the requirements for for something to be a chart Okay, but what I want to underline here now for example, we said The tangent space at a given point remember one proposition we proved the as a corollary of one proposition We proved that the tangent space at each point is spent by the vectors Vx du Vx dv So what are these vectors in this case? Well, x u is really the vector given by partial derivatives with respect to you. So it's simple. It's one zero you Okay Now an x x v Is the other one, but now here is what I wanted to say about this little confusion So these vectors are defined where? These vectors are defined on you in the general language on our two So give me you and me and I compute these two vectors So these are not Vectors on the surface in some sense Okay These are vectors on the domain of the chart So now if I take the point P What are the tangents? So and when I said something like these two vectors are a canonical basis of the tangent space What do I really mean? Well to be extremely precise? I should say well take a point P P will be X of some point on You so it will be X of u naught v naught For some choice of u naught and v naught So the tangent space at P to S is the span of the vectors x u at u naught v naught x v at u naught v naught Now these two things are vectors at the point of the surface Okay, I compute it at the corresponding point Okay, so you not v naught is the value here, but I compute it at x of that. Okay Okay, think about this I mean and you will I mean you should just think of it because now this this ambiguity Will come into the game often Okay, so sometimes we will be careful and sometimes not and you have to not to fall into the trap Okay, so let me erase this and let me go back to now normal vectors now Let's draw. Let's still draw a picture Now the problem is how many normal vectors do we have at a given point? Well, we have our surface But in fact how how many unit are in normal vectors do we have? We have a we pick a point P on S. We have a tangent space So the tangents we are in our three and this is a plane So how many unitary normal vectors do we have at P? To okay, I can pick this one or this one No other choice. There is no other way Okay Well, that's clear now The point is this was at a fixed point Now let's start moving the point and let's see what happens. I mean how many choices we have globally on a surface of Unitary normal vectors. Well, let's make an intermediate step So let's see first what happens not globally on the surface But on the piece of the surface which is covered by a chart Okay, so effects From u to S is a local chart then at there exists at least one then there exists at least one Unitary Normal vector field on on the part of the surface which is covered by this chart Now the part of the surface which is covered by a chart is it by itself a regular surface It's an open piece of a regular surface. So it's a regular surface. Okay So, you know what it means to for something to be differentiable on X of u Okay How do I prove it? Well think first to the point to be as fixed So you fix point the point point P. How do I construct one of these two? In our three well, that's simple if I have a basis of the tangent space If I have two vectors which span the tangent space Then the third one. I mean I complete it as an orthon or in particular doesn't really matter if they are orthonormal or not Okay, any basis I can complete it to a to a basis of our three with a property that this one is orthogonal to these two Okay, so in particular. There is a simple algebraic way to do it Give me these two vectors and take the vector product of these two and now depending on which one is which It will be either this or this Okay Up or down but one of them is what is I can write it like this But the point is that on X of u. I certainly have a basis of the tangent space automatically The one I just erased on the blackboard Okay, so notation just this is just to simplify So notation the vectors I wrote before the x d u the x d v So I in fact notation to that be better. So this is I write in short X u for this probably I've already done it but I mean and this is x v Okay partial derivatives. I put it that little down on the right. Okay Subscript on the right. So we know that x u x v is a basis Is a basis and here now on this lemma will be precise about the ambiguity I said before so it's a basis of what? It's a basis of t x of P So now well, let's call it. Let's call it q because usually we call q the point on The domain and p the point on the surface. I mean these are names. So You can change it whenever you want, but I mean so whenever q is a point on you okay, so This is what I tried to convince you before okay So you have to put this when you compute this at some point u v You really are taking the basis at the corresponding point x of u v okay now So then put define a vector and and let me pull it the sub a superscript x and x at the point q of u To be nothing but x u at q Wedge x so the cross product the vector product x v q now for what we said before this is certainly a normal vector But I wanted unitary so I divided by its norm. I know that this vector is non-zero because this is a basis Okay, so I'm dividing by something which is certainly not zero and this is a base So where is this map going so nx? written in this form is a map from you Takes a point on the domain of the chart two are three You see why I'm emphasizing this little ambiguity so strictly speaking and Is not a vector field on s or x of u, but it's a vex. I mean it's a vector field on you Okay, so we have to do something if you want to adjust it Okay, now for but first is it a vector field? I mean meaning is it a differentiable function? Well, it is because x is a smooth function These are partial derivatives the vector product is certainly a differentiable