Solution 5

Due on 10/26

Problem 1.
Proof. For (a): Given a trivialization {(Uα , hα )}α∈I of the vector bundle π : V → M , we define the trivial-
ization of Λtop (V ) as follows. For each α, the local sections (or local frame) of V is defined by
−1
sα k
i (p) = hα (p, ei ), for hα : V |Uα → Uα × R ,

here ei is the i-th standard basis of Rk for i = 1, · · · , k. Then locally over Uα , the wedge product sα α
1 ∧ · · · ∧ sk
defines a nowhere section of Λtop (V ) := Λk (V ) over Uα . Then we can trivialize Λtop (V ) over Uα by

h̃α : Λtop (V )|Uα → Uα × R, f (p)sα α

1 |p ∧ · · · ∧ sk |p 7→ (p, f (p)).

This defines a bundle isomorphism because it’s fiber-wise linear and an isomorphism for all p ∈ Uα .
To obtain the transition data for Λtop (V ), suppose that (Uβ , hβ ) and (Uβ , h̃β ) are trivializations of V and
Λtop (V ) over Uβ respectively. Then on the double overlaps Uα ∩ Uβ , the transition data gαβ : Uα ∩ Uβ →
GL(n, k) of V is given by

hα ◦ h−1 k k
β : Uα ∩ Uβ × R → Uα ∩ Uβ × R , (p, v) 7→ (p, gαβ (v)).
Pk β
This implies that for a fixed i, we have sα
i = j=1 gji sj , where gji is the (j, i)-th component of the (k × k)
∞
C (Uαβ )-valued matrix gαβ . Then we compute the data functions g̃αβ for Λk (V ) as follows
k
X Xk
sα α
1 ∧ · · · ∧ sk = ( gj1 sβj ) ∧ · · · ∧ ( gjk sβj )
j=1 j=1
X
= gj1 1 · · · · · gjk k sβj1 ∧ · · · ∧ sβjk
(j1 ,··· ,jk )∈Sk
X
= sgn(j1 , · · · , jk )gj1 1 · · · · · gjk k sβ1 ∧ · · · ∧ sβk
(j1 ,··· ,jk )∈Sk

= det(gαβ )sβ1 ∧ · · · ∧ sβk ,

where sgn(j1 , · · · , jk ) is the sign of the permutation which maps (j1 , · · · , jk ) to (1, 2, · · · , k). This shows
that det(gαβ ) is the transition data for Λk (V ).
For (b): First, for real vector spaces π −1 (p) = Vp and π −1 (p) = Wp of dimension k and l, we have a canonical
isomorphism
∼
=
Λk+l (Vp ⊕ Wp ) −
→ Λk (Vp ) ⊗ Λl (Wp ),
v1 ∧ · · · ∧ vk ∧ w1 ∧ · · · ∧ wk 7→ (v1 ∧ · · · ∧ vk ) ⊗ (w1 ∧ · · · ∧ wk ),
where {vi }ki=1
and {wi }li=1 are basis of Vp and Wp . This maps sends basis to basis so it defines an isomorphism
for each p ∈ M. Locally on each Uα , we can define a local vector bundle isomorphism
∼
=
Λtop (V ⊕ W ) −
→ Λk (V ) ⊗ Λk (W ),

sV1 ∧ · · · ∧ sVk ∧ sW W V V W W
1 ∧ · · · ∧ sl 7→ (s1 ∧ · · · ∧ sk ) ⊗ (s1 ∧ · · · ∧ sl ),

where {sVi }ki=1 and {sW l

i }i=1 are local frame of V and W over Uα as defined in part (a). To see, this vector
bundle isomorphism gives rise to a well-defined global vector bundle isomorphism, it suffice to check that

1
 V W
the transition data of Λtop (V ⊗ W ) and Λtop (V ) ⊗ Λtop (W ). Let gαβ and gαβ be the transition data of V
and W . Then we know the transition data for (V ⊕ W ) is
 V 
V ⊕W g 0
gαβ = αβ W
0 gαβ ,

