TABLE OF CONTENTS
1 MANIFOLDS AND FORMS 2
1.1 Submanifolds of Rn without boundary . . . . . . . . . . . . . 2
1.2 Notions of forms and elds . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 forms and vector elds on Rn . . . . . . . . . . . . . . 10
1.2.3 Integration over cubes and chains . . . . . . . . . . . . 15
1.3 Classical theorems of Green and Stokes . . . . . . . . . . . . . 18
1.3.1 Orientable Manifolds . . . . . . . . . . . . . . . . . . . 20
1.3.2 Riemannian Manifolds . . . . . . . . . . . . . . . . . . 27
2 EXAMPLES OF DIFFERENTIAL FORMS ON RIEMAN-
NIAN MANIFOLDS 28
2.1 Winding form and volume element associated to ellipsoids in
R2 and in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.1 Dierential forms on the 1-dimensional ellipsoid . . . . 28
2.1.2 Dierential forms on the 2-dimensional ellipsoid . . . . 33
2.2 Other quantities associated to R3 ellipsoid derived from Riemannian
structure, geodesics of R3 ellipsoid . . . . . . . . . . 39
2.2.1 The shape operator . . . . . . . . . . . . . . . . . . . . 39
2.2.2 Geodesics of the 2-dimensional ellipsoid . . . . . . . . . 42
2.3 Manifolds in higher dimensions: volume element, geodesics . . 48
2.3.1 Higher dimensional volume forms . . . . . . . . . . . . 48
2.3.2 Higher dimensional geodesics . . . . . . . . . . . . . . 52
Bibliography 58
iv
CHAPTER ONE
MANIFOLDS AND FORMS
1.1 Submanifolds of Rn without boundary
Denition 1.1.1.
A subset M of Rn is called a k-dimensional submanifold without boundary if
for each point p 2 M, there exist U; V open in Rn with p 2 U as well as a
dieomorphism : U ! V such that (U \M) is contained in the subspace
Rk Rn. In other words,
(U \M) = V \ (Rk f0gnô€€€k) = fy 2 V : yk+1 = = yn = 0g:
The pair (U; ) where U = U \ M is called a local chart around p and a
family of local charts covering all points of M is called an atlas on M. Thus
if fUi; igi2IN is an atlas on M, then M = [
i2I
Ui:
Remark: Because the dimension of M is k, we say that M has a local Rk
property and use this property to create parametrizations for the manifold,
which are basically dierentiable functions mapping from a subset of Rk onto
M. Parametrizations are needed for computational and analytical purposes
as we will see in chapter 2 on examples of dierential forms on Riemannian
manifolds.
If (U; ) is a local chart with p 2 U, we can identify p and the vector
(p) 2 Rn. The coordinates of (p) in Rn are called the local coordinates
2
of p in the local chart (U; ): For any two charts (Ui; i) and (Uj ; j) such
that Ui \ Uj is non-empty, we can dene the map,
i j
ô€€€1 : j(Ui \ Uj) ! i(Ui \ Uj)
which is called a chart transition from one chart to another. The sets
j(Ui \ Uj) and i(Ui \ Uj) are open sets of the coordinate space Rk and
and the transition function i j
ô€€€1 is a dieomorphism.
Alternatively, we may dene a submanifold of Rn without boundary as follows.
Denition 1.1.2.
Let U Rn be an open subset and let f : U ! Rnô€€€k be a smooth map.
Consider the set M = fx 2 U : f(x) = 0g:
If the gradient Df(x) has maximal rank (n-k) at each point x 2 M, then M
is a smooth k-dimensional submanifold of Rn without boundary.
Remark: This latter denition is derived from the former as a direct application
of the implicit function theorem, as we now brie y explain. For an
arbitrary point = (1; ; n) 2 M U, we have by the implicit function
theorem an open neighborhood A of (1; ; k) in Rk and a smooth function
g : A ! Rnô€€€k such that g(1; ; k) = (k+1; ; n) and f(; g()) = 0 for
all 2 A.
Hence, there exists an open neighborhood U of in U so that
U \M = fx 2 U : g(x1; ; xk) = (xk+1; ; xn)g:
Consider also the smooth function
ô€€€ggiven by
ô€€€g
: A Rnô€€€k U ! Rnô€€€k; x 7! g(x1; ; xk)
which belongs to the same dieomorphism class as fjU so that there also
exists a dieomorphism : U ! (U) Rn such that the map f ô€€€1
:
(U) ! Rnô€€€k is given by the formula
f ô€€€1
(x1; ; xn) = (xk+1; ; xn)
which implies that
f(U \M) = f ô€€€1
((U) \ (Rk f0gnô€€€k)) = 0;
3
i.e.
U \M = ô€€€1
((U) \ (Rk f0gnô€€€k))
since there are no other points in U whose image under f is 0.
This means (U \ M) = (U) \ (Rk f0gnô€€€k) and the local chart
(U \ M; ) needed around is thereby obtained, making M a smooth kdimensional
submanifold of Rn without boundary.
Let us consider the generalized case in Rn+1 of an ellipsoid with axial
symmetry.
Denition 1.1.3.
Dene
M(n) :=
(x1; ; xn+1) 2 Rn+1 :
x21
a2 +
x22
a2 + +
x2
n
a2 +
x2
n+1
b2 = 1
;
a; b 2 R+nf0g: M(n) is the generalized ellipsoid in Rn+1 with the xn+1 axis of
symmetry.
M(n) is a dierentiable submanifold of Rn+1 without boundary, having
dimension n. We may justify this statement using the latter description of
submanifolds without boundary given in denition 1.1.2.
Consider the function
f : Rn+1 ô€€€! R
dened by
x = (x1; ; xn+1) 7ô€€€! f(x) = kbxk2 ô€€€ a2b2 + (a2 ô€€€ b2)x2
n+1
M(n) = fx 2 Rn+1 : f(x) = 0g and
Df(x) = (2b2x1; 2b2x2; ; 2b2xn; 2a2xn+1):
The rank of the 1(n+1) matrix Df(x) is strictly 1 because the xi’s cannot
be simultaneously zero since the real constants a and b are positive. This
means that M(n) is a manifold of dimension (n + 1) ô€€€ 1 = n:
An atlas f(U1; 1), (U2; 2)g for M(n) is given as follows.