operation Dividing by the norm is certainly a differentiable operation as long as the norm inside the vector inside doesn't go to zero And we are sure that this will never happen. So this is a differentiable function okay and It has the property that nx So this is differentiable and in some sense solves the problem besides it has the wrong I mean forgetting the only problem that it has the wrong domain in which sense it solves the problem It is really you it's certainly unitary because it's a vector divided by its norm and it certainly so nx at the point q is orthogonal to the tangent space to x of q by definition By for what we construct okay Well, and let me just write it for Memory, but I mean an x of q at every point the norm is equal to one Okay, so this would be this looks like a unitary normal, but we have to change the domain Well, but to change the domain. It's easy. We compose it with a local chart. So put then define and N of p to be nothing like but and x Composed so I take the point p on the surface. I pull it back to the domain of the chart Okay, I don't draw it the usual picture now every time you see a think of a surface You should think of the usual picture and this is x inverse of Okay So this is just a way to change the domain the map in some sense. It's the same It's just defined on the two corresponding domains Okay via the map x or x inverse So very good. So this ends the story because now this is of course still a differentiable function is a differential It's a composition of two differentiable functions and at every point it's unitary and normal. Okay, so So this proves the lemma now of Course if there is one So here I just produced one In fact in the picture I don't know because I never said if this is the first or this is the second of course the order of the two vectors When you take the vector product is what is Telling you if you are going up or down Okay, so since I never said if it's this wedge this or this wedge this I don't know in the picture. I cannot tell but I certainly produced that one and Now it's defined on every point of the surf of the of the chart covered by the chart at every point Covered up by the charts. Now. I'm supposing. It's the one going up. I have this differentiable vector field, okay But now of course How many other choices I had well only another one so on X of you either I take this one or I take minus this one. There is no other possibility Because again at every point. It's a plane in R3. So there are only at every point. There are only two possibilities Okay, now And in fact, let's What I mean, let's be Let's prove it in some sense, but it's a simple topological argument You see choosing one of these two is essentially deciding from which side you are looking at the surface Are you looking from above or from below or in other terms from inside or from outside? Okay So another lemma if s is connected. I mean you need one simple topological Condition if you have two normal unitary normal vector fields, so let's say and one and two are both are unitary normal vector fields on s then There are only two possibilities either and one is equal to and two or and one is equal to minus and two Okay, this is the intuitive situation so that it is a certainly true at one point now the point is It's true on the whole surface Okay, well, how do you prove it? This is just topology because now For any P in s if I evaluate these two vector fields so and one at P and two at P are Two unitary normal vectors at that point so these equations hold at that point So either they are the same or they are opposite. There is no choice Okay, so these are unitary are unitary normals normals at P so And one at P is equal and two at P or and one at P is equal minus and two Okay, well Now the problem is If I change point, so this is true at every point So what is the only plus if you understand the problem you solve it immediately? What is the problem the problem is if I take now another another point Q Then it's still true that that Q and one of Q is equal and two of Q or but the problem is Are you falling in the same? case at P. I mean at so suppose at P. This was the was true Now take Q Maybe this one becomes true Is it possible or not? What the lemma is saying is not Okay, the lemma is saying if it's if something like this is true at one point It's true over the whole connected component of this point Okay, and how do you prove it? So suppose you have two points in the same connected component for which at what at P This is true and that Q. This is cute. This is true. So now a little intuition. I mean there is something funny So somewhere you would like to say if this was for example a curve No, I mean one they met somewhere in between. I Mean at some point in between something strange has to happen Now of course, this is a surface is not an interval So you cannot say well, this is a positive function here negative function here in between there is a zero This is more this is what you should have in mind But of course, this is not a one-dimensional problem and this is not a positive or negative. These are vectors Okay, but that's simple. How do you how do you argue called S? This is this and a these equations are true at every point So either this or this is true if I write S is equal a union B Where a is the set of points on? S of course such that the first one is true and One at P is equal and two at P And B is the other one the set of Q if you want to give it Such that and one at Q is equal minus and two at Q The above observe a simple observation tells you that S is the union of these two So every point falls either here or here Now now of course, I'm assuming it's connected So of course the claim is that these are this is an open the composition of S and Open non-empty The comp I mean of course the contradiction now you argue by