x y
Ux+ Ux−
Fig. 19.2.Two charts on the unit circle.
19.8 Exterior Differentiation Under a Pullback
The pullback of differential forms commutes with the exterior derivative.
Theorem 19.8.LetF:N −→Mbe a smooth map of manifolds. Ifω∈k(M), then dF∗ω=F∗dω.
Proof. We first check the casek=0 whenωis aC∞functionhonM. Forp∈N andXp∈TpN,
(dF∗h)p(Xp)=Xp(F∗h) (property (iii) ofd)
=Xp(h◦F ) (definition of the pullback of a function) and
(F∗dh)p(Xp)=(dh)F (p)(F∗Xp) (definition of the pullback of a 1-form)
=(F∗Xp)h (definition of the differentialdh)
=Xp(h◦F ) (definition ofF∗).
Now consider the general case of aC∞k-formω onM. It suffices to verify dF∗ω = F∗dω at an arbitrary point p ∈ N. This reduces the proof to a local computation. If(V , y1, . . . , ym)is a chart ofMatF (p), then onV,
ω=
aIdyi1∧ ã ã ã ∧dyik, I =(i1, . . . , ik), for someC∞functionsaIonV and
F∗ω=
(F∗aI)F∗dyi1∧ ã ã ã ∧F∗dyik (Proposition 18.7)
=
(aI ◦F ) dFi1∧ ã ã ã ∧dFik (F∗dyi =dF∗yi =d(yi ◦F )
=dFi).
So
dF∗ω=
d(aI ◦F )∧dFi1∧ ã ã ã ∧dFik. On the other hand,
F∗dω=F∗(
daI∧dyi1 ∧ ã ã ã ∧dyik)
=
F∗daI∧F∗dyi1∧ ã ã ã ∧F∗dyik
=
d(F∗aI)∧dFi1∧ ã ã ã ∧dFik (by the casek=0)
=
d(aI ◦F )∧dFi1∧ ã ã ã ∧dFik. Therefore,
dF∗ω=F∗dω.
Example19.9.LetUbe the open set(0,∞)×(0,2π )in the(r, θ )-planeR2. Define F:U ⊂R2−→R2by
(x, y)=F (r, θ )=(rcosθ, rsinθ ).
Compute the pullbackF∗(dx∧dy).
Solution. We first computeF∗dx:
F∗dx=dF∗x (Theorem 19.8)
=d(x◦F ) (definition of the pullback of a function)
=d(rcosθ )
=(cosθ ) dr−rsinθ dθ.
Similarly,
F∗dy =dF∗y =d(rsinθ )=(sinθ ) dr+rcosθ dθ.
Since the pullback commutes with the wedge product (Proposition 18.7), F∗(dx∧dy)=(F∗dx)∧(F∗dy)
=((cosθ ) dr−rsinθ dθ )∧((sinθ ) dr+rcosθ dθ )
=(rcos2θ+rsin2θ ) dr∧dθ
=r dr∧dθ.
Problems
19.1.* Extension of aC∞form Prove Proposition 19.6.
19.2. Transition formula for ann-form
Let(U, x1, . . . , xn)be a chart on a manifold andf1, . . . , fnsmooth functions onU. Prove that
df1∧ ã ã ã ∧dfn=det ∂fi
∂xj
dx1∧ ã ã ã ∧dxn.
19.8 Exterior Differentiation Under a Pullback 197 19.3. Pullback of a differential form
LetU be the open set(0,∞)×(0, π )×(0,2π )in the(ρ, φ, θ )-spaceR3. Define F:U −→R2by
(x, y, z)=F (ρ, φ, θ )=(ρsinφcosθ, ρsinφsinθ, ρcosφ).
Show thatF∗(dx∧dy∧dz)=ρ2sinφ dρ∧dφ∧dθ. 19.4. Pullback of a differential form
LetF:R2−→R2be given by
F (x, y)=(u, v)=(x2+y2, xy).
ComputeF∗(u du+v dv).
19.5. Pullback of a differential form by a curve
Let ω be the 1-form ω = (−y dx + x dy)/(x2 +y2) on R2 − {0}. Define c:R−→R2− {0}byc(t )=(cost,sint ). Computec∗ω.
