Notes on Smooth Manifolds

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Notes on Smooth Manifolds

Claudio Gorodski October 8, 2013

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Smooth manifolds

In order to motivate the definition of abstract smooth manifold, we first define submanifolds of Euclidean spaces. Recall from vector calculus and differential geometry the ideas of parametrizations and inverse images of regular values.

1.1 Submanifolds of Euclidean spaces

A smooth mapf :U →R^n+k, whereU ⊂Rⁿis open, is called animmersion atp, wherep ∈ U, ifdf_p : Rⁿ → R^n+k is injective. f is called simply an immersionif it is an immersion everywhere. An injective immersion will be called aparametrization.

A smooth map F : W → R^k, whereW ⊂ R^n+k is open, is called a submersion atp, wherep ∈W, ifdf_p :R^n+k → R^kis surjective. F is called simply asubmersionif it is a submersion everywhere. Forz₀ ∈R^k, ifF is a submersion along the level setF⁻¹(z₀), thenz₀is called aregular valueofF. Images of parametrizations and inverse images of regular values are thus candidates to be submanifolds of Euclidean spaces. Next we would like to explain why the second class is “better” than the first one. The argument involves the implicit function theorem, and how it is proved to be a consequence of the inverse function theorem.

Assume thenz₀ is a regular value of F as above. WriteM = F⁻¹(z₀) and considerp ∈ M. ThendF_p is surjective and, up to relabeling the coordinates, we may assume that (d₂F)_p, which is the restriction of dF_p to {0} ⊕R^k ⊂ R^n+k, is an isomorphism ontoR^k. Write p = (x₀, y₀) where x₀ ∈Rⁿ,y₀ ∈R^k. Define a smooth map

Φ :W →R^k, Φ(x, y) = (x, F(x, y)−z₀)

ThendΦ_(x0,y⁰)is easily seen to be an isomorphism, so the inverse function theorem implies that there exist open neighborhoodsU,V ofx₀,y₀inRⁿ, R^k, respectively, such thatΦis a diffeomorphism of U ×V onto an open

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subset of R^n+k, i.e. Φ is a smooth bijective map onto its image and the inverse map is also smooth. Now the fundamental fact is that

Φ(M∩(U×V)) = (Rⁿ× {0})∩Φ(U×V), as it follows from the form ofΦ; namely,Φ“rectifies”M.

Letϕ:M∩(U ×V) →Rⁿbe the restriction ofΦ. It also follows from the above calculation thatM∩(U ×V)is exactly the graph of the smooth mapf : U → V, satisfyingf(x₀) = y₀, given byf = projRk ◦ϕ⁻¹. Also, M∩(U×V)is the image of a parametrizationϕ⁻¹ :ϕ(M∩(U×V))→R^n+k which is a homeomorphism onto its image, where the latter is equipped with the topology induced fromR^n+k.

1.1.1 Definition (i) A subset M ⊂ R^n+k will be called a submanifold of dimension n of R^n+k if for every p ∈ M, there exists a diffeomorphism Φ from an open neighborhood U of p in R^n+k onto its image such that Φ(M ∩U) = (Rⁿ× {0})∩Φ(M). In this case we will say that(U,Φ) is a local chart ofR^n+kadapted toM.

(ii) A pair(U, f)whereU ⊂Rⁿis open andf :U →R^n+kis an injective immersion will be called animmersed submanifold of dimensionnofR^n+k. 1.1.2 Example Let(R, f)be the immersed submanifold of dimension1ofR², wheref :R → R² has imageM described in Figure??. ThenM is not a submanifold of dimension1ofR². In fact no connected neighborhood ofp can be homeomorphic to an interval ofRminus a point. Note thatf is not a homeomorphism onto its image.

1.1.3 Exercise Prove that the graph of a smooth map f : U → R^k, where U ⊂Rⁿis open, is a submanifold of dimensionnofR^n+k.

1.1.4 Exercise Letf,g: (0,2π) →R²be defined by

f(t) = (sint,sintcost), g(t) = (sint,−sintcost).

a. Check thatf,gare injective immersions with the same image.

b. Sketch a drawing of their image.

c. Write a formula forg◦f⁻¹ : (0,2π) →(0,2π).

d. Deduce that the identity map id : imf → img is not continuous, where imf, imgis equipped with the topology induced fromRvia f,g, respectively.

The algebraC^∞(M)of real smooth functions onM LetM be a submanifold ofR^n+k.

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1.2. DEFINITION OF ABSTRACT SMOOTH MANIFOLD 7 1.1.5 Definition A function f : M → R is said to be smooth at p ∈ M if f◦Φ⁻¹ : Φ(U)∩Rⁿ→Ris a smooth function for some adapted local chart (U,Φ)aroundp.

1.1.6 Remark (i) The condition is independent of the choice of adapted local chart aroundp. Indeed if(V,Φ)is another one,

f◦Φ⁻¹= (f◦Ψ⁻¹)◦(Ψ◦Φ⁻¹)

whereΨ◦Φ⁻¹ : Φ(U∩V)→Ψ(U ∩V)is a diffeomorphism and the claim follows from the the chain rule for smooth maps between Euclidean spaces.

(ii) A smooth function onM is automatically continuous.

1.2 Definition of abstract smooth manifold

LetM be a topological space. Alocal chart ofM is a pair(U, ϕ), whereU is an open subset ofM andϕis a homeomorphism fromU onto an open subset ofRⁿ. A local chartϕ:U →Rⁿintroduces coordinates(x₁, . . . , x_n) onU, namely, the component functions ofϕ, and that is why(U, ϕ)is also called asystem of local coordinatesonM.

A (topological) atlas for M is a family {(U_α, ϕ_α)} of local charts of M, where the dimension n of the Euclidean space is fixed, whose domains coverM, namely, ∪U_α = M. If M admits an atlas, we say that M is lo- cally modeled onRⁿandM is atopological manifold.

Asmooth atlasis an atlas whose local charts satisfy the additionalcom- patibility condition:

ϕ_β◦ϕ⁻_α¹ :ϕ_α(U_α∩U_β)→ϕ_β(U_α∩U_β)

is smooth, for allα,β. A smooth atlasAdefines a notion of smooth function onM as above, namely, a functionf : M → R issmooth iff ◦ϕ⁻¹ : ϕ(U)→Ris smooth for all(U, ϕ)∈ A. We say that two atlasA,BforMare equivalentif the local charts of one are compatible with those of the other, namely,ψ◦φ⁻¹ is smooth for all(U, ϕ) ∈ A,(V, ψ) ∈ B. In this case, it is obvious thatAandBdefine the same notion of smooth function onM.