4
U1 = M(n)nf(0; 0; ; 0; b)g and
1 : U1 ô€€€! Rn
x = (x1; ; xn+1) 7ô€€€! 1(x) =
x1
a
1 ô€€€ xn+1
b
;
x2
a
1 ô€€€ xn+1
b
; ;
xn
a
1 ô€€€ xn+1
b
:
U2 = M(n)nf(0; 0; ; 0;ô€€€b)g and
2 : U2 ô€€€! Rn
x = (x1; ; xn+1) 7ô€€€! 2(x) =
x1
a
1 + xn+1
b
;
x2
a
1 + xn+1
b
; ;
xn
a
1 + xn+1
b
:
These charts are obtained as compositions of stereographic projections (h1
and h2) of the unit sphere Sn := fx 2 Rn+1 : kxk = 1g onto Rn with the linear
map; T : M(n) ! Sn given by (x1; ; xn+1) 7!
x1
a
;
x2
a
; ;
xn
a
;
xn+1
b
.
Indeed, we have
h1 : Sn ô€€€ f(0; 0; ; 0; 1)g ! Rn;
with h1(x1; ; xn+1) =
x1
1 ô€€€ xn+1
; ;
xn
1 ô€€€ xn+1
;
h2 : Sn ô€€€ f(0; 0; ; 0;ô€€€1)g ! Rn;
with h2(x1; ; xn+1) =
x1
1 + xn+1
; ;
xn
1 + xn+1
;
so that 1 = h1 T and 2 = h2 T. Since h1 and h2 are surjective, the
maps 1 and 2 are onto Rn.
Parametrizations of the manifold are considered in the second chapter.
In the remainder of this section, we give important properties of M(n) in
connection with its submanifolds.
Denition 1.1.4.
Let M be a dierentiable manifold of dimension n. A submanifold of dimen-
sion d n of M is a subset W M such that for any point p 2 W, there
exists a local chart (
; ) around p such that (
\W) = U V;
U Rd; V Rnô€€€d and (
\W) = U f0gnô€€€d. Thus, there exists a system
5
of local coordinates (x1; ; xn) on
in which the submanifold W is locally
dened by the equations: xd+1 = xd+2 = = xn = 0.
Proposition 1.1.5.
M(d) is isometrically isomorphic to a submanifold of M(n) for d n.
Proof. Take an arbitrary point p 2 M(d) Rd+1; p = (x1; ; xd+1). Clearly,
Rd+1 is isometrically isomorphic to f0gnô€€€d Rd+1 Rn+1, where f0gnô€€€d is
the zero vector in Rnô€€€d. We label the associated isomorphism I and note it
acts as follows.
I : M(d) ô€€€! M(n); p 7ô€€€! I(p) := p0 = (
nô€€€d z }| {
0; 0; ; 0; x1; ; xd+1)
Hence, for each point p 2 M(d), the local chart around p0 = I(p) (which
is either (U1; 1) or (U2; 2) as specied above) has the following property,
i(Ui \ I(M(d))) = f0gnô€€€d Rd. This is to say that M(d) is isometrically
isomorphic to I(M(d)); a submanifold of M(n).
By a similar approach, we also see that the sphere a:Sd is a submanifold
of M(n) for d nô€€€1, where Sd = fx 2 Rd+1 : kxk = 1g is the unit sphere in
Rd+1 .
More precisely, we have that M(d)
=
M(n) \ (f0gnô€€€d Rd+1) and
a:Sd = M(n) \ (Rd+1 f0gnô€€€d), adhering to the axial orientation specied in
denition 1.1.3.
Cartesian products of manifolds may be dened when appropriate with
dim(A B) = dim(A) + dim(B) for manifolds A and B. Nevertheless, it is
clearer that we can obtain the ellipsoid M(n) as a manifold of revolution. We
specify how to obtain M(n) by revolution in the following proposition.
Proposition 1.1.6.
Let Kn+1 = fx 2 Rn+1 : x1 = 0 and x2; ; xn 0g. Then the manifold
M(n) is recovered by rotating its (n-1) dimensional submanifold M(n) \Kn+1
completely about the xn+1 axis for n 2.
Proof. Each point p 2 Rn+1 can be given a polar coordinate (rp; (1)p; ; (n)p)
where rp is the Euclidean distance from p to the origin and (k)p is the angular
position of p in the (xk; xn+1) plane measured counterclockwise from
6
the xk axis. Hence for a point p0 2 M(n) \Kn+1, we have f(1)p0 ; ; (n)p0g
as a subset of
ô€€€
2
;
2
.
Rotation of a manifold about the xn+1 axis entails rotation of its crosssections
about the xn+1 axis in each (xk; xn+1) plane. As such, by one full
revolution about the xn+1 axis, the angular positions of the points (mod 2)
are no longer restricted in any (xk; xn+1) plane for 2 k n. This eliminates
the restriction x2; ; xn 0 from the result of revolving M(n) \Kn+1
about the xn+1 axis; which is a submanifold of M(n) by virtue of its symmetry
about the xn+1 axis.
Moreover, by revolving an arbitrary point with Euclidean coordinates
(0; x0
2; x0
3; ; x0
n+1) about the xn+1 axis, the result is the sphere in
fx 2 Rn+1 : xn+1 = x0
n+1g centered at (0; 0; ; 0; x0
n+1) with radius
k(0; x0
2; x0
3; ; x0
n; 0)k2. But the x1 coordinates of this sphere are clearly
not restricted to zero as long as at least one of x0
2; x0
3; ; x0
n is not zero. If
(0; x0
2; x0
3; ; x0
n+1) 2 M(n) \Kn+1, then its sphere by revolution about the
axis of symmetry is a submanifold ofM(n), meaning that the x1 coordinates of
the manifold by revolution about the xn+1 axis are no longer restricted to 0.
In conclusion, by revolving M(n)\Kn+1 about the xn+1 axis of symmetry, we
get a submanifold of M(n) without the restrictions x1 = 0 and x2; ; xn 0,
which is necessarily a recovery of M(n).
