 In a previous video, we introduced spherical geometry as a model of elliptic geometry, the so-called two-point elliptic geometry. But I mentioned there were some problems with that because in spherical geometry, you don't have the line determination, which is probably, I would argue, the most important axiom of incidence geometry. So if spherical geometry, although it satisfies the elliptic parallel posh, it's not an incidence geometry. In the previous video for this lecture number nine here, we did introduce the real projective plane, which is in fact, an incidence geometry which satisfies the elliptic parallel posh. It's in fact a, it's a projective geometry, something we've defined previously. But I also made a comment that just like the sphere, the real projective plane is locally Euclidean. So what does that even mean? So this is the second patchwork that where I'm trying to talk to you about spherical geometry. It's not an incidence geometry, but what do we mean by non-Euclidean? Well, it satisfies, I should say, it doesn't satisfy the Euclidean parallel posh, but in some essence, spherical geometry is still very much like Euclidean geometry. And this leads to a very deep topological notion known as a manifold. So before I define that, let me say a little bit about topology. What is a topological space? We can define that axiomatically like we do affine geometry or incidence geometry or anything else like that. But I don't want to do that in this lecture. This is a topology course. This is a geometry course, but very loosely speaking, a topological space is a generalization of the notion of geometry, which axiomizes the notion of point. So we have this undefined term point here. I'll actually spell out the whole thing. But we also need some type of notion of closeness. What does it mean for two points to be close to each other? And so that's basically what a topology is trying to axiomatize. Now, typically this is done using the language of open sets, maybe closed sets, but really the idea of a topology is we have points and we have closeness. We have nearness. What does it mean for two points to be close to each other? No notion of a line is given to a topology. So that's why we're not really calling it a geometry, but the two things are similar because we're talking about this idea of points. And if you have a distance function, for example, you know what it means for two points to be close because you can look at their distance. But if you have a distance function, you can also use that to construct lines because you can say the line determined by two points is the path of shortest distance between them. That gives you a line segment and then you extend that to a line. So there's some technical terms going on there that I'm not gonna divulge at the moment. But while distance is one notion of closeness, you can form a topology using a distance function. You can also form a geometry using a distance function. This is called a metric space. And we'll say a little bit more about those later on in this lecture series. Topology typically uses the notion of an open set, like I said, to characterize what closeness means. And that essence, you might have like two points, we'll call the points X and Y. And there could be some, you know, there could be some open set that contains one, maybe then you have some other open set that's smaller and then another open set, oh, and now it separates the two. Now I'm kind of getting to separation axioms of topology right now, but nonetheless, open sets are this idea of when you say that two things are close to each other. So you can have like these shrinking neighborhoods. So you're getting closer and closer to X because the open sets that contain X gets smaller, smaller, smaller, smaller. All right, so this is a very, very broad notion. But in other words, topology gets this idea of closeness and why do we care about closeness? Cause this is where we can do calculus because if you have a notion of closeness, I can get closer and closer and closer to what? A limit, right? We can talk about sequences that converge. We can talk about Cauchy sequences. We can talk about limits. So essentially topology you could argue is the weakest form of geometry for which calculus can be done because you have points, you have closeness from that. You can talk about convergence. If you have convergence, you can talk about sequence. You can talk about limits and you can go from there. Now, in a calculus class that you might have taken before, you probably did talk about some notion of limits, like the precise definition of the limit. You can talk about epsilon and delta, things like that. The idea is you're constructing epsilon neighborhoods and delta neighborhoods, these balls, these circles of radii, epsilon and delta. And so as epsilon gets smaller, we shrink delta and things like that. So you're doing topology in a metric space when you talk about epsilon, delta, proofs of limits. But in the general topological setting, you don't need a metric. You don't need a distance function. You can still use open sets to talk about convergence. And so that's why people care about topology in sort of like a grand scheme of things, right? Topology is the geometry for which we