19.6. Coordinate functions and differential forms
Letf1, . . . , fnbeC∞functions on a neighborhoodUof a pointpin a manifold of dimensionn. Show that there is a neighborhoodWofpon whichf1, . . . , fnform a coordinate system if and only if(df1∧ ã ã ã ∧dfn)p=0.
19.7. Local operators
An operatorL:∗(M)−→∗(M)issupport-decreasingif suppL(ω)⊂suppωfor everyk-formω∈∗(M)for allk≥0. Show that an operator on∗(M)is local if and only if it is support-decreasing.
19.8. Derivations ofC∞functions are local operators
LetMbe a smooth manifold. The definition of a local operatorDonC∞(M)is similar to that of a local operator on ∗(M): D islocal if whenever a function f ∈C∞(M)vanishes identically on an open subsetU, thenDf ≡0 onU. Prove that a derivation ofC∞(M)is a local operator onC∞(M).
19.9. Global formula for the exterior derivative of a 1-form
Prove that ifωis aC∞1-form and X andY areC∞ vector fields on a manifold M, then
dω(X, Y )=Xω(Y )−Y ω(X)−ω([X, Y]).
19.10. A nowhere-vanishing form on a smooth hypersurface
(a) Letf (x, y)be aC∞function onR2and assume that 0 is a regular value off. By the regular level set theorem, the zero setMoff (x, y)is a one-dimensional submanifold ofR2. Construct a nowhere-vanishing 1-form onM.
(b) Letf (x, y, z)be aC∞function onR3and assume that 0 is a regular value off. By the regular level set theorem, the zero setMoff (x, y, z)is a two-dimensional submanifold ofR3. Letfx,fy,fzbe the partial derivatives off with respect to x,y,z, respectively. Show that the equalities
dx∧dy
fz =dy∧dz
fx = dz∧dx fy
hold on M whenever they make sense, and therefore piece together to give a nowhere-vanishing 2-form onM.
(c) Generalize this problem to a regular level set off (x1, . . . , xn+1)inRn+1. 19.11. Vector fields as derivations ofC∞functions
In Section 14.4 we showed that aC∞vector field X on a manifoldM gives rise to a derivation ofC∞(M). To distinguish the vector field from the derivation, we will temporarily denote the derivation arising fromX byϕ(X). Thus, for anyf ∈ C∞(M),
(ϕ(X)f )(p)=Xpf for allp∈M.
(a) LetF=C∞(M). Prove thatϕ:X(M)−→Der(C∞(M))is anF-linear map.
(b) Show thatϕis injective.
(c) IfDis a derivation ofC∞(M)andp∈M, defineDp:Cp∞(M)−→Cp∞(M)by Dp[f] = [Df˜] ∈Cp∞(M),
where [f] is the germ of f at p andf˜ is a global extension of f given by Proposition 19.6. Show thatDp[f]is well defined. (Hint: Apply Problem 19.8.) (d) Show thatDpis a derivation ofCp∞(M).
(e) Prove thatϕ:X(M)−→Der(C∞(M))is an isomorphism ofF-modules.
19.12. Twentieth-century formulation of Maxwell’s equations
In Maxwell’s theory of electricity and magnetism, developed in the late nineteenth century, the electric fieldE= E1, E2, E3and the magnetic fieldB= B1, B2, B3 in a vacuumR3with no charge or current, satisfy the following equations:
∇×E= −∂B
∂t, ∇×B=∂E
∂t , divE=0, divB=0.
By the correspondence in Section 4.6, the 1-formEonR3corresponding to the vector fieldEis
E=E1dx+E2dy+E3dz and the 2-formBonR3corresponding to the vector fieldBis
B =B1dy∧dz+B2dz∧dx+B3dx∧dy.
LetR4be space-time with coordinates(x, y, z, t ). Then bothEandB can be viewed as differential forms onR4. DefineF to be the 2-form on space-time
F =E∧dt+B.
Decide which two of Maxwell’s equations are equivalent to the equation dF =0.
