Riemannian Geometry - Examples, pullback Oxford Mathematics 4th Year Student Lecture

Welcome to Riemannian Geometry lecture 1c. In this lecture I want to continue to discuss examples, now slightly more sophisticated examples than last time, and I want to talk about the notion of pullback of Riemannian metric, which you have seen before in differentiable manifolds. But now I want to see how we can use the pullback to understand Riemannian metrics locally. So let's begin with a very simple but very important example. So suppose that I have my manifold which is contained in some Euclidean space, say Rn. Then there's a nice way to define a Riemannian metric on m by just restricting the Euclidean metric. So given any point in m and tangent vectors x and y to m at p, then those are also tangent vectors to Rn. So we can plug them into the Euclidean metric g0, and that will define for us a Riemannian metric. This makes sense. g at p at xy is just g0 at xy. And we call g the induced metric on m. So that's one of these examples where we drop the phrase Riemannian. Now in particular, if we take the example where m is just a surface in R3, then the induced metric is what you called the first fundamental form when you studied surfaces in R3. Now a particularly important example of the induced metric is if we take the n-sphere, which I remind you is the set of points in Rn plus 1 such that the sum of the squares of the xi's is 1, then we can define a Riemannian metric which is just the induced metric in the sense of the previous example. And we call this metric g the round metric, or sometimes standard round metric on the n-sphere. So this also shows that even though we are just restricting the Euclidean metric to points on m and tangent vectors to m, the properties of the metric are very different than the properties of the Euclidean metric. As we've seen in the previous differential manifolds course, some of the obvious things that we know about the Euclidean metric are certainly not true for the round metric on the sphere. So let's have a look at this example of the round sphere in a little bit of detail. Let's have a look at the dimension 2. So on the 2-sphere, we can choose our local coordinates theta and phi as follows using this map. So we define f of theta and phi to be sine theta cos phi sine theta sine phi cos theta. And that's for theta in this open interval 0 pi and phi in R. Then f is a local diffeomorphism onto its image, and we can define these two vector fields x1 and x2 by pushing forward partial theta and partial phi. So just as we saw before, the way that we do that is we differentiate this formula with respect to theta, and you're going to get cos theta cos phi partial 1 plus cos theta sine phi partial 2 minus sine theta partial 3. That's for the push forward of d theta. And if you want to compute the push forward of d phi, then you differentiate this with respect to phi. You're going to get minus sine theta sine phi partial 1 plus sine theta cos phi partial 2, and then nothing here because the final coordinate does not depend on pi. And because this is a local diffeomorphism, f and theta and phi are independent, which means that partial theta and partial phi are independent, you see that x1 and x2 are everywhere linearly independent on the image of f, which is just the 2-sphere minus the north and south poles. Those are the points where theta is equal to 0 and pi. With respect to the round metric g on the 2-sphere, we can compute what is g of x1 and x1. Well, that's just 1. We just take the sum of the squares here and we're going to get 1. Just cos squared theta cos squared phi plus cos squared theta sine squared phi. That's just cos squared theta plus sine squared theta. We can take the dot product of x1 and x2, and we can see that these are clearly orthogonal. It's easy to see, so g of x1 x2 is 0. And then we can compute g of x2 x2, and that's going to be sine squared theta sine squared phi plus sine squared theta cos squared phi, so that's sine squared theta. And so we see that the matrix of g with respect to this basis x1 x2 is the following matrix of smooth functions 1 0 0 sine squared theta. Again, this is clearly a symmetric matrix and it's positive definite if and only if sine theta is non-zero. But that's exactly the points that we excluded, because we're saying that theta is strictly between 0 and pi, and that means that indeed sine theta is non-zero. We're on the 2-sphere minus the poles. And again, we note that g is a well-defined Riemannian metric everywhere, but it's just this coordinate system that we've chosen goes bad at those points. It's not that the Riemannian metric degenerates at the north and south pole, it's just that our coordinate system does. So what about if we think about the round 3-sphere? What can we, what does the metric look like there? Well, on the 3-sphere we can define three vector fields as follows. So if we have coordinates x0 x1 x2 x3 on R4, then we can write down these three vector fields. Now these are all tangent to the 3-sphere, that's easy to check, and they're all independent from each other. In fact, we see that with respect to the round metric g, they're actually an orthonormal basis. If I take g of ei and ej, then I get the Kronecker delta, delta ij. So in fact, we have an orthonormal basis for the vector fields on S3 everywhere, and so globally we can view the round metric on the 3-sphere as the identity matrix. So