Differential Geometry - Claudio Arezzo - Lecture 01

so welcome all to the beginning of this course so my name is Claudia reso and I'm your lecturer in in a topic which is called differential geometry I hope you all know something about it but we will start from the from the very beginning so you are not supposed to know anything about this before ok so what is differentials you actually throw a few instructions before starting putting our hands on that 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 manfred odo karma 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 ok on this topic every two weeks I will give you a homework with a list of exercises 5 or 6 exercises and so there will be one week the exit of 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 okay 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 matter it's not part of the evaluation okay but it's important you try to write down things mathematics 150 percent of our job is having ideas the other 50% is being able to communicate the ideas okay so try to make an effort also to to learn how to write down things okay the first homework will come in 10 days or so because now of course it's too early okay so what is differential geometry about well you you are certainly you have seen certainly some geometry in Euclidean space and analysis in Euclidean space you use 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 okay 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 okay 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 okay 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 okay 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 object we will be quick about this because I assume probably you will all yours I've seen already something about this and then we will move quickly to surfaces so two dimension objects and already two-dimensional objects have already all the features that make things complicated and nice to go higher dimension to higher dimensions okay so let's start with one-dimensional objects curves okay and this will be the topic probably for the first three or four lectures of our course okay so the first thing to notice is that the first definition so what you would like to say that the curve is a set okay it's a specific set with some property but this is not what we are going to do okay and we will have to think about why we are not doing that but the most common definition of of a curve is in fact of a map okay 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 the image of the map and the map itself okay 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 C 1 C 2 C 3 but if I don't say anything if I just say differentiable it means smooth okay a differentiable curve is a map let's say alpha from an interval to r3 where I is an interval let's say with x3 a be a subset of R okay and that's it is it of course this is a smooth map okay this infinity map from from an interval to r3 of course what does it mean smooth well you know this is a map from a subset of our to a subset of our three so it's the 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 XY and z are smooth functions okay 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 order 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 curve a differentiable curve is a planar curve so alpha is called planner if there if there exists a plane so a two-dimensional plane is dimension to a plane p in our three such that the image of this map alpha is contained in this plane P okay now since by a rigid motion in our three okay so imagine you have your curve so this is our three x y&z okay and your curve lies in some playing P okay this is the image alpha I okay but by a rigid motion I can change coordinates of our three in such a way by a rigid motion P I can always think of P as the plane Z equal to zero okay I can always put a plane by a translation and the rotation I can always think of this plane P as to be Z equal to zero okay so a planar curve in some sense after a change a linear change of coordinate will always be of the form alpha T will be of the form X of T Y of T 0 okay so being zero the third coordinate I drop it okay so a planar curve is just given by two functions x and y okay be careful up to a change of coordinates okay okay so this is definition one now comments is this when you give a definition you have a whole ways 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 the 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 okay for example take as a planar curve now alpha here eg is a shortcut to say for example okay in case you okay for example a alpha of T equal well this is T cube minus 40 T squared minus four okay what okay if you if you just check you quick you can quickly click we quickly check that if you take the two values T 1 2 to be equal plus or minus 2 okay you get alpha of T 1 is equal alpha of T 2 is equal to 0 0 0 ok and actually you can have a feeling of what the graph of this curve is what is the image geometrically in the plane XY this is a planar curve you see I've already written just two coordinates and this is a curve which looks like this ok so when T comes from minus infinity you see that X comes from minus infinity and Y comes from plus infinity ok 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 T 1 is equal to minus 2 you are in the origin then you make this loop and at T 2 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 ok so first first comment a curve can have self intersections ok 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 two is that the graph of alpha in some sense the image of alpha alpha by could have corners so you see somehow you say alpha is made of smooth functions so you thought you might be able you might suspect that the image of alpha is some sense it's a smoothing without corners no that's the idea behind being smooth okay well that's not true alpha could have corners even if all the components are smooth okay 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 okay but what is the what is the image of this function here well you can think of it in many way 