contradiction. Okay, so suppose that these are Well, okay, how do you argue what are topologically what can you say immediately about a and B? They are closed. They are certainly closed because these are these are equations and Continuous equation in part the differentiable equation n is a differentiable vector field Okay, so the set of points where these two objects are the same is closed and Where they are opposite are closed. So these are both closed So at least one of them has to be empty Otherwise you contradict the connectedness Okay, so a and B are closed and this implies a or B Empty and that's it end of story Okay Now the problem is okay. So now what do we know? We know that at every point. There is a unitary normal On on the piece of a surface covered by a chart. There is a differentiable choice of a unitary normal in general On the whole surface or that is on the whole connected component I mean we usually assume even without saying that surfaces are connected but in principle they could be made of Pieces, okay, so on on the whole connected component. I have only two choices of Norm of unitary normal, but now the question is does there always exist a unitary normal on a surface? Remember locally. Yes Because every surface is X of you Okay, so on the piece of the surface which is given like X of you for some chart There is a unitary normal But who is who who is going to tell me if there is a unitary normal over the whole surface? Well, nobody and in fact it may be false Okay Wait a moment. Okay, so let me give a definition So the surface where a unitary normal exists are special and so let's give them a name definition if I mean a serve a regular surface S such that there exists a unitary normal globally, that's the point and We find over the whole of s to R3 is called Orientable well the name is is right because you see if you think of a the choice of a norm of a unitary normal As to choose the choice of a side where to look to the surface Okay, orientable means orientable surfaces. They have a side Okay I mean you cannot explain names I mean if you like it you like it or no you give it another name But I mean it is a meaningful name so example Examples one well as I said, I mean usually you should start there is a should ignore your everything There should be example zero the plane. Okay, so the plane is the simplest Regular surface of R3. So let's do it. I mean in fact, I mean I was going to start from example one But let's start from example zero But we even without writing anything take a plane in R3. Is it orientable a Plane in R3 you can actually take as unitary normal The normal vector to the plane one normal vector to the plane one of the two And this will work at every point so n in this case will be the constant vector Of course when I think when I draw it I draw it pointed at every point of the plane But actually as a vector. It's always the same vector Okay, so the plane you take could the constant vector field given by the norm one of the two normals Okay, and that's it. So now you know your list So example zero is the plane example one the sphere Okay, so is the sphere orientable or not? Doesn't really matter now. Let me simplify notations So let's take as to Well as to R Well as to R. Let's do the general case. Okay center with with a given center of the given radius Is it normal or not? Is it orientable or not? Well, let's draw a picture again. So This is the point P naught and this radius Okay, so at a given point P We should think of the tangent space at P and then try to imagine a normal vector But we know it. So what is what are the only two possible choices at P? P minus P naught or P naught minus P the one you prefer Okay, but then well if so that means and from S to P P naught R in two are three. That's that's that this is the object. I need to define. Okay a Unitary normal. So a vector field so differentiable Which at which at each point it's unitary and normal. So I take P and I define for example well, I would say we were saying P minus P naught well P minus P naught as a vector is Good in the sense that it's orthogonal to the tangent space at every point We know it. Okay. It's the characterization of the tangent space to the sphere But it is not normal a unitary Nobody's telling me how much in fact nobody's telling me somebody's telling me how much is the norm of this How much is the norm of this? R by definition of the sphere. Okay, so if I want to take a unitary normal I need to take this vector divided by R. Okay Now this is will be of length one and orthogonal to the tangent space Of course, I could have taken minus this Okay, it would be it would have been a generally good choice also this one for example say example to In your list of examples of regular surface the simplest next thing is what? graphs graphs of functions, so if s If s is something like the graph of a function f as the little example we did at the beginning Whatever f is are we able to produce and you need to normal? well You can rely on the lemma on the first lemma we proved because we know That on each piece of the serve of a surface which is covered by one chart You know how to define a unitary norm that was the content of the lemma, but then graphs Can be covered with one chart So take the proof of the lemma apply to this and that's it And in fact, that's what you do now x. What is x in this case x of u v is u v f of u v Okay, and what what what was the lemma telling you? construct nx so nx was the vector field The cross product of x u with x v normalized well, let's do it. This is 1 0 partial derivative of f with respect to u cross product wedge 0 1 f v Normalized Okay, so divided by its norm to have it Normal, okay, so let's do it pick your favorite way to compute the wedge product So this is the 1 0 1 0 f u 0 1 f v how much which vector is this? This is f u