More generally, if
K = fx 2 Rn+1 : x1 = x2 = = xk = 0 and xk+1; ; xn 0g;
then by a similar construction we recover M(n) by rotating its (n-k) dimensional
submanifold M(n) \ K completely about the xn+1 axis. The simplest
case is by rotating the 1-dimensional half ellipse given by
x 2 Rn+1 : x1 = x2 = = xnô€€€1 = 0; xn 0 and
x2
n
a2 +
x2
n+1
b2 = 1
about the xn+1 axis in the space Rn+1 to get M(n). This geometric property
of M(n) will be reconciled to dierential forms on the manifold in further
analytic and theoretic observations. (See section 1 of chapter 2 on the volume
element of M(2).)
7
1.2 Notions of forms and elds
1.2.1 Tensors
A mapping T from V , an n-dimensional vector space over R, to R is called
a k-tensor on V if T : V k ! R is k-linear. In other words, T is a k-tensor on
V i the following two conditions hold:
i) T(v1; ; vi+v0
i; ; vk) = T(v1; ; vi; ; vk)+T(v1; ; v0
i; ; vk),
ii) T(v1; ; avi; ; vk) = aT (v1; ; vi; ; vk):
The set of all k-tensors on V constitutes a co-vector space denoted =k(V ).
For S 2 =k(V ) and T 2 =l(V ), their tensor product S
T belongs to =k+l(V )
and is dened by
S
T(v1; ; vk; vk+1; ; vk+l) = S(v1; ; vk):T (vk+1; ; vk+l)
A k-tensor T is said to be alternating if
T(v1; ; vi; ; vj ; ; vk) = ô€€€T(v1; ; vj ; ; vi; ; vk):
The subset of alternating k-tensors in =k(V ) also constitutes a co-vector
space denoted
Vk(V ). For every T 2 =k(V ),we generally dene Alt(T) by
Alt(T)(v1; ; vk) =
1
k!
X
2Sk
sgn:T (v(1); ; v(k))
where Sk is the set of all permutations of the integers 1 to k.
We observe that Alt(T) 2
Vk(V ): For ! 2
Vk(V ); 2
Vl(V ), we dene
tVheir wedge product or exterior product denoted ! ^ which belongs to k+l(V ) by
! ^ =
(k + l)!
k!l!
Alt(!
):
We also give the following properties of the wedge product
1. (! + ) ^ = ! ^ + ^ 8!; 2
Vk(V ); 2
Vl(V )
2. ! ^ ( + ) = ! ^ + ! ^ 8! 2
Vk(V ); ; 2
Vl(V )
3. a! ^ = ! ^ (a) = a(! ^ ) 8a 2 R; ! 2
Vk(V ); 2
Vl(V )
8
4. (! ^ ) ^ = ! ^ ( ^ ) 8! 2
Vk(V ); 2
Vl(V ); 2
Vm(V )
5. ! ^ = (ô€€€1)kl ^ ! 8! 2
Vk(V ); 2
Vl(V )
=1(V ) is the set of all linear maps from V to R, which in this case coincides
with the dual V of V because V is nite dimensional. If (v1; ; vn)
is a basis for V and (‘1; ; ‘n) the corresponding dual basis then the set
of all k-fold tensor products ‘i1
‘ik : 1 i1; ; ik n is a basis for
=k(V ), hence having dimension nk. Note that ‘i(vj) = 0 when i 6= j and
‘i(vi) = 1.
The set of all ‘i1 ^ ^ ‘ik : 1 i1 < i2 < < ik n is a basis for
Vk(V ) which therefore has dimension
n
k
=
n!
k!(n ô€€€ k)!
For a dierentiable function f : Rn ! R;Df(p)p2Rn is an example of a
linear map from Rn to R so Df(p) 2
V1(Rn) = =1(Rn).
An inner product T : V V ! R is a bilinear functional or 2-tensor on
V which is symmetric, that is, T(v;w) = T(w; v) 8v;w 2 V and positive
denite, that is, T(v; v) > 0 if v 6= 0. Hence, T 2 =2(V ) and we recognize
h; i as the usual inner product on Rn.
A symplectic map A : V V ! R is another type of bilinear functional
on V which is anti-symmetric, that is, A(v;w) = ô€€€A(w; v) 8v;w 2 V and so
satises A(v; v) = 0 for all v 2 V . A is also non-degenerate, meaning that
A(u; v) = 0 for all v 2 V if and only if u = 0. Hence A 2
V2(V ) and we
specically identify such an alternating 2-tensor in section 1 of chapter 2.
Let us examine the vector space
Vn(V ) which has dimension
n
n
= 1.
Because of this singular dimension, each element of
Vn(V ) is simply a scalar
product of any other non-zero one. The determinant is clearly an alternating
n-tensor on V and we state this as det 2
Vn(V ). This fact comes to play in
the following theorem.
Theorem 1.2.1.
Let (v1; ; vn) be a basis for V and let ! 2
Vn(V ). If wi =
Pn
j=1 aijvj are
n vectors in V, then !(w1; ;wn) = det(aij):!(v1; ; vn).
9
Proof. We rst dene 2 =n(Rn) by
((a11; ; a1n); ; (an1; ; ann)) = !(
Pn
j=1 a1jvj ;
Pn
j=1 anjvj)
= !(w1; ;wn):
Then 2
Vn(Rn) so = :det for some constant 2 R. Now, applying
both sides to (e1; ; en), we get = (e1; ; en) = !(v1; ; vn). As such,
((a11; ; a1n); ; (an1; ; ann)) = :det(aij)
which implies
!(w1; ;wn) = det(aij):!(v1; ; vn):
As a consequence of theorem 1.2.1, a non-zero ! 2
Vn(V ) splits the bases
of V into two disjoint groups, those with !(v1; ; vn) > 0 and those with
!(v1; ; vn) < 0. If (v1; ; vn) and (w1; ;wn) are two bases such that
wi =
Pn
j=1 aijvj , then these two bases are in the same group i det(aij) > 0.