that's very different from the situation that we saw on the 2-sphere, where we didn't get the identity matrix, and in fact it's impossible to globally view it as identity matrix, because the 2-sphere is not parallelizable. We can't have everywhere independent vector fields, because every vector field on the 2-sphere has to vanish somewhere. Now, we can always do this same trick of finding a basis for the vector fields such that the remaining metric becomes the identity matrix, as long as our manifold is parallelizable. That's all that we've used here. We've used the fact that the 3-sphere is parallelizable, that means that we can do this trick of finding vector fields e1 up to en, an n-dimensional parallelizable manifold, and get that g of ei ej is delta ij. And of course, this reinforces the point that I made last time, that locally we can always do the same trick, because locally every manifold is parallelizable. Now let's have a look at a slightly different example. Let's look at the 2-torus, and look at the induced metric on the 2-torus in R3. So in other words, the first fundamental form of the 2-torus. So one way to embed the 2-torus, a standard way to embed it in R3, is using the following map. So this just gives you the usual torus that you can imagine in R3, the one that we drew in the very very first lecture. Here's the formula, and it's easy to compute what are the corresponding push forwards of partial theta and partial phi with respect to this local diffeomorphism f, and we can compute it here. So you just differentiate with respect to theta, you get this formula, you differentiate with respect to phi, and you get this formula. So let me just read it out. x1 is minus sine theta cos phi partial 1, minus sine theta sine phi partial 2, plus cos theta partial 3. Next 2, which is the push forward of partial phi, is minus 2 plus cos theta times sine phi partial 1, plus 2 plus cos theta times cos phi partial 2. And again, because partial theta and partial phi are linearly independent on R2, their push forwards are everywhere linearly independent on the 2-torus T2, which is the image of this map f. Now with respect to the induced metric g that we get on the 2-torus from the Euclidean metric on R3, we have the following formula. We can compute what is g of x1 x1. Well, we see it's sine squared theta cos squared phi plus sine squared theta sine squared phi, that's sine squared theta plus cos squared theta, so that's 1. So g of x1 x1 is 1. It's easy to see that g of x1 and x2 is 0, they're clearly orthogonal, and g of x2 x2, well it's 2 plus cos theta squared times sine squared phi plus cos squared phi. So it's 2 plus cos theta all squared. And so we can identify the induced metric g on this embedded 2-torus with the matrix 1 0 0 2 plus cos theta all squared. And what we see is that this matrix now is positive definite everywhere. 2 plus cos theta is never 0, so this thing is always a positive definite matrix, and so this is a globally well-defined formula for the induced metric on the 2-torus. And again, that points to the fact that we know that the 2-torus is parallelizable. So now I want to come on to remind you about the notion of pullback. So suppose I have a smooth map f between manifolds m and n, and that suppose that I have a Riemannian metric h on n, then I can define the pullback. So this is going to be an object defined on m, so it's a map that goes backwards from n to m, and it's called f star of h, f upper star of h. And the formula is that f star h at a point p in m, evaluated on tangent vectors x and y at p to m, is h at the point f of p, remember f of p is a point in n so that makes sense, evaluated on dfp, so the differential of f at p evaluated on x, and dfp evaluated on y. So this is a formula that makes sense, and defines for you a symmetric bilinear form f star h, so in fact this is a section of S2 T star m. Now it's going to be important for us that we want to always think about vector fields x and y rather than So if we have vector fields x and y on m, then f star h evaluated on x and y is just h evaluated on f star of x, f star of y, where f star is the push forward of x and y by f. Now this is all stuff you've seen before in differential manifolds, so I would urge you to go back and refresh your memory about push forward and pull back. Let m be a manifold, and let nh be a Riemannian manifold, and let f from m to n be an immersion, so that means that dfp is injective for all p in m. Then what we know is that this pullback of h, we can call it g because it is a Riemannian metric on m. So this pullback, it always makes sense for any smooth map, but it's going to be a Riemannian metric when f is an immersion. Now typically we are going to care about when f is a diffeomorphism or local diffeomorphism, but in those cases then the pullback of h is a Riemannian metric because the differential in that case is an isomorphism. So in all the cases that we've seen so far, f has been a local diffeomorphism, and so we've always got that the pullback is going to be an isomorphism. So secretly that's what we've been doing for a little while, is that we've been computing the pullback Riemannian metric. So let's just see how we can use the pullback to talk about Riemannian metrics. So one thing that we've already seen is the induced metric. So one way to think about that is suppose we take our manifold contained in Rn, and let's write little i for the