Sorry minus a few second Okay, this is the vector product. Okay, so this is the vector minus a few oh In fact since I hate to write to two minuses I change sign Why not I can pick n or minus 10. It's the same. I mean it's another unitary norm So this is f u f v minus 1 divided by its norm What is the norm of this? This is square root of 1 plus f u squared plus f v squared Okay, if you want if you want to Short it a little bit. This is of course the gradient square of f For example, as you want, okay So this is the vector nx and now if you really want to produce n You have to just to write down nx compose that x inverse And that's it Okay, example three the most sophisticated class of regular surface we know up to now Is the one given by regular values of functions The inverse image of regular values of functions. So What does it mean S now it's f inverse of a Of course under the assumption that a is a regular value But we know already what is the tangent space. We have compute as a little exercise What is that at a given point p? What is the tangent space to s? It's the kernel of the differential of f And when we discuss that we observed that you can write it in this form It's the set of vectors v in r3 such that The gradient the scalar product between the gradient of f at p And v is equal to zero This was an equivalent way to characterize the the kernel of this map okay so It's written here. What is a unit normal? I mean By the so this becomes the set of vectors which are orthogonal to the gradient So the gradient is a normal Okay by definition Okay, so n Equal grad f over the norm of grad f is a unit normal Okay Again grad f cannot be zero. Otherwise p was not otherwise a was not a regular value By definition of regular value. That's something where the differential is non-zero The differential and the gradient are represented by the same same matrix here. So Okay Well, so now here he starts looking like an accident We have defined the notion of an orientable surface and everything we know is orientable So conjecture Everything is orientable For what you know up to now. This is a very logical conjecture Okay, you will solve it You will disprove this conjecture In an exercise in the next homework But You certainly know why it's not true The typical example The simplest example you can think is this Take a rectangle If you glue it In this way along the side You get a cylinder Is a cylinder An orientable surface Yes Yes, for example, it falls here Okay, but you can also imagine also you can imagine pictorial geometrically There is no problem in going around The whole cylinder with the unit normal vector pointing in the same direction Okay But now you can play the other game Glue the two sides of the rectangle In a way, so here before to produce the cylinder you had this point Glued to the same point in the same the same distance from the long side to the other side Now do the opposite so take this point here and glue it to the opposite point to the symmetric point so the distance between This point and the and the long side here will be the same as the distance Of the corresponding point to the other long side Okay, now it's difficult to do it with the piece of paper because I mean with this type of paper because of course tends to break but you can do it Okay, and actually this was This is the source of many famous pictures Okay Ants are there because General people don't like to speak about unit unitary normals So the ant the insect here is your unitary normal Okay This is a famous picture by escher who was a kind of mathematical fanatic, okay of The end of the 19th century Now Now, of course the surface here is this net Okay And the way the ants are moving are telling is telling you exactly that you cannot define a unitary normal Okay, because If you move Along with the ants So instead of the ants put a little flag, which is really a more mathematical sensible unitary normal Okay, move you with the ants Okay, when you have done a loop You have come to the same point on the surface, but the ants is on the other side Okay, they will not meet They have to go around twice to meet Well, good question. First, is it a regular surface? Otherwise we are speaking about something else This will be the first question in the exercise. Okay It is right. It is actually a very simple surface You will see the analytic expression of this surface is very simple So and then the point is what is more delicate is really What are these ants showing you? Is just that the obvious choice doesn't work But we are mathematicians and they say why should be the obvious choice? Maybe there is another choice of n Okay, as mathematicians we are forced to give a proof that you cannot find a unitary normal vector field Not just that the simplest thing doesn't work Okay, and this is slightly more delicate Okay, but these objects exist. So not every surface is a regular is an orientable surface Okay So when we we are going to study surfaces, we have to specify explicitly whether This will be orientable or not on the other hand We have the first lemma the first lemma is telling us On the piece of a surface which is covered by a chart You can make a choice of n So also on this monster here, which is called the merbius strip. By the way, it has a famous name Okay, the merbius strip On the piece of on a piece of this surface, which is parametrized by an x There is n So now because now we are going on with the theory of surfaces Assuming that there exists n and so you might say well, but what kind of surfaces are we starting? So this will be the theory not of surfaces. This will be the theory of orientable surfaces That's true, but every surface is orientable locally Okay So whatever we are going to say from now on Will work globally on a surface If it's orientable or Locally on a surface if it's not orientable, but