Asmooth structure onM is an equivalence class [A]of smooth atlases onM. Finally, asmooth manifold is a topological space M equipped with a smooth structure [A]. In order to be able to do interesting analysis on M, we shall assume, as usual, thatthe topology ofM is Hausdorff and second countable.

1.2.1 Remark (a) It follows from general results in topology that (smooth) manifolds are metrizable. Indeed, manifols are locally Euclidean and thus locally compact. A locally compact Hausdorff space is (completely) regular, and the Urysohn metrization theorem states that a second countable regular space is metrizable.

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(b) The condition of second countability also rules out pathologies of the following kind. ConsiderR²with the topology with basis of open sets {(a, b)× {c} | a,b,c∈R,a < b}. This topology is Hausdorff but not second countable, and it is compatible with a structure of smooth manifold of dimension1(a continuum of real lines)!

1.2.2 Exercise LetMbe a topological space. Prove that two smooth atlases Aand B are equivalent if and only if their union A ∪ B is smooth atlas.

Deduce that every equivalence class of smooth atlases for M contains a unique representative which ismaximal(i.e. not properly contained in any other smnooth atlas in the same quivalence class).

LetM,N be smooth manifolds. A mapf : M → N is calledsmoothif for everyp ∈ M, there exist local charts (U, ϕ), (V, ψ) of M, N aroundp, f(p), resp., such thatf(U)⊂V andψ◦f◦ϕ⁻¹:ϕ(U)→ψ(V)is smooth.

1.2.3 Remark (i) The definition is independent of the choice of local charts.

(ii) The definition is local in the sense thatf :M → N is smooth if and only if its restriction to an open subsetU ofM is smooth (cf. Example??).

(iii) A smooth mapM →N is automatically continuous.

We have completed the definition of the categoryDIFF, whose objects are the smooth manifolds and whose morphisms are the smooth maps. An isomorphism in this category is usually called adiffeomorphism.

1.2.4 Exercise LetMbe a smooth manifold with smooth atlasA. Prove that any local chart(U, ϕ) ∈ Ais a diffeomorphism onto its image. Conversely, prove any mapτ :W →Rⁿ, wheren= dimM andW ⊂Mis open, which is a diffeomorphism onto its image belongs to a smooth atlas equivalent to A(in particular,(W, τ)∈ AifAis maximal).

1.2.5 Examples (i) Rⁿ has a canonical atlas consisting only of one local chart, namely, the identity map, which in fact is aglobalchart. This is the standard smooth structure onRⁿwith respect to which all definitions coin- cide with the usual ones. Unless explicit mention, we will always consider Rⁿwith this smooth structure.

(ii) Any finite dimensional real vector spaceV has a canonical structure of smooth manifold. In fact a linear isomorphismV ∼=Rⁿdefines a global chart and thus an atlas, and two such atlases are always equivalent since the transition map between their global charts is a linear isomorphism of Rⁿand hence smooth.

(iii) Submanifolds of Euclidean spaces (Definition 1.1.1(i)) are smooth manifolds.

(iv) Graphs of smooth maps defined on open subsets ofRⁿwith values onR^n+k are smooth manifolds. More generally, a subset ofR^n+k which

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1.2. DEFINITION OF ABSTRACT SMOOTH MANIFOLD 9 can be covered by open sets each of which is a graph as above is a smooth manifold.

(v) It follows from (iii) that then-sphere

Sⁿ={(x₁, . . . , x_n+1)∈Rⁿ⁺¹ :x²₁+· · ·+x²_n+1 = 1} is a smooth manifold.

(vi) If A is an atlas for M and V ⊂ M is open then A|V := {(V ∩ U, ϕ|V∩U) : (U, ϕ)∈ A}is an atlas forV. It follows that any open subset of a smooth manifold is a smooth manifold.

(vii) IfM,N are smooth manifolds with atlasesA,B, resp., thenA × B is an atlas for the Cartesian product M ×N with the product topology, and henceM×N is canonically a smooth manifold of dimensiondimM+ dimN.

(viii) It follows from (iv) and (vi) that then-torus Tⁿ=S¹× · · · ×S¹ (nfactors) is a smooth manifold.

(ix) Thegeneral linear groupGL(n,R)is the set of alln×nnon-singular real matrices. Since the set ofn×nreal matrices can be identified with aRⁿ² and as such the determinant becomes a continuous function,GL(n,R)can be viewed as the open subset ofRⁿ²where the determinant does not vanish and hence acquires the structure of a smooth manifold of dimensionn².

The following two examples deserve a separate discussion.

1.2.6 Example The map f : R → Rgiven by f(x) = x³ is a homeomorphism, so it defines a local chart around any point ofRand we can use it to define an atlas {f}forR; denote the resulting smooth manifold by R˜. We claim that R˜ 6= R as smooth manifolds, because C^∞( ˜R) 6= C^∞(R).

In fact, id : R → Ris obviously smooth, butid : ˜R → R is not, because id◦f⁻¹ :R→Rmapsxto √³

xso it is not differentiable at0. On the other hand, R˜ is diffeomorphic to R. Indeedf : ˜R → Rdefines a diffeomorphism since its local representationid◦f ◦f⁻¹is the identity.

1.2.7 Example Thereal projective space, denotedRPⁿ, as a set consists of all one-dimensional subspaces of Rⁿ⁺¹. We introduce a structure of smooth manifold of dimensionnonRPⁿ. Each subspace is spanned by a non-zero vectorv ∈ Rⁿ⁺¹. Let U_i be the subset ofRPⁿ specified by the condition that thei-th coordinate of vis not zero. Then{U_i}ⁿ⁺¹i=1 coversRPⁿ. Each line in U_i meets the hyperplanex_i = 1 in exactly one point, so there is a bijective mapϕi :Ui →Rⁿ⊂Rⁿ⁺¹. Fori6=j,ϕi(Ui∩Uj)⊂Rⁿ⊂Rⁿ⁺¹is precisely the open subset of the hyperplanex_i = 1defined byx_j 6= 0, and

ϕ_j◦ϕ⁻_i¹ :{x∈Rⁿ⁺¹:x_i= 1, x_j 6= 0} → {x∈Rⁿ⁺¹ :x_j = 1, x_i 6= 0}

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is the map

v7→ 1 x_jv,

thus smooth. So far there is no topology inRPⁿ, and we introduce one by declaring

∪ⁿ⁺¹i=1{ϕ⁻_i ¹(W) :W ⊂ϕ_i(U_i) =Rⁿis open}

to be a basis of open sets. It is clear that∅andM are open sets (since each U_iis open) and we have only to check that finite intersections of open sets are open. LetW_i ⊂ϕ_i(U_i)andW_j ⊂ϕ_i(U_j)be open. Then

ϕ⁻_i ¹(W_i)∩ϕ⁻_j¹(W_j) =ϕ⁻_j¹ ϕ_jϕ⁻_i ¹(W_i∩ϕ_i(U_i∩U_j))∩W_j . Sinceϕjϕ⁻_i ¹is a homeomorphism, a necessary and sufficient condition for the left hand side to describe an open set for alli, j, is thatϕ_i(U_i ∩U_j)be open for alli, j, but this does occur in this example. Now the topology is well defined, second countable, and theϕ_iare homeomorhisms onto their images. It is also clear that forℓ∈RPⁿthe sets

{ℓ^′ ∈RP :∠(ℓ, ℓ^′)< ǫ}

forǫ >0are open neighborhoods ofℓ. It follows that the topology is Haus- dorff.