can still do calculus. We might not be able to do triangle or trigonometry, but we can still do calculus or something that's similar to the notion of a calculus. All right, but the general idea of a topology, the axiomax of topology are very, very broad. If we wanna do calculus that resembles the calculus we do in classes like using SAU's number in math 12, 10, that's calculus one, math 12, 20, that's calculus two, math 22, 10, calculus three, if we wanna do calculus that resembles like the real analysis, the calculus we do in Euclidean space, then your geometry, I should say your topology has to resemble Euclidean space in some regard. And so that's where we introduce the idea of a manifold. So an N manifold M is a topological space which locally looks like n-dimensional Euclidean space Rn. So in particular, a two manifold is gonna be a topology that locally looks like R2, okay? And so what does that mean exactly? So for each point in the manifold, there exists an open set that contains it. This is what we call a neighborhood. A neighborhood appears an open set that contains it. Now this neighborhood U has to be isomorphic to Rn as a topology. So again, what does that exactly mean? We have a homeomorphism. So this is a bijective continuous map whose inverse is also a continuous map. Again, I don't wanna delve too much into the topology here. So each point needs to have a neighborhood that's locally isomorphic to Rn. And then these neighborhoods are gonna overlap each other and this has to be done so that transition happens well. And again, I don't wanna define exactly what that means, but it has to have well transitions, a good transition. Furthermore, we can talk about the idea of a smooth manifold, a smooth manifold. So when we talk about a manifold, this isomorphism is isomorphism topologies. So U and Rn are isomorphic as a topology. If we require that this isomorphism between U and R be an isomorphism of a geometry. Cause Rn is more than just a topology, it's more than just a vector space. There's a lot of structure going on there and particularly Euclidean geometry has all of that stuff interplaying with each other. If this isomorphism between the neighborhood U with Euclidean space, Rn, if this is a isomorphism of the geometry, so the two things are the same as a geometry, we call this a smooth manifold and this isomorphism is called a chart, right? And so I wanna give you an example to show you why spherical geometry is a two manifold. So this is the way we can try to think of spherical geometry. Okay, it's not an incidence geometry. If we want it to be an incidence geometry, we have to basically collapse the thing down to get to the real projective geometry, real projective plane, which is an incidence geometry, but it's really hard to visualize, right? What if we just keep spherical geometry, we don't change it, but we don't think of it as an incidence geometry, what if we think of it as a manifold, right? So the sphere S2 is in fact a smooth two manifold since locally a sphere resembles a plane, a Euclidean plane, right? So like if you take a neighborhood of some point, right? If you look at that disk, if you flatten it, can you tell the difference between you're on a sphere or you're in the Euclidean plane, right? Locally, it looks like Euclidean space. I mean, after all, it doesn't the legend go that the ancients believed that Earth could be flat? Why did they think it was flat? Because when you look around you, what do you see? Like flat, flat, flat, flat, flat, curved, nope, that was a mountain, it's flat. Everything is flat around you. And so locally, it looks like it is flat, but globally there is curvature, curvature that can be very difficult to see when you look at it really, really close. Some of the most best geometry that happened in the ancient world was used to prove that the Earth in fact was round. How do you find the curvature of the Earth when you don't have the technology to blast a rocket into space and take a picture from the moon or something, right? We have to do it while we're still in the geometry, okay? And so what are these charts when you think of the sphere as a smooth manifold? I claim it as a smooth manifold because in fact this geometry of a neighborhood is actually isomorphic to the geometry of R2. So we have this two-dimensional smooth manifold. So this diagram you see over here is what's often referred to as the stereographic projection. Let me write that on the board here. Stereographic projection. I want you to be able to look that up later on if you wanna study this topic in more details. There's a lot that can be said about the stereographic projection. So the idea is you have the sphere, right? It's actually subset of R3, all the points which are one unit away from the origin right here. And then let's take the XY plane in R3. So this is of course R2, right? If this is in fact a smooth manifold, there has to be charts, these isomorphism between neighborhoods and of course to the plane for which it basically is the following idea. The first chart what I want you to do is we're gonna take the open set that contains everything except for N. That is in fact an open set here. And so if you take any point on the sphere called capital Z, we can