## 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 f0gnk) = 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

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

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 ! Rnk 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 ! Rnk 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

ggiven by

g

: A Rnk U ! Rnk; 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 1

:

(U) ! Rnk is given by the formula

f 1

(x1; ; xn) = (xk+1; ; xn)

which implies that

f(U \M) = f 1

((U) \ (Rk f0gnk)) = 0;

3

i.e.

U \M = 1

((U) \ (Rk f0gnk))

since there are no other points in U whose image under f is 0.

This means (U \ M) = (U) \ (Rk f0gnk) 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 ! R

dened by

x = (x1; ; xn+1) 7! f(x) = kbxk2 a2b2 + (a2 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) 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 ! Rn

x = (x1; ; xn+1) 7! 1(x) =

x1

a

1 xn+1

b

;

x2

a

1 xn+1

b

; ;

xn

a

1 xn+1

b

:

U2 = M(n)nf(0; 0; ; 0;b)g and

2 : U2 ! Rn

x = (x1; ; xn+1) 7! 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 f(0; 0; ; 0; 1)g ! Rn;

with h1(x1; ; xn+1) =

x1

1 xn+1

; ;

xn

1 xn+1

;

h2 : Sn f(0; 0; ; 0;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 Rnd and (

\W) = U f0gnd. 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 f0gnd Rd+1 Rn+1, where f0gnd is

the zero vector in Rnd. We label the associated isomorphism I and note it

acts as follows.

I : M(d) ! M(n); p 7! I(p) := p0 = (

nd 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))) = f0gnd 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 n1, where Sd = fx 2 Rd+1 : kxk = 1g is the unit sphere in

Rd+1 .

More precisely, we have that M(d)

=

M(n) \ (f0gnd Rd+1) and

a:Sd = M(n) \ (Rd+1 f0gnd), 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

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 = = xn1 = 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) = 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. ! ^ = (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 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) = 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 [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) = (D2F3D3F2)(e1)p+(D3F1D1F3)(e2)p+(D1F2D2F1)(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

! = (1)k(nk)!:

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

! = (1)nk+n+1 d !:

Hence, we see that :

Vk(Rn

p) !

Vk1(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 = (1)1dxi ^ dxi = 0 and

d(! ^ ) = d! ^ + (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

(1)1

Z 1

0

tk1!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

tk1!i1; ;ik(tx)dt

dxi1 ^ ^ dxik

+

X

i1<<ik

Xk

=1

Xn

j=1

(1)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

(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

tk1!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; ; xi1; 0; xi; ; xn1) = (x1; ; xi1; 0; xi; ; xn1)

2) In

(i;1)(x) = In(x1; ; xi1; 1; xi; ; xn1) = (x1; ; xi1; 1; xi; ; xn1)

for x 2 [0; 1]n1.

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 (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

(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]k1

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 (1)j+

Z

[0;1]k1

Ik

(j;)

(fdx1 ^ ^ dxi ^ ^ dxk)

= (1)i+1

Z

[0;1]k

[f(x1; ; 1; ; xk) f(x1; ; 0; ; xk)]dx1 dxk

ZBesides,

Ik

d(fdx1 ^ ^ dxi ^ ^ dxk)

=

Z

[0;1]k

Difdxi ^ dx1 ^ ^ dxi ^ ^ dxk

= (1)i1

Z

[0;1]k

Dif

= (1)i1

Z 1

0

Z 1

0

Dif(x1; ; xk)dxidx1 dxi dxk (by Fubini’s

theorem)

= (1)i1

Z 1

0

Z 1

0

[f(x1; ; 1; ; xk)f(x1; ; 0; ; xk)]dx1 dxi dxk

(by the fundamental theorem of calculus in one-dimension)

= (1)i+1

Z

[0;1]k

[f(x1; ; 1; ; xk) 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

1 : V ! Rn (V Rk is open) around p = 1(a) for some a 2 V .

The k-dimensional vector space 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! = (f1)(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,

[(g1 f)((e1)a); ; (g1 f)((ek)a)] = [(e1)b; ; (e1)b] so that

det(g1 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

1. Since k2

(p) = 0, the set

2(U1\U2) then contains a point from @Hk = Rk1 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; ; vk1) 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 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; ; vk1 2 Tp(@M) so that we

have [n(p); v1; ; vk1] = p. If also [n(p);w1; ;wk1] = p then both

[v1; ; vk1] and [w1; ;wk1] 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; ; fnk : U ! R be smooth functions dened on an open subset

U Rn with df1 ^ ^ dfnk 6= 0 at each point. Then the k-dimensional

manifold Mk := fx 2 U : f1(x) = = fnk(x) = 0g is orientable.

Let

f : Rn+1nf0g ! R;

x = (x1; ; xn+1) 7! f(x) = kbxk2 a2b2 + (a2 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 D2)dx ^ dy =

Z Z

M

(

@

@x

@

@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 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 D2)dx ^ dy

=

Z Z

M

(

@

@x

@

@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 D2F3D3F2;D3F1D1F3;D1F2D2F1.

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 v3w2) 2(v1w3 v3w1) + 3(v1w2 v2w1):

dy ^ dz(v;w) = 2Alt(dy

dz(v;w)) = 2:

1

2!

(v2w3 v3w2)

=) 1dy ^ dz(v;w) = 1(v2w3 v3w2)

Likewise,

2dz ^ dx(v;w) = 2(v3w1 v1w3) and 3dx ^ dy(v;w) = 3(v1w2 v2w1):

Thus,

h; n(p)idA = 1dy ^ dz + 2dz ^ dx + 3dx ^ dy

and

h(r F); nidA

= (D2F3 D3F2)dy ^ dz +(D3F1 D1F3)dz ^ dx+(D1F2 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.

27

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