Prove your answer. (The other two are equivalent tod∗F =0 for a star-operator∗ defined in differential geometry. See [1, Section 19.1, p. 689].)
20
Orientations
20.1 Orientations on a Vector Space
OnR1an orientation is one of two directions (Figure 20.1).
Fig. 20.1.Orientations on a line.
OnR2an orientation is either counterclockwise or clockwise (Figure 20.2).
Fig. 20.2.Orientations on a plane.
OnR3 an orientation is either right-handed (Figure 20.3) or left-handed (Fig- ure 20.4). The right-handed orientation onR3is the choice of a Cartesian coordinate system so that if you hold out your right hand with the index finger curling from the x-axis to they-axis, then your thumb points in the direction of thez-axis.
How should one define an orientation onR4? If we analyze the three examples above, we see that an orientation can be specified by an ordered basis forRn. Let e1, . . . , enbe the standard basis forRn. ForR1an orientation could be given by either e1or−e1. ForR2the counterclockwise orientation is(e1, e2), while the clockwise
e2 e3=thumb
e1=index finger Fig. 20.3.Right-handed orientation(e1, e2, e3)onR3.
e2=index finger e3=thumb
e1
Fig. 20.4.Left-handed orientation(e1, e2, e3)onR3.
orientation is(e2, e1). For R3the right-handed orientation is (e1, e2, e3), and the left-handed orientation is(e2, e1, e3).
For any two ordered bases(u1, u2)and(v1, v2)forR2, there is a unique nonsin- gular 2 by 2 matrixA= [aij]such that
uj = 2
i=1
viaij, j =1,2,
called thechange of basis matrixfrom(v1, v2)to(u1, u2). In matrix notation, if we write ordered basis as row vectors, for example,[u1u2]for the basis(u1, u2), then
[u1u2] = [v1v2]A.
We say that two ordered bases areequivalent if the change of basis matrixAhas positive determinant. It is easy to check that this is indeed an equivalence relation on the set of all ordered bases forR2. It therefore partitions the ordered bases into two equivalence classes. Each equivalence class is called an orientation onR2.
20.2 Orientations andn-Covectors 203 The equivalence class containing the ordered basis(e1, e2)is the counterclockwise orientation and the equivalence class of(e2, e1)is the clockwise orientation.
The general case is similar. Two ordered basesu= [u1ã ã ãun]andv= [v1ã ã ãvn] of a vector spaceV are said to beequivalentifu=vAfor annbynmatrixAwith positive determinant. AnorientationonV is an equivalence class of ordered bases.
The zero-dimensional vector space{0}is a special case because it does not have a basis. We define an orientation on{0}to be one of the two numbers±1.
20.2 Orientations and n-Covectors
Instead of using an ordered basis, we can also use ann-covector to specify an orien- tation on ann-dimensional vector spaceV. This is based on the fact that the space .n
(V∗)ofn-covectors onV is one dimensional.
Lemma 20.1.Letu1, . . . , unandv1, . . . , vnbe vectors in a vector spaceV. Suppose uj =
n i=1
aijvi, j =1, . . . , n,
for a matrixA= [aij]of real numbers. Ifωis ann-covector onV, then ω(u1, . . . , un)=(detA) ω(v1, . . . , vn).
Proof. By hypothesis,
uj = aijvi. Sinceωisn-linear,
ω(u1, . . . , un)=ω
ai11vi1, . . . , ainnvin
=
ai11ã ã ãainnω(vi1, . . . , vin).
Forω(vi1, . . . , vin)to be nonzero,i1, . . . , inmust all be distinct. This meansi1, . . . , in is a permutation of 1, . . . , n. Sinceωis an alternatingn-tensor,
ω(vi1, . . . , vin)=(sgni)ω(v1, . . . , vn).
Thus,
ω(u1, . . . , un)=
i∈Sn
(sgni)ai11ã ã ãainnω(v1, . . . , vn)
=(detA) ω(v1, . . . , vn).
As a corollary,
sgnω(u1, . . . , un)=sgnω(v1, . . . , vn) iff detA >0
iffu1, . . . , unandv1, . . . , vnare equivalent ordered bases.
