di erential forms and applications
TRANSCRIPT
Differential Forms and Applications
A ThesisSubmitted to the African University of Science and Technology
Abuja-Nigeriain partial fulfilment of the requirements for
MASTER’S DEGREE IN PURE AND APPLIEDMATHEMATICS
By
Uchechukwu Michael Opara
Supervisor:
Doctor Guy Degla
African University of Science and Technologywww.aust.edu.ng
P.M.B 681, Garki, Abuja F.C.TNigeria.
December, 2011
Preface
This project deals mainly with Differential Forms on smooth Riemannianmanifolds and their applications through the properties of their classical Dif-ferential and Integral Operators.
The calculus of Differential Forms provides a simple and flexible alternativeto vector calculus. It is not dependent on any coordinate system, simplifiesor condenses variational principles, offers a more comprehensive means ofevaluating multivariate integrals, and is crucial in the analysis of the vari-ation of differentiable functions on smooth manifolds. Differential Formshave numerous applications within (and beyond) Differential Geometry andMathematical Physics.
Needless to mention, Differential Forms constitute the ingredients (test func-tions) of the Theory of k-current which is analogous to Distribution Theory,and so they offer diverse potential tools for research.
ii
Acknowledgements
I wish to give a special vote of thanks to God, my parents, and all supportingfaculty members of the African University of Science and Technology, par-ticularly Prof. Charles Chidume (AUST Acting President), Dr. Guy Degla,Dr. Ngalla Djite and Prof. Leonard Todjihounde (IMSP, Benin) for theirimmense roles in guiding me to the completion of this Master’s project.
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Contents
1 MANIFOLDS AND FORMS 21.1 Submanifolds of Rn without boundary . . . . . . . . . . . . . 21.2 Notions of forms and fields . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 forms and vector fields on Rn . . . . . . . . . . . . . . 101.2.3 Integration over cubes and chains . . . . . . . . . . . . 15
1.3 Classical theorems of Green and Stokes . . . . . . . . . . . . . 181.3.1 Orientable Manifolds . . . . . . . . . . . . . . . . . . . 201.3.2 Riemannian Manifolds . . . . . . . . . . . . . . . . . . 27
2 EXAMPLES OF DIFFERENTIAL FORMS ON RIEMAN-NIAN MANIFOLDS 282.1 Winding form and volume element associated to ellipsoids in
R2 and in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1.1 Differential forms on the 1-dimensional ellipsoid . . . . 282.1.2 Differential forms on the 2-dimensional ellipsoid . . . . 33
2.2 Other quantities associated to R3 ellipsoid derived from Rie-mannian structure, geodesics of R3 ellipsoid . . . . . . . . . . 392.2.1 The shape operator . . . . . . . . . . . . . . . . . . . . 392.2.2 Geodesics of the 2-dimensional ellipsoid . . . . . . . . . 42
2.3 Manifolds in higher dimensions: volume element, geodesics . . 482.3.1 Higher dimensional volume forms . . . . . . . . . . . . 482.3.2 Higher dimensional geodesics . . . . . . . . . . . . . . 52
Bibliography 58
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Introduction
This body of work introduces exterior calculus in Euclidean spaces and sub-sequently implements classical results from standard Riemannian geometryto analyze certain differential forms on a manifold of reference, which here isa symmetric ellipsoid in Rn.
We focus on the foundations of the theory of differential forms in a pro-gressive approach to present the relevant classical theorems of Green andStokes and establish volume (length, area or volume) formulas.
Furthermore, we introduce the notion of geodesics and show how to obtainthem with respect to the reference manifold.
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CHAPTER 1
MANIFOLDS AND FORMS
1.1 Submanifolds of Rn without boundary
Definition 1.1.1.A subset M of Rn is called a k-dimensional submanifold without boundary iffor each point p ∈ M , there exist U, V open in Rn with p ∈ U as well as adiffeomorphism φ : U → V such that φ(U ∩M) is contained in the subspaceRk ⊆ Rn. In other words,
φ(U ∩M) = V ∩ (Rk × 0n−k) = y ∈ V : yk+1 = · · · = yn = 0.
The pair (U∗, φ) where U∗ = U ∩M is called a local chart around p and afamily of local charts covering all points of M is called an atlas on M. Thusif Ui, φii∈I⊆N is an atlas on M, then M = ∪
i∈IUi.
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 differentiable functions mapping from a subset of Rk ontoM. Parametrizations are needed for computational and analytical purposesas we will see in chapter 2 on examples of differential forms on Riemannianmanifolds.
If (U∗, φ) is a local chart with p ∈ U∗, we can identify p and the vectorφ(p) ∈ Rn. The coordinates of φ(p) in Rn are called the local coordinates
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of p in the local chart (U∗, φ). For any two charts (Ui, φi) and (Uj, φj) suchthat Ui ∩ Uj is non-empty, we can define the map,
φi φj−1 : φj(Ui ∩ Uj)→ φi(Ui ∩ Uj)which is called a chart transition from one chart to another. The setsφj(Ui ∩ Uj) and φi(Ui ∩ Uj) are open sets of the coordinate space Rk andand the transition function φi φj−1 is a diffeomorphism.Alternatively, we may define a submanifold of Rn without boundary as fol-lows.
Definition 1.1.2.Let U ⊆ Rn be an open subset and let f : U → Rn−k be a smooth map.Consider the set M = x ∈ U : f(x) = 0.If the gradient Df(x) has maximal rank (n-k) at each point x ∈ M , then Mis a smooth k-dimensional submanifold of Rn without boundary.
Remark : This latter definition is derived from the former as a direct ap-plication of the implicit function theorem, as we now briefly explain. For anarbitrary point ξ = (ξ1, · · · , ξn) ∈ M ⊂ U , we have by the implicit functiontheorem an open neighborhood A of (ξ1, · · · , ξk) in Rk and a smooth functiong : A→ Rn−k such that g(ξ1, · · · , ξk) = (ξk+1, · · · , ξn) and f(ρ, g(ρ)) = 0 forall ρ ∈ A.
Hence, there exists an open neighborhood Uξ of ξ in U so that
Uξ ∩M = x ∈ Uξ : g(x1, · · · , xk) = (xk+1, · · · , xn).
Consider also the smooth function−g given by
−g : A× Rn−k ⊃ Uξ → Rn−k;x 7→ g(x1, · · · , xk)
which belongs to the same diffeomorphism class as f |Uξ so that there also
exists a diffeomorphism φξ : Uξ → φξ(Uξ) ⊂ Rn such that the map f φ−1ξ :
φξ(Uξ)→ Rn−k is given by the formula
f φ−1ξ (x1, · · · , xn) = (xk+1, · · · , xn)
which implies that
f(Uξ ∩M) = f φ−1ξ (φξ(Uξ) ∩ (Rk × 0n−k)) = 0,
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i.e.Uξ ∩M = φ−1
ξ (φξ(Uξ) ∩ (Rk × 0n−k))
since there are no other points in Uξ whose image under f is 0.
This means φξ(Uξ ∩M) = φξ(Uξ) ∩ (Rk × 0n−k) and the local chart(Uξ ∩M,φξ) needed around ξ is thereby obtained, making M a smooth k-dimensional submanifold of Rn without boundary.
Let us consider the generalized case in Rn+1 of an ellipsoid with axialsymmetry.
Definition 1.1.3.
Define
M(n) :=
(x1, · · · , xn+1) ∈ Rn+1 :
x21
a2+x2
2
a2+ · · ·+ x2
n
a2+x2n+1
b2= 1
;
a, b ∈ R+\0. M(n) is the generalized ellipsoid in Rn+1 with the xn+1 axis ofsymmetry.
M(n) is a differentiable submanifold of Rn+1 without boundary, havingdimension n. We may justify this statement using the latter description ofsubmanifolds without boundary given in definition 1.1.2.
Consider the functionf : Rn+1 −→ R
defined by
x = (x1, · · · , xn+1) 7−→ f(x) = ‖bx‖2 − a2b2 + (a2 − b2)x2n+1
M(n) = x ∈ Rn+1 : f(x) = 0 and
Df(x) = (2b2x1, 2b2x2, · · · , 2b2xn, 2a
2xn+1).
The rank of the 1× (n+1) matrix Df(x) is strictly 1 because the xi’s cannotbe simultaneously zero since the real constants a and b are positive. Thismeans that M(n) is a manifold of dimension (n+ 1)− 1 = n.
An atlas (U1, φ1), (U2, φ2) for M(n) is given as follows.
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U1 = M(n)\(0, 0, · · · , 0, b) and
φ1 : U1 −→ Rn
x = (x1, · · · , xn+1) 7−→ φ1(x) =
( x1a
1− xn+1
b
,x2a
1− xn+1
b
, · · · ,xna
1− xn+1
b
).
U2 = M(n)\(0, 0, · · · , 0,−b) and
φ2 : U2 −→ Rn
x = (x1, · · · , xn+1) 7−→ φ2(x) =
( x1a
1 + xn+1
b
,x2a
1 + xn+1
b
, · · · ,xna
1 + xn+1
b
).
These charts are obtained as compositions of stereographic projections (h1
and h2) of the unit sphere Sn := x ∈ Rn+1 : ‖x‖ = 1 onto Rn with the lin-
ear map; T : M(n) → Sn given by (x1, · · · , xn+1) 7→(x1
a,x2
a, · · · , xn
a,xn+1
b
).
Indeed, we haveh1 : Sn − (0, 0, · · · , 0, 1) → Rn,
with h1(x1, · · · , xn+1) =
(x1
1− xn+1
, · · · , xn1− xn+1
),
h2 : Sn − (0, 0, · · · , 0,−1) → Rn,
with h2(x1, · · · , xn+1) =
(x1
1 + xn+1
, · · · , xn1 + xn+1
),
so that φ1 = h1 T and φ2 = h2 T . Since h1 and h2 are surjective, themaps φ1 and φ2 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) inconnection with its submanifolds.
Definition 1.1.4.Let M be a differentiable manifold of dimension n. A submanifold of dimen-sion d ≤ n of M is a subset W ⊂ M such that for any point p ∈ W , thereexists a local chart (Ω, φ) around p such that φ(Ω ∩W ) = U × V,U ⊂ Rd, V ⊂ Rn−d and φ(Ω∩W ) = U ×0n−d. Thus, there exists a system
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of local coordinates (x1, · · · , xn) on Ω in which the submanifold W is locallydefined 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 ∈M(d) ⊆ Rd+1; p = (x1, · · · , xd+1). Clearly,
Rd+1 is isometrically isomorphic to 0n−d × Rd+1 ⊂ Rn+1, where 0n−d isthe zero vector in Rn−d. We label the associated isomorphism I and note itacts as follows.