Either of the two disjoint groups is called an orientation for V. The orientation
to which a basis (v1; ; vn) belongs is denoted [v1; ; vn] and the
other orientation is denoted ô€€€[v1; ; vn]. Notably, orientations are independent
of the element ! which acts, meaning that ! 6= 0 separates bases into
the same orientations. In Rn, the usual orientation is dened as [e1; ; en].
There is a unique ! 2
Vn(V ) such that !(v1; ; vn) = 1 whenever
v1; ; vn is an orthonormal basis such that [v1; ; vn] = . This unique
! is called the volume element of V determined by the orientation and an
inner product. The determinant is the volume element of Rn determined by
the usual inner product and usual orientation.
1.2.2 forms and vector elds on Rn
Let p 2 Rn. The set of all pairs (p; v) for v 2 Rn is denoted Rnp
and is called
the tangent space of Rn at p, i.e. Rnp
:= (p; v); v 2 Rn: This set is clearly
made a vector space by dening the following operations :
1. (p; v) + (p;w) = (p; v + w); v;w 2 Rn;
2. a(p; v) = (p; av); a 2 R:
10
A vector v 2 Rn can be seen as an arrow from 0 to v, and the vector
(p; v) 2 Rn
p is then seen as an arrow with the same direction and length, but
with initial point p. The vector (p; v) then goes from p to p+v and we write
(p; v) as vp and call it the vector v at p:
We dene the usual inner product h; ip for Rn
p by hvp;wpip = hv;wip and
assign to Rn
p its usual orientation := [(e1)p; ; (en)p].
A vector eld on Rn is a function F : Rn ! Rn
p such that F(p) 2 Rn
p
for each p 2 RnP . Hence any vector eld F can be written as F(p) = n
i=1 Fi(p):(ei)p thereby yielding n component functions Fi : Rn ! R.
A similar structure can be placed only on open subsets of Rn. For an open
subset U of Rn, we dene a vector eld as a function which assigns to each
point p 2 U a unique vector from the tangent space of Rn at p:
The divergence of F (divF) is dened as
Pn
i=1 DiFi. Employing the
notation r =
Pn
i=1 Di:ei, we may symbolize div F by hr; Fi. Note that
Di =
@
@xi
.
For n = 3, we can dene a vector eld called the curl of F or curl F, which
we symbolize r F in accordance with the notation for r. Hence,
(rF)(p) = (D2F3ô€€€D3F2)(e1)p+(D3F1ô€€€D1F3)(e2)p+(D1F2ô€€€D2F1)(e3)p.
Now, a function ! with !(p) 2
Vk(Rn
p) is called a k-form on Rn or a
dierential form. If ‘1(p); ; ‘n(p) is the dual basis to (e1)p; ; (en)p then
!(p) is of the appearance
X
i1<<ik
!i1 !ik’i1(p) ^ ^ ‘ik(p)
for certain functions or 0-forms !i1; ;ik . Functions written as f which map to
R are 0-forms. Suppose f : Rn ! R is dierentiable so that Df(p) 2
V1(Rn).
We then obtain an associated 1-form df, dened by df(p)(vp) = Df(p)(v).
Upon consideration of the projection maps i otherwise denoted dxi; for
(1 i n), we observe that these belong to the dual of Rn and
dxi(p)(ei)p = 1 ; dxi(p)(ej)p = 0 whenever i 6= j.
This immediately gives us that dx1(p); ; dxn(p) is the dual basis to (e1)p; ; (en)p,
so if !(p) is a k-form on Rn
p it can always be written as
11
X
i1<<ik
!i1 !ik(p)dxi1(p) ^ ^ dxik(p):
For a dierentiable map f : Rn ! R, df = D1f:dx1 + + Dnf:dxn.
For f : Rn ! Rm dierentiable, we have a linear map Df(p) : Rn ! Rm
to which is associated the linear transformation f : Rn
p ! Rm
f(p) dened
by f(vp) = (Df(p)(v))f(p):
The above induces another linear transformation called the pullback of
f, written f :
Vk(Rm
f(p)) !
Vk(Rn
p): Therefore if ! is a k-form on
Rm, we dene a k-form f! on Rn by (f!)(p) = f(!(f(p))): This simply
means that if v1; ; vk 2 Rn
p, then we have f!(p)(v1; ; vk) =
!(f(p))(f(v1); ; f(vk)). Let ! be a k-form, then the dierential operator
(d) acts on ! to produce a (k+1)-form d! which is called the dierential of !:
In general, if
! =
X
i1<<ik
!i1 !ikdxi1 ^ ^ dxik ;
then
d! =
X
i1<<ik
d(!i1 !ik) ^ dxi1 ^ ^ dxik
=
X
i1<<ik
Xn
=1
D(!i1 !ik)dx ^ dxi1 ^ ^ dxik
Denitions
The hodge operator, denoted , is a linear operator on
Vk(Rn
p) which assigns
an (n-k)-form to each k-form. It has the following property which describes
it concisely;
(dxi1(p) ^ ^ dxik(p)) = dxik+1(p) ^ ^ dxin(p);
where (i1; ; ik; ik+1; ; in) is an even permutation of the integers from 1
to n.
For ! 2
Vk(Rn
p), the (n-k)-form ! is called the hodge dual of !. Note that
! = (ô€€€1)k(nô€€€k)!:
12
An important application of the hodge operator is to dene the codierential
() of forms. For a k-form !, we have its codierential given by
! = (ô€€€1)nk+n+1 d !:
Hence, we see that :
Vk(Rn
p) !
Vkô€€€1(Rn
p).
The Laplace – Beltrami operator :
Vk(Rn
p) !
Vk(Rn
p) is given by
= d + d:
In separate outstanding considerations, the hodge operator, codierential
and Laplace – Beltrami operator are important tools used in the analysis of
Hodge theory.
Important properties of f; the pullback of f for f : Rn ! Rm;
(u1; u2; ; un) 7! (x1; x2; ; xm) dierentiable are listed below
1. f(dxi) =
Pn
j=1 Djfiduj = dfi
2. f(!1 + !2) = f(!1) + f(!2)
3. f(g !) = (g f)f!; for a functional g : Rm ! R
4. f(! ^ ) = f! ^ f
5. If n = m, then f(hdx1 ^ ^ dxn) = (h f)(detf0)du1 ^ ^ dun
6. f(d!) = d(f!)
