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Gluing manifolds together September 23, 2015 1 Introduction I talked in class Friday about a possible example of an integral curve that is defined only for some t < b, and cannot be further extended even if the manifold is enlarged. I stated that the example failed to exhibit this behavior, but what I said was incomplete and probably unconvincing. Here is a (somewhat simplified) version of the example from class. The manifold and vector field X=− M = (−∞, 0), 1 d , x dx (1.1a) the open half line; the vector field was chosen to blow up at the “endpoint” zero. We calculated integral curves for this vector field: they look like q γx0 (t) = − x20 − 2t (x0 < 0); (1.1b) then γx0 (0) = x0 , γx0 : (−∞, x20 /2) → M. (1.1c) That is, this integral curve is defined exactly for t < x20 /2, and it maps that interval (diffeomorphically) onto M . The question I asked in class was whether one might be able to extend this integral curve by enlarging M ; for example, replacing it by M̃ = (−∞, δ), in which M sits as an open subset. The reason someone suggested in class that this should be impossible is that the derivative dγx0 1 =p 2 dt x0 − 2t 1 blows up as γx0 approaches zero, so there is no extension of γx0 to a smooth map from (−∞, x20 /2 + ) into (−∞, δ). Another reason it’s impossible is that the vector field X does not extend smoothly to (−∞, δ) for any positive δ. These are both correct reasons that M̃ = (−∞, δ) does not work. It’s also true that a connected one-dimensional noncompact manifold must be diffeomorphic to an open interval (a, b) Nevertheless, I stated in class that it is possible to enlarge M so that γx0 can be extended. You should think for yourself a bit about how this could be true. Like an email in which you have to scroll down to see the punch line, these notes will come to an explanation slowly. Section 2 describes a general way to enlarge a manifold; Section 3 describes the general process of following integral curves as far as possible; Section 4 combines the ideas of the preceding two sections to extend integral curves very generally; and Section 5 describes very explicitly (following what came before) how to extend the vector field and integral curves of (1.1). It should be possible to read that last section without reading any of the others; they are meant to explain where to find the formulas in the last section. 2 Glue Suppose M1 ⊃ U1 , M2 ⊃ U2 (2.1a) are n-dimensional manifolds with open subsets U1 and U2 . We want to enlarge M1 by pasting to it a copy of M2 , gluing the open set U1 to the open set U2 . In order to do this, we need φ12 : U1 → U2 diffeomorphism. (2.1b) Then the enlarged manifold as a set is N = (M1 ∪ M2 )/ ∼, m1 ∼ m2 ⇐⇒ mi ∈ Ui , m2 = φ12 (m1 ). (2.1c) That is, N = M 1 ∪ M2 , M1 ∩ M2 = U1 ' U2 . The topological structure is U ⊂ N is open ⇐⇒ U ∩ Mi is open in Mi . 2 (2.1d) This makes each Mi an open subset of N ; an open set is the union of one in M1 and one in M2 . We define smooth functions by C ∞ (N ) = {f : N → R | f |Mi ∈ C ∞ (Mi )}. (2.1e) That is, a smooth function f on N is specified by a pair of smooth functions fi on Mi , with the property that f2 |U2 ◦ φ12 = f1 |U1 . Proposition 2.2. In the setting (2.1), every point in the topological space N has a neighborhood diffeomorphic to an open set in Rn . A vector field X on N may be specified by a vector fields Xi on Mi , subject to the requirement (m1 ∈ U1 , m2 = φ12 (m1 )). X2 (m2 ) = dφ12 (m1 )(X1 (m1 )) A smooth map γ : (a, b) → N is an integral curve for X if and only if γi = γ|γ −1 (Mi ) is an integral curve for Xi . (Since Mi is open in M , γ −1 (Mi ) must be an open subset of (a, b), and therefore a countable union of intervals.) 