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 okay there is no problem is the graph of the function 2 over 3 ok you see if you take if you think this this is why you divide by 3 you get X ok and then you see why corners can arise see because this is not a smooth function you see this is made of smooth functions but all together is the graph of a function which is not smooth ok 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 okay self-intersection like this is what we usually call it node and this is what we call a cusp okay how do you write it sorry can you cast sorry - which one are you referring oh yeah you're yeah yeah okay okay well it's enough to call this this is X and this is y okay now the third comment is even a bit more delicate and requires a little bit of topological side common three even if alpha is injective so even in the good case okay alpha could be not and homeomorphism onto his image okay now this requires few seconds of thought okay so what does it mean well you you might be led to think well after all alpha is what alpha is a particle moving in space so if it 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 okay and so the two things are not homeomorphic that's okay that's clear but then you might be led to think well this is the only possibility to introduce some topology in the in the image okay so now the comment I'm going to I'm making is no there is something more subtle than this okay how do you see that for example let me explain what what I mean with an example again suppose you take alpha now it's defined from minus 1 plus infinity okay into r2 again it's enough to take planar curves okay and you define it to be alpha T is equal 3t over 1 plus T cube 3 T squared one plus D cube okay how does this curve look like so here we have in our in the XY plane again so if T goes to plus infinity well well let's go first to minus one from from the right okay 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 zero okay so and it's going to zero from from the positive side so this is going to plus infinity okay so this curve from T equal to minus one starts here okay plus infinity plus infinity in some sense then it comes down I mean that's something when you go to zero when you reach T equal x zero what's going on well below now it's 1 1 and above it's 0 0 so you are passing through the origin okay see why why is always positive so you are always in the upper half of the of the graph of the 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 okay so that means the curve must do something like this okay well this is a famous curve so let's give it a name okay again you can also characterize it in terms of polynomial because it's the zero set this is just for your curiosity but I mean it's 3 X Y is equal X cube plus y cube okay so it which again shows you that this phenomenon can occur even as a 0 set of a beautiful function because it's a polynomial function okay equal to 0 like in the case of the cusp okay so but 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 around here see that the topology induced by the plane on the on this set how do you get it you get it by taking open set of the of the ambient intersect with the subset that's the induce topology that means that open set open neighbors of this point are of the form interval around this which is okay but plus something like this so you seen a bird of this point are always made like this and this topology is not homomorphic to the standard topology of the interval okay so this is kind of curious but shows you how delicate things are okay so I curve the image of a curve even though everything is defined on interval on intervals the which can be topologically more complicated than any interval and not just because you can make loops okay loops are okay that's a way to change the topology but there is something else that we have to be careful about okay 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 a good set of examples to to go on and in some sense up to now I've shown you pathologies which is typical of mathematicians okay you give the definition and you always you start by going to the most delicate things but let now let's stay to the most standard situations so what are the simplest curve okay but the simplest curve you can think of of course is the straight line how do you realize a straight line in this language well in some sense what we are doing now in this definition of differentiable curve is what you are used to call a parameterised curve probably now you have already seen something so how do you parameterize a straight line well you need a starting point and the die and the vector which gives the direction okay so you just write it something like this given two fixed vectors of r3 the straight line passing through the point V not with velocity or tangent vector or directional director or whatever you however you call it okay V is this one okay and here of course you see that T the parameter is free to move in the whole real line okay so this is of course the simplest I mean if if our theory does not cover lines and circles then our theory is to be thrown away so okay circles how do we parameterize circles well what do we need well now I'm parameterizing it as a planar curve okay because of course the circle is free to be on each on the plane that you want okay so again I pick the standard plane to be Z equal to zero and they give you the parameterization in X Y okay and if you want so you need a center and you need a radius okay so if you give me a center and the radius I just construct a curve like this okay now C will be the center of my circle so it's just any point in the plane it's just any positive number it's the radius and then T again it will be free to be any real number if I parameterize 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 okay 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 okay depends on the situation so the question is why I'm dividing tea by our just to have this question no I mean just to stimulate a question at the in this moment it's totally relevant if I'm