We say that the n-covector ωrepresents the orientation (v1, . . . , vn)if ω(v1, . . . , vn) >0. By the preceding corollary, this is a well-defined notion, independent of the choice of ordered basis for the orientation. Moreover, twon-covectorsωand ωonV represent the same orientation if and only ifω=aωfor some positive real numbera.
An isomorphism.n
(V∗) Ridentifies the set of nonzeron-covectors on V withR− {0}, which has two connected components. Two nonzeron-covectorsω andωonV are in the same component if and only ifω=aωfor some real number a >0. Thus, each connected component of.n
(V∗)− {0}represents an orientation onV.
Example20.2.Lete1, e2be the standard basis forR2andα1, α2its dual basis. Then the 2-covectorα1∧α2represents the counterclockwise orientation onR2since
α1∧α2
(e1, e2)=1>0.
Example20.3.Let∂/∂x|p, ∂/∂y|pbe the standard basis for the tangent spaceTp(R2), and(dx)p, (dy)pits dual basis. Then(dx)p∧(dy)prepresents the counterclockwise orientation onTp(R2).
We define an equivalence relation on the nonzeron-covectors on then-dimensional vector spaceV as follows:
ω∼ω iff ω=aω for somea >0.
Then an orientation onV is also given by an equivalence class of nonzeron-covectors onV.
20.3 Orientations on a Manifold
Every vector space of dimensionnhas two orientations, corresponding to the two equivalence classes of ordered bases or the two equivalence classes of nonzeron- covectors. To orient a manifoldM, we orient the tangent space at each pointp∈M.
This can be done by simply assigning a nonzeron-covector to each point ofM, in other words, by giving a nowhere-vanishingn-form onM. The assignment of an orientation at each point must be done in a “coherent’’ way, so that the orientation does not change abruptly in a neighborhood of a point. The simplest way to guarantee this is to require that then-form onMspecifying the orientation at each point beC∞. (It is enough to require that then-form be continuous, but we prefer working with C∞forms in order to apply the methods of differential calculus.)
20.3 Orientations on a Manifold 205 Definition 20.4.A manifoldMof dimensionnisorientableif it has aC∞nowhere- vanishingn-form.
Ifωis aC∞ nowhere-vanishingn-form onM, then at each pointp ∈ M the n-covectorωppicks out an equivalence class of ordered bases for the tangent space TpM.
Example20.5.The Euclidean spaceRnis orientable as a manifold, because it has the nowhere-vanishingn-formdx1∧ ã ã ã ∧dxn.
Ifωandωare twoC∞nowhere-vanishingn-forms on a manifoldMof dimen- sionn, thenω=f ωfor aC∞nowhere-vanishing functionfonM. On aconnected manifoldM, such a functionf is either everywhere positive or everywhere nega- tive. Thus, theC∞nowhere-vanishingn-forms on a connected manifoldMcan be partitioned into two equivalence classes:
ω∼ω iff ω=f ω withf >0.
We call either equivalence class anorientationon the connected manifold M. By definition a connected manifold has exactly two orientations.
If a manifold is not connected, each connected component can have one of two possible orientations. We call aC∞nowhere-vanishingn-form onMthat specifies the orientation ofManorientation form. Anoriented manifoldis a pair(M,[ω]), where Mis a manifold of dimensionnand[ω]is an orientation onM, i.e., the equivalence class of a nowhere-vanishingn-formωonM. We sometimes writeM, instead of (M,[ω]), for an oriented manifold, if it is clear from the context what the orientation is. For example, unless otherwise specified,Rnis oriented bydx1∧ ã ã ã ∧dxn. Remark20.6 (Orientations on a zero-dimensional manifold).A zero-dimensional manifold is a point. According to the definition above, a zero-dimensional manifold is always orientable. Its two orientations are represented by the two numbers±1.
A diffeomorphismF:(N,[ωN])−→(M,[ωM])of oriented manifolds is said to beorientation-preservingif[F∗ωM] = [ωN]; it isorientation-reversingif[F∗ωM] = [−ωN].