I : M(d) −→M(n), p 7−→ I(p) := p′ = (
n−d︷ ︸︸ ︷0, 0, · · · , 0, x1, · · · , xd+1)
Hence, for each point p ∈ M(d), the local chart around p′ = I(p) (whichis either (U1, φ1) or (U2, φ2) as specified above) has the following property,φi(Ui ∩ I(M(d))) = 0n−d × Rd. This is to say that M(d) is isometricallyisomorphic to I(M(d)); a submanifold of M(n).
By a similar approach, we also see that the sphere a.Sd is a submanifoldof M(n) for d ≤ n− 1, where Sd = x ∈ Rd+1 : ‖x‖ = 1 is the unit sphere inRd+1 .
More precisely, we have that M(d)∼= M(n) ∩ (0n−d × Rd+1) and
a.Sd ∼= M(n) ∩ (Rd+1×0n−d), adhering to the axial orientation specified indefinition 1.1.3.
Cartesian products of manifolds may be defined when appropriate withdim(A × B) = dim(A) + dim(B) for manifolds A and B. Nevertheless, it isclearer that we can obtain the ellipsoid M(n) as a manifold of revolution. Wespecify how to obtain M(n) by revolution in the following proposition.
Proposition 1.1.6.Let Kn+1 = x ∈ Rn+1 : x1 = 0 and x2, · · · , xn ≥ 0. Then the manifoldM(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 ∈ Rn+1 can be given a polar coordinate (rp, (θ1)p, · · · , (θn)p)where rp is the Euclidean distance from p to the origin and (θk)p is the an-gular position of p in the (xk, xn+1) plane measured counterclockwise from
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the xk axis. Hence for a point p′ ∈M(n) ∩Kn+1, we have (θ1)p′ , · · · , (θn)p′
as a subset of
[−π2,π
2
].
Rotation of a manifold about the xn+1 axis entails rotation of its cross-sections about the xn+1 axis in each (xk, xn+1) plane. As such, by one fullrevolution about the xn+1 axis, the angular positions of the points (mod 2π)are no longer restricted in any (xk, xn+1) plane for 2 ≤ k ≤ n. This elimi-nates 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 symmetryabout the xn+1 axis.
Moreover, by revolving an arbitrary point with Euclidean coordinates(0, x′2, x
′3, · · · , x′n+1) about the xn+1 axis, the result is the sphere in
x ∈ Rn+1 : xn+1 = x′n+1 centered at (0, 0, · · · , 0, x′n+1) with radius‖(0, x′2, x′3, · · · , x′n, 0)‖2. But the x1 coordinates of this sphere are clearlynot restricted to zero as long as at least one of x′2, x
′3, · · · , x′n is not zero. If
(0, x′2, x′3, · · · , x′n+1) ∈M(n)∩Kn+1, then its sphere by revolution about the
axis of symmetry is a submanifold of M(n), meaning that the x1 coordinates ofthe 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, weget 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 = x ∈ Rn+1 : x1 = x2 = · · · = xk = 0 and xk+1, · · · , xn ≥ 0,
then by a similar construction we recover M(n) by rotating its (n-k) dimen-sional submanifold M(n) ∩K completely about the xn+1 axis. The simplestcase is by rotating the 1-dimensional half ellipse given by
x ∈ Rn+1 : x1 = x2 = · · · = xn−1 = 0, xn ≥ 0 andx2n
a2+x2n+1
b2= 1
about the xn+1 axis in the space Rn+1 to get M(n). This geometric propertyof M(n) will be reconciled to differential forms on the manifold in furtheranalytic and theoretic observations. (See section 1 of chapter 2 on the volumeelement of M(2).)
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1.2 Notions of forms and fields
1.2.1 Tensors
A mapping T from V , an n-dimensional vector space over R, to R is calleda k-tensor on V if T : V k → R is k-linear. In other words, T is a k-tensor onV iff the following two conditions hold:
i) T (v1, · · · , vi+v′i, · · · , vk) = T (v1, · · · , vi, · · · , vk)+T (v1, · · · , v′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 ∈ =k(V ) and T ∈ =l(V ), their tensor product S⊗T belongs to =k+l(V )and is defined byS ⊗ 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-vectorspace denoted
∧k(V ). For every T ∈ =k(V ),we generally define Alt(T) by
Alt(T )(v1, · · · , vk) =1
k!
∑σ∈Sk
sgnσ.T (vσ(1), · · · , vσ(k))
where Sk is the set of all permutations of the integers 1 to k.
We observe that Alt(T ) ∈∧k(V ). For ω ∈
∧k(V ), η ∈∧l(V ), we define
their wedge product or exterior product denoted ω ∧ η which belongs to∧k+l(V ) by
ω ∧ η =(k + l)!
k!l!Alt(ω ⊗ η).
We also give the following properties of the wedge product
1. (ω + η) ∧ θ = ω ∧ θ + η ∧ θ ∀ω, η ∈∧k(V ); θ ∈
∧l(V )
2. ω ∧ (η + θ) = ω ∧ η + ω ∧ θ ∀ω ∈∧k(V ); η, θ ∈
∧l(V )
3. aω ∧ η = ω ∧ (aη) = a(ω ∧ η) ∀a ∈ R, ω ∈∧k(V ), η ∈
∧l(V )
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4. (ω ∧ η) ∧ θ = ω ∧ (η ∧ θ) ∀ω ∈∧k(V ), η ∈
∧l(V ), θ ∈∧m(V )
5. ω ∧ η = (−1)klη ∧ ω ∀ω ∈∧k(V ), η ∈
∧l(V )
=1(V ) is the set of all linear maps from V to R, which in this case coin-cides with the dual V ∗ of V because V is finite dimensional. If (v1, · · · , vn)is a basis for V and (ϕ1, · · · , ϕn) the corresponding dual basis then the setof all k-fold tensor products ϕi1 ⊗ · · · ⊗ ϕik : 1 ≤ i1, · · · , ik ≤ n is a basis for=k(V ), hence having dimension nk. Note that ϕi(vj) = 0 when i 6= j andϕi(vi) = 1.
The set of all ϕi1 ∧ · · · ∧ ϕik : 1 ≤ i1 < i2 < · · · < ik ≤ n is a basis for∧k(V ) which therefore has dimension
(nk
)=
n!
k!(n− k)!
For a differentiable function f : Rn → R, Df(p)p∈Rn is an example of a
linear map from Rn to R so Df(p) ∈∧1(Rn) = =1(Rn).
An inner product T : V × V → R is a bilinear functional or 2-tensor onV which is symmetric, that is, T (v, w) = T (w, v) ∀v, w ∈ V and positivedefinite, that is, T (v, v) > 0 if v 6= 0. Hence, T ∈ =2(V ) and we recognize〈, 〉 as the usual inner product on Rn.
A symplectic map A : V × V → R is another type of bilinear functionalon V which is anti-symmetric, that is, A(v, w) = −A(w, v) ∀v, w ∈ V and sosatisfies A(v, v) = 0 for all v ∈ V . A is also non-degenerate, meaning thatA(u, v) = 0 for all v ∈ V if and only if u = 0. Hence A ∈
∧2(V ) and wespecifically identify such an alternating 2-tensor in section 1 of chapter 2.
Let us examine the vector space∧n(V ) which has dimension
(nn
)= 1.
Because of this singular dimension, each element of∧n(V ) is simply a scalar
product of any other non-zero one. The determinant is clearly an alternatingn-tensor on V and we state this as det ∈
∧n(V ). This fact comes to play inthe following theorem.
Theorem 1.2.1.Let (v1, · · · , vn) be a basis for V and let ω ∈
∧n(V ). If wi =∑n
j=1 aijvj aren vectors in V, then ω(w1, · · · , wn) = det(aij).ω(v1, · · · , vn).
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Proof. We first define η ∈ =n(Rn) by
η((a11, · · · , a1n), · · · , (an1, · · · , ann)) = ω(∑n
j=1 a1jvj, · · ·∑n
j=1 anjvj)
= ω(w1, · · · , wn).
Then η ∈∧n(Rn) so η = λ.det for some constant λ ∈ R. Now, applying
both sides to (e1, · · · , en), we get λ = η(e1, · · · , en) = ω(v1, · · · , vn). As such,
η((a11, · · · , a1n), · · · , (an1, · · · , ann)) = λ.det(aij)
which impliesω(w1, · · · , wn) = det(aij).ω(v1, · · · , vn).
As a consequence of theorem 1.2.1, a non-zero ω ∈∧n(V ) splits the bases
of V into two disjoint groups, those with ω(v1, · · · , vn) > 0 and those withω(v1, · · · , vn) < 0. If (v1, · · · , vn) and (w1, · · · , wn) are two bases such thatwi =
∑nj=1 aijvj, then these two bases are in the same group iff det(aij) > 0.
Either of the two disjoint groups is called an orientation for V. The orien-tation to which a basis (v1, · · · , vn) belongs is denoted [v1, · · · , vn] and theother orientation is denoted −[v1, · · · , vn]. Notably, orientations are indepen-dent of the element ω which acts, meaning that ω 6= 0 separates bases intothe same orientations. In Rn, the usual orientation is defined as [e1, · · · , en].
There is a unique ω ∈∧n(V ) such that ω(v1, · · · , vn) = 1 whenever
v1, · · · , vn is an orthonormal basis such that [v1, · · · , vn] = µ. This uniqueω is called the volume element of V determined by the orientation µ and aninner product. The determinant is the volume element of Rn determined bythe usual inner product and usual orientation.
1.2.2 forms and vector fields on Rn
Let p ∈ Rn. The set of all pairs (p, v) for v ∈ Rn is denoted Rnp and is called
the tangent space of Rn at p, i.e. Rnp := (p, v), v ∈ Rn. This set is clearly
made a vector space by defining the following operations :
1. (p, v) + (p, w) = (p, v + w); v, w ∈ Rn,
2. a(p, v) = (p, av); a ∈ R.