letting T to be free if I call it T of T over R it's the same thing it's just a scaling okay it's usually convenient once one once we will fix a range and interval which will cover the circle only once it's usually convenient to have this range to be always to be radius independent you will see it it's a convenient it's just it's just a choice of after all yeah 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 hella C's and these are what let me give you first the parameterization then we will comment on the graph on the on the image of these curves so alpha of T I will call something to be an alux if if there are two numbers a and B such that so these are spaced curve now I really need three coordinates so these are not planar curves cause T over square root of a squared plus B squared a sine T over square root a squared plus B squared and then B T 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 sign so a and B are just real numbers of course you avoid both to be equal to zero otherwise this is nonsense okay so not let's say a and B are both different from zero otherwise they are degenerate okay 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 be s okay there will be you will see a similar convenience later for having this okay 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 okay geometric intuition so one way is to make projections and see what happens to the boat what are the projections on the coordinate planes okay so here we are XY and Z so how do I understand what is the projection on this 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 the origin because there is not a translation here ok and with radius a Monday I mean okay sometimes I will automatically assume that a and B are positive because otherwise you can switch things okay so on this 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 okay so what's going on well after all I already told you the name so you know you know what's going to be now 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're moving on a circle okay I'd fix speed so you should imagine something like this okay my drawing is terrible but more or less gives you the idea now so the projection you should imagine a cylinder no a cylinder on this cylinder here you draw this kind of hellix okay at fixed speed okay okay so this is a little series of examples nowadays the blackboard because now what is the first dramatic property of a curve that we want to learn well the first geometric thing is associated to a curve is its length so let's learn how to measure length lengths of curves the problem is 'm is again giving a good definition so giving a definition that corresponds to what we know because on how we feel that we know we have to mathematically formalise it ok you see this game what the reason why we are studying curves is to practicing this logical thing because I can tell you when you when we will study 2 or 3 or 4 or 6 dimensional objects it will not be even clear what what we have in our mind ok so let's do this let's play this game in dimension 1 when everything is easy it's also a little bit boring but we understand the method to do it in dimension 26 ok where it will not be clear at all what is the area of the volume or something of a of an object ok 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 ok so given a differentiable curve alpha in space ok and suppose we take a compact subset of the interval where it is defined alpha ok because after all we can measure lengths only of finite pieces of a curve otherwise we should expect the length to be infinite ok so the question is what is what is a good definition of so in this way I introduced the symbol that I want to use the length of the curve of the piece of the curve between a and B ok well the idea is to break the curve into pieces so suppose that this is your curve okay and let's suppose this is so this is alpha Y and let's say this is alpha a and this is alpha B so the idea to solve this problem is take a partition of the interval a B so what is a partition a partition is just ie a choice so P of numbers which starts from a and gets to the final one has to be B okay so remember this is so if this is alpha here you have a B on the interval a partition mean just okay let's call this T 0 let's call this T naught T N and let's put some numbers in between okay TI this is just a partition okay so then you go and look what are the corresponding point here alpha T so T 0 T 1 alpha T to alpha T 3 and so on okay you go and see the corresponding point and then you define the length so the definite this is a definition ok the length between a and B of the curve alpha subject to the partition P okay I'm not going to write it down so the same you read the symbols like the length between a and B of the curve alpha subject to the partition P okay to be what to be the sum from I equal to 1 to end of the length of the segments alpha TI minus alpha TI minus 1 okay think for a moment we are in Euclidian 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 this symbol here okay so I'm saying the length of the curve subject to P is essentially the length of this series of segments okay of course this can be a good approximation of what I think should be the length of the curve or Abed absorb proximation okay 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 of okay 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 okay and this norm becomes very small for the points in this partition are very close to each other okay so now so the theorem which will justify our next definition so theorem one is the following under these notations okay so give an alpha given a compact subset compact subinterval off the tip of the of the domain of alpha okay 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 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 V T this is less than Epsilon okay before proving a 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 is this up as disappeared okay and what is this is saying is that if P becomes I mean denser and denser in the interval now because P get the norm of P