Concerning the dierential operator d, there are yet some important observations
to make. We have d2 = 0, which is to say d(d!) = 0 for any
dierential form !. Also, dxi ^ dxi = (ô€€€1)1dxi ^ dxi = 0 and
d(! ^ ) = d! ^ + (ô€€€1)k! ^ d for a k-form ! and an l-form .
A form ! is closed if d! = 0 and exact if ! = d for some form . Every
exact form is closed since if ! = d then d! = d(d) = 0. The converse does
not necessarily hold. The next theorem gives a sucient condition for closed
forms to be exact.
Theorem 1.2.2. (Poincare Lemma)
Let W Rn be an open set star-shaped with respect to the origin, then every
closed form on W is exact. A set is said to be star-shaped with respect to the
origin if it includes the origin as well as the entire line segment connecting
the origin to each of its other points.
13
Proof. We dene a function I from k-forms to (k-1)-forms (for each k), such
that I(0) = 0 and ! = I(d!) + d(I!) for any form !. It follows that
! = d(I!) if d! = 0.
Let
! =
X
i1<<ik
!i1; ;ikdxi1 ^ ^ dxik :
Since A is star-shaped we can dene
I!(x) =
X
i1<<ik
Xk
=1
(ô€€€1)ô€€€1
Z 1
0
tkô€€€1!i1; ;ik(tx)dt
xidxi1^ ^dxi^ ^dxik
(The strikethrough beneath dxi indicates that it is omitted from the term.)
We now prove that ! = I(d!) + d(I!).
By Leibnitz’s rule,
d(I!) = k:
X
i1<<ik
Z 1
0
tkô€€€1!i1; ;ik(tx)dt
dxi1 ^ ^ dxik
+
X
i1<<ik
Xk
=1
Xn
j=1
(ô€€€1)ô€€€1
Z 1
0
tkDj(!i1; ;ik)(tx)dt
xi
dxj ^ dxi1 ^ ^ dxi ^ ^ dxik :
We also have
d! =
X
i1<<ik
Xn
j=1
Dj(!i1; ;ik)dxj ^ dxi1 ^ ^ dxik :
Applying I to the (k+1)-form d!, we obtain
I(d!) = A + B
where
A =
X
i1<<ik
Xn
j=1
Z 1
0
tkDj(!i1; ;ik)(tx)dt
xjdxi1 ^ ^ dxik
14
and
B =
X
i1<<ik
Xk
=1
Xn
j=1
(ô€€€1)
Z 1
0
tkDj(!i1; ;ik)(tx)dt
xidxj^dxi1^ ^dxi^ ^dxik
Adding, the triple sums cancel, and we obtain
d(I!) + I(d!) =
X
i1<<ik
k:
Z 1
0
tkô€€€1!i1; ;ik(tx)dt
dxi1 ^ ^ dxik
+
X
i1<<ik
Xn
j=1
Z 1
0
tkxjDj(!i1; ;ik)(tx)dt
dxi1 ^ ^ dxik
=
X
i1<<ik
Z 1
0
d
dt
[tk!i1; ;ik(tx)]dt
dxi1 ^ ^ dxik
=
X
i1<<ik
!i1; ;ikdxi1 ^ ^ dxik = !
An example worthy of note which outrightly incorporates these discussed
notions about dierential forms and vector elds in physics is Maxwell’s
equations of electromagnetism. The setting is R4 (a space – time manifold),
and performing relevant operations on the electromagnetic eld as an exact
dierential 2-form yields mathematical interpretations of profound physical
results. However, this is an illustration in Lorentzian geometry which diers
from Riemannian geometry by way of the metric.
1.2.3 Integration over cubes and chains
Essentially, dierential forms have to be integrated over domains where they
are dened in Rn. This gives rise to the use of singular k-cubes in domains of
Rn, which are suitable parametrizations of the domains for this purpose. A
singular k-cube in A Rn is a continuous function c mapping from [0; 1]k to
A. Any singular 1-cube is a curve, and singular 2-cubes are surfaces. Standard
n-cubes in Rn are often denoted In with In : [0; 1]n ! Rn dened by
In(x) = x for x 2 [0; 1]n.
A linear combination
P
i2IN aici ; ai 2 Z of singular k-cubes ci is referred
to as a singular k-chain. Each singular k-chain c has a boundary denoted @c
which is a (k-1) chain. To get the general formula of @c for an n-chain c,
we rst formulate @In. For i : 1 i n, dene the following singular (n-1)
15
cubes.
1) In
(i;0)(x) = In(x1; ; xiô€€€1; 0; xi; ; xnô€€€1) = (x1; ; xiô€€€1; 0; xi; ; xnô€€€1)
2) In
(i;1)(x) = In(x1; ; xiô€€€1; 1; xi; ; xnô€€€1) = (x1; ; xiô€€€1; 1; xi; ; xnô€€€1)
for x 2 [0; 1]nô€€€1.
In
(i;0) is called the (i,0)-face of In and In
(i;1) the (i,1)-face.
Now, @In :=
Pn
i=1
P
=0;1 (ô€€€1)i+In
(i;).
For a singular n-cube c; we dene its (i; )-face, c(i;) = c (In
(i;))
so that
@c :=
Xn
i=1
X
=0;1
(ô€€€1)i+c(i;):
Finally, the boundary of an n-chain
P
i2IN
aici is given by
@
X
i2I
aici
!
=
X
i2I
ai@(ci):
In R2 for instance, the boundary of I2 is depicted as follows.
x2
x1
I2
(2,1)
I2
(2,0)
I2
(1,1) I2
(1,0)
1
1
@I2 can be described as the sum of four singular 1-cubes arranged counter-
16
clockwise around the boundary of [0; 1]2.
A property of the boundary operator @ is @2 = 0 which is to say @(@c) = 0
for any singular n-chain c. Other properties and relations derived from the
boundary operator are highlighted next in classical theorems by Stokes and
Green. The orientations of domains of integration will also be considered,
without which integrands obtained over singular k-cubes can only be guaranteed
to be accurate up to sign.
