Tangent Bundle Definition. Let M be a differentiable manifold. The tangent bundle TM of the manifold M is the disjoint union of the tangent spaces TpM, p ∈ M,

TM = a TpM, p∈M

equipped with a natural projection map

π : TM → M with π(w)=p for w ∈ TpM,

which sends each vector in TpM to the “base point” p.

Lemma 4.1. For any smooth n-manifold M, the tangent bundle TM has a natural topology and smooth structure that make it into s 2n-dimensional smooth manifold. d • If ϕ: U → R is a chart for M, we let TU be the disjoint union of the TpM with p ∈ U and define the chart

dϕ : TU → Tϕ(U)(≡ [ Tϕ(p)ϕ(U)), ϕ(p)∈ϕ(U)

where Tϕ(U) carries the differentiable structure of ϕ(U) × Rd

w → dϕ(π(w))(w) ∈ Tϕ(π(w))ϕ(U).

• The transition maps dψ ◦ (dϕ)−1 then are differentiable. • π is locally represented as ϕ ◦ π ◦ dϕ−1

and this map maps (x0,v) to x0. With this structure, π : TM → M is a smooth map. Proof. (I) We begin by defining the maps that will become our smooth charts. Given any smooth chart (U, ϕ) for M, let (x1, ··· ,xn) denote the coordinate functions of ϕ, and define a map ϕ : π−1(U) → R2n by e

 i ∂  1 n 1 n ϕ v =(x (p), ··· ,x (p),v , ··· ,v ). ∂xi e p — Its image set is ϕ(U) × Rn, which is an open subset of Rn. — It is a bijection on its image, because its inverse can be written explicitly as

∂ −1 1 ··· n 1 ··· n i ϕ (x (p), ,x (p),v , ,v )=v i . ∂x − e ϕ 1(x) (II) Now suppose we are given two smooth charts (U, ϕ) and (V,ψ) for M, and let (π−1(U), ϕ), (π−1(V ), ψ) be the corresponding charts on TM. e e Typeset by AMS-TEX

1 2

— The sets

ϕ(π−1(U) ∩ π−1(V )) =ϕ(U ∩ V ) × Rn e ψ(π−1(U) ∩ π−1(V )) =ψ(U ∩ V ) × Rn e are both open in R2n, and the transitition map

ψ ◦ ϕ−1 : ϕ(U ∩ V ) × Rn → ψ(U ∩ V ) × Rn e e can be written explicitly using (3.9) as

1 n −1 1 n 1 n 1 n ∂x j ∂x j ψ ◦ ϕ (x , ··· ,x ,v , ··· ,v )=(x , ··· , x , e (x)v , ··· , e (x)v ) e e e e ∂xj ∂xj This is clearly smooth. (III) Choosing a countable cover {Ui} of M by smooth coordinate domains, we obtain −1 a countable cover of TM by coordinate domains {π (Ui)} satisfying conditions −1 2n (i) ∀i, ϕ(π (Ui)) is an open subset of R . −1 −1 −1 −1 2n (ii)∀i, je, ϕi(π (Ui) ∩ π (Uj )) and ϕi(π (Ui) ∩ π (Uj )) are open in R . −1 −1 (iii) Whenevere π (Ui) ∩ π (Uj ) =6 ∅e,

−1 −1 −1 −1 −1 ϕi ◦ ϕj : ϕj (π (Ui) ∩ π (Uj )) → ϕi(π (Ui) ∩ π (Uj )) e e e e is a diffeomorphism. — To check the Hausdorff condition, note that any two points in the same fiber of π lie in one chart, while if (p, X) and (q, Y ) lie in different fibers, there exist disjoint smooth coor- dinate domains U, V for M such that p ∈ U and q ∈ V , and then the sets in π−1(U) and π−1(V ) are disjoint smooth coordinate neigh- borhoods containing (p, X) and (q, Y ), respectively. (IV) To check that π is smooth, just note that its coordinate representation w.r.t. charts (U, ϕ) for M and (π−1(U), ϕ) for TM is π(x, v)=x.  e • Given any smooth chart (U, ϕ) for M with coordinate functions (xi), define a map Φ : π−1(U) → U × Rn by