it will work So whatever we are going to say depending on local things. It's completely general Okay, so we are not really restricting too much our study Okay now Because now we really have to face the key problem of this course The key problem of this course is what does it mean that something Is flat and what does it mean that something has some curvature? Okay, this is kind of once we understand this We have done 75 percent of what of our course Now, but let's follow intuition. So what is flat? Well, for example the plane whatever we are going to define the plane should be flat Okay, otherwise there's something really funny going on and how Which way we can characterize the plane In terms of the normal vector or the tangent space of course Knowing n and knowing the tangent space. It's the same thing in r3 Because if you give me the plane I have n if you give me n I think the orthogonal complement I have the tangent space. So Philosophically, it's the same amount of information It's more convenient to work with n because it's a single vector While tps is a plane in r3. So it's more difficult to handle I have on the other side. I have just one function Okay, n Now, how do you characterize a plane? We have seen it n is constant Is this a characterization Of the plane? So basically n is constant if and only if we have proved if it's a plane n is constant the other way around of course Meaning geometrically the tangent space is always the same If n is constant the tangent space is always the same plane Okay So first intuition maybe curved or non-curved or flat and non-flat depends on how much The tangent space changes Because you would like to say well if I start Doing something like this I introduce some curvature and in fact n Stops being a constant map Okay But we have we will be careful about this But certainly it's an intuition we have to follow at the beginning Okay, meaning you see I'm trying to convince you that it's time that we study derivatives of n So now the key point is n and its derivatives What kind of information we can extract out of them? so And a general warning that I will not do anymore I take n from s to r3 a unitary normal vector field That means either i'm taking an orientable surface or i'm taking a piece of a non-orientable surface But I will all I assume that n exists Okay Now In particular n doesn't just go to the whole of r3 I know a bit something more because since n of p the norm of n of p is equal to one n of p is a vector where? On the sphere so in fact that it's better to Underline it so n is really a map from s to the sphere of radius one and center the origin Okay And this is still a differentiable map remember the definition was a differentiable map here But this is still a differentiable map now between surfaces because It's the restriction of a differentiable map in r3, okay so now n Is called this map Written in this form is called the gauss map of s Okay But now it's a differentiable map between two surfaces and we are very interested in knowing how much n changes We know how to do it now You feel you have to you should feel satisfied of having paid the price before You just take you know what it means to compute the differential of this map Okay, and what where does it go? So where is the differential of the map defined? Well At the point p takes a tangent vector to the first surface At p and where does it go? It goes to the tangent space to the to the image surface at the image point Okay, so this will go to t n of p s 01 s 2 Okay, on the other hand, what is the tangent space at a given point to the sphere? It's the orthogonal To the point itself Now here, there is not p not no because p not is the origin So it's really the point itself But this this object here is canonically isomorphic to what? To the orthogonal complement Of this vector Obvious notations. Okay, this is the set of vectors orthogonal to n of p By which are the vectors orthogonal to n of p? By definition of n n is what? n is the is a vector orthogonal to the tangent space So what is the orthogonal complement of n the tangent space to the original surface? Remember in all these things what is important is that these isomorphism are natural Canonical because of course they these objects here are of course isomorphic. These are two dimensional vector spaces. So They are all isomorphic to r2 Here the key point is that these isomorphism are canonical natural. They do not depend on a choice of a basis Okay But then look what happened the differential of the gauss map At the end of the story takes A vector a tangent vector to s at a given point and gives me back a tangent vector to s at the same point Okay This just is an accident of the sphere. It's this It's this accident of the sphere which keeps on coming back. So it's an endomorphism of this vector space Well, but then let's study which kind of properties does it have? The theorem and this actually deserves the dignity Of the word theorem dnp Is a symmetric? endomorphism Endomorphism in just it's a it's a nom nomomorphism of vectors of this vector space to itself Okay inverting I mean The vector space to itself and the symmetric endomorphism of tps Remember what does it mean symmetric? I I reminded you In the little exercise about the differential of the map That means that if you take dnp Of a vector v and you take the scalar product with another vector w This is equal to v dnp w From for every choice of v and w in the vector space that you are looking at so for any v and w in tps Okay, so this is formula In formula that what that's what you need Well, let's prove it and actually as usually in the proof we will find interesting formula I need a bit more of space here Well, if you want to check that something is a symmetric endomorphism in a vector space It's enough to check it on a basis Okay, if you have a