17
1.3 Classical theorems of Green and Stokes
If ! is a k-form on [0; 1]k, then ! = fdxi1 ^ ^ dxik for a unique 0-form f.
We then have
Z
[0;1]k
! =
Z
[0;1]k
fdx1 ^ ^ dxk =
Z
[0;1]k
f(x1; ; xk)dx1 dxk
For ! a k-form on A Rn and c a singular k-cube in A, we dene
Z
c
! :=
Z
[0;1]k
c!
recalling that c! is an induced k-form on [0; 1]k. The integral of a form !
over a k-chain c =
P
i2I aici is given by
R
c! =
P
i2I ai
R
ci
!.
The integral of a 1-form over a 1-chain is called a line integral and the integral
of a 2-form over a singular 2-chain is called a surface integral.
Hitherto observations made permit a clear breakdown of the proof of a
theorem by Stokes, which is popularly recognized as the fundamental theorem
of calculus in higher dimensions.
Theorem 1.3.1. (Stokes’ Theorem (a))
If ! is a (k-1) form on an open subset A Rn and c is a k-chain in A,
then
R
cd! =
R
@c!.
Proof. We rst take c to be the standard k-cube Ik, and ! to be a (k-1) form
on [0; 1]k:
In this case, ! can be written as the sum of (k-1) forms of the type
fdx1 ^ ^ dxi ^ ^ dxk
(the strikethrough beneath dxi indicates that this 1-form is excluded from
the term), and we suciently prove the theorem for each of these. Note that
Z
[0;1]kô€€€1
Ik
(j;)
(fdx1^ ^dxi^ ^dxk) = ij
Z
[0;1]k
f(x1; ; ; ; xk)dx1 dxk;
where ij =
1 if j = i
0 otherwise
18
Thus,
Z
@Ik
fdx1 ^ ^ dxi ^ ^ dxk
=
Pk
j=1
P
=0;1 (ô€€€1)j+
Z
[0;1]kô€€€1
Ik
(j;)
(fdx1 ^ ^ dxi ^ ^ dxk)
= (ô€€€1)i+1
Z
[0;1]k
[f(x1; ; 1; ; xk) ô€€€ f(x1; ; 0; ; xk)]dx1 dxk
ZBesides,
Ik
d(fdx1 ^ ^ dxi ^ ^ dxk)
=
Z
[0;1]k
Difdxi ^ dx1 ^ ^ dxi ^ ^ dxk
= (ô€€€1)iô€€€1
Z
[0;1]k
Dif
= (ô€€€1)iô€€€1
Z 1
0
Z 1
0
Dif(x1; ; xk)dxidx1 dxi dxk (by Fubini’s
theorem)
= (ô€€€1)iô€€€1
Z 1
0
Z 1
0
[f(x1; ; 1; ; xk)ô€€€f(x1; ; 0; ; xk)]dx1 dxi dxk
(by the fundamental theorem of calculus in one-dimension)
= (ô€€€1)i+1
Z
[0;1]k
[f(x1; ; 1; ; xk) ô€€€ f(x1; ; 0; ; xk)]dx1 dxk
Hence, Z
Ik
d! =
Z
@Ik
!:
Now, let c be an arbitrary singular k-cube, then
Z
c
d! =
Z
Ik
c(d!) =
Z
Ik
d(c!) =
Z
@Ik
c! =
Z
@c
!:
Finally, if c is a k-chain
P
i2I
aici, then
Z
c
d! =
X
i2I
ai
Z
ci
d! =
X
i2I
ai
Z
@ci
! =
Z
@c
!:
Before presenting the other theorems, we brie y view the structures of
elds and forms on dierentiable manifolds.
19
Let M be a k-dimensional manifold in Rn and the local chart around
a point p 2 M be (U; ). Then we can dene a local coordinate system
ô€€€1 : V ! Rn (V Rk is open) around p = ô€€€1(a) for some a 2 V .
The k-dimensional vector space ô€€€1
(Rk
a) is denoted TpM, and is called the
tangent space of M at p. This space is independent of which local coordinate
system is used to derive it. A function which assigns a vector in TpM to each
point p 2 M is called a vector eld on M. A function which assigns an alternating
k-tensor in
Vk(TpM) to each p 2 M is called a k-form on M. Hence,
given a vector eld F on M, F : M !
S
p2MTpM and a 1-form !p : TpM ! R,
we may obtain the composition !p(F) = !F(p) which is a mapping from M
to R. Inadvertently, dierential 1-forms constitute the dual to vector elds
on a given manifold.
If f : W Rk ! Rn is a coordinate system, ! a k-form on M, then f!
is a k-form on W and we say ! is dierentiable if f! is. A k-form ! can be
written ! =
P
i1<<ik
!i1 !ikdxi1 ^ ^ dxik .
Since the functions !i1 ; ; !ik may be dened only on M, the previous
denition given for d! may not be valid here, as Dj(!i1 ; ; !ik) would have
no meaning. However, the relation f(d!) = d(f!) still holds, so we dene
the dierential of ! as d! = (fô€€€1)(d(f!)).
1.3.1 Orientable Manifolds
It is often important to choose, if possible, an orientation p for each tangent
space TpM of a manifold M. These choices are called consistent if, given a
coordinate system f : W ! Rn and a; b 2 W, then we have
[f((e1)a); ; f((ek)a)] = f(a) () [f((e1)b); ; f((ek)b)] = f(b)
If orientations p have been chosen consistently and f : W ! Rn is a coordinate
system such that [f((e1)a); ; f((ek)a)] = f(a) for one and hence
every a 2 W, then f is called orientation – preserving. If f is not orientation
– preserving and T : Rk ! Rk is a linear transformation such that detT is
negative, then f T is orientation – preserving. Hence, as long as orientations
can be chosen consistently, there exists an orientation – preserving coordinate
system around each point.