 i ∂  1 n Φ v =(p, (v , ··· ,v )). ∂xi p The composite map

ϕ×Id n π−1(U) −−−−→Φ U × Rn −− − − −→R ϕ(U) × Rn

ie equal to the coordinate map ϕ constructed above. e 3

Another Proof. • For each chart (U, ϕ)ofM. define a “chart” (bijective, no regularity) (π−1(U), Φ) for TM by ◦ −1 Φ(ξ)=(ϕ π(ξ),θU,ϕ,π(ξ)(ξ)) • We now equip TM with a topology such that all the Φ are homeomorphisms, deciding that a fundamental system of nbhds of ξ ∈ π−1(U) is given by the inverse iamges by Φ of the nbhds of Φ(ξ)inϕ(U) × R2n. — One must check that (1) this topology is well-defined; and (2) the topology is Hausdorff. TM has now a structure of topological C0 manifold. • We now turn to the transition functions. —If(U, ϕ) and (V,ψ) are two charts for M around m, the transition function for the associated charts (π−1(U), Φ) and (π−1(V ), Ψ) is

Ψ ◦ Φ−1 = ϕ(U ∩ V ) × Rn → ψ(U ∩ V ) × Rn

with −1 −1 −1 Ψ ◦ Φ (p, u)=(ψ ◦ ϕ (p),Dp(ψ ◦ ϕ ) · u); it is Cp−1. 

• The coordinates (xi,vi) defined in this lemma will be called standard coordi- nates for TM. 4

Vector Fields on Manifolds Definition. If M is a smooth manifold, a vector field on M is a section of the map π : TM → M. • More concretely, a vector field is a continuously map Y : M → TM, usually written p 7→ Yp, with the property that

(4.1) π ◦ Y = IdM , or equivalently, Yp ∈ TpM, ∀p ∈ M.

• We will be primarily interested in smooth vector fields, the ones that are smooth as maps from M to TM. • In addition, for some purposes it is useful to consider maps from M to TM that would be vector fields except that they might not be continuous. Definition. A rough vector field on M is a (not necessarily continuous) map Y : M → TM satisfying (4.1).

• If Y : M → TM is a rough vector field and (U, (xi)) is any smooth coordinate chart for M, we can write the value of Y at any point p ∈ U in terms of the coordinate basis vectors:

i ∂ (4.2) Yp = Y (p) . ∂xi p This defines n functions Y i : U → R, called the component functions of Y in the given chart. Lemma 4.2 (Smooth Criterion for Vector fields). Let M be a smooth man- ifold, and let Y : M → TM be a rough vector field. If (U, (xi)) is any smooth coordinate chart on M, then Y is smooth on U iff its component functions w.r.t. this chart are smooth. Proof. Let (xi,vi) be the standard coordinates on π−1(U) ⊂ TM associated with the chart (U, (xi)). By definition of standard coordinates, the coordinate represen- tation of Y : M → TM on U is Y (x)=(x1, ··· ,xn,Y1(x), ··· ,Yn(x)), b where Y i is the ith component function of Y in xi-coordinates. It follows imme- diately that smoothness of Y in U is equivalent to smoothness of the component functions. 

Example 4.3. If (U, (xi)) is any smooth chart on M, the assignment ∂ p 7→ ∂xi p determines a smooth vector field on U, called the ith coordinate vector field ∂ and denoted by ∂xi . (It is smooth because its component functions are constants). 5

Lemma 4.5. Let M be a smooth manifold. If p ∈ M and X ∈ TpM, there is a smooth vector field X on M such that X = X. e ep Proof. Let (xi) be smooth coordinates on a nbhd U of p, and let

∂ X = Xi ∂xi p be the coordinate expression for X.Ifψ is a smooth bump function supported in U and with ψ(p) = 1, the vector field X defined by e

 i ∂ ψ(q)X i ,q∈ U, X =  ∂x eq q  0,q/∈ supp ψ, is easily seen to be a smooth vector field with X = X.  ep Definition. The support of a vector field Y is defined to be the closure of the set {p ∈ M : Yp =06 }. • A vector field is said to be compactly supported if its support is a compact set.