Proposition 20.7.LetU and V be open subsets of Rn. AC∞ mapF: U −→ V is orientation-preserving if and only if the Jacobian determinantdet[∂Fi/∂xj]is everywhere positive onU.
Proof. Letx1, . . . , xn andy1, . . . , ynbe the standard coordinates onU ⊂Rn and V ⊂Rn. Then
F∗(dy1∧ ã ã ã ∧dyn)=d(F∗y1)∧ ã ã ã ∧d(F∗yn)
=d(y1◦F )∧ ã ã ã ∧d(yn◦F )
=dF1∧ ã ã ã ∧dFn
=det ∂Fi
∂xj
dx1∧ ã ã ãdxn (by Problem 19.2).
Thus,F is orientation-preserving if and only if det[∂Fi/∂xj]is everywhere positive onU.
20.4 Orientations and Atlases
Definition 20.8.An atlas onM is said to be oriented if for any two overlapping charts (U, x1, . . . , xn)and(V , y1, . . . , yn)of the atlas, the Jacobian determinant det[∂yi/∂xj]is everywhere positive onU∩V.
Proposition 20.9.A manifoldMof dimensionnhas aC∞nowhere-vanishingn-form ωif and only if it has an oriented atlas.
Proof.
(⇐)Given an oriented atlas{(Uα, xα1, . . . , xαn)}α∈A, let{ρα}be aC∞partition of unity subordinate to{Uα}. Define
ω=
ραdxα1∧ ã ã ã ∧dxαn. (20.1) For anyp∈ M, there is an open neighborhoodUpofpthat intersects only finitely many of the sets suppρα. Thus (20.1) is a finite sum onUp. This shows thatωis defined andC∞at every point ofM.
Let(U, x1, . . . , xn)be one of the charts aboutpin the oriented atlas. OnUα∩U, by Problem 19.2,
dxα1∧ ã ã ã ∧dxαn=det ∂xαi
∂xj
dx1∧ ã ã ã ∧dxn, where the determinant is positive because the atlas is oriented. Then
ω=
ραdxα1∧ ã ã ã ∧dxαn= ραdet
∂xαi
∂xj
dx1∧ ã ã ã ∧dxn. In the last sumρα ≥0 and det[∂xαi/∂xj]>0 atpfor allα. Moreover,ρα(p) >0 for at least oneα. Hence,
ωp=(positive number)ì(dx1∧ ã ã ã ∧dxn)p =0.
Aspis an arbitrary point ofM, then-formωis nowhere-vanishing onM.
(⇒)Supposeωis aC∞nowhere-vanishingn-form onM. Given an atlas forM, we will useωto modify the atlas so that it becomes oriented. Without loss of generality, we may assume that all the open sets of the atlas are connected.
On a chart(U, x1, . . . , xn),
ω=f dx1∧ ã ã ã ∧dxn
for aC∞functionf. Sinceωis nowhere-vanishing andf is continuous,f is either everywhere positive or everywhere negative onU. If f > 0, we leave the chart
20.4 Orientations and Atlases 207 alone; iff <0, we replace the chart by(U,−x1, x2, . . . , xn). After all the charts have been checked and replaced if necessary, we may assume that on every chart (V , y1, . . . , yn),
ω=h dy1∧ ã ã ã ∧dyn
withh > 0. This is an oriented atlas, since if(U, x1, . . . , xn)and(V , y1, . . . , yn) are two charts, then onU∩V
ω=f dx1∧ ã ã ã ∧dxn=h dy1∧ ã ã ã ∧dyn
withf, h >0. By Problem 19.2,f/ h=det[∂yi/∂xj]. It follows that det[∂yi/∂xj]
>0 onU∩V.
Definition 20.10.Two oriented atlases{(Uα, φα)}and{(Vβ, ψβ)}on a manifoldM are said to beequivalentif the transition functions
φα ◦ψβ−1:ψβ(Uα∩Vβ)−→φα(Uα∩Vβ) have positive Jacobian determinant for allα, β.
It is not difficult to show that this is an equivalence relation on the set of oriented atlases onM(Problem 20.1).