Suppose that f and g are orientation – preserving and p = f(a) = g(b);
20
then
[f((e1)a); ; f((ek)a)] = p = [g((e1)b); ; g((ek)b)]. Therefore,
[(gô€€€1 f)((e1)a); ; (gô€€€1 f)((ek)a)] = [(e1)b; ; (e1)b] so that
det(gô€€€1 f)0 > 0.
A manifold for which orientations p can be chosen consistently is orientable
and a choice for p is called an orientation of the manifold. A manifold
M together with an orientation is called an oriented manifold.
One of the most known examples of a non-orientable manifold in R3 is
the Mobius strip.
Manifolds with Boundary
If we have M Rn to be a k-dimensional manifold – with – boundary, then
for each point p 2 M, either
1. there exist open sets U and V with p 2 U Rn; V Rn and a
dieomorphism : U ! V such that (U \M) = V \ (Rk f0g), OR
2. there exist open sets U and V, with p 2 U Rn; V Rn and a
dieomorphism : U ! V such that
(U\M) = V \(Hkf0g) = fy 2 V : yk 0 and yk+1 = = yn = 0g,
and (p) has kth component equal to 0.
Hk = fx 2 Rk : xk 0g is called a half-space of Rk.
Conditions (1) and (2) cannot be satised by the same point p 2 M.
Assuming on the contrary that there is a point which satises (1) and (2),
then there would exist dieomorphisms 1 : U1 ! V1 and 2 : U2 ! V2 such
that 1(U1 \M) = V1 \ Rk and 2(U2 \M) = V2 \ Hk; k2
(p) = 0.
The set 1(U1 \ U2) would then be an open subset in Rk mapped onto
2(U1 \ U2) by the dieomorphism 2 1
ô€€€1. Since k2
(p) = 0, the set
2(U1\U2) then contains a point from @Hk = Rkô€€€1 and so it cannot be open
in Rk. This is a contradiction to the inverse function theorem.
A point p 2 M which satises (2) is called a boundary point of M and
we denote by @M the boundary of M, which is the set of all boundary points
of M. If M is a k-dimensional manifold with boundary, then @M is a (k –
1) dimensional submanifold without boundary. Let M M be a smooth
k-dimensional manifold extended from M at its boundary. If p 2 @M, then
Tp(@M) is a (k – 1) dimensional subspace of the k-dimensional space TpM.
21
As a result, there are exactly 2 unit vectors in TpM which are perpendicular
to Tp(@M). If (v1; ; vk) is an orthonormal basis for TpM such that
(v1; ; vkô€€€1) is a basis for Tp(@M), then vk 2 TpM is one of the unit vectors
perpendicular to Tp(@M) and the other clearly is ô€€€vk.
If f : W ! Rn is a coordinate system with W Hk and f(0) = p 2 @M,
then only one of these unit vectors is f(v0) for some v0 2 W with (v0)k < 0.
This unit vector is called the outward unit normal n(p) and it is independent
of the coordinate system f used to obtain it.
Suppose that is an orientation of the k-dimensional manifold – with
– boundary M. If p 2 @M, we choose v1; ; vk􀀀1 2 Tp(@M) so that we
have [n(p); v1; ; vkô€€€1] = p. If also [n(p);w1; ;wkô€€€1] = p then both
[v1; ; vkô€€€1] and [w1; ;wkô€€€1] are the same orientation for Tp(@M), either
of which is denoted by (@)p. If M is orientable, then @M is also orientable
and an orientation for M determines an orientation @ for @M called the
induced orientation.
The ellipsoid M(n), as we recall from denition 1.1.3, is an n-dimensional
manifold in Rn+1 without boundary and it is the boundary for the (n+1) –
dimensional manifold with boundary
L(n+1) :=
(x1; ; xn+1) 2 Rn+1 :
x21
a2 +
x22
a2 + +
x2
n
a2 +
x2
n+1
b2
1
of Rn+1. As such, if for p 2 M(n) we have [v1; ; vn] = p, we obtain the
outward unit normal to M(n) at p; (p) 2 Rn+1
p so that (p) is a unit vector
prependicular to TpM(n) and [ (p); v1; ; vn] is the orientation of Rn+1
p
which induces p. Note that for an interior point a 2 L(n+1), the vector space
Rn+1
a coincides with TaL(n+1). A direct explanation for the orientability of
M(n) is drawn from an alternative denition given as follows.
Let f1; ; fnô€€€k : U ! R be smooth functions dened on an open subset
U Rn with df1 ^ ^ dfnô€€€k 6= 0 at each point. Then the k-dimensional
manifold Mk := fx 2 U : f1(x) = = fnô€€€k(x) = 0g is orientable.
Let
f : Rn+1nf0g ! R;
x = (x1; ; xn+1) 7! f(x) = kbxk2 ô€€€ a2b2 + (a2 ô€€€ b2)x2
n+1
Then
df = 2b2x1dx1+2b2x2dx2+ +2b2xndxn+2a2xn+1dxn+1 6= 0 on Rn+1nf0g.
22
Hence, M(n) = fx 2 Rn+1nf0g : f(x) = 0g is orientable.
We now state further theorems utilizing the concepts of boundaries of
manifolds and their orientations.
Theorem 1.3.2. (Stokes’ Theorem (b))
If M is a compact oriented k-dimensional manifold – with – boundary and
! is a (k – 1) form on M, then
R
Md! =
R
@M! where @M is given the induced
orientation.
The proof of this theorem incorporates a standard tool required in the
theory of integration called partitions of unity.
Lemma 1.3.3.
For A Rn and O an open cover of A, there is a collection of C1 func-
tions ‘ dened in an open set containing A called a C1 partition of unity
for A subordinate to the cover O, with the following properties:
(1) For each x 2 A we have 0 ‘(x) 1.
(2) For each x 2 A there is an open set V containing x such that all but
nitely many ‘ 2 are 0 on V.
(3) For each x 2 A, we have
P
‘2
‘(x) = 1. By (2) for each x this sum
is nite in some open set containing x.
(4) For each ‘ 2 there is an open set U in O such that ‘ = 0 outside
of some closed set contained in U.