The argument in Example 1.2.7 is immediately generalized to prove the following proposition.

1.2.8 Proposition LetMbe a set and letnbe a non-negative integer. A countable collection{(U_α, ϕ_α)} of injective mapsϕ : U_α → Rⁿ whose domains coverM satisfying

a. ϕα(Uα)is open for allα;

b. ϕ_α(U_α∩U_β)is open for allα,β;

c. ϕ_βϕ⁻_α¹ :ϕ_α(U_α∩U_β)→ϕ_β(U_α∩U_β)is smooth for allα,β;

defines a second countable topology and smooth structure on M (the Hausdorff condition is not automatic and must be checked in each case).

1.3 Tangent space

As a motivation, we first discuss the case of a submanifoldM ofR^n+k. Fix p∈Mand take an adapted local chart(U,Φ)aroundp. Recall that we get a parametrization ofMaroundpby settingϕ:= projRn◦Φ|M∩U and taking

ϕ⁻¹:Rⁿ∩Φ(U)→R^n+k.

It is then natural to define thetangent spaceofM atpto be the image of the differential of the parametrization, namely,

T_pM :=d(ϕ⁻¹)_ϕ(p)(Rⁿ).

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1.3. TANGENT SPACE 11 If(V,Ψ)is another adapted local chart aroundp,ψ:= projRⁿ◦Ψ|M∩V and ψ⁻¹ :Rⁿ∩Ψ(V)→R^n+kis the associated parametrization, then

d(ϕ⁻¹)_ϕ(p)(Rⁿ) = d(ψ⁻¹)_ψ(p)d(ψϕ⁻¹)_ϕ(p)(Rⁿ)

= d(ψ⁻¹)_ϕ(p)(Rⁿ)

since d(ψϕ⁻¹)_ϕ(p) : Rⁿ → Rⁿ is an isomorphism. It follows thatT_pM is well defined as a subspace of dimensionnofR^n+k.

Note that we have the following situation:

v∈TpM

a∈Rⁿ

d(ψϕ⁻¹)_ϕ(p)>

dϕ⁻_ϕ(p)¹ >

b∈Rⁿ dψ_ψ(p)⁻¹

<

Namely, the tangent vectorv∈T_pMis represented by two different vectors a,b ∈ Rⁿ which are related by the differential of the transition map. We can use this idea to generalize the construction of the tangent space to an abstract smooth manifold.

LetM be a smooth manifold of dimensionn, and fixp ∈ M. Suppose thatAis an atlas defining the smooth structure ofM. Thetangent space of M atpis the setT_pM of all pairs(a, ϕ)— wherea∈Rⁿand(U, ϕ) ∈ Ais a local chart aroundp— quotiented by the equivalence relation

(a, ϕ) ∼(b, ψ) if and only if d(ψ◦ϕ⁻¹)_ϕ(p)(a) =b.

It follows from the chain rule inRⁿthat this is indeed an equivalence relation, and we denote the equivalence class of(a, ϕ) by [a, ϕ]. Each such equivalence class is called atangent vectoratp. For a fixed local chart(U, ϕ) aroundp, the map

a∈Rⁿ7→[a, ϕ]∈T_pM

is a bijection, and it follows from the linearity of d(ψ ◦ϕ⁻¹)_ϕ(p) that we can use it to transfer the vector space structure of Rⁿ toT_pM. Note that dimTpM = dimM.

1.3.1 Exercise Let M be a smooth manifold and let V ⊂ M be an open subset. Prove that there is a canonical isomorphism T_pV ∼= T_pM for all p∈V.

1.3.2 Exercise Let N be a smooth manifold and let M be a submanifold ofN. Prove thatT_pM is canonically isomorphic to a subspace ofT_pN for everyp∈M.

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Let(U, ϕ= (x₁, . . . , x_n))be a local chart ofM, and denote by{e₁, . . . , e_n} the canonical basis ofRⁿ. Thecoordinate vectorsatpare with respect to this chart are defined to be

∂

∂x_i

p= [e_i, ϕ].

Note that (1.3.3)

∂

∂x₁

p, . . . , ∂

∂x_n p

is a basis ofT_pM.

In the case of Rⁿ, for eachp ∈ Rⁿ there is a canonical isomorphism Rⁿ→T_pRⁿgiven by

(1.3.4) a7→[a,id],

whereidis the identity map ofRⁿ. Usually we will make this identification without further comment. In particular, T_pRⁿ andT_qRⁿ are canonically isomorphic for everyp,q ∈Rⁿ. In the case of a general smooth manifold M, obviously there are no such canonical isomorphisms.

Tangent vectors as directional derivatives

LetMbe a smooth manifold, and fix a pointp∈M. For each tangent vector v∈T_pM of the formv = [a, ϕ], wherea∈Rⁿand(U, ϕ)is a local chart of M, and for eachf ∈ C^∞(U), we define thedirectional derivative off in the direction ofvto be the real number

v(f) = d dt

t=0(f◦ϕ⁻¹)(ϕ(p) +ta)

= d(f ◦ϕ⁻¹)(a).

It is a simple consequence of the chain rule that this definition does not depend on the choice of representative ofv.

In the case ofRⁿ, _∂r^∂_i

pf is simply the partial derivative in the direction e_i, theith vector in the canonical basis ofRⁿ. In general, ifϕ= (x₁, . . . , x_n), thenx_i◦ϕ⁻¹ =r_i, so

v(x_i) =d(r_i)_ϕ(p)(a) =a_i, wherea=P_n

i=1a_ie_i. Sincev= [a, ϕ] =P_n

i=1a_i[e_i, ϕ], it follows that

(1.3.5) v=

n

X

i=1

v(xi) ∂

∂x_i p.

Ifvis a coordinate vector _∂x^∂_i andf ∈C^∞(U), we also write

∂

∂xi

pf = ∂f

∂xi

p.

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1.3. TANGENT SPACE 13 As a particular case of (1.3.5), take nowvto be a coordinate vector of another local chart(V, ψ = (y₁, . . . , y_n))aroundp. Then

∂

∂yj

p =

n

X

i=1

∂x_i

∂yj

p

∂

∂xi

p.