So the transition data for Λtop (V ⊕ W ) is

V ⊕W V W
det(gαβ ) = det(gαβ ) det(gαβ ) (0.1)

On the other hand, we use the fact that the transition data of a tensor product is the product of the transition
V W
data (proof omitted), that is, det(gαβ ) det(gαβ ) as well. This shows that the vector bundle isomorphism is
well-defined.
For (c): By definition of the normal bundle to immersion, we have that
˜)∼
NRn f ⊕ Im(df = NRn f ⊕ T M ∼
= f ∗ (T Rn ),
˜ is given by the universal property of pullback. By part (b), we have that as line bundles
where df

Λtop (f ∗ T Rn ) = Λtop (NRn f ) ⊗ Λtop (T M ).

Now since T Rn ∼ = Rn × Rn is trivial, we have f ∗ T Rn is also trivial and the transition function associated to
∗ n
f T R is in fact given by multiplication by ·1 for all α, β ∈ I. The fact that M is orientable is equivalent
to Λn (T M ) is trivial. This implies the transition functions of the line bundle Λn (T M ) can be chosen to be
0
gαβ (p) > 0 for all α, β ∈ I and p ∈ Uα ∩ Uβ . Hence the transition function gαβ associated to NRn f satisfies
0
gαβ (p) > 0 for all α, β ∈ I and p ∈ Uα ∩ Uβ due to equation (0.1). Therefore, we conclude that NRn f is
orientable if and only if M is orientable.

2
 Problem 2.
(a) Let ι : S n → Rn+1 be the inclusion map and ã : Rn+1 → Rn+1 , x = (x0 , · · · , xn ) 7→ −x =
(−x1 , · · · , −xn ) be the antipodal map on Rn+1 . Then the antipodal map on S n is defined by the
commutative diagram
Sn
a / Sn

ι ι
 
Rn+1
ã / Rn+1
The standard volume form on Rn+1 is given by Ω = dx0 ∧ dx1 ∧ · · · ∧ dxn . One can obtain the standard
volume form on S n by considering α := ι∗ (iX Ω), where iX Ω is the contraction of Ω by the radial vector
n
X ∂
field X = xi . This in fact defines a non-vanishing n-form on S n because the defining function
i=0
∂x i
of S n
n
X
f : Rn+1 → R, (x0 , · · · , xn ) 7→ x2i
i=0
n n+1 n
Pnon S ∂⊂ R
is constant Pnhave2 that dp f (v) = 0 for all vn ∈ Tp S . Whereas
. We / Tp S n because
Xp (f ) ∈
n
X(f ) = i=0 xi ∂xi (f ) = 2 i=0 xi , so we know Xp ∈ / Tp S for all p ∈ S . The non-vanishing of Ωp
for all p ∈ Rn+1 means that Ωp evaluates to nonzero on any set of (n + 1) linearly independent vectors
in Tp Rn+1 ∼= Rn+1 . In particular, one can take the first vector to be Xp and the remaining n vectors
to be a basis of Tp S n , then we conclude that (ι∗ (iX Ω))p is non-vanishing for all p ∈ S n and hence α is
a volume form on S n .
Easy computations show that ã∗ Ω = (−1)n+1 Ω and dã(X) = X on Rn+1 . By commutativity of the
above diagram, we have that
a∗ α = a∗ ι∗ (iX Ω) = ι∗ ã∗ (idã(X) Ω) = ι∗ (iX (ã∗ Ω)) = (−1)n+1 ι∗ (iX Ω) = (−1)n+1 α, (0.2)
here we used the fact that contraction commutes with pullback by diffeomorphisms, i.e. f ∗ (idf (X) ω) =
iX (f ∗ ω) (and it does not commute with pullback by arbitrary smooth maps). Equation (0.2) implies
that a : S n → S n is orientation-preserving if and only if (n + 1) is even, i.e., n is odd.
(b) Before investigating orientability of RP n for different n, we first make the following general observation.
Let M̃ be a smooth manifold M and G be a group that acts on M properly discontinuously by
diffeomorphisms. In Problem 3 of Homework 1, we have shown that the quotient M := M̃ /G has a
smooth structure making q : M̃ → M a smooth map. We first prove the following Lemma.
Lemma 0.1. There is a bijection between sets of (G-invariant) differential forms
Ω∗ (M̃ )G := {α̃ ∈ Ω∗ (M̃ ) | g ∗ α̃ = α̃, ∀g ∈ G} = {q ∗ α | α ∈ Ω∗ (M )}.