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A vector v ∈ Rn can be seen as an arrow from 0 to v, and the vector(p, v) ∈ Rn
p is then seen as an arrow with the same direction and length, butwith 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 define the usual inner product 〈, 〉p for Rnp by 〈vp, wp〉p = 〈v, w〉p and
assign to Rnp its usual orientation := [(e1)p, · · · , (en)p].
A vector field on Rn is a function F : Rn → Rnp such that F (p) ∈ Rn
p
for each p ∈ Rn. Hence any vector field F can be written as F (p) =∑ni=1 F
i(p).(ei)p thereby yielding n component functions F i : Rn → R.A similar structure can be placed only on open subsets of Rn. For an opensubset U of Rn, we define a vector field as a function which assigns to eachpoint p ∈ U a unique vector from the tangent space of Rn at p.
The divergence of F (divF ) is defined as∑n
i=1DiFi. Employing the
notation ∇ =∑n
i=1Di.ei, we may symbolize div F by 〈∇, F 〉. Note that
Di =∂
∂xi.
For n = 3, we can define a vector field called the curl of F or curl F , whichwe symbolize ∇× F in accordance with the notation for ∇. Hence,(∇×F )(p) = (D2F
3−D3F2)(e1)p+(D3F
1−D1F3)(e2)p+(D1F
2−D2F1)(e3)p.
Now, a function ω with ω(p) ∈∧k(Rn
p) is called a k-form on Rn or adifferential form. If ϕ1(p), · · · , ϕn(p) is the dual basis to (e1)p, · · · , (en)p thenω(p) is of the appearance∑
i1<···<ik
ωi1 · · ·ωikϕi1(p) ∧ · · · ∧ ϕik(p)
for certain functions or 0-forms ωi1,··· ,ik . Functions written as f which map toR are 0-forms. Suppose f : Rn → R is differentiable so that Df(p) ∈
∧1(Rn).We then obtain an associated 1-form df, defined by df(p)(vp) = Df(p)(v).
Upon consideration of the projection maps πi otherwise denoted dxi, for(1 ≤ i ≤ n), we observe that these belong to the dual of Rn anddxi(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 ω(p) is a k-form on Rn
p it can always be written as
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∑i1<···<ik
ωi1 · · ·ωik(p)dxi1(p) ∧ · · · ∧ dxik(p).
For a differentiable map f : Rn → R, df = D1f.dx1 + · · ·+Dnf.dxn.
For f : Rn → Rm differentiable, we have a linear map Df(p) : Rn → Rm
to which is associated the linear transformation f∗ : Rnp → Rm
f(p) definedby f∗(vp) = (Df(p)(v))f(p).
The above induces another linear transformation called the pullback off , written f ∗ :
∧k(Rmf(p)) →
∧k(Rnp). Therefore if ω is a k-form on
Rm, we define a k-form f ∗ω on Rn by (f ∗ω)(p) = f ∗(ω(f(p))). This sim-ply means that if v1, · · · , vk ∈ Rn
p, then we have f ∗ω(p)(v1, · · · , vk) =ω(f(p))(f∗(v1), · · · , f∗(vk)). Let ω be a k-form, then the differential operator(d) acts on ω to produce a (k+1)-form dω which is called the differential of ω.
In general, if
ω =∑
i1<···<ik
ωi1 · · ·ωikdxi1 ∧ · · · ∧ dxik ,
then
dω =∑
i1<···<ik
d(ωi1 · · ·ωik) ∧ dxi1 ∧ · · · ∧ dxik
=∑
i1<···<ik
n∑α=1
Dα(ωi1 · · ·ωik)dxα ∧ dxi1 ∧ · · · ∧ dxik
DefinitionsThe hodge operator, denoted ∗, is a linear operator on
∧k(Rnp) which assigns
an (n-k)-form to each k-form. It has the following property which describesit concisely;
∗(dxi1(p) ∧ · · · ∧ dxik(p)) = dxik+1(p) ∧ · · · ∧ dxin(p),
where (i1, · · · , ik, ik+1, · · · , in) is an even permutation of the integers from 1to n.For ω ∈
∧k(Rnp), the (n-k)-form ∗ω is called the hodge dual of ω. Note that
∗ ∗ ω = (−1)k(n−k)ω.
12
An important application of the hodge operator is to define the codifferential(δ) of forms. For a k-form ω, we have its codifferential given by
δω = (−1)nk+n+1 ∗ d ∗ ω.
Hence, we see that δ :∧k(Rn
p)→∧k−1(Rn
p).
The Laplace - Beltrami operator ∆ :∧k(Rn
p)→∧k(Rn
p) is given by
∆ = dδ + δd.
In separate outstanding considerations, the hodge operator, codifferentialand Laplace - Beltrami operator are important tools used in the analysis ofHodge theory.
Important properties of f ∗; the pullback of f for f : Rn → Rm;(u1, u2, · · · , un) 7→ (x1, x2, · · · , xm) differentiable are listed below
1. f ∗(dxi) =∑n
j=1 Djfiduj = df i
2. f ∗(ω1 + ω2) = f ∗(ω1) + f ∗(ω2)
3. f ∗(g · ω) = (g f)f ∗ω; for a functional g : Rm → R
4. f ∗(ω ∧ η) = f ∗ω ∧ f ∗η
5. If n = m, then f ∗(hdx1 ∧ · · · ∧ dxn) = (h f)(detf ′)du1 ∧ · · · ∧ dun
6. f ∗(dω) = d(f ∗ω)
Concerning the differential operator d, there are yet some important ob-servations to make. We have d2 = 0, which is to say d(dω) = 0 for anydifferential form ω. Also, dxi ∧ dxi = (−1)1dxi ∧ dxi = 0 andd(ω ∧ η) = dω ∧ η + (−1)kω ∧ dη for a k-form ω and an l-form η.
A form ω is closed if dω = 0 and exact if ω = dη for some form η. Everyexact form is closed since if ω = dη then dω = d(dη) = 0. The converse doesnot necessarily hold. The next theorem gives a sufficient condition for closedforms to be exact.
Theorem 1.2.2. (Poincare Lemma)Let W ⊂ Rn be an open set star-shaped with respect to the origin, then everyclosed form on W is exact. A set is said to be star-shaped with respect to theorigin if it includes the origin as well as the entire line segment connectingthe origin to each of its other points.
13
Proof. We define a function I from k-forms to (k-1)-forms (for each k), suchthat I(0) = 0 and ω = I(dω) + d(Iω) for any form ω. It follows thatω = d(Iω) if dω = 0.
Letω =
∑i1<···<ik
ωi1,··· ,ikdxi1 ∧ · · · ∧ dxik .
Since A is star-shaped we can define
Iω(x) =∑
i1<···<ik
k∑α=1
(−1)α−1
(∫ 1
0
tk−1ωi1,··· ,ik(tx)dt
)xiαdxi1∧· · ·∧dxiα∧· · ·∧dxik
(The strikethrough beneath dxiα indicates that it is omitted from the term.)
We now prove that ω = I(dω) + d(Iω).By Leibnitz’s rule,
d(Iω) = k.∑
i1<···<ik
(∫ 1
0
tk−1ωi1,··· ,ik(tx)dt
)dxi1 ∧ · · · ∧ dxik
+∑
i1<···<ik
k∑α=1
n∑j=1
(−1)α−1
(∫ 1
0
tkDj(ωi1,··· ,ik)(tx)dt
)xiα
dxj ∧ dxi1 ∧ · · · ∧ dxiα ∧ · · · ∧ dxik .
We also have
dω =∑
i1<···<ik
n∑j=1
Dj(ωi1,··· ,ik)dxj ∧ dxi1 ∧ · · · ∧ dxik .
Applying I to the (k+1)-form dω, we obtain
I(dω) = A+B
where
A =∑
i1<···<ik
n∑j=1
(∫ 1
0
tkDj(ωi1,··· ,ik)(tx)dt
)xjdxi1 ∧ · · · ∧ dxik
14
and
B =∑
i1<···<ik
∑k
α=1
∑n
j=1(−1)α
(∫ 1
0
tkDj(ωi1,··· ,ik)(tx)dt
)xiαdxj∧dxi1∧· · ·∧dxiα∧· · ·∧dxik
Adding, the triple sums cancel, and we obtain
d(Iω) + I(dω) =∑
i1<···<ik
k.
(∫ 1
0
tk−1ωi1,··· ,ik(tx)dt
)dxi1 ∧ · · · ∧ dxik
+∑
i1<···<ik
∑n
j=1
(∫ 1
0
tkxjDj(ωi1,··· ,ik)(tx)dt
)dxi1 ∧ · · · ∧ dxik
=∑
i1<···<ik
(∫ 1
0
d
dt[tkωi1,··· ,ik(tx)]dt
)dxi1 ∧ · · · ∧ dxik
=∑
i1<···<ikωi1,··· ,ikdxi1 ∧ · · · ∧ dxik = ω
An example worthy of note which outrightly incorporates these discussednotions about differential forms and vector fields in physics is Maxwell’sequations of electromagnetism. The setting is R4 (a space - time manifold),and performing relevant operations on the electromagnetic field as an exactdifferential 2-form yields mathematical interpretations of profound physicalresults. However, this is an illustration in Lorentzian geometry which differsfrom Riemannian geometry by way of the metric.
1.2.3 Integration over cubes and chains
Essentially, differential forms have to be integrated over domains where theyare defined in Rn. This gives rise to the use of singular k-cubes in domains ofRn, which are suitable parametrizations of the domains for this purpose. Asingular k-cube in A ⊆ Rn is a continuous function c mapping from [0, 1]k toA. Any singular 1-cube is a curve, and singular 2-cubes are surfaces. Stan-dard n-cubes in Rn are often denoted In with In : [0, 1]n → Rn defined byIn(x) = x for x ∈ [0, 1]n.A linear combination
∑i∈I⊆N aici ; ai ∈ Z of singular k-cubes ci is referred
to as a singular k-chain. Each singular k-chain c has a boundary denoted ∂cwhich is a (k-1) chain. To get the general formula of ∂c for an n-chain c,we first formulate ∂In. For i : 1 ≤ i ≤ n, define the following singular (n-1)
15
cubes.
1) In(i,0)(x) = In(x1, · · · , xi−1, 0, xi, · · · , xn−1) = (x1, · · · , xi−1, 0, xi, · · · , xn−1)
2) In(i,1)(x) = In(x1, · · · , xi−1, 1, xi, · · · , xn−1) = (x1, · · · , xi−1, 1, xi, · · · , xn−1)
for x ∈ [0, 1]n−1.
