The Theory of Manifolds Lecture 4 a Vector Field on an Open Subset, U, of R N Is a Function, V, Which Assigns to Each Point, P

The theory of manifolds Lecture 4

A vector field on an open subset, U, of Rn is a function, v, which assigns to each n point, p ∈ U, a vector, v(p) TpR . This definition makes perfectly good sense for manifolds as well. R Definition 1. Let X ⊆ RN be an n-dimensional manifold. A vector field on X is a function, v, which assigns to each point, p ∈ X, a vector, v(p) ∈ TpX.

N By deﬁnition, TpX is a vector subspace of TpR and since

n N TpR = {(p, v) , v ∈ R }

v(p) is an (n + 1)-tuple v(p) = (p, v1(p), . . . , vN (p)) . Let vi : X → R

be the function, p ∈ X → vi(p).

∞ ∞ Deﬁnition 2. We will say that v is a C vector ﬁeld if the vi’s are C functions.

Hence by Munkres, §16, exercise 3 we can ﬁnd a neighborhood, U, of X in RN ∞ and functions, wi ∈ C (U) such that wi = vi on X. Let w be the vector ﬁeld,

N ∂ w = w , (1) i ∂x i=1 i X on U. This vector ﬁeld has the property that for every p ∈ X, w(p) = v(p) and in particular, w(p) ∈ TpX . (2) A vector ﬁeld, w, on U with the property (2) for every p ∈ X is said to be tangent to X, and we can summarize what we’ve shown above as the assertion:

Theorem 1. If v is a C∞ vector ﬁeld on X, then there exists a neighborhood, U, of X in RN and a C∞ vector ﬁeld, w, on U with the properties

i. w is tangent to X.

ii. w|X = v.

1 Given a vector field, v, on X there are often some particularly nice choices for this extended vector field, w. For instance if X is compact then by choosing a bump ∞ function, ρ ∈ C0 (U) with ρ = 1 on a neighborhood of X, and replacing wi by ρwi one can arrange that w be compactly supported. More generally we will say that the vector field, v, is compactly supported if the set

supp v = {p ∈ X , v(p) =6 0}

is compact. Suppose this is the case. Then by choosing the bump function ρ above to be 1 on a neighborhood of supp v we can again arrange that the extension, w, of v to U be compactly supported. Another interesting choice of an extension is the following. Fix p ∈ X and let V be a neighborhood of p in RN and

fi(x) = 0 , i + 1, . . . , k (3)

an independent set of deﬁning equations for X. (See lecture 2. Here k has to be ∞ N − n and the fi’s have to be in C (V ).) Proposition 2. There exists a vector ﬁeld, w, on V such that v(q) = w(q) at points, q ∈ X ∩ V and Lwfi = 0 i = 1, . . . , k . (4)

Since the fi’s are independent the k × N matrix ∂f i (p) , i ≤ j ≤ k , 1 ≤ j ≤ N ∂x j is of rank k and hence some k × k minor of this matrix is of rank k. By permuting the xi’s if necessary we can assume that the k × k matrix ∂f i (q) , 1 ≤ i ≤ k , i ≤ j ≤ k (5) ∂x j is of rank k at p = q, and hence (shrinking V if necessary) is of rank k for all q ∈ V . Now let u be any vector ﬁeld on V extending v and let gi = Lufi. Since the matrix (5) is non-singular, the system of equations

k ∂f i (q)a (q) = g (q) (6) ∂x j i j=1 j X ∞ is solvable for the aj’s in terms of the gi’s, and the solutions depend on C on q. Moreover, if q ∈ X

gi(q) = (Lufi)(q) = (dfi)q(u(q)) = (dfi)q(v(q))

2 and since fi = 0 on X and vi(q) ∈ TqX, the right hand side of this equation is zero. Thus the gi’s are zero on X and so, by (6), the ai’s are zero on X. Now let

k ∂ w = u − a . j ∂x j=1 j X

Since the ai’s are zero on X, w = u = v on X and by deﬁnition

∂fi Lwfi = Lufi − aj = gi − gi = 0 . ∂xj X Q.E.D. We will now show how to generalize to manifolds a number of vector ﬁeld results that we discussed in the Vector Field segment of these notes. To simplify the statement of these results we will from now on assume that X is a closed subset of RN .

1 Integral curves

Let I be an open interval. A C∞ map, γ : I → X is an integral curve of v if, for all t ∈ I and p = γ(t), dγ p , = v(p) . (7) dt We will show in a moment that the basic existence and uniqueness theorems for integral curves that we proved in Vector Fields, Lecture 1, are true as well for vector ﬁelds on manifolds. First, however, an important observation.