SupposeMis a connected orientable manifold. To each oriented atlas{(Uα, φα)} and partition of unity{ρα}subordinate to{Uα}, we associate the nowhere-vanishing n-form
ω=
ραdxα1∧ ã ã ã ∧dxαn
on M as in (20.1). In this way, equivalent oriented atlases give rise to equiva- lent nowhere-vanishing n-forms (Problem 20.2). Since there are two equivalence classes of oriented atlases and two equivalence classes of nowhere-vanishing n- forms on M, this construction is a map from {±1}to{±1}. If the oriented atlas {(Uα, x1α, xα2, . . . , xαn)} gives rise to the n-form ω, then by switching the sign of just one coordinate, we get an oriented atlas, for example,{(Uα,−xα1, xα2, . . . , xαn)}, that gives rise to then-form−ω. Hence, the map: {±1} −→ {±1} is surjective and therefore a bijection. This shows that an orientation on a connectedM may also be specified by an equivalence class of oriented atlases. By considering each connected component in turn, we can extend to an arbitrary orientablen-manifold the correspondence between eqivalence classes of oriented atlases and equivalence classes of nowhere-vanishingn-forms. More formally, we say that anoriented atlas {(Uα, φα)} = {(Uα, xα1, . . . , xαn)}givesorspecifiesthe orientation of an orientedn- manifold(M,[σ])if for everyα, there is an everywhere positive functionfαonUα such that
σ =fαφα∗(dr1∧ ã ã ã ∧drn)=fαdx1α∧ ã ã ã ∧dxαn. Herer1, . . . , rnare the standard coordinates on the Euclidean spaceRn.
Ifωis a nowhere-vanishingn-form that orients a manifoldM, then on anycon- nectedchart(U, x1, . . . , xn), there is by continuity an everywhere positive or every- where negative functionf such that
ω=f dx1∧ ã ã ã ∧dxn.
Thus, on an oriented manifold the orientation at one point of a connected chart deter- mines the orientation at every point of the chart.
Example20.11 (The open Mửbius band). LetRbe the rectangle R= {(x, y)∈R2|0≤x ≤1, −1< y <1}
(see Figure 20.5). The open Mửbius bandM(Figure 20.5 and 20.6) is the quotient of the rectangleRby the equivalence relation
(0, y)∼(1,−y). (20.2)
The interior ofRis the open rectangle
U= {(x, y)∈R2|0< x <1, −1< y <1}.
e1 e2
e1 e2
e1 e2
Fig. 20.5.Nonorientability of the Mửbius band.
An orientation onMrestricts to an orientation onU. To avoid confusion with an ordered pair of numbers, in this example we write an ordered basis without the parentheses. Without loss of generality we may assume the orientation onU to be e1, e2. By continuity the orientation at the points(0,0)and(1,0)are alsoe1, e2. But under the identification (20.2), the ordered basise1, e2at(1,0)maps toe1,−e2
at (0,0). Thus, at (0,0)the orientation has to be bothe1, e2 ande1,−e2. This contradiction proves that the Mửbius band is not orientable.
Example20.12.By the regular level set theorem, if 0 is a regular value of a C∞ functionf (x, y, z)onR3, then the setM=f−1(0)=Zero(f )is aC∞manifold.
In Problem 19.10 we constructed a nowhere-vanishing 2-form onM. Thus, Mis orientable. Combined with Example 20.11 it follows that an open Mửbius band cannot be realized as a regular level set of aC∞function onR3.
Problems
20.1. Equivalence of oriented atlases
Show that the relation in Definition 20.10 is an equivalence relation.
20.4 Orientations and Atlases 209
Fig. 20.6.Mửbius band.
20.2.* Equivalent nowhere-vanishingn-forms
Show that equivalent oriented atlases give rise to equivalent nowhere-vanishingn- forms.
20.3. Orientation-preserving diffeomorphisms
Let F: (N, ωN) −→ (M, ωM) be an orientation-preserving diffeomorphism. If {(V , ψ )} = {(V , y1, . . . , yn)}is an oriented atlas onMthat specifies the orienta- tion ofM, show that{(F−1V , F∗ψ )} = {(F−1V , F1, . . . , Fn)}is an oriented atlas onNthat specifies the orientation ofN, whereFi =yi ◦F.
