Manifolds Ed Segal Autumn 2016 Contents 1 Introduction2 2 Topological manifolds and smooth manifolds3 2.1 Topological manifolds........................3 2.2 Smooth atlases............................6 2.3 Smooth structures.......................... 11 2.4 Some more examples......................... 15 3 Submanifolds 19 3.1 Definition of a submanifold..................... 19 3.2 A short detour into real analysis.................. 23 3.3 Level sets in Rn ............................ 24 4 Smooth functions 32 4.1 Definition of a smooth function................... 32 4.2 The rank of a smooth function................... 36 4.3 Some special kinds of smooth functions.............. 39 5 Tangent spaces 43 5.1 Tangent vectors via curves...................... 43 5.2 Tangent spaces to submanifolds................... 48 5.3 A second definition of tangent vectors............... 51 6 Vector fields 53 6.1 Definition of a vector field...................... 53 6.2 Vector fields from their transformation law............ 57 6.3 Flows................................. 59 7 Cotangent spaces 62 7.1 Covectors............................... 62 7.2 A third definition of tangent vectors................ 66 7.3 Vector fields as derivations...................... 70 8 Differential forms 75 8.1 One-forms............................... 75 8.2 Antisymmetric multi-linear maps and the wedge product..... 81 8.3 p-forms................................ 89 8.4 The exterior derivative........................ 92 1 9 Integration 97 9.1 Orientations.............................. 98 9.2 Partitions-of-unity and integration................. 101 9.3 Stokes' Theorem........................... 106 A Topological spaces 109 B Dual vector spaces 113 C Bump functions and the Hausdorff condition 115 D Derivations at a point 116 E Vector bundles 119 F Manifolds-with-boundary 125 1 Introduction A manifold is a particular kind of mathematical space, which encodes an idea of `smoothness'. They're the most general kind of space on which we can easily do calculus - differentiation and integration. This makes them very important, and they're fundamental objects in geometry, topology, and analysis, as well as having lots of uses in applied maths and theoretical physics. The simplest example of a manifold is the real vector space Rn, for any n. More generally, a manifold is a space that `locally looks like Rn', so if you zoom in close enough, you can't tell that you're not in Rn. Example 1.1. The surface of the Earth is approximately a 2-dimensional sphere, a space that we denote S2. There's a myth that people used to think the Earth was flat - the myth is obviously false, in fact the ancient Greeks had a decent estimate of the radius of the Earth! But the story has a grain of plau- sibility, because `close up' the Earth does look flat, and we could imagine that we're living on the surface of the plane R2. Hence the sphere S2 is an example of a 2-dimensional manifold. Example 1.2. The surface of a ring doughnut is a space we call a (2-dimensional) torus, and denote T 2 (see Figure1). If you were a very small creature sitting on the doughnut, it wouldn't be immediately obvious that you weren't sitting on R2. So T 2 is another example of a 2-dimensional manifold. Let's go down a dimension: Example 1.3. A circle, sometimes denoted S1, is an example of a 1-dimensional manifold. A small piece of a circle looks just like a small piece of the real line R. Here's an example of a different flavour: Example 1.4. Let Mat2×2(R) be the set of all 2 × 2 real matrices, this is a 4-dimensional real vector space so it's isomorphic to R4. Now let GL2(R) ⊂ Mat2×2(R) 2 Figure 1: A torus. denote the subset of invertible matrices. If M is an invertible matrix, and N is any matrix whose entries are sufficiently small numbers, then the matrix M + N will still be invertible. So every matrix `nearby' M also lies in GL2(R) (i.e. GL2(R) is an open subset). This means that a small neighbourhood of ∼ 4 M looks exactly like a small neighbourhood of the origin in Mat2×2(R) = R . Hence GL2(R) is an example of a 4-dimensional manifold. This is an example of a Lie group, a group that is also a manifold. Lie groups are very important, but they won't really be covered in this course. We often picture manifolds as being subsets of some larger vector space, e.g. we think of S2 or T 2 as smooth surfaces sitting inside R3. This is very helpful for our intuition, but the theory becomes much more powerful when we can talk about manifolds abstractly, without reference to any ambient vector space. A lot of the hard work in this course will involve developing the necessary machinery so that we can do this. 2 Topological manifolds and smooth manifolds 2.1 Topological manifolds We now begin formalizing the concept of a manifold. The full definition is rather complicated, so we begin with a simpler version, called a topological manifold. Definition 2.1. Let X be a topological space. A co-ordinate chart on X is the data of: • An open set U ⊂ X. • An open set U~ ⊂ Rn, for some n. • A homeomorphism f : U −!