basis you can check it on the basis if it works. That's it Okay, so it's it is enough To check it on a basis the point the problem is which basis do we have a basis? well Yes, we pick a point p And we pick as usual a local chart chart around p Once you have done this you have a basis x u x v okay The problem is Of course if I want to check a formula like this, so you see now v and w will be x v x u x v So it's clear that the whole point of the proof is are you able to tell me something about this I mean can you compute it in some way? Of course and the same for x v Well, let's go back to the definition. What is the differential of the map applied to a vector? As usual, what do I have to do? I have to take A curve Passing through alpha Passing through the point p at time zero with velocity the vector that i'm looking at okay So alpha of t Now will be a curve x of u of t v of t So now here this is one of the points where you have to be careful about this some ambiguity I was telling you at the beginning because now i'm writing So this is alpha is a curve on s Okay, but actually i'm already i'm automatically taking the inverse image of the curve on the domain of the chat Okay, automatically we have done it already three or four times and We are old enough to do it By default, okay So and the only thing i'm asking is alpha of zero is equal to p at the moment So how much is d and p? So let me write it in general And now depending on the choice of alpha will impose that this is equal to this or this is equal to the other one that i care Okay, how much is this? Well, first I can use chain rule To express alpha prime at zero you see alpha is the composition of two things No, it's a curve in the plane composed with x So this is first I can do this d and p of what of u prime so the derivative of u with respect to t times the derivative of x with respect to u Okay chain rule now. This is equal to bilinearity The n the differential is always a linear map. So this sum splits as the Sum of the images and these are scalar numbers. So they go out. So this is u prime zero d and p x u Plus v prime zero d and p x v Now, how do I get x u and x v? Well x u is by definition It's all what is the partial derivative in one direction If this is u and this is v and you have a function which depends on u and v What is the partial derivative in the direction u at a given point? Well by definition means you restrict your function to the line passing through this point parallel To the direction where you are taking the partial derivative And you take the derivative in the usual sense in the old sense of a function of one variable Okay, but but that means that this object here is automatically The partial derivative of the map n Actually here probably should have been more nx in the language, but I'm using that mistake. I was telling you n derivative with respect to u Plus v prime zero n derivative with respect to v Okay It's quite convince yourself This is automatic by the definition of the differential and by the definition of the partial derivative in one direction Okay So this object here becomes automatically the partial derivative of the map In the direction that you are using here, okay now So see so this is general picture now. What what is the symmetric condition asking us? Well So written in this form now that we know This so this here this is asking n of u Times x v Is it equal So equal with question mark because this is the content of the theorem Is it is it equal to x u and v? Okay, do you agree that now this is the problem? now How do I get to this equation? Well, I take the only thing I know x u x v is a basis of the tangent space and n Is a normal vector. What does it mean? It means that n scalar product x u is equal to zero at every point Because this is tangent and this is normal and of course also, maybe Yeah, it's better to write it in another line and x v Is equal to zero for the same reason one is tangent and one is normal Then what how do you get here? You see Well, let's do it. It's simpler to do it and to justify it I mean I take the the derivative of this Equation with respect to the missing coordinate here v So let's take d in d v of this equation. Of course, if I differentiate zero I get zero And take d in d u Of this well actually here's strictly speed. I should put the partial derivative. Okay How much is this? Well the first one As usual scalar product as differentiated as a standard product. So derivative of the first n v x u plus n x u v Second derivatives. Okay with obvious notation. And the other one becomes what? n u x v Plus n Well, I should say x v u But then standard calculus tells me that the order is irrelevant x v u is equal x u v Okay, and so I'm done Why am I done? Because both are zero So take this up. So that means x n v x u is equal to minus this But n u x v is equal to minus the same thing. So in particular they are the same Okay They are the same and in the future I might be interested in knowing that they are not just the same, but they are actually so this proves That this is not a question mark But also that they are both equal to minus n x u v Which could be useful It's an extra thing that was not required in the statement of the theorem, but why not We learned it. We take it Okay Right. So what does a mathematician do when you have a when he or she has a symmetric Endomorphism of a vector space it looks to the associated quadratic form It's always the same thing. So let's do it also in this case Even though the knowledge of the endomorphism and the knowledge of the associated quadratic form is the same Again, so that you you have the same amount of information. It's just a question of what you prefer So since Gauss preferred the quadratic form we shut up and we Study the quadratic form, okay So In this particular case You put you give this strange