getting smaller and smaller means that the number of point is increasing and they are becoming very well distributed in the interval then the length of this segment the sum of the lengths of these segments is converging to this number here okay so this is the good definition of length at least this is what I was suspecting as a good definition of length before starting no I mean you would say if you pick 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 okay 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 okay 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 a it's an application of the mean value theorem so this is the statement let's write here the proof so let's define the function f in this f from three times I so I cross I plus I into R so this takes three numbers t1 t2 t3 into R and gives me the square root of x prime T 1 squared plus y prime t2 squared plus Z prime T 3 squared okay now this is not completely crazy Allah what is the 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 f prime of T is the vector X prime T Y prime T Z prime T ok so what is the norm of the vector but 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 ok so here the only trick is to split for each coordinate I use a different parameter C when t1 is equal to t2 is equal to t3 this function is exactly giving me the norm of the tangent vector ok so this is just a trick to decouple the variables ok now F is clearly a continuous function isn't maybe not more than continuous no 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 certainly continues so hence it is uniformly continues on compact subsets on C we are restricting ourselves to a compact sub piece of I so adab a be okay in fact let me call it a B cross a because a B with some abuse of notation this is by definition Kb Cuba okay just to shortcut okay so so what does it mean it's uniformly continuous well so for any epsilon there exists Delta such that if it 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 a B Cuba which differ from each component with t1 minus s1 less than Delta T Q 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 the D image is not too far away so that f of T 1 T 2 T 3 minus F of s 1 s 2 s 3 is less I would say less than epsilon here for the proof is convenient to say epsilon over B minus a which is just another epsilon okay 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 okay by the mean value theorem I know that the length of the model the absolute that the length of this vector TI minus alpha TI minus 1 this is equal actually to F of beta I gamma I delta x TI minus TI minus 1 okay for some batai gamma and delta in this interval TI minus 1 TI okay and this is simply the observation I was doing before okay the relationship between F and the norm of the tangent vector okay just convince yourself that I mean it's immediate okay 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 now I take a partition and I take the sum of the of the lengths of these segments so using this property I can write it as the sum from I 1 to N of F of beta I gamma I dealt i TI minus 1 TI minus t 1 minus TI minus 1 okay so this is one side remember we 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 on the D what 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 our additives in the in the extremes okay so if I have a partition this becomes just the sum of the integrals between TI minus 1 and TI of the length okay this is just a partition okay but then again by the mean value theorem each of them so now I apply the mean value theorem to this function here and this is equal has to be equal so this is some of the value itself at some point XII okay TI minus TI minus one okay but what is alpha prime of CI by the initial observation in the definition of f this is f of e IX i IX I I so this is equal to the sum from 1 to N of F say I say I say i TI minus TI minus 1 okay for some of course here what I mean is for some XII in the interval TI minus 1 TI okay 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 okay so now if we take a partition with norm less than Delta that means that all these TIA the length of these intervals are all less than Delta by definition okay but then what happens if TI so this implies TI minus TI minus 1 is less than Delta for nei okay but then beta X ayah beta ixi I are both in the same interval so they must be Delta close they are inside an interval of amplitude Delta okay so this implies beta I minor minus ki less than Delta and the same for gamma and the same for Delta of course don't be confused about Delta I and Delta okay I just had few so gamma minus ki is less than Delta and Delta minus Delta is less than them sorry minus ki is less than Delta okay because they they all fall in the same interval and this interval is small by assumption okay so now that's it put things together and you get the conclusion okay there is nothing left but what do I mean by two things together remember we were evaluating the difference between this and this in absolute value so now you just take this this of this formula and this formula so you are taking this - 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 be tanks I to the gamma ixi and so on so basically the T will be xie xie xie xie and the s will be beta gamma delta okay and here there is written exactly what you want okay so this is the end of the proof I mean it's formerly not very beautiful I have to admit but it's it's simple okay it's just the mean value theorem applied with a certain ingenuity okay 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 definition of length but now we have to be careful what is this definition taking us so remember this theorem is justifying and giving us also a way to explicitly compute for for hundreds even thousands of years this was the way to compute length of curves so by really putting denser and denser partitions okay for example all the I mean now we know the the value of pi' up to few trillions digits okay but the way Greeks for example were estimating PI was because of course they knew that the the length of a circle was two pi