Note that the preceding formula shows that even ifx₁ =y₁we do not need to have _∂x^∂₁ = _∂y^∂₁.

The differential

Letf :M → N be a smooth map between smooth manifolds. Fix a point p ∈ M, and local charts (U, ϕ) of M around p, and (V, ψ) of N around q=f(p). Thedifferentialortangent mapoff atpis the linear map

dfp:TpM →TqN given by

[a, ϕ]7→[d(ψ◦f◦ϕ⁻¹)_ϕ(p)(a), ψ].

It is easy to check that this definition does not depend on the choices of local charts. Using the identification (1.3.4), one checks thatdϕ_p : T_pM → Rⁿ anddψ_q:T_pM →Rⁿare linear isomorphisms and

df_p = (dψ_q)⁻¹◦d(ψ◦f ◦ϕ⁻¹)_ϕ(p)◦dϕ_p.

1.3.6 Proposition (Chain rule) LetM,N,P be smooth manifolds. Iff :M → N andg:N →P are smooth maps, theng◦f :M →P is a smooth map and

d(g◦f)_p =dg_f_(p)◦df_p forp∈M.

1.3.7 Exercise Prove Proposition 1.3.6.

Iff ∈ C^∞(M, N),g ∈ C^∞(N)andv ∈ TpM, then it is a simple matter of unravelling the definitions to check that

df_p(v)(g) =v(g◦f).

Now (1.3.5) together with this equation gives that df_p

∂

∂x_j p

=

n

X

i=1

df_p ∂

∂x_j p

(y_i) ∂

∂y_i p

=

n

X

i=1

∂(y_i◦f)

∂x_j p

∂

∂y_i p.

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The matrix

∂(yi◦f)

∂x_j p

is called theJacobian matrix offatprelative to the given coordinate systems.

Observe that the chain rule (Proposition 1.3.6) is equivalent to saying that the Jacobian matrix ofg◦f at a point is the product of the Jacobian matrices ofgandf at the appropriate points.

Consider now the case in which N = Randf ∈ C^∞(M). Thendf_p : T_pM → T_f(p)R, and upon the identification between T_f(p)R and R, we easily see that dfp(v) = v(f). Applying this to f = xi, where (U, ϕ = (x₁, . . . , x_n))is a local chart aroundp, and using again (1.3.5) shows that (1.3.8) {dx₁|p, . . . , dx_n|p}

is the basis ofT_pM^∗dual of the basis (1.3.3), and hence df_p =

n

X

i=1

df_p ∂

∂x_i p

dx_i|p =

n

X

i=1

∂f

∂x_i dx_i|p.

Finally, we discuss smooth curves onM. Asmooth curveinM is simply a smooth map γ : (a, b) → M where(a, b) is an interval of R. One can also consider smooth curvesγinMdefined on a closed interval[a, b]. This simply means thatγ admits a smooth extension to an open interval(a− ǫ, b+ǫ)for someǫ >0.

Ifγ : (a, b)→M is a smooth curve, thetangent vectortoγatt∈(a, b)is

˙

γ(t) =dγ_t ∂

∂r t

∈T_γ(t)M,

where r is the canonical coordinate of R. Note that an arbitrary vector v =∈ T_pM can be considered to be the tangent vector at 0 to the curve γ(t) = ϕ⁻¹(ta), where(U, ϕ) is a local chart around p withϕ(p) = 0 and dϕ_p(v) =a.

In the case in which M = Rⁿ, upon identifying T_γ(t)Rⁿ andRⁿ, it is easily seen that

˙

γ(t) = lim

h→0

γ(t+h)−γ(t)

h .

The inverse function theorem

It is now straightforward to state and prove the inverse function theorem for smooth manifolds.

1.3.9 Theorem (Inverse function theorem) Letf :M →N be a smooth map between two smooth manifoldsM, N, and let p ∈ M andq = f(p). If df_p :

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1.4. SUBMANIFOLDS OF SMOOTH MANIFOLDS 15 T_pM → T_qN is an isomorphism, then there exists an open neighborhood W ofp such thatf(W)is an open neighborhood ofq andf restricts to a diffeomorphism fromW ontof(W).

Proof. The proof is really a transposition of the inverse function theorem for smooth mappings between Euclidean spaces to manifolds using local charts. Note that M andN have the same dimension, say, n. Take local charts(U, ϕ)ofM aroundpand(V, ψ)ofN aroundqsuch thatf(U) ⊂V. Setα = ψ◦f ◦ϕ⁻¹. Thendα_ϕ(p) : Rⁿ → Rⁿ is an isomorphism. By the inverse function theorem for smooth mappings ofRⁿ, there exists an open subsetW˜ ⊂ϕ(U)withϕ(p)∈W˜ such thatα( ˜W)is an open neighborhood ofψ(q)andαrestricts to a diffeomorphism fromW˜ ontoα( ˜W). It follows that f = ψ⁻¹ ◦ α ◦ϕ is a diffeomorphism from the open neighborhood W =ϕ⁻¹( ˜W)ofponto the open neighborhoodψ⁻¹(α( ˜W))ofq.

A smooth mapf :M → N satisfying the conclusion of Theorem 1.3.9 at a point p ∈ M is called a local diffeomorphism at p. It follows from the above and the chain rule thatf is a local diffeomorphism atpif and only if df_p : T_pM → T_qN is an isomorphism. In this case, there exist local charts (U, ϕ) ofM aroundpand(V, ψ)of N aroundf(p) such that the local rep- resentationψ◦f ◦ϕ⁻¹ of f is the identity, owing to Problem 1.2.4, after enlarging the atlas ofM, if necessary.

1.3.10 Exercise Let f : M → N be a smooth bijective map that is a local diffeomorphism everywhere. Show thatf is a diffeomorphism.

1.4 Submanifolds of smooth manifolds

Similar to the situation of submanifolds of Euclidean spaces, some manifolds are contained in other manifolds in a natural way (compare Defini- tion 1.1.1). LetN be a smooth manifold of dimensionn+k. A subsetM ofN is called asubmanifoldofN of dimensionnif, for everyp ∈ M, there exists a local chart(V, ψ)ofN such thatψ(V ∩M) =ψ(V)∩Rⁿ, where we identifyRⁿwithRⁿ× {0} ⊂Rⁿ×R^k=R^n+k. We say that(V, ψ)is a local chart ofNadaptedtoM. A submanifoldM ofN is a smooth manifold in its own right, with respect to the relative topology, in a canonical way. In fact an atlas ofM is furnished by the restrictions of the local charts ofN toM. Namely, if{(V_α, ψ_α)} is an atlas ofN, then{(V_α∩M, ψ_α|Vα∩M)}becomes an atlas ofM. Note that the compatibility condition for the local charts of M follows automatically from the compatibility condition forN.