Proof. To see ⊃: For α ∈ Ω∗ (M ), because q ◦ g = q for all g ∈ G we have that

q ∗ α = (q ◦ g)∗ α = g ∗ (q ∗ α),
which implies that {q ∗ α | α ∈ Ω∗ (M )} ⊂ Ω∗ (M̃ )G .
To see ⊂: Suppose α̃ ∈ Ω∗ (M̃ )G , we define α ∈ Ω∗ (M ) such that q ∗ α = α̃ as follows.
If p ∈ M is a point in the quotient, we choose p̃ in the preimage q −1 (p) of p under q and we choose
neighborhoods U ⊂ M and Ũ ⊂ M̃ of p and p̃ such that q : Ũ → U restricts to a diffeomorphism. Now
use the fact that q|Ũ is a diffeomorphism, we define αp0 ∈ Λ∗ Tp∗0 M by

(q|Ũ )∗ (αp0 ) = α̃p̃0 for all p0 ∈ U and p̃0 ∈ q −1 (p0 ) ⊂ Ũ .

This definitions depends on the choice of p̃. One can check that if p̃0 ∈ q −1 (p) is an another point in
the preimage, there exists g ∈ G such that g p̃0 = p̃, We define Ũ 0 = g −1 (Ũ ). Since q ◦ g = q for all
g ∈ G, we have that
(q|Ũ 0 )∗ (αp0 ) = (q|Ũ ◦ g)∗ (αp0 ) = g ∗ (q|∗Ũ (αp0 )) = g ∗ (α̃p̃0 ) = α̃p̃00 for p̃0 , p̃00 ∈ q −1 (p0 ).

3
This implies that α is well-defined. Because the map q|Ũ is a diffeomorphism, we know αp0 varies
smoothly and the form α is a smooth form such that q ∗ α = α̃ by construction. This completes the
proof of Lemma

Now in the case of RP n , the antipodal map a : S n → S n satisfies a ◦ a = idS n . It defines a smooth
Z/2Z-action and the quotient RP n := S n /Z/2Z is the real projective space. If n is odd, the n-form
a∗ α = (−1)n+1 α = α. By the above lemma , we conclude that α is a Z/2Z-invariant form and it
defines a non-zero n-form on the quotient RP n . So RP n is orientable when n is odd. If n is even, we
will show that there does not exist a non-zero n-form β ∈ Ωn (S n )Z/2Z . Suppose by contradiction that
such β ∈ Ωn (S n )Z/2Z exits, then because α ∈ Ωn (S n ) we can write β = f α for some f ∈ C ∞ (S n ) and

f α = β = a∗ β = (f ◦ a)a∗ α = −(f ◦ a)α =⇒ f ◦ a = −f,

which implies f must vanish somewhere on S n and therefore β cannot be non-vanishing. So RP n is

non-orientable for n even.

