Manifolds #1 - Introducing Manifolds

[Music] okay so now let's give a fairly lights and non-technical introduction to manifolds so first of all what is a manifold well the essential idea is that a manifold is topological space we have some kind of idea about what that means it's just a set with a topology that's understood and certain to turn this topological space into a manifold we have to cover it with what are known as charts so we'll see what all of this terminology means in a second I'll be loosely using terminology throughout this video that we're going to formalize eventually but for now let's just stay vague and intuitive a lot of these ideas come from a very intuitive place so as useful to have these notions in the back of your mind when thinking about these things so what is a topological space first of all well we know a few examples so the circle or s1 is a topological space what do we mean by this well we mean all of the points that are on this circle lie in the set s1 and they'll usually think about circles as being round we know this is just an artifact of how it's embedded in this two-dimensional plane the circle can be stretched and deform as much as we like we could deform it into something that looks like this and topologically these two things are completely equivalent or even anything anything we want really as long as we don't cut or pinch the circle these are all equivalent as topological spaces so we've mentioned this previously it's the notion of these topological spaces being what's called homeomorphic so all of these images which I've drawn are homeomorphic as topological spaces so we're not talking about the geometry yet we're just talking about elements of abstract sense which we realize as topological spaces geometry comes much later when we define things like metrics and connections we'll get there eventually but for now all of these shapes are considered to be equivalent so there are lots of other top logical spaces we could construct such as the sphere which is the two-dimensional topological space and as I've drawn it here this round sphere we usually understand it as being embedded in some higher dimensional so in this case we've embedded the sphere in our three three-dimensional real flat space and we know that our three as a set or topological space usually carries the standard topology and now this embedded topological space or embedded manifold this sphere s2 is going to inherit his apology from the embedding space so all sort of carry a standard topology so again this is the topological space s2 I've drawn it as the round sphere I could have drawn some any kind of three-dimensional you'll be object it would be equivalent to this runs fair because we know that we can continuously deform one into the other without cutting or tearing and this is the notion of the two spaces being homeomorphic however we could have another type of topological space called the torus this is not homeomorphic to the sphere because it has a hole there's no way we can continuously deform this space into here we would have to punch a hole in it so this notion of homeomorphism we're going to see will carry over to manifolds when we're talking about land quads in fact it will be upgraded to the stronger notion of Adithya morphism we'll get to all of this eventually but for now I want to just talk about what what we are actually doing when we construct a manifold and why we want to construct a manifold so all of these topological spaces which I've drawn are perfectly understandable as abstract sets we can draw them in this way that we don't really have any way to talk about the points in the set so the essential idea behind a manifold is that we want to cover we want to cover our topological space in what are known as sharks so these are two pieces of terminology which we'll come back to but the essential idea is that by covering the space in chance we give coordinates to the space so what these charts do is they map a small portion of our manifold into some subset of the real numbers to the deef power where this D is going to be the dimension of the manifold so in this case of the torus it would be r2 since this is a two dimensional manifold but what these charts do is they give coordinates to what is known as a coordinate patch in the manifold so this can be viewed as saying that locally on a small enough scale the manifold looks like some subset of the real numbers because we can map it into the real numbers in some particular way and then we cover the whole month all in these charts and that allows us to talk about the elements of a set just in terms of D tuples of real numbers or just a list of D real numbers which we understand as being the coordinates so that was a bit of an abstract discussion let's consider a concrete example so if you consider the surface of the earth which is a sphere to be on manifold we know that what we understand this manifold is being the set of all the possible locations on the earth so this is just an abstract set that contains all of the locations on the air and now we know that we can produce now a literal map a literal map of portions of the earth and the collection of all the maps we call an atlas that's terminology we'll see shortly but one is a map well it's essentially just a two-dimensional image of some portion of our original manifold so this can be viewed as saying we have our manifold and we map from this abstract set of locations into some flat two-dimensional image now we can then on this two-dimensional image concretely talk about the points of our manifold since we can label the points on our image by coordinates and we can easily construct these in a flat two-dimensional plane and now you can tell me by giving me the coordinates of any location on the map I can then translate that back into the original manifold and work out where the original location is so this is the essential idea of what we do when we construct a manifold just we have some abstract topological space you will just need a set of elements and then we map that set of elements in into a subset of the real numbers which just allows us to talk about the elements of the manifold concretely in terms of lists of numbers or coordinates so the collection of all the maps is known as the atlas and now in your atlas there may be portions of the map which overlap they might be covering the same region in the manifold now in order to be a good map they should agree on whether the charts overlap so if the maps were drawn accurately enough you should be able to tear out all the pages from your atlas and then if you stitch all of the pages back together so that you align although but overlapping regions where charts overlap you should from that set of flat pages that you ripped out you should reconstruct a speck or object so this is again emphasizing that locally small regions of our what might globally be not are to the sphere is obviously not r2 but locally on the sphere we can represent it as a portion of our tin so if you ever meet a flat earther all you need to do is explain that whilst locally the earth place to be homeomorphic to r2 globally the structure is wildly different is in fact a sphere so move the manifolds that I've kind of been talking about loosely here feel very geometrical objects all of these can be realizes pictures embedded in some kind of real flat space we will see however that there can be many more abstract types of manifold anything that's a topological space can be turned into a manifold so we'll see other examples like a special type of group known as a lead group which is a continuous group that can be realized as a man thought so what's all of this intuition is easiest to picture in terms of these geometric objects one of the machinery which we're going to develop can carry over into more abstract situations so just to summarize then this kind of like informal introduction a manifold is some topological space or we feel like we understand what that is from which we cover up or we cover the topological space in these circle charts and these charts are viewed as giving coordinates to some local patch of the man thought and then we should be able to from the set of all these charts reconstruct the manifold all the points in our manifold by effectively sewing them together in a particular way [Music] [Music]

Channel: WHYB maths

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Keywords: Maths, Math, Physics, Tensor, Vector, Algebra, Abstract Algebra, Algebraic Structure, Group, Differential Geometry, Linear Algebra, Multilinear Algebra, Undergraduate, Education, Topology, Topological Spaces, Manifolds, Continuity, Maps, Sets, Coordinates, Homeomorphism, Homeomorphic, Relativity, General Relativity, Spacetime

Id: GqRoiZgd6N8

Length: 12min 37sec (757 seconds)

Published: Tue Oct 29 2019