In(i,0) is called the (i,0)-face of In and In(i,1) the (i,1)-face.
Now, ∂In :=∑n
i=1
∑α=0,1 (−1)i+αIn(i,α).
For a singular n-cube c; we define its (i, α)-face, c(i,α) = c (In(i,α))so that
∂c :=n∑i=1
∑α=0,1
(−1)i+αc(i,α).
Finally, the boundary of an n-chain∑
i∈I⊆Naici is given by
∂
(∑i∈I
aici
)=∑i∈I
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) = 0for any singular n-chain c. Other properties and relations derived from theboundary operator are highlighted next in classical theorems by Stokes andGreen. The orientations of domains of integration will also be considered,without which integrands obtained over singular k-cubes can only be guar-anteed to be accurate up to sign.
17
1.3 Classical theorems of Green and Stokes
If ω is a k-form on [0, 1]k, then ω = fdxi1 ∧ · · · ∧ dxik for a unique 0-form f .We then have∫
[0,1]kω =
∫[0,1]k
fdx1 ∧ · · · ∧ dxk =
∫[0,1]k
f(x1, · · · , xk)dx1 · · · dxk
For ω a k-form on A ⊆ Rn and c a singular k-cube in A, we define∫c
ω :=
∫[0,1]k
c∗ω
recalling that c∗ω is an induced k-form on [0, 1]k. The integral of a form ωover a k-chain c =
∑i∈I aici is given by
∫cω =
∑i∈I ai
∫ciω.
The integral of a 1-form over a 1-chain is called a line integral and the inte-gral 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 atheorem by Stokes, which is popularly recognized as the fundamental theo-rem of calculus in higher dimensions.
Theorem 1.3.1. (Stokes’ Theorem (a))
If ω is a (k-1) form on an open subset A ⊆ Rn and c is a k-chain in A,then
∫cdω =
∫∂cω.
Proof. We first take c to be the standard k-cube Ik, and ω to be a (k-1) formon [0, 1]k.In this case, ω 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 fromthe term), and we sufficiently prove the theorem for each of these. Note that∫
[0,1]k−1Ik(j,α)
∗(fdx1∧· · ·∧dxi∧· · ·∧dxk) = δij
∫[0,1]k
f(x1, · · · , α, · · · , xk)dx1 · · · dxk,
where δij =
1 if j = i
0 otherwise
18
Thus,∫∂Ikfdx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxk
=∑k
j=1
∑α=0,1 (−1)j+α
∫[0,1]k−1
Ik(j,α)∗(fdx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxk)
= (−1)i+1
∫[0,1]k
[f(x1, · · · , 1, · · · , xk)− f(x1, · · · , 0, · · · , xk)]dx1 · · · dxk
Besides,∫Ikd(fdx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxk)
=
∫[0,1]k
Difdxi ∧ dx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxk
= (−1)i−1
∫[0,1]k
Dif
= (−1)i−1
∫ 1
0
· · ·∫ 1
0
Dif(x1, · · · , xk)dxidx1 · · · dxi · · · dxk (by Fubini’s
theorem)
= (−1)i−1
∫ 1
0
· · ·∫ 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
∫[0,1]k
[f(x1, · · · , 1, · · · , xk)− f(x1, · · · , 0, · · · , xk)]dx1 · · · dxk
Hence, ∫Ikdω =
∫∂Ikω.
Now, let c be an arbitrary singular k-cube, then∫c
dω =
∫Ikc∗(dω) =
∫Ikd(c∗ω) =
∫∂Ikc∗ω =
∫∂c
ω.
Finally, if c is a k-chain∑i∈Iaici, then
∫c
dω =∑i∈I
ai
∫ci
dω =∑i∈I
ai
∫∂ci
ω =
∫∂c
ω.
Before presenting the other theorems, we briefly view the structures offields and forms on differentiable manifolds.
19
Let M be a k-dimensional manifold in Rn and the local chart arounda point p ∈ M be (U, φ). Then we can define a local coordinate systemφ−1 : V → Rn (V ⊆ Rk is open) around p = φ−1(a) for some a ∈ V .The k-dimensional vector space φ−1
∗ (Rka) is denoted TpM , and is called the
tangent space of M at p. This space is independent of which local coordinatesystem is used to derive it. A function which assigns a vector in TpM to eachpoint p ∈M is called a vector field on M. A function which assigns an alter-nating k-tensor in
∧k(TpM) to each p ∈ M is called a k-form on M. Hence,given a vector field F on M, F : M →
⋃p∈MTpM and a 1-form ωp : TpM → R,
we may obtain the composition ωp(F ) = ωF (p) which is a mapping from Mto R. Inadvertently, differential 1-forms constitute the dual to vector fieldson a given manifold.
If f : W ⊆ Rk → Rn is a coordinate system, ω a k-form on M, then f ∗ωis a k-form on W and we say ω is differentiable if f ∗ω is. A k-form ω can bewritten ω =
∑i1<···<ik
ωi1 · · ·ωikdxi1 ∧ · · · ∧ dxik .
Since the functions ωi1 , · · · , ωik may be defined only on M, the previousdefinition given for dω may not be valid here, as Dj(ωi1 , · · · , ωik) would haveno meaning. However, the relation f ∗(dω) = d(f ∗ω) still holds, so we definethe differential of ω as dω = (f−1)
∗(d(f ∗ω)).
1.3.1 Orientable Manifolds
It is often important to choose, if possible, an orientation µp for each tangentspace TpM of a manifold M. These choices are called consistent if, given acoordinate system f : W → Rn and a, b ∈ W , then we have
[f∗((e1)a), · · · , f∗((ek)a)] = µf(a) ⇐⇒ [f∗((e1)b), · · · , f∗((ek)b)] = µf(b)
If orientations µp have been chosen consistently and f : W → Rn is a coor-dinate system such that [f∗((e1)a), · · · , f∗((ek)a)] = µf(a) for one and henceevery a ∈ W , then f is called orientation - preserving. If f is not orientation- preserving and T : Rk → Rk is a linear transformation such that detT isnegative, then f T is orientation - preserving. Hence, as long as orientationscan be chosen consistently, there exists an orientation - preserving coordinatesystem 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)] = µp = [g∗((e1)b), · · · , g∗((ek)b)]. Therefore,[(g−1 f)∗((e1)a), · · · , (g−1 f)∗((ek)a)] = [(e1)b, · · · , (e1)b] so thatdet(g−1 f)′ > 0.
A manifold for which orientations µp can be chosen consistently is ori-entable and a choice for µp is called an orientation of the manifold. A mani-fold M together with an orientation µ is called an oriented manifold.
One of the most known examples of a non-orientable manifold in R3 isthe Mobius strip.
Manifolds with BoundaryIf we have M ⊆ Rn to be a k-dimensional manifold - with - boundary, thenfor each point p ∈ M, either
1. there exist open sets U and V with p ∈ U ⊆ Rn, V ⊆ Rn and adiffeomorphism φ : U → V such that φ(U ∩M) = V ∩ (Rk × 0), OR
2. there exist open sets U and V, with p ∈ U ⊆ Rn, V ⊆ Rn and adiffeomorphism φ : U → V such thatφ(U∩M) = V ∩(Hk×0) = y ∈ V : yk ≥ 0 and yk+1 = · · · = yn = 0,and φ(p) has kth component equal to 0.
Hk = x ∈ Rk : xk ≥ 0 is called a half-space of Rk.
Conditions (1) and (2) cannot be satisfied by the same point p ∈ M .Assuming on the contrary that there is a point which satisfies (1) and (2),then there would exist diffeomorphisms φ1 : U1 → V1 and φ2 : U2 → V2 suchthat φ1(U1 ∩M) = V1 ∩ Rk and φ2(U2 ∩M) = V2 ∩Hk, φk2(p) = 0.The set φ1(U1 ∩ U2) would then be an open subset in Rk mapped ontoφ2(U1 ∩ U2) by the diffeomorphism φ2 φ1
−1. Since φk2(p) = 0, the setφ2(U1∩U2) then contains a point from ∂Hk = Rk−1 and so it cannot be openin Rk. This is a contradiction to the inverse function theorem.
A point p ∈ M which satisfies (2) is called a boundary point of M andwe denote by ∂M the boundary of M, which is the set of all boundary pointsof 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 smoothk-dimensional manifold extended from M at its boundary. If p ∈ ∂M , thenTp(∂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 perpendic-
ular to Tp(∂M). If (v1, · · · , vk) is an orthonormal basis for TpM∗ such that
(v1, · · · , vk−1) is a basis for Tp(∂M), then vk ∈ TpM∗ is one of the unit vec-tors 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 ∈ ∂M ,then only one of these unit vectors is f∗(v0) for some v0 ∈ W with (v0)k < 0.This unit vector is called the outward unit normal n(p) and it is independentof the coordinate system f used to obtain it.
Suppose that µ is an orientation of the k-dimensional manifold - with- boundary M. If p ∈ ∂M , we choose v1, · · · , vk−1 ∈ Tp(∂M) so that wehave [n(p), v1, · · · , vk−1] = µp. If also [n(p), w1, · · · , wk−1] = µp then both[v1, · · · , vk−1] and [w1, · · · , wk−1] are the same orientation for Tp(∂M), eitherof which is denoted by (∂µ)p. If M is orientable, then ∂M is also orientableand an orientation µ for M determines an orientation ∂µ for ∂M called theinduced orientation.
The ellipsoid M(n), as we recall from definition 1.1.3, is an n-dimensionalmanifold in Rn+1 without boundary and it is the boundary for the (n+1) -dimensional manifold with boundary
L(n+1) :=
(x1, · · · , xn+1) ∈ Rn+1 :
x21
a2+x2
2
a2+ · · ·+ x2
n
a2+x2n+1
b2≤ 1
of Rn+1. As such, if for p ∈ M(n) we have [v1, · · · , vn] = µp, we obtain theoutward unit normal to M(n) at p; ψ(p) ∈ Rn+1
p so that ψ(p) is a unit vec-tor prependicular to TpM(n) and [ψ(p), v1, · · · , vn] is the orientation of Rn+1
p
which induces µp. Note that for an interior point a ∈ L(n+1), the vector spaceRn+1
a coincides with TaL(n+1). A direct explanation for the orientability ofM(n) is drawn from an alternative definition given as follows.