3 Despite the assertions in the proposition, the space N may not be a manifold. Following curves In this section we will describe a general example of pasting a manifold to itself. Suppose M is a manifold with a vector field X. For each x ∈ M , we can therefore find a maximal integral curve for X starting at x: γx : (a(x), b(x)) → M, γx0 (t) = X(γx (t)), γx (0) = x. (3.1a) Here −∞ ≤ a(x) < 0, 0 < b(x) ≤ +∞. For any > 0, we can define V≤ = {x ∈ M | b(x) > } = {x | γx () is defined}. 3 (3.1b) The proof of the existence theorem for integral curves makes clear that V≤ is open in M . For formal reasons [ V = M. (3.1c) 0 < =⇒ V ⊂ V0 , >0 Similarly, we define U≥− = {x ∈ M | a(x) < −} = {x | γx (−) is defined}. (3.1d) The proof of the existence theorem for integral curves makes clear that U≥− is open in M . For formal reasons [ 0 < =⇒ U≥− ⊂ U≥−0 , U≥− = M. (3.1e) >0 It is notationally reasonable to define also U≥0 = M = V≤0 . Example 3.2. In the example described in (1.1), we have M = (−∞, 0), Therefore a(x) = −∞, √ V≤ = (−∞, − 2), b(x) = x2 /2. U≥− = (−∞, 0). We would like to define a diffeomorphism Γ from M to itself by following an integral curve of X for time . The difficulty is that, since integral curves need not be defined for all t, the diffeomorphism Γ is not defined on all of M . Here are the details. Definition 3.3. The one-parameter group of diffeomorphisms attached to the vector field X consists of the following diffeomorphism: for = 0, Γ0 : M → M, Γ0 (x) = γx (0) = x (x ∈ M ); for > 0, Γ : V≤ → U≥− , Γ (x) = γx (); and finally Γ− : U≥− → V≤ , Γ− (x) = γx (−). 4 Here is why the terminology “one parameter group” makes sense. If 0 is another strictly positive number, then it is easy to see that Γ0 : V≤+0 → V≤ ∩ U≥−0 , (3.4a) Γ : V≤ ∩ U≥−0 → U≥−(+0 ) , (3.4b) Γ ◦ Γ0 = Γ+0 . (3.4c) and that (The preceding two displays show that the two sides have matching domains and ranges). Similarly Γ− ◦ Γ−0 = Γ−−0 . (3.4d) The situation for composing a positive with a negative translation is slightly more subtle. Suppose 0 > ; we want to compute Γ ◦ Γ−0 , and the answer we would like to get is Γ−0 . This cannot work out precisely, because the domain of Γ−0 is the (large) open set U≥−0 + , and the domain of Γ−0 is the (smaller) open set U≥−0 . So we must first restrict to this smaller open set. Then we find Γ−0 : U≥−0 → V≤0 , (3.4e) Γ : V≤0 → V≤(−+0 ) ∩ U≥− , (3.4f) Γ ◦ Γ−0 = Γ−0 |U≥−0 ; (3.4g) and that each side is a diffeomorphism U≥−0 → V≤(−+0 ) ∩ U≥− . In the same way we find Γ0 ◦ Γ− = Γ−+0 |U≥− ∩V≤(−+0 ) ; (3.4h) each side is a diffeomorphism U≥− ∩ V≤(−+0 ) → U≥−0 . What these relations say is that Γs0 ◦ Γs = Γs0 +s (s, s0 ∈ R) (3.4i) with the provision that the relations must be restricted to a small enough open set for all relevant Γ to be defined. These open sets may be empty, but as s and s0 approach 0, they grow toward all of M . I’ll conclude this section by mentioning an important fact relating the diffeomorphisms Γ to the vector field X. 5 Definition 3.5. Suppose M and N are manifolds with vector fields X and Y , and that φ : M → N is a smooth map. We say that X and Y are φ-related if for every m ∈ M , Y (φ(m)) = dφ(m)(X(m)). This is an important notion, but (unlike most important notions) there is practically nothing “functorial” about it: if we start with X on M , there may be no vector field on M that is φ-related to X, or there may be infinitely many. The same is true if we start with Y and try to find X. Proposition 3.6. In the setting of (3.1), the vector field X (restricted to the open set V≤ ) is Γ -related to X (restricted to the open set U≥− ). 