CHAPTER 1: SMOOTH MANIFOLDS DAVID GLICKENSTEIN 1. Introduction This semester we will focus primarily on the basics of smooth manifold theory. We will spend some time on what a manifold is and what its properties are. This involves a discussion of the di¤erential theory of manifolds, including vector …elds and tensor …elds. We will then discuss a bit about integration on manifolds, speci…- cally the integration of di¤erential forms. Finally, the relationship between the two is given by Stokes’Theorem. Next semester we will tackle algebraic topology and its relation to di¤erential forms and the work from this semester. Reminder: the qualifying exam will also cover the basics of complex analysis. This will not be covered in this class, though I hope to give some questions about it throughout the semester. You are expected to have a good grounding in multivariable calculus and point- set topology already. You may want to review a bit. 2. Topological manifolds We de…ne a topological manifold as follows: De…nition 1. A topological space M is a manifold of dimension n if (1) M is Hausdor¤, and (2) M is second countable, and (3) M is locally Euclidean of dimension n. This de…nition depends on the following de…nitions: De…nition 2. A topological space X is Hausdor¤ if for all x; y X such that x = y; there exist open sets U; V such that x U; y V and U V 2= : 6 2 2 \ ; De…nition 3. A topological space X is second countable if it has a countable basis for the topology, i.e., there exists a countable collection of open sets U such N f g 2 that for any open set U X containing a point x; there exists a N such that 2 x U U: 2 De…nition 4. A topological space X is locally Euclidean of dimension n if for each x X; there exists an open set U X containing X and a map : U Rn such that2 : U (U) is a homeomorphism (in particular, (U) is an open! subset of ! Rn). The …rst two conditions are essentially technical conditions, with the third con- dition giving the main condition on being a manifold. Date: September 2, 2010. 1 2 DAVIDGLICKENSTEIN Remark 1. We sometimes use the terminology M n is a manifold to mean that M is a manifold of dimension n: We also say M is an n-dimensional manifold. The set and map in the de…nition of locally Euclidean is called a coordinate chart. De…nition 5. A coordinate chart on M is a pair (U; ) where U M is open and : U (U) Rn is a homeomorphism. The set U is called a coordinate domain ! or coordinate neighborhood or coordinate patch. If (U) is a ball in Rn;U is called a coordinate ball. A coordinate chart (U; ) is centered at p if (p) = 0: Theorem 6. If M is a manifold, every point x M is contained in a coordinate ball centered at x. 2 Proof. Since M is locally Euclidean, x must be contained in a coordinate chart (U; ) : Since (U) is an open set containing (x) ; by the topology of Rn there must be an open ball B containing (x) and contained in (U) : The appropriate 1 coordinate ball is (B) ; 1 : If we compose with a translation taking j (B) (x) to 0; we have completed the proof. Example 1. The graph of a continuous function f : U Rk; where U Rn; is a manifold: ! n k (f) = (x; y) R R : x U and y = f (x) : 2 2 n+k (f) is given the subspace topology (of R ), so it is automatically Hausdor¤ and second countable. It has a single coordinate chart given by ( (f) ; 1) where 1 is projection onto the …rst coordinate. The inverse of 1 is the map x (x; f (x)) : This map is continuous if and only if f is continuous. ! Example 2. Spheres are manifolds. An n-sphere is de…ned as n n+1 2 S = x R : x = 1 : 2 j j n n o Since it is a subspace of R , it is Hausdor¤ and second countable. We can de…ne coordinate charts as follows + n U = x S : xk > 0 ; k f 2 g + 1 2 k 1 k k+1 n+1 k = x ; x ; : : : ; x ; x ; x ; : : : ; x ; k k n where x denotes that x is not there (so ck : Uk R ). The inverse of k is ! n c 1 n 1 2 k 1 j 2 k n k y ; : : : ; y = y ; y ; : : : ; y ; 1 (y ) ; y ; : : : ; y 0 v 1 u j=1 u X @ t A (since xk > 0). To get charts that cover the sphere, we also need the corresponding Uk; k charts. Remark 