Let f1, · · · , fn−k : U → R be smooth functions defined on an open subsetU ⊆ Rn with df1 ∧ · · · ∧ dfn−k 6= 0 at each point. Then the k-dimensionalmanifold Mk := x ∈ U : f1(x) = · · · = fn−k(x) = 0 is orientable.
Letf : Rn+1\0 → R;x = (x1, · · · , xn+1) 7→ f(x) = ‖bx‖2 − a2b2 + (a2 − b2)x2
n+1
Thendf = 2b2x1dx1 + 2b2x2dx2 + · · ·+ 2b2xndxn + 2a2xn+1dxn+1 6= 0 on Rn+1\0.
22
Hence, M(n) = x ∈ Rn+1\0 : f(x) = 0 is orientable.
We now state further theorems utilizing the concepts of boundaries ofmanifolds and their orientations.
Theorem 1.3.2. (Stokes’ Theorem (b))
If M is a compact oriented k-dimensional manifold - with - boundary andω is a (k - 1) form on M, then
∫Mdω =
∫∂Mω where ∂M is given the induced
orientation.
The proof of this theorem incorporates a standard tool required in thetheory of integration called partitions of unity.
Lemma 1.3.3.For A ⊆ Rn and O an open cover of A, there is a collection Φ of C∞ func-tions ϕ defined in an open set containing A called a C∞ partition of unityfor A subordinate to the cover O, with the following properties:
(1) For each x ∈ A we have 0 ≤ ϕ(x) ≤ 1.
(2) For each x ∈ A there is an open set V containing x such that all butfinitely many ϕ ∈ Φ are 0 on V.
(3) For each x ∈ A, we have∑ϕ∈Φ
ϕ(x) = 1. By (2) for each x this sum
is finite in some open set containing x.
(4) For each ϕ ∈ Φ there is an open set U in O such that ϕ = 0 outsideof some closed set contained in U.
Proof of Theorem 1.3.2 Commencing the proof this theorem, supposethat there is an orientation - preserving singular k-cube c in M − ∂M suchthat ω = 0 outside of c([0, 1]k). By Theorem 1.3.1 and the definition of dωwe have ∫
c
dω =
∫[0,1]k
c∗(dω) =
∫[0,1]k
d(c∗ω) =
∫∂Ikc∗ω =
∫∂c
ω.
23
Then, ∫M
dω =
∫c
dω =
∫∂c
ω = 0,
since ω = 0 on ∂c. On the other hand,∫∂Mω = 0 since ω = 0 on ∂M .
Suppose next that there is an orientation-preserving singular k-cube inM such that c(k,0) is the only face in ∂M , and ω = 0 outside of c([0, 1]k). Then∫
M
dω =
∫c
dω =
∫∂c
ω =
∫∂M
ω.
We may now consider the general case. There is an open cover O of Mand a partition of unity Φ for M subordinate to O such that for each ϕ ∈ Φthe form ϕ · ω is of one of the two sorts just considered. We have
0 = d(1) = d
(∑ϕ∈Φ
ϕ
)=∑ϕ∈Φ
dϕ,
so that ∑ϕ∈Φ
dϕ ∧ ω = 0.
Since M is compact, this is a finite sum and we have∑ϕ∈Φ
∫M
dϕ ∧ ω = 0.
Therefore, ∫M
dω =∑ϕ∈Φ
∫M
ϕ · dω
=∑ϕ∈Φ
∫M
dϕ ∧ ω + ϕ · dω
=∑ϕ∈Φ
∫M
d(ϕ · ω)
=∑ϕ∈Φ
∫∂M
ϕ · ω
=
∫∂M
ω.
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 α, β : M → R are differentiable. Then∫
∂M
αdx+ βdy =
∫M
(D1β −D2α)dx ∧ dy =
∫ ∫M
(∂β
∂x− ∂α
∂y)dxdy,
where M is given the usual orientation and ∂M the induced orientation, oth-erwise called the counterclockwise orientation.
Proof. We find the differential of the 1-form (αdx+ βdy) to be
d(αdx+ βdy) = dα ∧ dx+ dβ ∧ dy= (D1αdx+D2αdy) ∧ dx+ (D1βdx+D2βdy) ∧ dy= D2αdy ∧ dx+D1βdx ∧ dy= (D1β −D2α)dx ∧ dy
We now apply theorem 1.3.2 directly to get∫∂M
αdx+ βdy =
∫M
d(αdx+ βdy)
=
∫M
(D1β −D2α)dx ∧ dy
=
∫ ∫M
(∂β
∂x− ∂α
∂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 orientationof M. Let ∂M have the induced orientation. Let G be the vector field on ∂Mwith ds(G) = 1 and F be a differentiable vector field in an open set containing
25
M. Then∫M〈(∇× F ), n〉dA =
∫∂M〈F,G〉ds
(dA and ds are respectively referred to as element of area and element ofarclength.)
Proof. Define η on M by η = F 1dx + F 2dy + F 3dz. Recall the curl of F,∇×F respectively has componentsD2F
3−D3F2, D3F
1−D1F3, D1F
2−D2F1.
For a two-dimensional manifold, the element of volume is the element of areadA ∈
∧2(TpM) and
dA(v, w) = det
vwn(p)
∀v, w ∈ TpM ,
where n(p) is the outward unit normal, since dA(v, w) is 1 if v and w forman orthonormal basis for TpM with [v, w] = µp.Note that dA(v, w) = 〈v × w, n(p)〉.Let ξ ∈ R3
p, observing that v × w = α.n(p), α = ±‖v × w‖ ∈ R.〈ξ, n(p)〉dA(v, w) = 〈ξ, n(p)〉α = 〈ξ, αn(p)〉 = 〈ξ, v × w〉
The above scalar triple product equals∣∣∣∣∣∣ξ1 ξ2 ξ3
v1 v2 v3
w1 w2 w3
∣∣∣∣∣∣ = ξ1(v2w3 − v3w2)− ξ2(v1w3 − v3w1) + ξ3(v1w2 − v2w1).
dy ∧ dz(v, w) = 2Alt(dy ⊗ dz(v, w)) = 2.1
2!(v2w3 − v3w2)
=⇒ ξ1dy ∧ dz(v, w) = ξ1(v2w3 − v3w2)
Likewise,ξ2dz ∧ dx(v, w) = ξ2(v3w1 − v1w3) and ξ3dx ∧ dy(v, w) = ξ3(v1w2 − v2w1).
Thus,〈ξ, n(p)〉dA = ξ1dy ∧ dz + ξ2dz ∧ dx+ ξ3dx ∧ dy
and〈(∇× F ), n〉dA= (D2F
3−D3F2)dy ∧ dz+ (D3F
1−D1F3)dz ∧ dx+ (D1F
2−D2F1)dx∧ dy
= dη.
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 ∈∂M , since G(p) is a basis for Tp(∂M). Therefore, on ∂M we have
26
〈F,G〉ds = F 1G1ds+ F 2G2ds+ F 3G3ds
= F 1dx+ F 2dy + F 3dz
= η
By theorem 1.3.2, we get∫M
〈(∇× F ), n〉dA =
∫M
dη
=
∫∂M
η
=
∫∂M
〈F,G〉ds
1.3.2 Riemannian Manifolds
As illustrations for the theoretical content of this chapter, we will considerspecific examples of differential forms on Riemannian manifolds in chapter 2.We define a Riemannian manifold as a manifold M, equipped with a Riem-manian metric g. At each point p of M, the metric gp must have the followingproperties.
1. gp : (TpM × TpM)→ R is bilinear.
2. gp(v, w) = gp(w, v) ∀v, w ∈ TpM , which is to say gp is symmetric.
3. gp(v, v) > 0 ∀v ∈ TpM : v 6= 0, which is to say gp is positive definite.
4. The coefficients gij in every local chart
gp =∑i,j
gij(p) · dxi|p ⊗ dxj|p
are differentiable functions, where dxi ⊗ dxj(a, b) = dxi(a) · dxj(b).
For further remarks on the Riemannian metric tensor, see section 3 ofchapter 2 on manifolds in higher dimensions.
27
CHAPTER 2
EXAMPLES OF DIFFERENTIAL FORMS ON
RIEMANNIAN MANIFOLDS
2.1 Winding form and volume element asso-
ciated to ellipsoids in R2 and in R3
2.1.1 Differential forms on the 1-dimensional ellipsoid
Consider the ellipse M(1) ⊂ R2 given by
(x, y) ∈ R2 :
x2
a2+y2
b2= 1
where
a and b are positive real constants. The eccentricity ε of ellipse M(1) is given
by√
1− (ab)2 for b > a and ε =
√1− ( b
a)2
for a > b. As a one-dimensional
manifold, we can only define one-forms (and zero forms) on it. The first formwhich we will apply here is referred to as the winding form, which we denoteω1. It is given by
ω1 =−y
x2 + y2dx+
x
x2 + y2dy
and defined on R2\0 ⊃M(1).
This one-form is an example of a closed form which is not exact. Indeed,
28
the differential of ω1 is
dω1 =(x2 + y2)D1(−y) + yD1(x2 + y2)
(x2 + y2)2 dx ∧ dx
+(x2 + y2)D2(−y) + yD2(x2 + y2)
(x2 + y2)2 dy ∧ dx
+(x2 + y2)D1(x)− xD1(x2 + y2)
(x2 + y2)2 dx ∧ dy
+(x2 + y2)D2(x)− xD2(x2 + y2)
(x2 + y2)2 dy ∧ dy
=−(x2 + y2) + 2y2
(x2 + y2)2 dy ∧ dx+(x2 + y2)− 2x2
(x2 + y2)2 dx ∧ dy
=x2 − y2
(x2 + y2)2dx ∧ dy +y2 − x2
(x2 + y2)2dx ∧ dy
= 0
Thus, ω1 is closed.