4
 Problem 3.
Proof. Let {(Uα ), ϕα }α∈I be a collection of charts that defines the smooth structure of M . For each α, we
have local coordinates ϕα = (x1 , · · · , xn ) : Uα → Rn on Uα , then Ωtop (M )−0 has two components containing
ν α := dx1 ∧ · · · ∧ dxn and −ν α := −dx1 ∧ · · · ∧ dxn . We define an equivalence class on Ωtop (M ) − 0 by
claiming that α ∼ β if α = f β for some smooth function f > 0. Then we see that as sets M̃ |Uα = π −1 (Uα ) ∼ =
Uα × {±1}, where +1 denotes the orientation class corresponding to [ν α ] and −1 denotes the orientation
class corresponding to [−ν α ]. One can define a topology on M̃ by declaring π is a local homeomorphism.
One checks that M̃ is a second countable and Hausdorff space because M is. Then the smooth structure
on M̃ can be defined as follows. For any p ∈ M̃ , there is a chart (Uα , ϕα ) of M which contains π(p). Pick
the component of π −1 (p) containing p and call it Ũα . We define the chart (Ũα , ϕ̃α ) of M̃ containing p by
setting ϕ̃α = ϕα ◦ π. With respect to this smooth structure, the projection π is smooth and in fact a local
diffeomorphism since π|Ũα : Uα × {+1} → Uα and π|Ũα : Uα × {−1} → Uα are diffeomorphisms in either
cases. This makes π into a smooth 2-sheeted covering space.
In fact, as a smooth manifold M̃ is diffeomorphic to the sphere bundle S(Λn (T ∗ M )) of Λn (T ∗ M ) defined
as follows. By Proposition 0.3 proved in Problem 4 below, every real vector bundle V admits a Riemannian
metric. Then the sphere bundle is defined by
S(V ) = {v ∈ V |hv, vig = 1}.
This is called the sphere bundle because when the rank of V is k, the fiber of this (fiber) bundle is diffeo-
morphic to S k−1 . In the case that the vector bundle V is the top exterior line bundle Λtop (M ) := Λn (T ∗ M ),
its fiber at p is given by S(Λtop (M ))p ∼
= S0 ∼= {±1}. [Explicitly, one can check that there are local diffeo-
morphisms
Ψα : M̃ |Uα → S(Λtop (M ))|Uα , (p, ±1) 7→ (p, (±νpα )/||νpα ||g ).
The transition data associated to M̃ and S(Λtop (M )) is given by
det : Uα ∩ Uβ → Z/2Z ∼
= {±1},
where ±1 ∈ O(1) denotes the multiplication by ±1. One checks Ψα gives a well-defined diffeomorphism on
double overlaps Uα ∩ Uβ (details omitted here)] To see M̃ ∼ = S(Λtop (M )) is orientable, it suffices to observe
that Λtop (S(V )) has a nowhere-zero smooth section defined locally by
(
α νpα /||νpα ||, if q = +1;
ν(p,q) =
−νpα /||νpα ||, if q = −1, for p ∈ Uα
Because this local section is invariant under the Z/2Z-action on S(Λtop (M )) given by (p, q) 7→ (p, −q), it
implies that ν α and ν β agrees on double overlaps Uα ∩ Uβ and it gives a well-defined nonzero glocal section
of Λtop (S(V )).
To finish, we prove the following statement.
Proposition 0.2. Let M be a connected smooth manifold and π : V → R be a real line bundle. The sphere
bundle S(V ) is not connected if and only V is orientable.
The case of V = Λtop (M ) = Λn (T ∗ M ) follows from the Proposition because the smooth manifold M is
orientable if and only if Λtop (M ) is orientable by definition.

We prove the Proposition (0.2) as follow. Suppose that V is orientable, then V ∼

= Λtop (V ) as a line
bundle is isomorphic to V ∼
= M × R. This implies that
a
S(V ) = S(M × R) = M × S 0 = M M
is not connected. Conversely, if M is connected and S(V ) is not connected, then we denote by S(V )+
one of the components of S(V ). Because π : S(V ) → M is a 2-sheeted covering map, the restriction to
−1
π|S(V )+ : S(V )+ → M is a bijective smooth covering map, hence it is a diffeomorphism. The inverse πS(V )+
determines a nowhere-zero section of V . Thus, V is isomorphic to the trivial line bundle M × R. This gives
the proof of the Proposition.

5
 Problem 4. [Hodge star operation & the divergence Theorem for Riemannian manifolds]
(a) We will prove the following proposition and part (a) follows as a special case.
Proposition 0.3. Every real vector bundle π : V → M admits a Riemannian metric.