Proof of Theorem 1.3.2 Commencing the proof this theorem, suppose
that there is an orientation – preserving singular k-cube c in M 􀀀 @M such
that ! = 0 outside of c([0; 1]k). By Theorem 1.3.1 and the denition of d!
we have
Z
c
d! =
Z
[0;1]k
c(d!) =
Z
[0;1]k
d(c!) =
Z
@Ik
c! =
Z
@c
!:
23
Then, Z
M
d! =
Z
c
d! =
Z
@c
! = 0;
since ! = 0 on @c. On the other hand,
R
@M! = 0 since ! = 0 on @M.
Suppose next that there is an orientation-preserving singular k-cube in
M such that c(k;0) is the only face in @M, and ! = 0 outside of c([0; 1]k). Then
Z
M
d! =
Z
c
d! =
Z
@c
! =
Z
@M
!:
We may now consider the general case. There is an open cover O of M
and a partition of unity for M subordinate to O such that for each ‘ 2
the form ‘ ! is of one of the two sorts just considered. We have
0 = d(1) = d
X
‘2
‘
!
=
X
‘2
d’;
so that X
‘2
d’ ^ ! = 0:
Since M is compact, this is a nite sum and we have
X
‘2
Z
M
d’ ^ ! = 0:
Therefore,
Z
M
d! =
X
‘2
Z
M
‘ d!
=
X
‘2
Z
M
d’ ^ ! + ‘ d!
=
X
‘2
Z
M
d(‘ !)
=
X
‘2
Z
@M
‘ !
=
Z
@M
!:
24
Now, we give some practical versions of Stokes’ Theorem.
Theorem 1.3.4. Green’s Theorem
Let M R2 be a compact 2-dimensional manifold – with – boundary.
Suppose that ; : M ! R are dierentiable. Then
Z
@M
dx + dy =
Z
M
(D1 ô€€€ D2)dx ^ dy =
Z Z
M
(
@
@x
ô€€€
@
@y
)dxdy;
where M is given the usual orientation and @M the induced orientation, oth-
erwise called the counterclockwise orientation.
Proof. We nd the dierential of the 1-form (dx + dy) to be
d(dx + dy) = d ^ dx + d ^ dy
= (D1dx + D2dy) ^ dx + (D1dx + D2dy) ^ dy
= D2dy ^ dx + D1dx ^ dy
= (D1 ô€€€ D2)dx ^ dy
We now apply theorem 1.3.2 directly to get
Z
@M
dx + dy =
Z
M
d(dx + dy)
=
Z
M
(D1 ô€€€ D2)dx ^ dy
=
Z Z
M
(
@
@x
ô€€€
@
@y
)dxdy
Theorem 1.3.5. (Stokes’ Theorem (c))
Let M R3 be a compact oriented two-dimensional manifold – with –
boundary and n the unit outward normal on M determined by the orientation
of M. Let @M have the induced orientation. Let G be the vector eld on @M
with ds(G) = 1 and F be a dierentiable vector eld in an open set containing
25
M. Then
R
Mh(r F); nidA =
R
@MhF;Gids
(dA and ds are respectively referred to as element of area and element of
arclength.)
Proof. Dene on M by = F1dx + F2dy + F3dz. Recall the curl of F,
rF respectively has components D2F3ô€€€D3F2;D3F1ô€€€D1F3;D1F2ô€€€D2F1.
For a two-dimensional manifold, the element of volume is the element of area
dA 2
V2(TpM) and
dA(v;w) = det
0
@
v
w
n(p)
1
A8v;w 2 TpM,
where n(p) is the outward unit normal, since dA(v;w) is 1 if v and w form
an orthonormal basis for TpM with [v;w] = p.
Note that dA(v;w) = hv w; n(p)i.
Let 2 R3
p, observing that v w = :n(p); = kv wk 2 R:
h; n(p)idA(v;w) = h; n(p)i = h; n(p)i = h; v wi
The above scalar triple product equals
1 2 3
v1 v2 v3
w1 w2 w3
= 1(v2w3 ô€€€ v3w2) ô€€€ 2(v1w3 ô€€€ v3w1) + 3(v1w2 ô€€€ v2w1):
dy ^ dz(v;w) = 2Alt(dy
dz(v;w)) = 2:
1
2!
(v2w3 ô€€€ v3w2)
=) 1dy ^ dz(v;w) = 1(v2w3 ô€€€ v3w2)
Likewise,
2dz ^ dx(v;w) = 2(v3w1 ô€€€ v1w3) and 3dx ^ dy(v;w) = 3(v1w2 ô€€€ v2w1):
Thus,
h; n(p)idA = 1dy ^ dz + 2dz ^ dx + 3dx ^ dy
and
h(r F); nidA
= (D2F3 ô€€€D3F2)dy ^ dz +(D3F1 ô€€€D1F3)dz ^ dx+(D1F2 ô€€€D2F1)dx ^ dy
= d:
Also, since ds(G) = 1 on @M we have G1ds = dx;G2ds = dy;G3ds = dz.
These equations are easily checked by applying both sides to G(p) for p 2
@M, since G(p) is a basis for Tp(@M): Therefore, on @M we have
26
hF;Gids = F1G1ds + F2G2ds + F3G3ds
= F1dx + F2dy + F3dz
=
By theorem 1.3.2, we get
Z
M
h(r F); nidA =
Z
M
d
=
Z
@M
=
Z
@M
hF;Gids
1.3.2 Riemannian Manifolds
As illustrations for the theoretical content of this chapter, we will consider
specic examples of dierential forms on Riemannian manifolds in chapter 2.
We dene a Riemannian manifold as a manifold M, equipped with a Riemmanian
metric g. At each point p of M, the metric gp must have the following
properties.
1. gp : (TpM TpM) ! R is bilinear.
2. gp(v;w) = gp(w; v) 8v;w 2 TpM, which is to say gp is symmetric.
3. gp(v; v) > 0 8v 2 TpM : v 6= 0, which is to say gp is positive denite.
4. The coecients gij in every local chart
gp =
X
i;j
gij(p) dxijp
dxj jp
are dierentiable functions, where dxi
dxj(a; b) = dxi(a) dxj(b).
For further remarks on the Riemannian metric tensor, see section 3 of
chapter 2 on manifolds in higher dimensions.
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