identify it with a point in the plane called lower cases E by taking the unique line in Euclidean three space determined by the North Pole and Z, right? So R3 is in fact an affine geometry. We have unique lines in that situation. Take the line from the North Pole to that point on the sphere and then that line extended onward will eventually intersect the plane at a unique point. And so we identify that point on the sphere with that point on the plane like so. Now, that point could be exterior to the sphere like in this example. That's gonna happen if your point is in the Northern Hemisphere. If your point Z is in the Southern Hemisphere, then the line that connects the North Pole to that point will actually intersect the plane inside of the sphere. And so that's what happens. The Southern Hemisphere gets identified with the inside of this circle which is the intersection between the sphere and the plane. And then the Northern Hemisphere gets exploded outward to be everything outside that. And so you have this isomorphism, a geometric isomorphism between all the points on the sphere with all the points on the plane. Now, I should correct that. It's not all the points on the sphere, it's only the points on the neighborhood, okay? So there was a neighborhood where we excluded the North Pole, but I can do the same trick where I take the map, excuse me, I take the open set that excludes everything except for the South Pole, I can redo this thing. And so even the North Pole and the South Pole do have a map, they do have a chart that tells them how to get mapped onto it. Because with a smooth manifold, you'll have numerous charts when they put them all together. We call this an atlas, it's a course of play on words right there. And so we can locally make the sphere into using the stereographic projection here, we can make it into Euclidean space here. I should mention that the real project of plan we talked about in the previous video is also an example of a two-manifold, I'm not gonna go through all the details of that. But essentially, as we talk about non-Euclidean geometry in this video, what we're really interested in are these two manifolds. We want geometries that are almost Euclidean, that is, they're locally Euclidean. So locally, it looks like lines are parallel to each other in a unique manner. But when we look at the global behavior, it could be that lines in fact always intersect like on the sphere. And so as people study these non-Euclidean geometries, what we're really interested in when people talk about non-Euclidean is it doesn't just mean a geometry other than the Euclidean geometry, because our finite geometries are technically non-Euclidean in that sense. But the idea is when people are talking about non-Euclidean geometry, they really mean we want to look at manifolds, two manifolds or higher if necessary. We're looking for manifolds, these things that are locally Euclidean. So their local geometry is in fact Euclidean, but globally we get a different situation. Why might someone care about such a thing? After all, a manifold is locally Euclidean, it's locally flat, but globally could have some curvature to it, just like the sphere does right here, which things like curvature, we can make very explicit using geometric axioms, although we're not gonna do that in this video. Lecture nine is meant just to be an exploration of things. We're not gonna go into the details of everything of course. So a space could locally look like R2, thus being a two manifold with maybe curvature to it. But that curvature could be elliptic, it could be hyperbolic, which we'll talk about more hyperbolic stuff in just a second. But as such, the study of non-Euclidean geometry, essentially, especially when we talk about higher dimensional non-Euclidean geometry, is looking at these locally Euclidean objects, manifold, but globally could have a different curvature to it. If people used to think the earth was flat, they just recognized we live on a manifold, but it took some time, but we discovered that manifold has curvature into sphere. What if we blow that same question up to the universe that we live in, outer space? When you look at it, it looks like three dimensional Euclidean space. That just means it's a manifold. Is there curvature to that space? Very likely there is, because that curvature, even if we can't see it, because we might have to exit the geometry to see it, that's very difficult to do. Maybe one day we can blast a rocket outside of the universe to take a look at the curvature, like we did with the earth and the moon, but assuming that's even possible, in the meanwhile, how are we gonna do something like that? Can we see the curvature while we're inside the geometry? Well, geometers were able to do that with planet earth, maybe astrophysicist and geometers of the future will be able to do the same thing with our universe and see its curvature and discover, oh, we're not in Euclidean space. We're probably not. Maybe we're in elliptic space. Maybe we're in hyperbolic space. By no means am I an astrophysicist whatsoever, but it's very, very likely that the curvature of the universe is hyperbolic. It has that curvature, which we'll talk about hyperbolic geometry in our next short video.