M(1) can be parametrized by the function f ;
f : [0, 2π] −→ R2
α 7−→ f(α) = (a cosα, b sinα),
or by the singular 1-cube γ in M(1);
γ : [0, 1] −→ R2
t 7−→ γ(t) = (a cos(2πt), b sin(2πt))
The parametrization f is orientation - preserving when R2 is endowedwith its usual orientation because as α runs from 0 to 2π, the points f(α) ∈M(1) run counterclockwise about the origin starting from (a, 0). Likewise, γis orientation - preserving because the map T : R → R with T (t) = 2πt isorientation - preserving and γ = f T . Hence,
29
∫γ
ω1 =
∫[0,1]
γ∗ω1
=
∫ 1
0
ω1(γ)
=
∫ 1
0
(2πabsin2(2πt) + 2πabcos2(2πt)
a2cos2(2πt) + b2sin2(2πt)
)dt
= 2π
∫ 1
0
(ab
a2cos2(2πt) + b2sin2(2πt)
)dt
=
∫ 2π
0
(ab
a2cos2α + b2sin2α
)dα
= ab
∫ 2π
0
(1
b2 + (a2 − b2)cos2α
)dα
We may apply the Cauchy residues theorem to evaluate this integral,which would involve complexification of the R2 space. Briefly, to evaluate∫ 2π
0
R(sinα, cosα)dα, we compute 2π∑|zi|<1
Rez(g; zi) for all singular points zi,
where g(z) =1
zR(z − z−1
2i,z + z−1
2).
In our case,
g(z) =1
z
1
b2 + (a2 − b2)((z + z−1)/2)2
=4z
(a2 − b2)z4 + (2a2 + 2b2)z2 + a2 − b2
The singular points of g are the four roots of its denominator which are
z1 := i
√|a− b|a+ b
, z2 := −i√|a− b|a+ b
, z3 := i
√a+ b
|a− b|, z4 := −i
√a+ b
|a− b|.
They are all simple poles and
|z1| = |z2| =√|a− b|a+ b
|z3| = |z4| =√
a+ b
|a− b|
30
Since a and b are positive, |z1| = |z2| < 1 < |z3| = |z4|.
∫ 2π
0
(ab
a2cos2α + b2sin2α
)dα = 2πab(Rez(g; z1) +Rez(g; z2))
= 2πab( limz→z1
(z − z1).g(z) + limz→z2
(z − z2).g(z))
= 2πab(A+B)
where, A = limz→z1
4z
(a2 − b2)(z − z2)(z − z3)(z − z4)=
1
2aband
B = limz→z2
4z
(a2 − b2)(z − z1)(z − z3)(z − z4)=
1
2ab.
Hence, the integral of the winding form ω1 over the one dimensionalmanifold or curve M(2), equals ∫
γ
ω1 = 2π.
The integral of the winding form along a closed curve in R2 surrounding theorigin is a measure of how often the curve turns around the origin. Manysources also give the winding form the notation dθ, where θ is an angle mea-sure.
Assume that ω1 were the differential of a smooth function, say h. Then∫γ
ω1 =
∫[0,1]
γ∗ω1 =
∫ 1
0
γ∗(dh)
=
∫ 1
0
d(h γ)
=
∫ 1
0
d(h γ)(t)
dtdt
= h(γ(1))− h(γ(0))
Hence, the line integral of an exact 1-form over a closed curve vanishessince γ(1) equals γ(0) in this case. This is how we see that the winding formω1 is not exact, as its integral over the closed curve M(1) is not zero.
31
Volume form on the one dimensional ellipsoidAnother 1-form we can apply to M(1) is the volume form which in this casecoincides with the element of arclength since M(1) is a closed curve in R2.Integrating the volume form over a manifold yields its Lebesgue measure.We denote the element of arclength ds; (ds)2 = (dx)2 + (dy)2.
∫γ
ds =
∫[0,1]
γ∗ds, (for the singular 1-cube γ in M1 given in this section)
=
∫ 1
0
√4π2a2sin2(2πt) + 4π2b2cos2(2πt) dt
=
∫ 2π
0
√a2sin2α + b2cos2α dα
= b
∫ 2π
0
√1 +
(a2
b2− 1
)sin2α dα
= b
∫ 2π
0
√1− ε2sin2α dα
= 4b
∫ π2
0
√1− ε2sin2α dα
The antiderivative of the above integrand cannot be expressed in terms ofelementary functions. However, we recognize the final expression as a com-plete elliptic integral which precisely has the following power series expansion
2πb∞∑n=0
−1
2n− 1
((2n)!
(2nn!)2
)2
ε2n;
where ε2 = 1− (ab)2.
By the ratio test, this series converges for all values of ε ∈ R for whichε < 1. The element of arclength is not exact (despite its given notation ds)since its integral over the closed curve M(1) is not zero.
32
2.1.2 Differential forms on the 2-dimensional ellipsoid
Now, for a, b ∈ R+\0, consider the ellipsoid M(2) ⊆ R3 given by
M(2) =
(x, y, z) ∈ R3 :
x2
a2+y2
a2+z2
b2= 1
.
We may define differential forms up to the second order on M(2) becauseit is a two-dimensional manifold. Let us begin with a 2-form (ω2) which isanalogous to the winding form ω1. The form ω2 is given as
ω2 =xdy ∧ dz + ydz ∧ dx+ zdx ∧ dy
r3
where r =√
(x2 + y2 + z2).
This 2-form is defined on R3\0 ⊃M(2) and it is a closed form which isnot exact. The differential of ω2 is
d(ω2) =r3(3dx ∧ dy ∧ dz)− (xdy ∧ dz + ydz ∧ dx+ zdx ∧ dy) ∧ d(r3)
r6
=r3(3dx ∧ dy ∧ dz)− 3r2(xdy ∧ dz + ydz ∧ dx+ zdx ∧ dy) ∧ d(r)
r6
=r3(3dx ∧ dy ∧ dz)− 3r(x2dx ∧ dy ∧ dz + y2dy ∧ dz ∧ dx+ z2dz ∧ dx ∧ dy)
r6
=3r3dx ∧ dy ∧ dz − 3r · r2dx ∧ dy ∧ dz
r6= 0
As such, ω2 is a closed 2-form.
M(2) can be parametrized by the function Φ;
Φ : [0, 2π]× [−π2, π
2] −→ M(2) ⊂ R3
(α, β) 7−→ (a cosα cos β, a sinα cos β, b sin β),
or by the singular 2-cube c in M(2);
c : [0, 1]× [0, 1] −→ R3
(t, u) 7−→ (a cos(2πt) sin(πu), a sin(2πt) sin(πu),−b cos(πu)).
33
These parametrizations yield the following partial derivatives∂Φ
∂α:= Φ∗(e1)(α,β) = (−a sinα cos β, a cosα cos β, 0);
∂Φ
∂β:= Φ∗(e2)(α,β) = (−a cosα sin β,−a sinα sin β, b cos β);
∂c
∂t:= c∗(e1)(t,u) = (−2πa sin(2πt) sin(πu), 2πa cos(2πt) sin(πu), 0);
∂c
∂u:= c∗(e2)(t,u) = (πa cos(2πt) cos(πu), πa sin(2πt) cos(πu), bπ sin(πu)).
Take the point p0 = (−a, 0, 0) ∈ M(2) at which the outward unit normaln(p0) to the manifold is clearly (−1, 0, 0) when R3 is endowed with its usualorientation. We have Φ(π, 0) = (−a, 0, 0) =: p0, with this point having noother pre-images under Φ. Let ν0 = (π, 0) so that Φ(ν0) = p0.
Φ∗(e1)(ν0) = (0,−a, 0) and Φ∗(e2)(ν0) = (0, 0, b).If Φ is an orientation - preserving parametrization, then
[n(p0),Φ∗(e1)(ν0),Φ∗(e2)(ν0)] must be the usual orientation of R3.
Det
n(p0)Φ∗(e1)(ν0)
Φ∗(e2)(ν0)
=
∣∣∣∣∣∣−1 0 00 −a 00 0 b
∣∣∣∣∣∣ = ab > 0
Hence, Φ is an orientation-preserving parametrization when R3 is en-dowed with its usual orientation and so is the succeeding singular two-cubec because in the affine transformation, T : R2 −→ R2(
tu
)7−→
(2π 00 π
)(tu
)+
(0−π2
)
we see that c = Φ T and the matrix
(2π 00 π
)is positive definite. We
may proceed to integrate ω2 over an orientation - preserving parametrization.∫c
ω2 =
∫[0,1]2
c∗ω2 =
∫U
Φ∗ω2; where U = [0, 2π]× [−π2, π
2].
With respect to the parametrization Φ we get the differential 1-forms:
dx = −a sinα cos βdα− a cosα sin βdβ ;dy = a cosα cos βdα− a sinα sin βdβ ;dz = b cos βdβ.
34
Hence by substituting in Φ∗ω2(e1, e2) = ω2(Φ)(Φ∗(e1),Φ∗(e2)), we haveas our integrand∫
U
Φ∗ω2 =
∫U
(η1 + η2 + η3
(a2cos2β + b2sin2β)32
)where
η1 = xdy ∧ dz = a cosα cos β(ab cosαcos2βdα ∧ dβ)
η2 = ydz ∧ dx = a sinα cos β(−ab sinαcos2βdβ ∧ dα)
η3 = zdx∧dy = b sin β(a2sin2α sin β cos βdα∧dβ−a2cos2α sin β cos βdβ∧dα).
Factorizing trigonometric expressions, we then have∫U
(a2b cos β(cos2αcos2β + sin2αcos2β + sin2αsin2β + cos2αsin2β)dα ∧ dβ
(a2cos2β + b2sin2β)32
)
=
∫U
(a2b cos β(cos2β + sin2β)dα ∧ dβ
(a2cos2β + b2sin2β)32
)= −
∫U
a2b cos β
(a2cos2β + b2sin2β)32
dβ ∧ dα
= −∫
U
d
(b sin β
(a2cos2β + b2sin2β)12
)∧ dα
= −∫
∂U
b sin β
(a2cos2β + b2sin2β)12
dα by Theorem 1.3.2
Finally, by setting
g(α, β) =−αb sin β√
a2cos2β + b2sin2β
we get the above integrand to equal
g(α, −π2
)|α=2πα=0 + g(α, π
2)|α=2πα=0 + g(0, β)|β=π
2
β=−π2
+ g(2π, β)|β=π2
β=−π2
=α|2π0 − α|2π0 + 0−[
2πb sin β√a2cos2β + b2sin2β
]β=π2
β=−π2
= −4π.
35
Observe that the closed 2-form ω2 is non-degenerate so that it is a sym-plectic form on M(2) and we may say that the pair (M(2), ω2) is a symplecticmanifold. The closed property is the final requirement of symplectic formsnot mentioned in section 2 of chapter 1. However, ω2 should not be confusedwith the canonical symplectic structure, which is related to volume formsin symplectic geometry, differing from Riemannian geometry by way of themetric.
Volume form on orientable surfaces in R3
The volume form on an orientable surface in R3 is usually also referred to asthe element of area. We denote the element of area dA and describe how toobtain it for an oriented two-dimensional manifold or surface in R3.