symbol It's a second It's a latin second with a p Okay So this is the quadratic form. What does it mean a quadratic form? It has to take two vectors in the vector space So it goes from the cross the cartesian product of the tangent space to the surface to itself And gives me a real number And how you do it? Well, you take two vectors two tangent vectors v and w and actually here There is a usual source of mistakes Also inside my lectures So part of the exercise is to check that I sometimes don't forget this minus sign Okay, you put minus dnp dw Okay This is completely general how to associate a quadratic form to an endomorphism of a vector space A symmetric endomorphism of vector space Okay, now this is called The second fundamental form second fundamental form form of s At p and basically it will be the the topic of our study up to the end of the course But now the first question when you define the second fundamental form is where is the first? Okay, so the first we actually used it without even giving giving it a name But why not I mean let's give it a name. So the first fundamental form Is again, it's another quadratic form So it takes again two tangent vectors And gives a real number Takes v w and just gives me the scalar product We never felt it the necessity of giving it another name different from scalar products, but Okay, so and this is called the first fundamental form first fundamental form form At p is just the scalar product restricted to the tangent space the scar product of r3 restricted to the tangent space to the surface Okay, and now here let's start with a list of definitions because Since The differential of the gauss map is a symmetric endomorphism in particular. It is diagonalizable Okay In this case, you don't even need what we so the np Symmetric Implies diagonalizable now they are going now lies about of course not diagonal I mean But diagonalizable so So that means that there are two numbers which are more interesting Than other numbers and two vectors which are more interesting than the other vectors The two numbers are the eigenvalues so You we define k1 of p and k2 of p i the eigenvalues of minus dnp I mean of course if dn is symmetric or so minus dn is symmetric. Okay, this is just a sign convention And how do I decide so of course this is diagonalizable? So it has two eigenvalues countered with multiplicity. I mean in principle they could be the same So how do I choose? Which one I call k1 and which one I call k2 So the assumption is that k1 of p is less than or equal is the smaller one If the smaller one whenever it makes sense Because if they are the same but on the other hand if they are the same, who cares Which one you call k1 and k2? They are the same okay So whenever they are different one is smaller and the smaller gets the one Okay becomes the first In any case, so this is the notation, but we call them. So these are called principal curvature Principal curvatures of s at p What do you do with the eigenvalues of a symmetric operator? many things But in particular Though which are kind of the characteristic numbers of a symmetric operator in dimension to the determinant and the trace These are the only two things that you can do The only two symmetric functions of the eigenvalues Okay, so it's likely they are likely to be important. So capital k at p is the product Times the product of the two eigenvalues, which of course if you want is the the the the determinant of the matrix This is called Gauss curvature or gaussian curvature is the same gauss From time to time gaussian curvature Of course of s at p. I don't repeat it five the trace The trace is what and here this is another source of usual mistakes because It's the average or is the sum From time to time you forget the half But in the beginning you put it And this is called the mean curvature, of course Of s at p That's it about numbers, but now about vectors We have to give names to some special directions So the directions so here you have to be I mean it's kind of long to say but the directions associated to The eigenvalue eigenvectors Because you see a single eigenvector is nothing I mean Twice an eigenvector is an which eigenvector do I choose? There is no choice of a better one The only interesting thing is the direction of the eigenvector Okay, so that's why I don't give it. I don't give a name to to a one eigenvector I give the name to the span of the so to the eigenspace in some sense. Okay, so the directions Associated to eigenvectors, but now I have to order again. I order them v1 v2 associated to k1 k2 So k1 will be associated to v1 k2 will be associated to v2. These are called Principal curvature principal directions, of course Remember everything depends on the point. So at a given point you have two numbers the product the mean These directions. Okay, but if you change point you have Change numbers. Okay, and now final thing Final name a curve alpha from some interval to s is called a line of curvature if It's tangent its velocity its tangent vector So this is in particular a tangent vector to the surface s at alpha of t So of course, there will be a special class of curves Those for which the tangent vector at every point is A principal direction Okay, so a line of curvature is a principal direction principal direction Now, let me be of s at Alpha of t at every at the corresponding point, of course for any t. Okay Do you have a question? No, it's okay encouraged This is a bunch of names Okay, nothing to be the key point the important the the heart of the problem of the point here Is that the differential of the gauss map is a symmetric endomorphic? That's the mathematical point then everything else is just Giving names to the associated objects. Okay examples Let's compute some of these things in the in the examples we have Okay, for example zero the plane Sorry, can