2. There are other important coordinate charts for spheres, notably stere- ographic projection. See problems. n Example 3. Real projective space RP is a manifold. We de…ne real projective n space RP to be the set of lines in Rn+1: We can represent it more formally, by SMOOTHMANIFOLDS 3 writing a line in Rn+1 as an equivalence class of points v Rn+1 0 such that 2n n f g v v0 if and only if v = tv0 for some t0 R 0 : Then RP is the quotient 2 n f g n+1 R 0 = : n f g n We can then give RP the quotient topology (i.e., a set is open if and only if its pre-image under the quotient map is open). We write x0 : x1 : : xn for an element in the quotient. We can de…ne coordinate charts 0 1 i 1 i+1 n 0 1 n x x x x x i x : x : : x = ; ;:::; ; ;:::; ; xi xi xi xi xi where the domain is 0 1 n i Ui = x : x : : x : x = 0 : 6 We can see that this map is continuous by seeing that the quotient map composed n with this map is continuous. Notice that the image i (Ui) = R and its inverse is 1 1 n 1 i 1 i+1 n y ; : : : ; y = y : : y : 1 : y : : y : i (Check this: it is not totally obvious!) This map is also continuous. One can check that the space is Hausdor¤ and second countable. n n Remark 3. It is not too hard to show that RP is compact. You can rewrite RP n as a quotient of Sn; showing that RP is the continuous image of a compact space, and hence compact. Example 4. Products of manifolds are manifolds. If we have manifolds M m and N n; then we can give M N the product topology. If we have coordinate charts (U; ) for M and (V; ) for N; then there is a coordinate chart (U V; ) for M N: Given charts which cover M and N; we can construct charts which cover M N: By induction we can show that any …nite product of manifolds is a manifold as well. Example 5. The torus S1 S1 is a manifold. So are other tori S1 S1 S1. Finally, we give some topological properties of manifolds that may come in useful. Theorem 7. Every manifold has a countable basis of coordinate balls. Corollary 8. Every manifold is locally compact (i.e., every point has a neighbor- hood contained in a compact set). Remark 4. We will use neighborhood of a point to mean an open set containing that point. Some authors use neighborhood to mean any set which contains an open set containing that point. Thus, we will always assume that a neighborhood is open. De…nition 9. A topological space X is connected if there do not exist two disjoint, nonempty sets whose union is X: The space X is path connected if every two points are connected by a path (i.e., for any x; y X; there exists a continuous map : [0; 1] X such that (0) = x and (1) =2y). A topological space is locally path connected! if X has a basis of path connected sets (i.e., every point has a path connected neighborhood). Note that connected need not imply path connected, though the reverse impli- cation is true. In a manifold, however, they are equivalent. 4 DAVIDGLICKENSTEIN Theorem 10. A connected manifold is connected if and only if it is path connected. Furthermore, the components of a manifold are the same as its path components. Proof. Since path connected implies connected, we need only prove that connected implies path connected. Consider a manifold M n and let x M: Let S be the set of all points y in M for which there exists a path from x2to y: We will show that S is open and closed. Suppose y S: Then there is a coordinate chart (U; ) 2 around y (i.e., U is a neighborhood of y) and so (U) Rn: Since (U) is open, there exists a path from (y) to z for every z in a su¢ ciently small open ball 1 B around (y) :B is an open set, so (B) is open in M; and contained in S 1 (since we can extend the continuous path from x to y by ). Thus S is open. Now suppose y is a limit point of S: Then for every open neighborhood of y there is a point x0 S: Take a coordinate ball (U; ) centered at y: Then there exists 2 x0 S U: Since x0 S; there is a path from x to x0: Furthermore, since (U) is 2 \ 2 1 a ball, there is a path from (x0) to 0; so by juxtaposing with ; there is a path from x to y; thus y S and S is closed.