Immersions and embeddings

Another class of submanifolds can be introduced as follows. Letf :M → N be a smooth map between smooth manifolds. The map f is called an

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immersionatp∈M ifdf_p :T_pM →T_f_(p)N is injective. Iff is an immersion everywhere it is simply called animmersion. Now call the pair(M, f) an immersed submanifoldofN iff :M →N is an injective immersion.

LetM be a submanifold ofN and consider the inclusionι : M → N.

The existence of adapted local charts implies that ιcan be locally represented around any point ofM by the standard inclusionx7→ (x,0),Rⁿ→ R^n+k. Since this map is an immersion, alsoιis an immersion. It follows that (M, ι)is an immersed submanifold ofN. This shows that every submanifold of a smooth manifold is an immersed submanifold, but the converse is not true.

1.4.1 Example Take the2-torusT² = S¹×S¹ viewed as a submanifold of R²×R² =R⁴and consider the smooth map

F :R→R⁴, F(t) = (cosat,sinat,cosbt,sinbt),

wherea,bare non-zero real numbers. Note that the image ofF lies inT². Denote by(r₁, r₂, r₃, r₄)the coordinates onR⁴. Choosing any two out ofr₁, r2,r3,r4gives a system of coordinates defined on some open subset ofT². It follows that the induced mapf : R → T² is smooth. Sincef^′(t) never vanishes, this map is an immersion. We claim that if b/ais an irrational number, thenf is injective and (R, f)is an immersed submanifold which isnota submanifold ofT². In fact, the assumption on b/aimplies thatM is a dense subset ofT², but a submanifold of another manifold is always locally closed.

We would like to further investigate the gap between immersed submanifolds and actual submanifolds.

1.4.2 Lemma (Local form of an immersion) Let M and N be smooth mani- folds of dimensionsnandn+k, respectively, and suppose thatf :M →N is an immersion atp∈M. Then there exist local charts ofM andNsuch that the local expression off atpis the standard inclusion ofRⁿintoR^n+k.

Proof. Let(U, ϕ) and(V, ψ) be local charts ofM and N aroundp and q =f(p), respectively, such thatf(U) ⊂V, and setα =ψ◦f ◦ϕ⁻¹. Then dα_ϕ(p) : Rⁿ → R^n+k is injective, so, up to rearranging indices, we can assume that d(π₁ ◦α)_ϕ(p) = π₁ ◦ dα_ϕ(p) : Rⁿ → Rⁿ is an isomorphism, whereπ₁ : R^n+k = Rⁿ×R^k → Rⁿ is the projection onto the first factor.

By the inverse function theorem, by shrinkingU, we can assume thatπ₁◦α is a diffeomorphism fromU₀ = ϕ(U) onto its image V₀; let β : V₀ → U₀ be its smooth inverse. Now we can describeα(U₀)as being the graph of the smooth mapγ = π₂ ◦α◦ β : V₀ ⊂ Rⁿ → R^k, whereπ₂ : R^n+k = Rⁿ×R^k → R^k is the projection onto the second factor. By Exercise 1.1.3, α(U₀)is a submanifold ofR^n+kand the mapτ :V₀×R^k→V₀×R^kgiven

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1.4. SUBMANIFOLDS OF SMOOTH MANIFOLDS 17 byτ(x, y) = (x, y−γ(x))is a diffeomorphism such thatτ(α(U₀)) =V₀×{0}. Finally, we putϕ˜=π₁◦α◦ϕandψ˜=τ◦ψ. Then(U,ϕ)˜ and(V,ψ)˜ are local charts, and forx∈ϕ(U˜ ) =V₀we have that

ψ˜◦f ◦ϕ(x) =˜ τ ◦ψ◦f ◦ϕ⁻¹◦β(x) =τ◦α◦β(x) = (x,0).

1.4.3 Proposition If f : M → N is an immersion at p ∈ M, then there ex- ists an open neighborhood U ofp inM such thatf|U is injective andf(U)is a submanifold ofN.

Proof. The local injectivity off atpis an immediate consequence of the fact that some local expression off atpis the standard inclusion ofRⁿinto R^n+k, hence, injective. Moreover, in the course of proof of Lemma 1.4.2, we have produced a local chart(V,ψ)˜ ofN adapted tof(U).

A smooth map f :M → N is called anembeddingif it is an immersion and a homeomorphism fromM ontof(M)with the induced topology.

1.4.4 Proposition LetN be a smooth manifold. A subsetP ⊂Nis a submanifold ofN if and only if it is the image of an embedding.

Proof. Letf :M →N be an embedding withP =f(M). To prove that P is a submanifold ofN, it suffices to check that it can be covered by open sets in the relative topology each of which is a submanifold ofN. Owing to Proposition 1.4.3, any point ofP lies in a set of the formf(U), whereU is an open subset ofM andf(U)is a submanifold ofN. Sincef is an open map intoP with the relative topology,f(U)is open in the relative topology and we are done. Conversely, ifP is a submanifold ofN, it has the relative topology and thus the inclusionι : P → N is a homeomorphism onto its image. Moreover, we have seen above thatιis an immersion, whence it is

an embedding.

Recall that a continuous map between locally compact, Hausdorff topological spaces is calledproper if the inverse image of a compact subset of the target space is a compact subset of the domain. It is known that proper maps are closed. Also, it is clear that if the domain is compact, then every continuous map is automatically proper. An submanifoldM of a smooth manifoldN is calledproperly embeddedif the inclusion map is proper.

1.4.5 Proposition Iff :M →N is an injective immersion which is also a proper map, then the imagef(M)is a properly embedded submanifold ofN.

Proof. LetP =f(M)have the relative topology. A proper map is closed.

Sincef viewed as a mapM → P is bijective and closed, it is an open map and thus a homeomorphism. Due to Proposition 1.4.4,P is a submanifold

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ofN. The properness of the inclusionP →N clearly follows from that off.

1.4.6 Exercise Give an example of a submanifold of a smooth manifold which is not properly embedded.

1.4.7 Exercise Decide whether a closed submanifold of a smooth manifold is necessarily properly embedded.

Exercise 1.1.4 dealt with a situation in which a smooth mapf :M →N factors through an immersed submanifold (P, g) of N (namely, f(M) ⊂ g(P)) and the induced mapf₀ :M →P (namely,g◦f₀ =f) is discontinu- ous.

1.4.8 Proposition Suppose thatf :M →Nis smooth and(P, g)is an immersed submanifold ofN such thatf(M)⊂g(P). Consider the induced mapf0 :M → Pthat satisfiesg◦f₀ =f.

a. Ifgis an embedding, thenf₀is continuous.

b. Iff₀is continuous, then it is smooth.