4 Extending curves We now combine what we have learned in Section 3 about the one-parameter group of diffeomorphisms with the construction of Section 2. We begin with M and X, and a fixed > 0. Define M1 = M2 = M, X1 = X2 = X. (4.1a) For open sets and a diffeomorphism between them, we take U1 = U≥− ⊂ M = M1 , U2 = U≤ ⊂ M = M2 (4.1b) φ12 = Γ− : U1 → U2 . (4.1c) N = (M1 ∪ M2 )/ ∼ (4.1d) Now construct as in (2.1c). According to Proposition 2.2, we can get a vector field X on N by piecing together X on M1 = M and X on M2 = M ; the compatibility condition required is true because of Proposition 3.6. An integral curve of X on N is obtained in one of three ways: 1. as an integral curve of X in M1 = M γ1 : (a, b) → M1 = M with b − a ≤ ; or 2. as an integral curve of X in M2 = M γ2 : (a, b) → M2 = M with b − a ≤ ; or 6 3. as the union of an integral curve γ1 : (a, b) → M1 = M with b − a > , and the integral curve γ2 : (a + , b + ) → M2 = M, γ2 (t) = γ1 (t − ). The reason is that the condition in the first item is precisely the condition for the curve γ1 not to meet U1 ; and the condition in the second item is precisely the condition for the curve γ2 not to meet U2 . Proposition 4.2. In the setting (4.1), suppose that γ : (a, b) → N is a maximal integral curve of X in N . Then either 1. b − a ≤ , and γ takes values either in M1 or in M2 ; or 2. b − a > 2, and γ1 : (a, b − ) → M1 is a maximal integral curve in M1 = M . That is, every maximal integral curve in M1 = M (defined on an interval of length at least and having a finite right endpoint) can be strictly extended (in the larger manifold N ) by units to the right. If we apply the proposition to the example in the introduction, we see that every integral curve γx0 (defined on (−∞, x20 /2) as a curve in M ) can be extended to (−∞, x20 /2 + ) as a curve in N . 5 So how does that look exactly? Here is a very concrete description of how to extend the example in the introduction. As a topological space, the manifold is √ N = (−∞, 2) ⊃ (−∞, 0) = M. (5.1a) I just need to tell you smooth charts. At any n 6= 0, you choose a little neighborhood Un of n, not containing zero, and use the identity map to 7 identify Un with the open set Un in R. What’s interesting is n = 0. There I will choose the neighborhood √ √ U0 = (− 2, 2), (5.1b) √ and identify it with the open set (−2 , 0) in R using the chart ( √ √ − 2 + n2 (− 2 < n ≤ 0) √ √ φ0 (n) = − 2 − n2 (0 ≤ n < 2) (5.1c) Then√you √ can compute (by differentiating φ0 ) that a smooth vector field Φ on (− 2, 2) is one of the form ( p p √ ∂ (− 2 < n < 0), f (−n 1 + 2/n2 ) 2/n2 + 1 ∂n p p √ (5.1d) Φ(n) = ∂ (0 < n < 2), f (−n 2/n2 − 1) 2/n2 − 1 ∂n √ for some function f ∈ C ∞ (−2 , 0). Taking for f the smooth function −1/x, we find a smooth vector field ( − 1 ∂ (n < 0), Φ(n) = 1 n∂ ∂n (5.1e) (n > 0) n ∂n (The apparent “singularity” at n = 0 is because the coordinate n is not a smooth coordinate at n = 0.) A (non-maximal) integral curve for this vector field is ( √ − −2t (− < t ≤ 0), γ(t) = √ (5.1f) 2t (0 ≤ t < ) Again the apparent lack of smoothness at t = 0 is because the coordinate n is not smooth; if we compose with the smooth coordinate chart φ0 at zero, we get √ (− < t < ), (5.1g) φ0 ◦ γ(t) = − 2 − 2t now a perfectly smooth function of t. 8