∼ U~ When we want to specify a co-ordinate chart we always need to specify this triple (U; U;~ f), but often we'll be lazy and just write (U; f), leaving the U~ implicit. The key distinguishing property of manifolds is that co-ordinate charts exist! 3 Definition 2.2. Let X be a topological space, and fix a natural number n 2 N. We say that X is an n-dimensional topological manifold iff for any point x 2 X we can find a co-ordinate chart ∼ n f : U −! U~ ⊂ R with x 2 U. In words, this says that at any point in X we can find an open neighbourhood which is homeomorphic to some open set in Rn. A concise way to say this is that X is `locally homeomorphic' to Rn (some people use the term `locally Euclidean'). It's possible to prove that an open set in Rn cannot be homeomorphic to an open set in Rm unless n = m, so the dimension of a topological manifold is unambiguous. The proof of this fact is not difficult, but it uses some algebraic topology that isn't in this course. Remark 2.3. There are two more conditions that are usually part of the defini- tion of a topological manifold, namely that the space X should be: • Hausdorff, and • second-countable. These are technical conditions used to rule out certain `pathological' examples (see AppendixA). Every space we see in this course will be Hausdorff and second-countable, and we're going to avoid mentioning these conditions as far as possible. Example 2.4. The circle S1 is a 1-dimensional topological manifold. Let's prove this carefully. Firstly, let's define S1 to be the subset 1 2 2 2 S = (x; y); x + y = 1 ⊂ R and equip it with the subspace topology. Next we need to find some co-ordinate charts, we'll do this using stereographic projection. Let (x; y) be a point in S1, not equal to (0; −1). Draw a straight line through (x; y) and the point (0; −1), and letx ~ 2 R be the point where this line crosses the x-axis, so: x x~ = 1 + y This sets up a bijection between points in S1 (apart from (0; −1)) and points in the x-axis. So let's set 1 U1 = S n (0; −1) and note that this is an open set, since it's the intersection of S1 with the open 2 ~ set fy 6= −1g ⊂ R . Now set U1 = R, and f1 : U1 ! U~1 x (x; y) 7! x~ = 1 + y 4 Figure 2: Stereographic projection. Then f1 is continuous, since it's the restriction to U1 of a continuous function 2 defined on fy 6= −1g ⊂ R . To show that f1 is a bijection we write down the inverse function: −1 ~ f1 : U1 ! U1 2~x 1 − x~2 x~ 7! ; 1 +x ~2 1 +x ~2 −1 An elementary calculation shows that f1 (~x) really does lie in U1 for anyx ~ 2 R, −1 −1 and that f1 and f1 really are inverse to each other. Also f1 is continuous (since it's evidently continuous when viewed as a function to R2), so we conclude that f1 is a homeomorphism. The triple (U1; U~1; f1) defines our first co-ordinate chart. For our second co-ordinate chart, we use the same trick but we project from 1 ~ the point (0; 1) instead. So we define U2 = S n (0; 1) and U2 = R, and: ∼ f2 : U2 −! U~2 x (x; y) 7! 1 − y We repeat the previous arguments to check that this is also a co-ordinate chart. 1 Now any point in S lies in either U1 or U2 (most points lie in both) so we have proved that S1 is a 1-dimensional topological manifold. Now let's do the same thing for the n-dimensional sphere Sn. Example 2.5. Let n n X 2 o n+1 S = (x0; :::; xn); xi = 1 ⊂ R 5 with the subspace topology. We get our first co-ordinate chart using sterero- graphic projection from the point (0; :::; 0; −1) (the \south pole"). So we define n U1 = S n (0; :::; 0; −1) ~ n U1 = R and f1 : U1 ! U~1 x0 xn−1 (x0; :::; xn) 7! ; :::; 1 + xn 1 + xn We can prove that f1 is a homeomorphism using the arguments from the pre- vious example, in particular the inverse to f1 is the function P 2 −1 2~x0 2~xn−1 1 − x~i f1 : (~x0; :::; x~n−1) 7! P 2 ; ::: ; P 2 ; P 2 1 + x~i 1 + x~i 1 + x~i For our second co-ordinate chart we project from the point (0; :::; 0; 1) (the \north pole"), i.e.