• If U is any open subset of M, the fact that TpU is naturally identified with TpM for each p ∈ U allows us to identufy TU with the subset π−1(U) ⊂ TM. — Therefore, a vector field on U can be thought of either as a map from U to TU or as a map from U to TM, whichever is more convenient.

—IfY is a vector field on M, its restriction Y is a vector field on U, which is U smooth if Y is. Definition. Let T (M) be the set of all smooth vactor field on M. •T(M) is a vector space under pointwise addition and scalar multiplication:

(aY + bZ)p = aYp + bZp.

– The zero element of this vector space is the zero vector field, whose value at each p ∈ M is 0 ∈ TpM. • In addition, smooth vector fields can be multiplied by smooth real-valued func- tions: If f ∈ C∞(M) and Y ∈T(M), we define fY : M → TM by

(fY)p = f(p)Yp.

For example, the basis expression (4.2) for a vector field Y can also be written as an equation between vector fields instead of an equation between vectors at a point: ∂ Y = Y i , ∂xi where Y i is the ith component function of Y in the given coordinates. 6

• An essential property of vector fields is that they define operators on the space of smooth real-valued functions. • If Y ∈T(M) and f is a snooth real-valued function on an open set U ⊂ M,we obtain a new function Yf : U → R, defined by

Yf(p)=Ypf. — Because the action of a tangent vector on a function is determined by the values of the function in an arbitrarily small nbhd, it follows that Yf is locally determined.

– In particular, for any open set V ⊂ U,(Yf) = Y (f ). V V • This way of viewing vector fields yields another useful criterion for a vector field to be smooth. Lemma 4.6. Let M be a smooth manifold, and let Y : M → TM be a rough vector field. Then Y is smooth iff for every open set U ⊂ M and every f ∈ C∞(U), the function Yf : U → R is smooth. Proof. (⇐) Suppose Y is a rough vector field for which Yf is smooth whenever f is smooth. —If(xi) is any smooth local coordinates on U ⊂ M, we can think of each coordinate xi as a smooth function on U. – Applying Y to one of these functions, we obtain ∂ Yxi = Y i (xi)=Y i. ∂xj Because Yxi is smooth by assumption, it follows that the components functions of Y are smooth, so Y is smooth. (⇒) Suppose Y an smooth and let f be a smooth real-valued function defined in an open set U ⊂ M. — ∀p ∈ U, we can choose smooth coordinates (xi) on a nbhd W ⊂ U of p. – Then for x ∈ W , we can write ∂ ∂f Yf(x)=Y i(x)
f = Y i(x) (x). ∂xi ∂xi x – Since the component functions Y i are smooth on W by Lemma 4.2, it follows that Tf is smooth on W . — Since the same is true in a nbhd of each point of U, Yf is smooth on U. 

• One consequence of the preceeding lemma is that a smooth vector field Y ∈ T (M) defines a map from C∞(M) to itself by f 7→ Yf. — This map is clearly linear over R. — Moreover, the product rule (3.4) for tangent vectors translate into the following product rule for vector fields (4.3) Y (fg)=fYg + gY f, as you can easily check by evaluating both sides at an arbitrary point p ∈ M. 7

Definition. A map Y : C∞(M) → C∞(M) is called a derivation if it is linear over R and satisfies (4.3) for all f, g ∈ C∞(M).