Proof. Let π : V → M be a real vector bundle of rank k. Because M is paracompact by Lemma 1.9 in
Warner. We can choose a locally finite open cover U = {Uα }α∈I of M such that Uα is diffeomorphic
to Rn . We fix a trivializations {(Uα , hα )}α∈I of V . Let {ϕα }α∈I be the partition of unity subordinate
to the open cover U , that is, a collection of smooth maps ϕα satisfying
X
ϕα : M → [0, 1] and ϕα (p) ≡ 1, ∀p ∈ M
α∈I

and supp(ϕα ) = {p ∈ M | ϕα (p) 6= 0}.

Pk
We denote by h·i the standard inner-product on Rk given by hx, yi := xT · y = i=1 xi · yi for
x = (x1 , · · · , xn ) and y = (y1 , · · · , yn ). Then the inner product on V |Uα can be defined by
(
ϕα hhα (x), hα (y)i, if x, y ∈ Vp := π −1 (p), p ∈ Uα ;
hx, yiα := .
0, if x, y ∈ Vp , p ∈ M − supp(ϕα )

Since h, iα varies smoothly over the open sets Uα , M − supp(ϕα ) by construction, and h, iα agrees on
the overlaps Uα ∩ Uβ for all α, β ∈ I. We conclude that h, iα glues to a well-defined smooth inner
product on V . Explicitly, we define
X
hx, yi = hx, yiα , ∀x, y ∈ Vp and p ∈ M.
α∈I

We check that this in fact defines a Riemannian metric on V . Because by definition h, i is a symmetric
bilinear form. To see the degeneracy
P condition hx, xiα ≥ 0 for all α ∈ I and the sum is locally finite,
we conclude that hx, xi = α∈I hx, xiα ≥ 0. Also for all p ∈ M and x ∈ Vp − {0}, we have that
X
hx, xiα ≥ hx, xiβ > 0 for some β such that ϕβ (p) > 0,
α∈I

where we used the fact that hx, xiβ > 0 for x ∈ Vp − {0}. This completes the proof.

(b) (1) We prove that if e1 , · · · , en is an orthonormal basis for V , then we have

[
{1} ∪ {ei1 ∧ · · · ∧ eik }1≤i1 <···<ik ≤n
1≤k≤n

is an orthonormal basis for Λ(V ).

Proof. First, we notice that if the two basis vectors are of different degree i.e. if k 6= l, then
hei1 ∧ · · · ∧ eik , ei1 ∧ · · · ∧ eil i = 0 by definition. For k = l, we have that all basis vector are of unit
length because

hei1 ∧ · · · ∧ eik , ei1 ∧ · · · ∧ eik i := det((heir , eis i)r,s=1,··· ,k ) = det(δir ,is )r,s=1,··· ,k = det(Ik ) = 1.

If (i1 , · · · , ik ) 6= (j1 , · · · , jk ), we have that

hei1 ∧ · · · ∧ eik , ej1 ∧ · · · ∧ ejk i := det((heir , ejs i)r,s=1,··· ,k ) = det(δir ,js )r,s=1,··· ,k .

Suppose R is the smallest number such that iR 6= jR . If iR < jR , then we know that δiR ,is for
all s = 1, · · · , k. This is because for s < R, we have is < iR and by definition of R one can find
js such that js = is < iR . This implies that δiR ,is = 0 for s < R. Similarly if R ≥ s, we know

6
 that iR < jR ≤ js and δiR ,is = 0 for s ≥ R. In summary, if iR < jR , then the entries in the
R-th row of (δir ,js )r,s=1···k are all zero. Similarly, if iR > jR , then we show that the entries in the
R-th column are all zero. Since in either case we have an entire row (resp. column) in the matrix
(δis ,jr )s,r=1···k are zero, hence the determinant is zero, that is,

hei1 ∧ · · · ∧ eik , ej1 ∧ · · · ∧ ejk i = det(δir ,js )r,s=1,··· ,k = 0.