Proof. (a) In this casegis a homeomorphism ontog(P)with the relative topology. IfV ⊂ P is open, theng(V) = W ∩g(P) for some open sub- setW ⊂ N. By continuity off, we have that f₀⁻¹(V) = f₀⁻¹(g⁻¹(W)) = f⁻¹(W)is open inM, hence alsof₀is continuous.

(b) Letp ∈ M andq = f₀(p) ∈ P. By Proposition 1.4.3, there exists a neighborhoodU ofq and a local chart(V, ψ) ofNⁿ adapted tog(U), with g(U)⊂V. In particular, there exists a projectionπfromRⁿonto a subspace obtained by setting some coordinates equal to0such thatτ =π◦ψ◦gis a local chart ofP aroundq. Note thatf₀⁻¹(U)is a neighborhood ofpinM.

Now

τ ◦f0|f0⁻¹(U)=π◦ψ◦g◦f0|f0⁻¹(U)=π◦ψ◦f|f0⁻¹(U),

and the latter is smooth.

An immersed submanifold(P, g)ofN with the property thatf0 :M → P is smooth for every smooth mapf :M →N withf(M) ⊂g(P)will be called aninitial submanifold.

1.4.9 Exercise Use Exercise 1.3.10 and Propositions 1.4.4 and 1.4.8 to deduce that an embeddingf : M → N induces a diffeomorphism from M onto a submanifold ofN.

1.4.10 Exercise For an immersed submanifold(M, f)ofN, show that there is a natural structure of smooth manifold onf(M)and that(f(M), ι)is an immersed submanifold ofN, whereι:f(M)→N denotes the inclusion.

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1.4. SUBMANIFOLDS OF SMOOTH MANIFOLDS 19 Submersions

The mapf is called asubmersionat pifdf_p : T_pM → T_f(p)N is surjective.

Iff is a submersion everywhere it is simply called a submersion. A point q ∈ N is called a regular value of f if f is a submersion at all points in f⁻¹(q); otherwiseqis called asingular valueoff.

1.4.11 Lemma (Local form of a submersion) Let M an N be smooth mani- folds of dimensions n+k and k, respectively, and suppose that f : M → N is a submersion atp∈M. Then there exist local charts ofM andN such that the local expression off atpis the standard projection ofR^n+kontoR^k.

Proof. Let (U, ϕ) and(V, ψ) be local charts ofM andN aroundp and q=f(p), respectively, and setα =ψ◦f◦ϕ⁻¹. Thendα_ϕ(p) :R^n+k→R^kis surjective, so, up to rearranging indices, we can assume thatd(α◦ι₂)_ϕ(p)= dα_ϕ(p)◦ι₂ :R^k→R^kis an isomorphism, whereι₂:R^k→R^n+k =Rⁿ×R^k is the standard inclusion. Define α˜ : ϕ(U) ⊂ Rⁿ ×R^k → Rⁿ×R^k by

˜

α(x, y) = (x, α(x, y)). Sincedα_ϕ(p)◦ι₂ is an isomorphism, it is clear that d˜α_ϕ(p) :Rⁿ⊕R^k → Rⁿ⊕R^kis an isomorphism. By the inverse function theorem, there exists an open neighborhoodU₀ ofϕ(p) contained inϕ(U) such thatα˜is a diffeomorphism fromU0 onto its imageV0; letβ˜:V0 → U0

be its smooth inverse. We putϕ˜= ˜α◦ϕ. Then(ϕ⁻¹(U₀),ϕ)˜ is a local chart ofM aroundpand

ψ◦f◦ϕ˜⁻¹(x, y) =ψ◦f ◦ϕ⁻¹◦β(x, y) =˜ α◦β(x, y) =˜ y.

1.4.12 Proposition Letf : M → N be a smooth map, and let q ∈ N be such thatf⁻¹(q) 6= ∅. Iff is a submersion at all points ofP = f⁻¹(q), thenP is a submanifold ofM of dimensiondimM −dimN. Moreover, for p ∈ P we have T_pP = kerdf_p.

Proof. It is enough to construct local charts of M that are adapted to P and whose domains coverP. So supposedimM = n+k, dimN = k, let p ∈ P and consider local charts (W := ϕ⁻¹(U0),ϕ)˜ and (V, ψ) as in Theorem 1.4.11 such thatp ∈ U andq ∈ V. We can assume thatψ(q) = 0. Now it is obvious that ϕ(W˜ ∩P) = ˜ϕ(W) ∩Rⁿ, so ϕ is an adapted chart around p. Finally, the local representation off atp is the projection R^n+k → R^k. This is a linear map with kernelRⁿ. It follows thatkerdf_p =

(dϕ˜⁻¹)_ϕ(p)(Rⁿ) =T_pP.

1.4.13 Examples (a) LetA be a non-degenerate real symmetric matrix of order n+ 1 and define f : Rⁿ⁺¹ → R by f(p) = hAp, pi where h,i is the standard Euclidean inner product. Thendf_p : Rⁿ⁺¹ → Ris given by df_p(v) = 2hAp, vi, so it is surjective ifp6= 0. It follows thatfis a submersion

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onRⁿ⁺¹\ {0}, and thenf⁻¹(r)forr ∈ Ris an embedded submanifold of Rⁿ⁺¹of dimensionnif it is nonempty. In particular, by takingAto be the identity matrix we get a manifold structure forSⁿwhich coincides with the one previously constructed.

(b) Denote bySym(n,R) the vector space of real symmetric matrices of ordern, and definef : M(n,R) → Sym(n,R)byf(A) = AA^t. This is map between vector spaces whose local representations’ components are quadratic polynomials. It follows that f is smooth and that df_A can be viewed as a mapM(n,R) → Sym(n,R) for all A ∈ M(n,R). We claim that I is a regular value of f. For the purpose of checking that, we first compute forA∈f⁻¹(I)andB ∈M(n,R)that

df_A(B) = lim

h→0

(A+hB)(A+hB)^t−I h

= lim

h→0

h(AB^t+BA^t) +h²BB^t h

= AB^t+BA^t.

Now givenC∈Sym(n,R), we havedf_A(¹₂CA) =C, and this proves thatf is a submersion atA, as desired. Hencef⁻¹(I) ={ A∈M L(n,R)|AA^t = I }is a submanifold ofM(n,R)of dimension

dimM(n,R)−dimV =n²−n(n+ 1)

2 = n(n−1)

2 .

Note thatf⁻¹(I)is a group with respect to the multiplication of matrices;

it is called theorthogonal groupof ordernand is usually denoted byO(n).

It is obvious thatO(n)⊂GL(n,R).

We close this section by mentioning a generalization of Proposition 1.4.12.