inclusion map from m into Rn. Well then i is an immersion because it's the inclusion map, and so the pullback of the Euclidean metric by the immersion i inclusion map is a Riemannian metric. And this Riemannian metric is nothing mysterious, it is the induced metric that we already saw. So another way, instead of that formula that I wrote down before, one way to rewrite that formula is just to say that the induced metric is the pullback of the Euclidean metric by the inclusion map, if you prefer that way of saying it. Now before I continue, I want to make a little definition which will be useful for us going forward. Now we saw that if we have a chart u phi on an n-dimensional Riemannian manifold, then we can write down n coordinate vector fields x1 up to xn, and these are the push forwards of the standard vector fields on Rn. Then what we do is we call the basis x1 up to xn, this local basis of vector fields, the coordinate frame field in u phi. And what we're going to do is write down the following family of smooth functions on u. It's g sub ij and it's just g of xi xj. So let me just remind you, this is something that you'll have seen in differential manifolds, but let me say it again, if I have a Riemannian metric g and I have these vector fields xi and xj, then the object g of xi xj is a smooth function where those vector fields are defined. So these gijs are smooth functions on u. Why did I bother to make this definition? Well we see that if we look at the metric g and we pull it back by phi inverse, what's that going to do? Well phi inverse, that goes from Euclidean space into the manifold. So that means that the pull back phi inverse star, that will define something on Euclidean space. So phi inverse star g, that is a Riemannian metric on phi of u, which is defined in Euclidean space. Okay, so let me say that again, phi inverse star of g is a Riemannian metric on phi of u, defined as an open subset of Euclidean space. So I can evaluate it on the standard vector fields, partial i and partial j, and whatever answer I get, that will completely define this Riemannian metric. Now as we saw, the definition of pullback means that this phi inverse star of g, evaluated on partial i and partial j, is the same as g evaluated on the push forwards, phi inverse star of partial i, phi inverse star of partial j. And these are by definition the coordinate vector fields, x i and x j. So this is just g of x i x j, and that's what we were just calling g sub i j. In total, what that means is that we can write the Riemannian metric phi inverse star of g on phi of u in the following way. We can write it as the sum over i and j of g i j d x i d x j, where these g i j's are exactly these functions that are defined here. And this is what I mean by it's sufficient to understand Riemannian metrics defined on just open subsets of Rn, because this says that any Riemannian metric, once you pull it back using the chart map, you get yourself a Riemannian metric on Rn, which is defined by these local functions, this symmetric positive definite matrix of functions g i j. So these g i j's, they'll be super useful and they'll appear many times going forward. Now let me just finish up this lecture by discussing some examples. So the first example that I want to discuss is one that we're going to see quite a few times. It's of super importance. It's the n-dimensional upper half space Hn. So it's the set of points in Rn, x1 up to xn, such that the last coordinate of xn is positive. And we take what's called the hyperbolic metric g on Hn, and it's written in the following way. It's dx1 squared plus dot dot dot plus dxn squared, all over xn squared. Now if you don't recall this notation, dx i squared and so on, then I urge you to look back at the relevant part of the differentiable manifolds course. And as I've written here, this metric is extremely important, both in geometry but also topology, with links also to complex analysis and even to number theory. Now a much simpler example, but nonetheless important, is to think about polar coordinates. So here is our usual polar coordinate map, f of r theta is r cos theta r sin theta. Well then this is a local diffeomorphism, so we can pull back the Euclidean metric, and we saw that we get the metric dr squared plus r squared d theta squared. This is just another way of saying that matrix that I wrote, 1 0 0 r squared. And this should be a formula that looks familiar to you. Similarly, if we take the local coordinates that we saw before on S2 with the round metric, so f of theta phi is the usual formula, then the pullback by f of the round metric g, f star g, is d theta squared plus sin squared theta d phi squared. This again is just another way of saying that matrix 1 0 0 sin squared theta that I wrote down before. These are all metrics that will be important. And in general, as I've said in the differentiable manifolds course, and I'll say it again here, I think it's important in any subject to have a good wealth of examples that you understand really well, and that will help you to understand the general theory much better. To sum up, what we've seen in this lecture is a number of different examples of Riemannian metrics, and we've seen the notion of pullback of Riemannian metric, which using the coordinate frame field allows us to describe any Riemannian metric locally using smooth functions forming a symmetric positive definite. This concludes lecture 1c.