This completes the proof of part (1).

(2) We prove that ∗∗ = (−1)k(n−k) on Λk (V ).

Proof. We first show that the star operator ∗ : Λ(V ) → Λ(V ) as defined on the set of orthonormal
basis of Λ(V ) exits and is well-defined. Since we have fixed an orientation, that is, we fix a unique
unit vector ν in the chosen component of Λn (V ) − 0. With respect to this orientation, we can
define a bilinear map

A : Λ(V ) × Λ(V ) → R, v ∧ w = A(v, w)ν if v ∧ w ∈ Λn (V ) and A(v, w) = 0 otherwise.

This bilinear pairing is non-degenerate (non-singular) since for subsets I, J ⊂ {1, 2, · · · , n}, we
have that
eI ∧ (∗eJ ) = ±δIJ ν, or A(eI , ∗eJ ) = ±δIJ .
Such non-degenerate pairing defines an isomorphism between Λ(V ) and its dual vector space by

FA : Λ(V ) → (Λ(V ))∗ , FA (w)(v) = A(v, w), for v, w ∈ Λ(V ).

Also, the vector space Λ(V ) itself is equipped with an inner product

hwi1 ∧ · · · ∧ wik , vj1 ∧ · · · ∧ vjk i = det((hwir , vjs )r,s=1···k i).

One can define another non-degenerate bilinear pairing on Λ(V ) by

B : Λ(V ) × Λ(V ) → R, B(v, w) = hv, wi,

and it induces another isomorphism between Λ(V ) and Λ(V )∗

FB : Λ(V ) → Λ(V )∗ , (FB w)(v) = B(v, w), ∀v, w ∈ Λ(V ).

We claim that the star operator ∗ is defined by ∗ := FA−1 ◦ FB , explicitly,

FA−1 ◦ FB (eI ) = ∗eI , ∀eI := ei1 ∧ · · · ∧ eik ∈ Λ(V ) and all ordered subsets I ⊂ {1, 2, · · · , n}.

To prove the claim, it suffices to show that for all orthonormal basis vectors eI and eJ for Λ(V ),
we have

FA (∗eI ) = FB (eI ) ∈ Λ(V )∗ , or equivalently FA (∗eI )(eJ ) = FB (eI )(eJ ) ⇐⇒ A(eJ , ∗eI ) = heJ , eI i.

For any given k ≤ n and I = (i1 , · · · , ik ), after possible relabeling and reordering one can assume
(i1 , · · · , ik ) = (1, 2, · · · , k) as in Warner’s definition. For instance, the volume element ν can be
assumed to be e1 ∧ e1 ∧ · · · ∧ en after reordering. Then one verifies that if eJ = ej1 ∧ · · · ∧ ejk ,
then we have
eJ ∧ (∗eI ) = δJI e1 ∧ · · · ∧ en = δIJ ν,
for J = (ji , · · · , jk ) and I = (1, 2, · · · , k). This implies that A(eJ , ∗eI ) = δIJ because heJ , eI i = δIJ
by part (1). If eJ ∈ / Λk (V ), then trivially we have

heJ , eI i = 0, eJ ∧ (∗eI ) ∈
/ Λn (V ),

which implies A(eJ , ∗eI ) = 0. This proves the claim. In summary, we have shown that the star
operation is well-defined linear operator on Λ(V ) given by ∗eI := FA−1 ◦ FB (eI ) for orthonormal

7
 basis vectors eI , and it’s independent of the set of orthonormal basis that we choose for Λ(V ).
Now suppose that {ei }ni=1 is an oriented orthonormal basis of V , we check the sign as follows

ν = e1 ∧ · · · ∧ en = (ek+1 ∧ · · · ∧ en ) ∧ ((−1)k(n−k) e1 ∧ · · · ∧ ek )
=⇒ ∗ ∗ e1 ∧ · · · ∧ ek = ∗(ek+1 ∧ · · · ∧ en ) = (−1)k(n−k) e1 ∧ · · · ∧ ek
=⇒ ∗∗ = (−1)k(n−k) : Λk (V ) → Λk (V ).