• The next propositipn shows that derivations of C∞(M) can be identified with smooth vector fields. Proposition 4.7. Let M be a smooth manifold. A map

Y : C∞(M) → C∞(M)

is a deviation iff it is of the form Yf = Yffor some smooth vector field Y ∈T(M). Proof. (⇐)We just showed that every smooth vector field induces a derivation. (⇒) Suppose Y : C∞(M) → C∞(M) is a derivation. We need to concoct a vector field Y such that Yf = Yf for all f. From the discussion above, it is clear that if there is such a vector field, its value at p ∈ M must be the derivation at p whose action on any smooth real-valued function f is given by Ypf =(Yf)(p). – The linearity of Y guarantees that this expression depends linearly on f. – Evaluating (4.3) at p yields the product rule (3.4) for tangent vectors. ∞ Thus the map Yp : C (M) → R so defined is indeed a tangent vecor, i.e. a derivation of C∞(M)atp. To show that the assignment p 7→ Yp is a smooth vector field, we use Lemma 4.6. —Iff ∈ C∞(M)isaglobally defined smooth function, then Yf = Yf is certainly smooth; we need to show that the same thing holds for a smooth function defined only on an open subset of M. — Suppose therefore that U ⊂ M is open and f ∈ C∞(U). – ∀p ∈ U, let ψ be a smooth bump function that is equal to 1 in a nbhd of p and supported in U, and define f = ψf, e extended to be zero on M \ supp ψ. – Then Y f = Yf is smooth, e e and Yf = Y f in a nbhd of p by Proposition 3.6. e — This shows that Yf is smooth in a nbhd of each point of U. 

• Because of this result, we will sometimes identify smooth vector fields on M with derivations of C∞(M), using the same letter for both (1) the vector field (thought of as a smooth map from M to TM), and (2) the derivation (thought of as a linear map from C∞(M) to itself). 8

Pushforwards of Vector Fields • If F : M → N is a smooth map and Y is a vector field on M, then for each point p ∈ M, we obtain a vector F∗Yp ∈ TF (p)N by pushing forward Yp. • However, this does not in general define a vector field on N. — For example, if F is not surjective, there is no way to assign to a point q ∈ N \ F (M). —IfF is not injective, then for some points of N there may be several different vectors obtained as pushforwards of Y from different points of M. Definition. Let F : M → N be smooth and Y be a vector field on M. If a vector field Z on N has the property that for each p ∈ M, F∗Yp = ZF (p), then the vector fields Y and Z are said to be F -related.

• Here is a useful criterion for checking that two vector fields are F -related. Lemma 4.8. Suppose F : M → N is a smooth map, Y ∈T(M), and Z ∈T(N). Then Y and Z are F -related iff every smooth real-valued function f defined on an open subset of N,

(4.4) Y (f ◦ F )=(Zf) ◦ F.

Proof. Fro every p ∈ M and any smooth real-valued f defined near F (p),

Y (f ◦ F )(p)=Yp(f ◦ F )=(F∗Yp)f, while (Zf) ◦ F (p)=(Zf)(F (p)) = ZF (p)f.

Thus (4.4)is true for all f ⇔ F∗Yp = ZF (p) for all p, i.e. Y and Z are F -related.

Example 4.9. Let F : R → R2 be the smooth map F (t) = (cos t, sin t). Then d ∈T ∈T 2 dt (R)isF -related to the vector field Z (R ) defined by

∂ ∂ Z = x − y . ∂y ∂x

• It is important to remember that for a given smooth map F : M → N and vector field Y ∈T(M), there may not be any vector field on N that is F -related to Y . • There is one special case, however, in which there is always such a vector field, as the next proposition shows. Proposition 4.10. Suppose F : M → N is a diffeomorphism. For every Y ∈ T (M), there is a unique smooth vector field on N that is F -related to Y . Proof. For Z ∈T(N)tobeF -related to Y means that

F∗Yp = ZF (p) ∀p ∈ M. 9

If F is a deffeomorphism, we define Z by

Zq = F∗(YF −1(q)).

— It is clear that Z is the unique (rough) vector field that is F -related to Y . — To see that it is smooth, we just expand the definition in smooth local coordi- nates, using (3.6) for the pushforward:

j ∂F −1 i −1 ∂ Zq = (F (q))Y (F (q)) . ∂xi ∂yj q The component function of Z are smooth by composition. 

Definition. In the situation of the preceeding lemma, we denote the unique vector field that is F -related to Y by F∗Y , and call it the pushforward of Y by F .

• Remember, it is only when F is a diffeomorphism that F∗Y is defined.