I mean of radius one okay and then how you compute the length of a circle well you put 100 points 200 points 300 points okay and that was the way you do it okay it so in practice you could it could have been even useful now of course with our computational skills this is getting a bit but I mean still so but in any case let me remind you we got to this point okay so the just this was justifying the length the definition of length of a curve between two points has the integral between a and B of the length of the norm of the tangent vector okay 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 okay Jo I mean let me this is a key point so so what does it mean it has to be invariant under isometries we we have defined this length only using the our ability of measuring lengths of segments okay now what our eyes Amma trees of r3 so we have our curve alpha in R 3 and the point is if we have an isometry over 3 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 okay now this is a quick question because if it has change 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 the things that we are using in violence we are using the measurement of length of segments isometry that is the right thing okay so this is a good geometric object if is it invariant by a geometries otherwise it's not a good geometric object okay now the point it's all 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 and I solve this by definition something with respect distances okay then you prove that for r3 translate it's it's a composition of translations and rotations okay so well is it clear to to all of you because so let me let me speculate a moment on this okay so an isometry and isometry of our three of course here we mean our three with the usual way of measuring distances okay so our three has so many structures because it's a group it's a vector space it's it's everything okay 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 property the key property of r3 that we are interested in is that we are able to measure distances between points okay in some sense in a canonical way okay an isometry of r3 is a map is a transformation is a transformation over three such that the distance between two points is the same as the transformation let's say five over three such that the distance between two points P and Q is equal to the distance between Phi of P and Phi of Q okay so it respects distances of course this is for any choice of P and Q now it's a nice exercise not easier of standard classical geometry to prove that out of this condition you get that this implies that Phi is up to 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 atomization is a linear map whose Associated matrix a satisfies a a transpose equal the identity okay so it's what it's called an orthogonal transformation okay so in fact strictly speaking is not a composition of a translation and the rotation now because there could be also a reflection okay out of this equation you cannot decide that the determinant of a is equal to one the determinant of a is equal to plus 1 or minus 1 ok if it's plus 1 it's called direct isometry if it's minus 1 is called an inverse I geometry sometimes but it doesn't really matter okay this is just names ok ok this is enough for us this observation is enough for us to conclude the tower definition of length of a curve is invariant because of course length of segments is preserved by a geometry 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 okay and basic so being our definition of curve of length essentially the limit s of lengths of segments then it's okay okay 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 which gives also the first exercise I want to leave you is the following if I leave the definition of length there okay 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 okay and of course again remember I know it might sound a bit boring at the beginning but these are we are building again I'm saying again we are building a method for situations where the solution will be highly non-trivial okay so we want to check that we are building a theory even when we already know the answer okay 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 okay but there will be situation where we have absolutely no clue of what will be the answer okay so let's let's spend two minutes on this even if it has to be simple okay 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 v and alpha of B okay so we want to argue that the length of the segment which is just alpha of B minus alpha v the norm of this vector is less than or equal to the length between a and B of the curve alpha okay well is this true yes 30 seconds proof okay what is alpha B minus alpha way this is just fundamental theorem of calculus okay this is I can say that this is the absolute value of the integral of alpha prime without the norm okay but then by the simple properties of the integral I can put to the norm inside bite and then I get less than or equal okay and this is exactly the length okay so very simple it is true segments minimize lengths in Euclidean space okay so exercise if if you get the quality 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 T V Plus V naught so I guess a segment could be by definition just this between with T in some fixed interval okay so this could be one way to say what is a segment but then this will not be true okay think about this you see out of the leg 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 okay 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 12 of B will go in the uniform motion way or it going what it will go for example like a pendulum okay the length of this curve is the same I mean if the particle for some reason is stupid or I mean has some forces as some alien wants to go back from time to time and go on at the end the length of the curve is always the same thing okay so here you have to adjust I mean is the segment up to a parameterization I mean if you parameterize it well I mean this is a geometrically meaningful question okay and in fact this is exactly the problem we we are facing