This completes the proof of part (2).

(3) We prove that hv, wi = ∗(v ∧ ∗w) = ∗(w ∧ ∗v) for all v, w ∈ V, W.

Proof. For all v, w ∈ Λ(V ), we check that

∗(v ∧ ∗w) = A(v, ∗w) = (FA (∗w))(v) = (FB (w))(v) = B(v, w)

= hv, wi = hw, vi = B(w, v) = ∗(w ∧ ∗v),

where we used the fact that h, i is a symmetric bilinear form.

Given a Riemannian manifold (M, g), we define the Hodge star operator ∗ : Ωk (M ) → Ωn−k (M )
by (∗α)p = ∗(αp ) for α ∈ Ωk (M ). Then parts (1), (2) and (3) implies that the Hodge star satisfies

∗∗ = (−1)k(n−k) on Ωk (M ),

and defines a symmetric bilinear form on Ωk (M ) by

Z
Ωk (M ) × Ωk (M ) → R, (α, β) 7→ α ∧ ∗β.
M

Let (M, g) be a Riemannian manifold and let D ⊂ M be a regular domain with boundary ∂D in M .
We first show that there are well-defined Riemannian volume element dVg and dVg̃ on D and ∂D with
respect to Riemannian metrics g and g̃.
(c) Let U = {Uα }α∈I be a locally finite open cover such that each Uα is a local coordinate chart. Let
(xα α
1 , · · · , xn ) be the local coordinate on Uα . We know that the tangent bundle T M |Uα has a basis given
α n
by {∂/∂xi }i=1 . To obtain a local orthonormal frames of the tangent bundle, we apply the Gram-
Schmidt algorithm to the coordinate frame {∂/∂xα n α n
i }i=1 and obtain a new local frame {Ei }i=1 defined
inductively by the formula
Pj=1 α α
∂/∂xαi −
α
i=1 hEj , Ei ig Ei
Ejα = n=1 .
|∂/∂xα α α α
P
j − i=1 hEj , Ei ig Ei |g

Then the local Riemannian volume form is given by ν α := E1α ∧ E2α ∧ · · · ∧ Enα ∈ Ωn (Uα ). Moreover,
one can change of the sign of ν α if necessary and assume that ν α lies in the chosen component of
Ωn (Uα ) − 0 (here 0 is the zero section). Let {ϕα }α∈I be a partition of unity subordinate to the locally
finite open cover U , we define the Riemannian volume form by
X
dVg := ϕα ν α .
α∈I

Given a regular domain D ⊂ M , there is an induced smooth structure on D making it a smooth

manifold with boundary ∂D. Similarly, the interior of the domain D has an induced Riemannian
manifold structure given by

ι∗ g(X, Y ) = (g|int(D) )(X, Y ) = g(dι(X), dι(Y )) = g(X, Y ), X, Y ∈ Tp M, p ∈ int(D),

where ι : int(D) → M is the open immersion of int(D). The boundary of D as a codimension one
submanifold of M has an induced Riemannian metric

g̃(X, Y ) = (f ∗ g)(X, Y ) = g(df (X), df (Y )), for the inclusion f : ∂D → M, and X, Y ∈ Tp ∂D.