20.4. Orientability of a regular level set inRn+1
Supposef (x1, . . . , xn+1)is aC∞function onRn+1with 0 as a regular value. Show that the zero set off is an orientable surface inRn+1.
20.5. Orientability of a Lie group
Show that every Lie groupGis orientable by constructing a nowhere-vanishing top form onG.
20.6. Orientability of a parallelizable manifoldShow that a parallelizable manifold is orientable. (In particular, this shows again that every Lie group is orientable.) 20.7. Orientability of the total space of the tangent bundle
LetM be a smooth manifold andπ: T M −→ M its tangent bundle. Show that if {(U, φ)}is any atlas onM, then the atlas{(T U,φ)˜ }onT M, withφ˜defined in equation (12.1), is oriented. This proves that the total spaceT Mof the tangent bundle is always orientable, regardless of whether or notMis orientable.
21
Manifolds with Boundary
The prototype of a manifold with boundary is theclosed upper half-space Hn = {(x1, . . . , xn)∈Rn|xn≥0},
with the subspace topology ofRn. The points(x1, . . . , xn)∈Hnwithxn >0 are called theinterior pointsofHn, and the points withxn=0 are called theboundary pointsof Hn. These two sets are denoted int(Hn) and∂(Hn), respectively (Fig- ure 21.1).
xn
int(Hn)
∂(Hn)
Fig. 21.1.Upper half-space.
In the literature the upper half-space often means the open set {(x1, . . . , xn)∈Rn|xn>0}.
We require that Hn include the boundary in order for it to serve as a model for manifolds with boundary.
21.1 Invariance of Domain
To discussC∞functions on a manifold with boundary, we need to extend the domain of definition of aC∞function to nonopen subsets.
212 21 Manifolds with Boundary
Definition 21.1.Let S ⊂ Rn be an arbitrary subset. A functionf: S −→ Rm is smooth at a point p in S if there exist a neighborhood U of p inRn and a C∞ functionf˜:U −→Rmsuch thatf˜=f onU∩S. The function issmooth onSif it is smooth at each point ofS.
With this definition it now makes sense to speak of an arbitrary subsetS ⊂Rn being diffeomorphic to an arbitrary subsetT ⊂Rm; this will be the case if and only if there are smooth mapsf:S−→T andg:T −→Sthat are inverse to each other.
Exercise 21.2 (Smooth functions on a nonopen set).Using a partition of unity, show that a functionf:S−→RmisC∞onSif and only if there exists an open setUinRncontainingS and aC∞functionf˜:U−→Rmsuch thatf = ˜f|S.
Theorem 21.3 (C∞invariance of domain). LetU⊂Rnbe an open subset,S⊂Rn an arbitrary subset, andf:U−→Sa diffeomorphism. ThenSis open inRn.
A diffeomorphismf:U −→Stakes an open set inUto an open set inS. Thus, a priori we know only thatf (U )is open inS, not thatf (U )is open inRn. Because f is ontof (U )=S.
Proof. Letp ∈ U. Sincef:U −→ S is a diffeomorphism, there is an open setV containingSand aC∞mapg:V −→Rnsuch thatg|S=f−1. Thus,
U −→f V −→g Rn satisfies
g◦f =1U:U −→U⊂Rn, the identity map onU. By the chain rule,
g∗,f (p)◦f∗,p=1TpU :TpU −→TpUTp(Rn),
the identity map on the tangent spaceTpU. Hence,f∗,pis invertible. By the inverse function theorem, f is locally invertible at p. This means there are open neigh- borhoodsUp of p inU andVf (p) of f (p) inV such that f:Up −→ Vf (p) is a diffeomorphism. It follows that
Vf (p)⊂f (U )=S.
Hence,Sis open inRn.
Proposition 21.4.LetU and V be open subsets ofHn and f:U −→ V a diffeo- morphism. Thenf maps interior points to interior points and boundary points to boundary points.
Proof. Letp∈Ube an interior point. Thenpis contained in an open ballB, which is actually open inRn(not just inHn). By the invariance of domain,f (B)is open in Rn(again not just inHn). Since