in the last 10 minutes okay what is the effect after all we want to do geometry okay 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 parameterize the same geometric thing in an infinite number of ways so how do I go back to the geometric properties of the original object okay which quantities are independent of the way the particle moves okay you understand this is a key point so so in sometimes 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 is that many things so many many geometric things can be described in completely different analytical ways okay and since I'm really interested in the geometry of this subject I need to have some way to kill this freedom okay of the possible parameterizations okay that's what I want to try to do now so what is a de film or FISMA now I'm assuming I and J are both open intervals okay I never really said if interval forming for me meant closed semi closed doesn't really matter up to now now I require ing to be open just to avoid nonsense okay so what is a de fumer field and if your f is a map between two open intervals smooth which is smooth invertible with smooth inverse okay okay so every time we have a deformer physicai and we have a curve so given now a parameterized curve has moved curve alpha from I to a 3 we can construct a new curve that I call beta simply composing with this map Phi and this will go from J right because of course I wrote the intervals in the wrong way okay I'll have to composite with Phi inverse so let me correct this okay otherwise I have to put Phi minus 1 which is okay let me go from J to I and so beta will go from J to our 3s 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 R 3 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 on the other hand as a job 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 in the image of alpha are the same so beta is called a reaper a metallization of alpha okay now first observation to proposition if you want it's really up to you to decide what it what is the dignity of a theorem now somehow there is a proposition it's a bit less I mean it's a bit less important or game so Phi from J to I D for Murphy's Alpha a smooth curve and a be our closed sub interval of j5 let's say that Phi maps disclosed some interval into another closed interval that we call C D then the length between a and B of alpha compost Phi is equal to the length between C and D of alpha okay so rip aramet rising so changing parameters if you know these properties okay this is this is crucial you need to have this 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 raise everything we did up to now and start from scratch okay because we are trying to measure something which is a geometric property of the image and these two object have the same image so if this was not true it means we are on the wrong track track okay so these are just indications that we are doing the right thing okay let's prove it so 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 okay the lengths are given by integral so what is the norm of alpha composed by Prime at T well I computed this by chain rule now it's a composition so this is the norm of Alpha Prime at the point Phi of T norm x its 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 okay we use it the same symbol but now so this is in general now if I see the few Murphy's what does it mean it's the film or fish it's invertible with smooth inverse okay in particular that means that Phi prime can never be zero okay so it's either always positive or it's always negative okay on everywhere so call s is equal Phi of T okay so let's see what are we comparing now we are comparing the integral between a and B okay 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 compos Phi prime T norm in DT okay we did the first computation above this is equal the integral between a and B of Alpha Prime at Phi of T times the absolute value of Phi prime of T in DT okay and now now there is the only delicacy here because if Phi prime is always pop was always positive this absolute value is equal to the Phi prime itself okay and this so suppose for a moment if Phi prime is positive what can I say that this object here is actually d s okay and this object here is D in D s so this is exactly the integral between so by the change of variables in the integral this is the integral between C and D right of D in D s alpha of s in this okay now if I prime was negative this what this formula was also true and you have to convince yourself in the sense that of course you have to also to switch the so this this formula holds if you always assume that C is less than D now so you have to to switch sign-ons okay excellent that's it for today so what do we do so summary of the of the of the of the lecture of today definition start having a feeling of which kind of objects are recovering self-intersection cusps smoothness in some sense how smoothness is covered by our definition few examples and length length and how does it depend on isometry x' meaning does not depend on isometries of the ambient space and how does it depend on reaper ammeter ization of the domain and again does not depend on the repair metallization 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 parameterization is there a best parameterization I mean that's I mean maybe there is a canonical parameterization 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 okay and this is kind of the accident in dimension one the answering dimension one you will see in two days is yes four curves there is always a canonical way to move there is uniform motion okay velocity one okay but this exists only in dimension one okay so we will use it we will be happy the theory of curves will be almost over because of this accident in me almost immediately but then when we go to dimension two we will face the problem that this theorem fails okay there is no canonical parameterization of a surface okay so that's for Thursday you you

Channel: ICTP Mathematics

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Length: 89min 39sec (5379 seconds)

Published: Thu Nov 17 2016