8
 This equips ∂D with a Riemannian manifold structure (∂D, g̃) . The above construction applies to ∂D
and gives rise to a Riemannian volume element dVg̃ with respect to the fixed orientation on ∂D induced
by those of M and the outward-pointing unit vector field N . One can verify (or see Proposition 10.37
in Lee’s book) that the Riemannian volume form dVg̃ satisfies
dVg̃ = f ∗ (iÑ (dVg )), where f : ∂D → M,
where Ñ is a smooth extension of N to M . Precisely, we have that N and Ñ are f -related, i.e.
df (N ) = f ∗ (Ñ ) and df is given by the universal property of pullback here.
For a Riemannian manifold (M, g), the divergent operator div : Γ(M, T M ) → C ∞ (M ) is defined by the
composition div(X) = ∗−1 d(iX (dVg )). (Here we used the definition of divergent operator as in John
Lee’s since it gives rise to a cleaner proof of the Divergent Theorem). Also, for an oriented Riemannian
manifold, the top forms Ωn (M ) ∼ = C ∞ (M ) · dVg . Under this isomorphism, the Hodge star defines a
canonical isomorphism
∗ : C ∞ (M ) → Ωn (M ), f 7→ f dVg .
We prove the orientable case of divergent Theorem as follows.
Theorem 0.4 (The Divergent Theorem). Let M be a compact oriented Riemannian manifold. Given
a regular domain D in M and a smooth vector field X with compact support containing D, we have
that Z Z
(divX)dVg = hX, N idVg̃ ,
D ∂D
where N is the outward-pointing unit normal vector field along ∂D.

Proof. By definition of div : Γ(M, T M ) → C ∞ (M ), we have that

Z Z Z Z
(div(X))dVg = ∗div(X) = d(iX (dVg )) = iX (dVg ),
D D D ∂D
R
where we apply the Stoke’s Theorem in the last step and the notation ∂D iX (dVg ) should be understood as
f ∗ iX (dVg ) for the closed embedding f : ∂D → M . To prove the divergent Theorem, it suffices to prove
R
∂D
that Z Z
∗
f (iX dVg ) = hX, N idVg̃ , or equivalently f ∗ iX (dVg ) = hX, N i|∂D f ∗ (iÑ (dVg )).
∂D ∂D
Lemma 0.5. Let f : ∂D → M and N be a fixed outward-pointing unit normal vector field along ∂D, we
have that
f ∗ iX (dVg ) = hX, N i|∂D f ∗ (iÑ (dVg )) ∈ Ωtop (∂D) − 0 for X ∈ Γ(T M ) (0.3)
For each p ∈ ∂D, since D is a smooth manifold with boundary, there is a chart φ̃ = (x1 , · · · , xn−1 , xn ) : U →
Rn of M which restricts to φ : U ∩ D → Hn and φ = (x1 , · · · , xn−1 ) : ∂D → Rn−1 . We denote by Xi = ∂/∂xi
the local coordinate frame of T ∂D|U for i = 1, · · · , n − 1, it suffices to check the left hand side of equation
(0.3)
(f ∗ (iX dVg ))(X1 , · · · , Xn−1 ) = (iX dVg )(df (X1 ), · · · , df (Xn−1 )).
Because f is an injective immersion, one can find smooth extension X̃i ∈ Γ(U, T M ) of Xi such that they
are f -related, that is, df (Xi ) = f ∗ (X̃i ). In the case that we have a manifold with boundary, we extend Xi
defined on U ∩ ∂D to X̃i = ∂/∂xi defined on U for i = 1, · · · , n − 1. Then the above equation becomes
(dVg )(X, X̃1 , · · · , X̃n−1 ).
For each p ∈ ∂D, the fact that N is outward pointing implies that Ñ , X̃1 , · · · , X̃n−1 form an ordered
basis of T pU when the neighborhood U of p is chosen to be sufficiently small. Hence we can write X =
a0 Ñ |+ a1 X̃1 + · · · an−1 X̃n−1 for some smooth functions ai on U ∩ ∂D and i = 0, 1 · · · , n − 1. By alternating
nature of dVg , we have that
(dVg )(a0 Ñ + a1 X̃1 + · · · + an−1 X̃n−1 , X̃1 , · · · , X̃n−1 ) = a0 · dVg (Ñ , X̃1 , · · · , X̃n−1 ),
when restricting this smooth function to ∂D, we obtain hX, N i|∂D f ∗ (iÑ dVg )(X1 , · · · , Xn−1 ) as a0 |∂D =
hX, N i|∂D . This completes the proof.