I marry example So let's start with zero s is equal to a plane Well, what is the gauss map? n I'm going to write it here n from s to s one meaning sorry s to Zero one. Okay, we we decided that if I don't write anything it's with center the origin and the radius one so We have already said it I mean this is the constant map which takes at every point associates the normal vector to the plane very good So how much is dn How many votes for zero? You don't reach majority. That's now that's How do you do it? Think of what is this you should take a tangent vector to the plane whatever Take a curve in your mind alpha such that alpha of zero is p and alpha prime of zero is v Take n composed alpha in general the function composed alpha and take the derivative at time zero Whatever alpha is n composed alpha Is always the same vector because n is constant over the whole place or whatever alpha is n composed alpha is the is the is a constant vector So the derivative at time zero Will be zero. Okay. I don't have to make any computation But so of course zero is a symmetric endomorphism What are the eigenvalues? So principal curvatures of the plane both zero Gauss curvature zero mean curvature zero Which are the principal directions? There are infinitely many in this case You see because of course in general you give the definition thinking Well, I have one eigenvector eigenvalue I associate one eigenvector and that's it But of course if the two eigenvalues are the same The eigenspace must be the whole tangent space, but that means that every vector is an eigenvector Okay, so every every direction is a principal direction. So which are the lines of curvature? Any line any curve any curve on a plane is a line of curvature Okay, because there are no restriction on the tangent space on the tangent on the velocity Okay, well, this is of course example zero For a good reason, okay One example one is always the sphere So now let's take the sphere with something okay p naught of radius r, okay We have written before one hour ago. What is the norm one choice now remember Remember and actually this is important and interesting n is not uniquely defined So when I compute these numbers Or these directions or whatever I have to tell you which choice of n I'm making It's not something that comes with the surface by itself I know the only thing I know and it's important is that actually I give you my choice of n And you could take another choice, but actually there is not too much freedom among us So you either pick mine or you pick minus mine There are always two only two possibilities Okay But actually this minus can have an effect may have an effect on these numbers So in fact, it has an effect on these numbers. So I have to tell you not just s But which n I mean this was irrelevant here because anyway, it was a it would have been a constant pick mine or minus wine Mine it was in both cases a constant vector So whatever we said in the example zero was was okay Now it's important. So n from s To p naught r I have to tell you which normal vector I choose We I wrote one possibility before and I take that one Okay Now with this choice, how much is dn? Well, now you should start being able to do it Take alpha With alpha of zero is equal to p with alpha prime is equal to v Take n composed alpha Which just means Substitute at p you put alpha of t And take the derivative at time zero But the derivative of this subject at time zero is of course alpha prime of zero divided by r Alpha prime of zero is v So this subject is just one over r v Okay Now please you have to become I mean, I hope I've made enough examples. Okay to to me to make you quick So what is the second fundamental form? At a given point. Well, it's the quadratic form which takes two vectors v and w and gives you I should I should say I should write minus d and p So in fact, maybe I want to write minus one over r v double of a nice r Okay In any case, so this is the second fundamental form which are the eigenvalues of Minus d and p remember k1 and the principal curvature are the eigenvalues of minus d and p It's written here that the the differential of the gauss map Is the multiple of the identity And the multiple and the factor is one over r So the eigenvalues are One over r both. I mean there is only one eigenvalue with multiplicity two Okay, so k1 Of course, this is true for d and p. I'm looking at minus d and p So k1 of p is actually equal k2 of p Is actually minus one over r for any p So what's going on now this time the principal curvature are not zero But still they are constants. They don't depend on p in any case. These are the numbers. So Of course, what is the gauss curvature? Is the product So it's one over r squared What is the mean? The mean of two it's the same It doesn't change And what else and which are the principal directions? Sorry, you can take any vector Any vector is an eigenvector So again, we we fall in the same situation as the plane Any direction any tangent direction is a principal direction And so any curve is a line of curvature So actually this is a good Point where to check what if some of you prefer minus mine gauss map It would have been perfectly logical and Okay, there is not a right choice There are just two choices So if we take minus this Here I get the minus So the differential becomes minus So k1 k2 Change sign They become positive one over r So the mean curvature changes sign The gauss curvature Doesn't care, okay Now think is this a special situation of the sphere? It's a totally totally general situation So the choice of the orientation where we introduce another word obvious, okay, so which means the choice of n The choice of the orientation has the effect of switching sign to h k1 and k2 And it's irrelevant for the gauss curvature, okay Okay, next next example is a bit long. So we stop here