Let M be our oriented surface in R3, p ∈M and the outward unit normalto M at p be n(p). As we have seen in the proof of theorem 1.3.5, definingω ∈
∧2(TpM) by
ω(v1, v2) = det
v1
v2
n(p)
,
we get ω(v1, v2) = 1 if v1 and v2 form an orthonormal basis for TpM with[v1, v2] = µp. Hence ω is the element of area dA, and letting v1, v2 ∈ TpMarbitrarily, we have
dA(v1, v2) = ω(v1, v2) = 〈(v1 × v2), n(p)〉 = ±‖v1 × v2‖
since n(p) is perpendicular to the vectors v1 and v2.To compute the area of M, we integrate the 2-form dA over the surface
which is equivalent to evaluating∫
[0,1]2c∗(dA) for an orientation-preserving
singular 2-cube c in M. Letc : [0, 1]2 −→M ⊂ R3
ν 7−→ c(ν) = p = (c1(ν), c2(ν), c3(ν))
Then,c∗(dA)((e1)ν , (e2)ν) = dA(c∗((e1)ν), c∗((e2)ν))= ‖c∗((e1)ν)× c∗((e2)ν)‖= ‖(D1c
1(ν), D1c2(ν), D1c
3(ν))× (D2c1(ν), D2c
2(ν), D2c3(ν))‖
=
((D2c
3(ν)D1c2(ν)−D2c
2(ν)D1c3(ν))
2+ (D2c
3(ν)D1c1(ν)−D2c
1(ν)D1c3(ν))
2
+(D2c2(ν)D1c
1(ν)−D2c1(ν)D1c
2(ν))2
) 12
=
(([D1c
1(ν)]2 + [D1c2(ν)]2 + [D1c
3(ν)]2)([D2c1(ν)]2 + [D2c
2(ν)]2 + [D2c3(ν)]2)
−(D1c1(ν)D2c
1(ν) +D1c2(ν)D2c
2(ν) +D1c3(ν)D2c
3(ν))2
) 12
36
We may set ν = (t, u) ∈ [0, 1] × [0, 1] in order to rewrite the above ex-pression as√〈∂c∂t,∂c
∂t〉 · 〈 ∂c
∂u,∂c
∂u〉 − 〈∂c
∂t,∂c
∂u〉2
The three inner products involved above constitute the so-calledRiemannian structure of M. The Riemannian structure of an n-dimensionalmanifold P parametrized by f : U ⊆ Rn → P ;u = (u1, u2, ..., un) 7→ f(u)
is the set of all inner products 〈 ∂f∂ui
,∂f
∂uj〉 for 1 ≤ i, j ≤ n. Henceforth, we
denote 〈∂c∂t,∂c
∂t〉 =
∥∥∥∥∂c∂t∥∥∥∥2
by E; 〈 ∂c∂u,∂c
∂u〉 =
∥∥∥∥ ∂c∂u∥∥∥∥2
by G and 〈∂c∂t,∂c
∂u〉 by F.
Finally, the area of the surface M is obtained as∫ 1
0
∫ 1
0
√EG− F 2dtdu.
The Riemannian structure of M(2) is given as:E = 4π2a2sin2(πu)F = 0G = π2a2cos2(πu) + π2b2sin2(πu),by imputing from the orientation-preserving 2-cube c in M(2) given earlier inthis section.
Area (M(2))
=
∫ 1
0
∫ 1
0
|2πa sin(πu)|√π2a2cos2(πu) + π2b2sin2(πu)dtdu
= 2π2a
∫ 1
0
| sin(πu)|√a2cos2(πu) + b2sin2(πu)du
= 2π2a
∫ 1
0
sin(πu)√a2cos2(πu) + b2sin2(πu)du
= 2πa
∫ 1
−1
√a2ϑ2 + b2(1− ϑ2)dϑ ϑ = − cos(πu)
= 4πa
∫ 1
0
√a2ϑ2 + b2(1− ϑ2)dϑ
We may confirm this integral by using the conventional formula for sur-faces of revolution in R3. Recall from proposition 1.1.6 that M(2) is obtained
by revolving the curve C := (x, y, z) ∈ R3 : x = 0, y ≥ 0,y2
a2+z2
b2= 1 by
2π radians about the z-axis. C is then parametrized by
37
f : [−π2, π
2] → R3
α 7→ f(α) = (0, a cosα, b sinα)
The surface area of revolution for C about the z-axis is∫ π2
−π2
2πy
√(dy
dα
)2
+
(dz
dα
)2
dα
=
∫ π2
−π2
2πa cosα√a2sin2α + b2cos2αdα
=
∫ 1
−1
2πa√a2ϑ2 + b2(1− ϑ2)dϑ ϑ = sinα
= 4πa
∫ 1
0
√a2ϑ2 + b2(1− ϑ2)dϑ
The results from both formulae agree although computation by the Rie-mannian structure will have to be used for a manifold which is not obtainableby revolution. Upon evaluation of the above integral we get two possible out-comes.
Case 1: a > b. In this case, area of M(2)
= 2πa
[ϑ√ϑ2(a2 − b2) + b2 +
b2
√a2 − b2
ln(ϑ√a2 − b2 +
√ϑ2(a2 − b2) + b2 )
]1
ϑ=0
= 2πa
(a+
b2
√a2 − b2
ln
(a+√a2 − b2
b
))= 2πa2 +
πb2
εln
(1 + ε
1− ε
)Case 2: b > a. In this case, area of M(2)
= 2πa
[ϑ√b2 − ϑ2(b2 − a2) +
b2
√b2 − a2
arcsin
(ϑ√b2 − a2
b
)]1
ϑ=0
= 2πa
(a+
b2
√b2 − a2
arcsin
(√b2 − a2
b
))= 2πa2 +
2πab
εarcsin(ε)
In either case, ε is the eccentricity of the generating half-ellipse C, whichis also defined to be the eccentricity of M(2). Case 1 is a revolution about theminor axis of C which generates an OBLATE ELLIPSOID, while case 2 ofrevolution about the major axis of C generates a PROLATE ELLIPSOID.
38
2.2 Other quantities associated to R3 ellip-
soid derived from Riemannian structure,
geodesics of R3 ellipsoid
2.2.1 The shape operator
The ellipsoid in R3 is of particular interest for several practical reasons. Ingeodetics for example, the shape of the Earth’s surface is modelled as anoblate ellipsoid. M(2) ⊂ R3 is the ellipsoid of least dimension for which wecan discuss critical points of the arclength function within the manifold. Tothis effect, we will obtain standard values associated to M(2) derived fromthe Riemannian structure.
In the co-ordinate system c used in the previous chapter,c : [0, 1]× [0, 1] −→ R3
(t, u) 7−→ c(t, u) = (a cos(2πt) sin(πu), a sin(2πt) sin(πu),−b cos(πu)),the image of a fixed line t = t0 under c is called a meridian on M(2), while theimage of a fixed line u = u0 under c is called a parallel on M(2). Parallels andmeridians on the globe are respectively lines of latitude and longitude. Wecompute the Gauss map N from c, where N(c(ν))p gives the outward unitnormal vector to M(2) at p = c(ν) ∀ ν ∈ [0, 1]× [0, 1].
∂c
∂t× ∂c
∂u= (−2πa sin(2πt) sin(πu), 2πa cos(2πt) sin(πu), 0)×
(πa cos(2πt) cos(πu), πa sin(2πt) cos(πu), bπ sin(πu))
= 2π2a sin(πu)(b cos(2πt) sin(πu), b sin(2πt) sin(πu),−a cos(πu))∥∥∥∥∂c∂t × ∂c
∂u
∥∥∥∥ = 2π2a sin(πu)√b2sin2(πu) + a2cos2(πu)
Hence, the Gauss map is derived;N : M(2) → S2
N(c(t, u))p =(b cos(2πt) sin(πu), b sin(2πt) sin(πu),−a cos(πu))√
b2sin2(πu) + a2cos2(πu)
Note that the vectors∂c
∂tand
∂c
∂uare perpendicular from the Riemannian
structure since their inner product F is 0. Hence∂c
∂t|ν and
∂c
∂u|ν form an
39
orthogonal basis for the tangent plane Tc(ν)M(2) to M(2) at c(ν).
∂2c
∂t2= (−4π2a cos(2πt) sin(πu),−4π2a sin(2πt) sin(πu), 0)
∂2c
∂t∂u= (−2π2a sin(2πt) cos(πu), 2π2a cos(2πt) cos(πu), 0)
∂2c
∂u2= (−π2a cos(2πt) sin(πu),−π2a sin(2πt) sin(πu), bπ2 cos(πu))
〈∂2c
∂t2, N〉 =
−4π2absin2(πu)√b2sin2(πu) + a2cos2(πu)
=: h11
〈 ∂2c
∂t∂u,N〉 = 0 =: h12 =: h21
〈 ∂2c
∂u2, N〉 =
−π2ab√b2sin2(πu) + a2cos2(πu)
=: h22
The matrix [hij] for 1 ≤ i, j ≤ 2 is called the matrix of the secondfundamental form. A necessary and sufficient characterization of orientablesurfaces in R3 is differentiability of the Gauss map. The negative differentialof the Gauss map of an orientable surface is called the matrix of the shapeoperator of the surface. Its computation involves the matrix of the first fun-damental form of the surface: (
E FF G
).
The matrix of the shape operator is obtained with respect to an orthogonalbasis for the tangent plane at each point of the manifold.
In our case, the matrix of the shape operator −dN with respect to the
basis
(∂c
∂t,∂c
∂u
)of TpM(2) is given by
(E FF G
)−1(h11 h12
h21 h22
)=
−b
a(b2sin2(πu) + a2cos2(πu))32
(b2sin2(πu) + a2cos2(πu) 0
0 a2
).
40
Hence we derive the following three quantities for M(2):
i.) The principal curvatures λ1 and λ2, which are the eigenvalues of theshape operator
λ1 =−b
a√b2sin2(πu) + a2cos2(πu)
; λ2 =−ab
(b2sin2(πu) + a2cos2(πu))32
ii.) The Gaussian curvature, which is the product of the two eigenvaluesof the shape operator
λ1λ2 =b2
(b2sin2(πu) + a2cos2(πu))2
iii.) The mean curvature, which is the mean of the two eigenvalues of theshape operator
1
2(λ1 + λ2) =
−(a2b+ a2bcos2(πu) + b3sin2(πu))
2a(b2sin2(πu) + a2cos2(πu))32
Notably, from the preceding derivations, the Gauss map is a diffeomor-phism between the unit sphere S2 and M(2). We can define other such dif-feomorphisms, two of which are given below.