Letf :M → N be a smooth map and letQbe a submanifold ofN. We say thatf istransversetoQ, in symbolsf ⋔Q, if

df_p(T_pM) +T_f_(p)Q=T_f(p)N for everyp∈f⁻¹(Q).

1.4.14 Exercise Letf :M →N be a smooth map and letq ∈N. Prove that f ⋔{q}if and only ifqis a regular value off.

1.4.15 Proposition If f : M → N is a smooth map which is transverse to a submanifoldQofN of codimensionkandP =f⁻¹(Q)is non-empty, thenPis a submanifold ofM of codimensionk. MoreoverT_pP = (df_p)⁻¹(T_f_(p)Q)for every p∈P.

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1.5. PARTITIONS OF UNITY 21 Proof. For the first assertion, it suffices to check thatP is a submanifold of M in a neighborhod of a pointp ∈ P. Let (V, ψ) be a local chart ofN adapted to Q aroundq := f(p). Then ψ : V → R^n+k and ψ(V ∩Q) = ψ(V)∩Rⁿ, where n = dimQ. Let π₂ : R^n+k = Rⁿ ×R^k → R^k be the standard projection and putg =π₁◦ψ. Theng :V → R^kis a submersion andg⁻¹(0) =V ∩Q. Moreover

d(g◦f)_p(T_pM) = dg_q◦df_p(T_pM)

= dg_q(T_qN)

= Rⁿ

where, in view ofkerdg_q = T_qQ, the second equality follows from the as- sumptionf ⋔Q. Nowh :=g◦f :f⁻¹(V)→ R^kis a submersion atpand h⁻¹(0) =f⁻¹(V ∩Q) =f⁻¹(V)∩P andf⁻¹(V)is an open neighborhood ofpinM, so we can apply Proposition 1.4.12. All the assertions follow.

As a most important special case, two submanifolds M, P of N are calledtransverse, denotedM ⋔P, if the inclusion mapι:M → N is transverse toP.

1.4.16 Corollary IfM andP are transverse submanifolds ofN thenM ∩Pis a submanifold ofN and

codim(M∩P) = codim(M) + codim(P).

1.5 Partitions of unity

Many important constructions for smooth manifolds rely on the existence of smooth partitions of unity. This technique allows for a much greater flexibility of smooth manifolds as compared, for instance, with real analytic or complex manifolds.

Bump functions

We start with the remark that the function f(t) =

e⁻^1/t, ift >0 0, ift≤0 is smooth everywhere. Therefore the function

g(t) = f(t) f(t) +f(1−t)

is smooth, flat and equal to0on(−∞,0], and flat and equal to1on[1,+∞).

Finally,

h(t) =g(t+ 2)g(2−t)

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is smooth, flat and equal to 1 on[−1,1] and its support lies in (−2,2); h is called abump function. We can also make ann-dimensional version of a bump function by setting

k(x1, . . . , xn) =h(x1)· · ·h(xn),

and we can rescalek by precomposing withx 7→ r⁻¹x to have a smooth function onRⁿwhich is flat and equal to1on a closed ball of radiusrand with support contained in an open ball of radius2r.

Bump functions are very useful. As one application, note that for a given smooth manifold M so far we do not know whether the algebra C^∞(M) of smooth functions onM contains functions other than the con- stants (of course, the components of local charts are smooth, but these are notgloballydefined onM). We claim thatC^∞(M)is indeed in general huge.

In fact, let(U, ϕ)be a local chart ofMand take a bump functionk:Rⁿ→R whose support lies inϕ(U). Then

f(x) :=

k◦ϕ(x) if∈U, 0 ifx∈M\U

is a smooth function onM: this is clear for a pointp ∈ U; ifp 6∈ U, then we can find a neighborhoodV of pwhich does not meet the compact set ϕ⁻¹(supp(k)), sof|V = 0and thusfis smooth atp.

Partitions of unity

LetMbe a smooth manifold. Apartition of unityonMis a collection{ρ_i}i∈I

of smooth functions onM, whereI is an index set, such that:

(i) ρ_i(p)≥0for allp∈M and alli∈I;

(ii) the collection of supports{supp(ρ)}i∈I is locally finite (i.e. every point ofM admits a neighborhood meetingsupp(ρ_i)for only finitely many indicesi);

(iii) P

i∈Iρi(p) = 1for allp∈M (the sum is finite in view of (ii)).

Let{U_α}α∈Abe a cover ofM by open sets. We say that a partition of unity {ρ_i}i∈I issubordinateto{U_α}α∈Aif for everyi∈I there is someα∈Asuch thatsupp(ρ_i) ⊂U_α; and we say{ρ_i}i∈I isstrictly subordinateto{U_α}α∈Aif I =Aandsupp(ρ_α)⊂U_αfor everyα∈A.

Partitions of unity are used to piece together global objects out of local ones, and conversely to decompose global objects as locally finite sums of locally defined ones. For instance, suppose{U_α}α∈Ais an open cover ofM and{ρ_α}α∈Ais a partition of unity strictly subordinate to{U_α}. If we are givenf_α∈C^∞(U_α)for allα∈A, thenf =P

α∈Aρ_αf_αis a smooth function onM. Indeed for p ∈M andα ∈ A, it is true that eitherp ∈ U_α and then f_αis defined atp, orp 6∈ U_αand thenρ_α(p) = 0. Moreover, since the sum is locally finite,f is locally the sum of finitely many smooth functions and

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1.5. PARTITIONS OF UNITY 23 hence smooth. Conversely, if we start withf ∈C^∞(M)thenf =P

α∈Af_α for smooth functionsf_αwithsupp(f_α)⊂U_α, namely,f_α:=ρ_αf.

1.5.1 Exercise LetC be closed inM and letU be open inM withC ⊂ U. Prove that there exists a smooth functionλ∈C^∞(M)such that0≤λ≤1, λ|C = 1andsuppλ⊂U.

If M is compact, it is a lot easier to prove the existence of a partition of unity subordinate to any given open cover{U_α}ofM. In fact for each x ∈ U_α we construct as above a bump functionλ_x which is flat and equal to 1 on a neighborhoodV_x of x and whose (compact) support lies in U_α. Owing to compactness ofM, we can extract a finite subcover of{Vx}and thus we get non-negative smooth functionsλ_i :=λ_x_i fori= 1, . . . , nsuch thatλ_iis1onV_x_i. In particular, their sum is positive, so

ρ_i:= λ_i Pn

i=1λ_i

fori= 1, . . . , nyields the desired partition of unity.

1.5.2 Theorem (Easy Whitney embedding theorem) LetMbe a compact smooth manifold. Then there exists an embedding ofMintoR^mformsuffciently big.