Theorem 3. Let U be a neighborhood of X in RN and u a vector ﬁeld on U extending v. Then if γ : I → U is an integral curve of u and γ(t0) = p0 ∈ X for some t0 ∈ I, γ is an integral curve of v.

Proof. We will first of all show that γ(I) ⊆ X. To verify this we note that the set {t ∈ I , γ(t) ∈ X} is a closed subset of X, so to show that it’s equal to I, it suffices to show that it’s open. Suppose as above that t0 is in this set and γ(t0) = p0. Let N V be a neighborhood of p0 in R and (3) a system of defining equations for X ∩ V . By Proposition 2 there exists a vertex field, w, on V extending v and satisfying the equations Lwfi = 0 , i = 1, . . . , k .

However these equations say that the fi’s are integrals of the vector ﬁeld w, so if

γ˜(t) , t0 − < t < t0 ,

3 is an integral curve of w with γ˜(t0) = p0 ∈ X then equations (3) are deﬁning equations for X ∩ V , so this tells us that γ˜(t) ∈ X for t0 − < t < t0 + , and, in particular, if γ˜(t) = p, dγ˜(t) p, w(p) = v(p) = u(p) , dt so γ˜ is an integral curve of u and hence since γ˜(t0) = p0 = γ(t0), it has to coincide with γ on the integral − + t0 < t < + t0. This shows that the set of points

{t ∈ I , γ(t) ∈ X}

is open and hence is equal to I. Finally, to conclude the proof of Theorem 3 we note that since γ(I) ⊆ X, it follows that for any point t ∈ I and p = γ(t)

dγ p, = w(p) = v(p) dt since v = w on X and p ∈ X. Hence γ is an integral curve of v as claimed.

From the existence and uniqueness theorems for integral curves of w we obtain similar theorems for v.

Theorem 4 (Existence). Given a point, p0 ∈ X and a ∈ R there exists an interval, I = (a − T, a + T ), a neighborhood, U0, of p0 in X and, for every p ∈ U, an integral curve, γp : I → X with γp(q) = p.

Theorem 5 (Uniqueness). Let γi : Ii → X i = 1, 2 be integral curves of v. Suppose that for some a ∈ I, γ1(a) = γ2(a). Then γ1 = γ2 on I1 ∩ I2 and the curve, γ : I1 ∪ I2 → X deﬁned by γ1(t), t ∈ I1 γ(t) = (γ2(t), t ∈ I2 is an integral curve of v.

2 One parameter groups of diﬀeomorphisms

Suppose that v is compactly supported. Then, as we pointed out above, we can ﬁnd a neighborhood, U, of X and a vector ﬁeld, w, on U which is also compactly supported and extends v. Since w is compactly supported, it is complete, so for every p ∈ U there exists an integral curve

γp(t) , −∞ < t < ∞ (8)

4 with γp(0) = 0, and the maps

gt : U → U , gt(p) = γp(t)

are a one-parameter group of diﬀeomorphisms. In particular if p ∈ X the curve (8) is, as we’ve just shown, an integral curve of v and hence gt(p) = γp(t) ∈ X i.e.,

def ft = gt|X

∞ −1 −1 is a C map of X into X. Moreover, gt = g−t, so ft = f−t, so the ft’s are a one- parameter group of diﬀeomorphisms of X with the deﬁning property: For all p ∈ X the curve γp(t) = ft(p) , −∞ < t < ∞

is the unique integral curve of v with γp(σ) = p. If X itself is compact then v is automatically compactly supported so we’ve proved Theorem 6. If X is compact, every vector ﬁeld, v, is complete and generates a one-parameter group of diﬀeomorphisms, ft.

3 Lie diﬀerentiation

Let ϕ : X → R be a C∞ function. As in Vector Fields, Lecture 1, we will deﬁne the Lie derivative ∞ Lvϕ ∈ C (X) (9) by the recipe Lvϕ = dϕp(v(p)) (10) at p ∈ X. (If U is an open neighborhood of X in RN and w and ψ extensions of v and ϕ to U then Lvϕ is the restriction of Lwψ to Xj, so this shows that (9) is indeed in ∞ C (X).) If Lvϕ = 0 then ϕ is an integral of v and is constant along integral curves, γ(t), of v as is clear from the identity d ϕ(γ(t)) = L ϕ(γ(t)) = 0 . (11) dt v The following we’ll leave as an exercise. Suppose v is complete and generates a one-parameter group of diﬀeomorphisms

ft : X → X , −∞ < t < ∞ then d ∗ L ϕ = f ϕ| =0 (12) v dt t t ∗ where ft ϕ = ϕ ◦ ft.