Proof. SinceM is compact, there exists an open covering {V_i}^ai=1 such that for eachi,V¯_i ⊂ U_i where(U_i, ϕ_i)is a local chart ofM. For eachi, we can findρ_i ∈C^∞(M)such that0≤ρ_i ≤1,ρ_i|V^¯i = 1andsuppρ_i ⊂U_i. Put

f_i(x) =

ρ_i(x)ϕ_i(x), ifx∈U_i, 0, ifx∈M \U_i.

Then f_i : M → Rⁿ is smooth, where n = dimM. Define also smooth functions

g_i= (f_i, ρ_i) :M →Rⁿ⁺¹ and g= (g₁, . . . , g_a) :M →R^a(n+1). It is enough to check thatg is an injective immersion. In fact, on the open set Vi, we have that gi = (ϕi,1) is an immersion, so g is an immersion.

Further, ifg(x) = g(y)forx,y ∈ M, thenρ_i(x) = ρ_i(y)andf_i(x) = f_i(y) for alli. Take an indexj such thatρ_j(x) =ρ_j(y) 6= 0. Thenx, y ∈ U_j and ϕj(x) =ϕj(y). Due to the injectivity ofϕj, we must havex=y. Hencegis

injective.

1.5.3 Remark In the noncompact case, one can still use partitions of unity and modify the proof of Theorem 1.5.2 to prove thatM properly embedds intoR^m for somem. Then a standard trick involving Sard’s theorem and projections into lower dimensional subspaces of R^m allows to find the boundm ≤ 2n+ 1, where n = dimM. A more difficult result, thestrong Whitney embedding theoremasserts that in factm≤2n.

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In general, a reasonable substitute for compactness is paracompactness.

A topological space is calledparacompactif every open covering admits an open locally finite refinement. It turns out that every locally compact, second countable, Hausdorff space is paracompact. Hence manifolds are paracompact. Now the above argument can be extended to give the following theorem, for whose proof we refer the reader to [?].

1.5.4 Theorem (Existence of partitions of unity) Let M be a smooth mani- fold and let{U_α}α∈Abe an open cover ofM. Then there exists a countable parti- tion of unity{ρ_i :i= 1, 2, 3, . . .}subordinate to{U_α}withsupp(ρ_i)compact for eachi. If one does not require compact supports, then there is a partition of unity{ϕ_α}α∈Astrictly subordinate to{U_α}with at most countably many of the ρ_αnot zero.

1.6 Vector fields

LetM be a smooth manifold. Avector fieldonM is an assigment of a tangent vectorX(p)inT_pMfor allp∈M. Sometimes, we also writeX_pinstead ofX(p). So a vector field is a mapX :M → T M whereT M = ˙∪^p^∈^MTpM (disjoint union), and

(1.6.1) π◦X = id

where π : T M → M is the natural projection π(v) = p if v ∈ T_pM. In account of property (1.6.1), we say thatXis asectionofT M.

We shall need to talk about continuity and differentiability of vector fields, so we next explain thatT M carries a canonical manifold structure induced from that ofM.

The tangent bundle

LetM be a smooth manifold and consider the disjoint union T M =[˙

p∈MT_pM.

We can view the elements ofT M as equivalence classes of triples(p, a, ϕ), wherep∈M,a∈Rⁿand(U, ϕ)is a local chart ofM such thatp∈U, and

(p, a, ϕ)∼(q, b, ψ) if and only ifp=qandd(ψ◦ϕ⁻¹)_ϕ(p)(a) =b.

There is a natural projectionπ :T M →M given byπ[p, a, ϕ] =p, and then π⁻¹(p) =TpM.

SupposedimM =n. Note that we havendegrees of freedom for a point pinMandndegrees of freedom for a vectorv∈T_pM, so we expectT Mto

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1.6. VECTOR FIELDS 25 be2n-dimensional. We will use Proposition 1.2.8 to simultaneously introduce a topology and smooth structure onT M. Let{(U_α, ϕ_α)}be a smooth atlas forM with countably many elements (recall that every second countable space is Lindel ¨of). For eachα,ϕ_α :U_α →ϕ_α(U_α)is a diffeomorphism and, for each p ∈ U_α, d(ϕ_α)_p : T_pU_α = T_pM → Rⁿ is the isomorphism mapping[p, a, ϕ]toa. Set

˜

ϕ_α:π⁻¹(U_α)→ϕ_α(U_α)×Rⁿ, [p, a, ϕ]→(ϕ_α(p), a).

Thenϕ˜_αis a bijection andϕ_α(U_α)is an open subset ofR²ⁿ. Moreover, the maps

˜

ϕ_β◦ϕ˜⁻_α¹:ϕ_α(U_α∩U_β)×Rⁿ→ϕ_β(U_α∩U_β)×Rⁿ are defined on open subsets ofR²ⁿand are given by

(x, a)7→(ϕ_β◦ϕ⁻_α¹(x), d(ϕ_β◦ϕ⁻_α¹)_x(a)).

Sinceϕ_β ◦ϕ⁻_α¹ is a smooth diffeomorphism, we have thatd(ϕ_β◦ϕ⁻_α¹)_x is a linear isomorphism andd(ϕ_β ◦ϕ⁻_α¹)x(a)is also smooth onx. It follows that{(π⁻¹(U_α),ϕ˜_α)}defines a topology and a smooth atlas forM and we need only to check the Hausdorff condition. Namely, letv,w ∈ T M with v6=w. Note thatπis an open map. Ifv,w∈T Mandπ(v)6=π(w), we can use the Hausdorff property ofM to separatevandwfrom each other with open sets ofT M. On the other hand, ifv,w∈T_pM, they lie in the domain of the same local chart ofT M and the result also follows.

Iff ∈C^∞(M, N), then we define thedifferential off to be the map df :T M →T N

that restricts to df_p : T_pM → T_f(p)N for each p ∈ M. Using the above atlases forT M andT N, we immediately see thatdf ∈C^∞(T M, T N).

1.6.2 Remark The mapping that associates to each manifoldM its tangent bundle T M and associates to each smooth map f : M → N its tangent map df : T M → T N can be thought of a functorDIFF → VBfrom the category of smooth manifolds to the category of smooth vector bundles. In fact,d(id_M) = id_{T M}, andd(g◦f) =dg◦dffor a sequence of smooth maps M −→^f N −→^g P.

Smooth vector fields

A vector field X onM is called smooth (resp. continuous) if the map X : M →T M is smooth (resp. continuous).

More generally, letf :M →N be a smooth mapping. Then a (smooth, continuous)vector field alongf is a (smooth, continuous) mapX :M →T N such thatX(p) ∈ T_f(p)N for p ∈ M. The most important case is that in

Notes on Smooth Manifolds

Notes on Smooth Manifolds

Contents

Smooth manifolds