5 4 f-relatedness

Let X and Y be manifolds and f : X → Y a C∞ map. Given vector ﬁelds, v and w, on X and Y , we’ll say they are f-related if, for every p ∈ X and q = f(p)

dfp(v(p)) = w(q) . (13)

For “f-relatedness” on manifolds one has analogues of the theorems we proved earlier for open subsets of Euclidean space. In particular one has

Theorem 7. If γ : I → X is an integral curve of v, f ◦ γ : I → Y is an integral curve of w.

Theorem 8. If v and w are generators of one-parameter groups of diﬀeomorphisms

ft : X → Y and

gt : Y → Y

then gt ◦ f = f ◦ ft.

Theorem 9. If ϕ ∈ C∞(Y ) and f ∗ϕ ∈ C∞(X) is the function ϕ ◦ f,

∗ ∗ Dvf ϕ = f Dwϕ .

The proofs of these three results are, more or less verbatim, the same as before, and we’ll leave them as exercises. If X and Y are manifolds of the same dimension and f : X → Y is a diffeomor- phism then, given a vector field, v, on X one can define, as we did before, a unique f-related vector field, w, on Y by the formula

w(q) = dfp(v(p)) (14)

−1 where p = f (q). We’ll denote this vector field by f∗v and, as before, call it the push-forward of v by f. We’ll leave for you to show that if v is a C ∞ vector field the vector field defined by (14) is as well. Our final exercise: Let Xi, i = 1, 2, 3 be manifolds, vi a vector field on Xi and ∞ f : X1 → X2 and g : X2 → X3 C maps. Prove

Theorem 10. If v1 and v2 are f-related and v2 and v3 are g-related, then v1 and v3 are g ◦ f-related.

Hint: Chain rule.

6 Problem Set

3 2 2 1. Let X ⊆ R be the paraboloid, x3 = x1 + x2 and let w be the vector ﬁeld

∂ ∂ ∂ w = x1 + x2 + ∂x3 . ∂x1 ∂x2 ∂x3

(a) Show that w is tangent to X and hence deﬁnes by restriction a vector ﬁeld, v, on X. (b) What are the integral curves of v?

2 2 2 2 3 2. Let S be the unit 2-sphere, x1 + x2 + x3 = 1, in R and let w be the vector ﬁeld ∂ ∂ w = x1 − x2 . ∂x2 ∂x1

(a) Show that w is tangent to S2, and hence by restriction deﬁnes a vector ﬁeld, v, on S2. (b) What are the integral curves of v?

3. As in problem 2 let S2 be the unit 2-sphere in R3 and let w be the vector ﬁeld

∂ ∂ ∂ ∂ w = − x3 x1 + x2 + x3 ∂x3 ∂x1 ∂x2 ∂x3 (a) Show that w is tangent to S2 and hence by restriction deﬁnes a vector ﬁeld, v, on S2. (b) What do its integral curves look like?

1 2 2 2 1 1 4 4. Let S be the unit sphere, x1 + x2 = 1, in R and let X = S × S in R with deﬁning equations

2 2 f1 = x1 + x2 − 1 = 0

2 2 f2 = x3 + x4 − 1 = 0 .

(a) Show that the vector ﬁeld

∂ ∂ ∂ ∂ w = x1 − x2 + λ x4 − x3 , ∂x2 ∂x1 ∂x3 ∂x4 λ ∈ R, is tangent to X and hence deﬁnes by restriction a vector ﬁeld, v, on X.

7 (b) What are the integral curves of v?

5. For the vector ﬁeld, v, in problem 4, describe the one-parameter group of dif- feomorphisms it generates.

2 6. Let X and v be as in problem 1 and let f : R → X be the map, f(x1, x2) = 2 2 (x1, x2, x1 + x2). Show that if u is the vector ﬁeld,

∂ ∂ u = x1 + x2 , ∂x1 ∂x2

then f∗u = v.

7. Verify (11).

8. Let X be a submanifold of X in RN and let v and w be the vector ﬁelds on X and U. Denoting by ι the inclusion map of X into U, show that v and w are ι-related if and only if w is tangent to X and its restriction to X is v.

9. Verify that the vector ﬁeld (14) is C∞.

10.* An elementary result in number theory asserts

Theorem. A number, λ ∈ R, is irrational if and only if the set

{m + λn , m and n intgers}

is a dense subset of R.

Let v be the vector ﬁeld in problem 4. Using the theorem above prove that if λ/2π is irrational then for every integral curve, γ(t), −∞ < t < ∞, of v the set of points on this curve is a dense subset of X.