1) f1 : S2 →M(2)
(x, y, z) 7−→ T (x, y, z) = (ax, ay, bz)
2) f2 : S2 →M(2)
(x, y, z) 7−→ (abx, aby, abz)√b2(1− z2) + a2z2
The map f1 is linear while f2 radially maps each point of the unit sphereto the proximate point of M(2) which is collinear with the sphere’s point andthe origin.
41
2.2.2 Geodesics of the 2-dimensional ellipsoid
We will now present vital facts in connection with geodesics on the ellipsoidM(2). Any regular curve α drawn on a regular surface M ⊂ R3 is charac-terized by the equation α′′(s) = kg(s)n + kN(s)NM where α is parametrizedby arclength := s. The variables kg(s), kN(s),NM and n are respectivelythe geodesic curvature of α, normal curvature of α, Gauss map or outwardunit normal to M, and a unit normal to the curve α lying on the tangentplane to M. The curve α is a geodesic of M iff its geodesic curvature vanisheseverywhere. Hence, α′′(s) = kN(s)NM for a geodesic α ⊂M .
Because α is parametrized by arclength, α′′(s) = kNα, where k is thecurvature of α and Nα is the principal unit normal vector to α. Hence,for a geodesic curve α ⊂ M , we have kNα = kNNM . This implies thatthe principal unit normal to a geodesic curve at any point coincides with theoutward unit normal to M at that point. Minimizers of the arclength functionon M are smooth curves called minimal geodesics. In R3, the arclengthfunction is given by
I =
∫ds =
∫ √(dx)2 + (dy)2 + (dz)2
where ds is the element of arclength one-form, which represents an infinitesi-mal increment in the arclength function. We have x = x(t, u), y = y(t, u), z =
z(t, u) so that dx =∂x
∂tdt+
∂x
∂udu and
(dx)2 =∂x
∂t
2
(dt)2 + 2∂x
∂t
∂x
∂udtdu+
∂x
∂u
2
(du)2.
Obtaining similar expansions for (dy)2 and (dz)2, we get I =∫ √[∂x
∂t
2
+∂y
∂t
2
+∂z
∂t
2
](dt)2 + 2[∂x
∂t
∂x
∂u+∂y
∂t
∂y
∂u+∂z
∂t
∂z
∂u]dtdu+ [
∂x
∂u
2
+∂y
∂u
2
+∂z
∂u
2
](du)2
It is clear that the coefficients in this expression constitute the Riemann-nian structure;
∂x
∂t
2
+∂y
∂t
2
+∂z
∂t
2
=
∥∥∥∥∂f∂t∥∥∥∥2
= E
∂x
∂t
∂x
∂u+∂y
∂t
∂y
∂u+∂z
∂t
∂z
∂u= 〈∂f
∂t,∂f
∂u〉 = F
∂x
∂u
2
+∂y
∂u
2
+∂z
∂u
2
=
∥∥∥∥∂f∂u∥∥∥∥2
= G,
42
where f is the parametrization,
f : V ⊆ R2 −→ M ⊆ R3
(t, u) 7−→ (x, y, z)
Now, letdu
dt= u′,
dt
du= t′, to re-express I.
I =
∫ √E + 2Fu′ +G(u′)2dt,
or equivalently
I =
∫ √E(t′)2 + 2Ft′ +Gdu.
Let L =√E(t′)2 + 2Ft′ +G so that I =
∫Ldu
∂L
∂t=
1
2(E(t′)2 + 2Ft′ +G)
−12 (∂E
∂t(t′)2 + 2
∂F
∂tt′ +
∂G
∂t)
∂L
∂t′=
1
2(E(t′)2 + 2Ft′ +G)
−12 (2E(t′) + 2F )
We may now directly apply the Euler - Lagrange differential equationwhich minimizes functionals of the type I. Hence, we must have
d
dt
(∂L
∂u′
)=∂L
∂uand
d
du
(∂L
∂t′
)=∂L
∂t.
Since none of the terms from the Riemannian structure depend on t in thecase of our parametrization for M(2), the second equation will yield a clearerresult as taking the partial derivative with respect to t (on the right) willyield zero. We then have a geodesic equation given as
d
du
(E(t′) + F )√E(t′)2 + 2Ft′ +G
=
1
2(E(t′)2 + 2Ft′ +G)
−12 (∂E
∂t(t′)2 + 2
∂F
∂tt′ +
∂G
∂t) = 0
=⇒ E(t′) + F = γ√E(t′)2 + 2Ft′ +G for a constant γ
=⇒ t′ =dt
du=
√γ2G
E(E − γ2). . . eq(2.1) since F is zero.
43
Using the Riemannian structure obtained from the parametrization c, weget
dt
du=
√γ2(a2cos2(πu) + b2sin2(πu))
4a2sin2(πu)(4π2a2sin2(πu)− γ2)
⇒ t =γ
2a
∫ √a2cos2(πu) + b2sin2(πu)
sin2(πu)(4π2a2sin2(πu)− γ2)du
The appearance of each geodesic curve depends largely on the constantγ, which may either be determined by the starting point and azimuth orby starting and ending points of the curve. The azimuth is simply the anglebetween the path and meridian measured clockwise from the meridian on thetangent plane at the starting point. Assume we are given starting point p andazimuth β, we can always obtain the corresponding vector vp ∈ TpM(2) which
under the used parametrization (c) has a pre-image, say ψν ∈ Tν(]0, 1[2), aslong as ν is an interior point of ([0, 1]× [0, 1]). To be more precise, we havec∗ψν = vp, whenever ν ∈ ]0, 1[×]0, 1[. Note that the only points of M(2)
whose pre-images under c do not belong to ]0, 1[×]0, 1[ constitute the merid-ian t = 0 ≡ 1. If the starting point belongs to the meridian t = 0 (whichis a µ - null subset of M(2)) , then computations are easily improvised. Ob-
taining the values of dt(ψν) := ψ1 and du(ψν) := ψ2 we have that t′|ν =ψ1
ψ2
.
Substituting this value in our geodesic equation (2.1) immediately yields γ,which here is unique.
On the other hand, if we are given the starting and ending points of thegeodesic curve then we use Clairaut’s geodesic equation, which is applicableto 2-dimensional manifolds of revolution in R3. Clairaut’s equation statesr sin(β) = constant, where r is the radius of the parallel of latitude given by
a cos(πu)
(1− ε2sin2(πu))12
, β is the azimuth, ε is the eccentricity of M(2) and a is the
radius of the equator. It must be noted that in this latter case, the value ofγ obtained might not be unique, hence nullifying the uniqueness of geodesiccurves given starting and ending points.
In the geodesic equation (2.1), we may express t in terms of the eccen-tricity of M(2) for the oblate and prolate cases.
If M(2) is oblate, then a > b and 1− ε2 =b2
a2in which case
44
t =γ
2
∫ √cos2(πu) + (1− ε2)sin2(πu)
sin2(πu)(4π2a2sin2(πu)− γ2)du
If M(2) is prolate then b > a and 1− ε2 =a2
b2in which case
t =γ
2
∫ √(1− ε2)cos2(πu) + sin2(πu)
sin2(πu)(4π2a2sin2(πu)− γ2)du
The antiderivates of the above integrands for ellipsoidal geodesics cannotbe expressed in terms of elementary functions. In any event, the meridiansand equator of M(2) can be computationally proven to be geodesics.
For a meridian t = t0 fixed, we havedt
du= 0 making the constant γ to be 0
in our derived geodesic equation.
We will now also confirm that for a meridian α = c(t0, u), the outwardunit normal to the manifold and the principal unit normal to the meridianas a curve (resp. NM and Nα) coincide.
For any regular curve α in R3, we have
κNα = α′′(s) = T ′(s) =dT
du
du
ds,
where T =α′(u)
‖α′(u)‖is the unit tangent to α, κ its curvature and s its ar-
clength.α = (aδ sin(πu), aζ sin(πu),−b cos(πu)) where δ = cos(2πt0), ζ = sin(2πt0) =±√
1− δ2. As such,
α′(u) = (aπδ cos(πu), aπζ cos(πu), bπ sin(πu))
‖α′(u)‖ =√a2π2cos2(πu) + b2π2sin2(πu)
s(u) =
∫ u
u0
‖α′(v)‖dv
=
∫ u
u0
√a2π2cos2(πv) + b2π2sin2(πv)dv
ds
du= π
√a2cos2(πu) + b2sin2(πu)
T =(aπδ cos(πu), aπζ cos(πu), bπ sin(πu))
π√a2cos2(πu) + b2sin2(πu)
dT
du=
(−πb2aδ sin(πu),−πb2aζ sin(πu), πba2 cos(πu))
(a2cos2(πu) + b2sin2(πu))32
45
dT
du
du
ds=
(−b2aδ sin(πu),−b2aζ sin(πu), ba2 cos(πu))
(a2cos2(πu) + b2sin2(πu))2 = λ2NM .
Recall λ2 to be one of the principal curvatures of M(2).
We similarly confirm the coincidence of NM and Nφ for the equator φ,
φ = c(t, 12) = (a cos(2πt), a sin(2πt), 0)
φ′(t) = (−2πa sin(2πt), 2πa cos(2πt), 0)
‖φ′(t)‖ = 2πa
s(t) =
∫ t
t0
2πadv = 2πat− 2πat0
ds
dt= 2πa
T = (− sin(2πt), cos(2πt), 0)
κNφ =dT
dt
dt
ds
= (−2π cos(2πt),−2π sin(2πt), 0)1
2πa
= (−1a
cos(2πt), −1a
sin(2πt), 0)
The Gauss map evaluated on the equator, NM = (cos(2πt), sin(2πt), 0)
and the corresponding prinicpal curvature, λ1 =−1
a.
Hence, we observe that κNφ = λ1NM .
Other geodesic curves besides meridians and the equator appear in partic-ular patterns. For instance, in the case of the oblate ellipsoid which has beenextensively analyzed, geodesics appear as periodic curves oscillating aboutthe equator. They do not repeat after a complete revolution.
As a physical application, geodesics are the only curves on a surface alongwhich a particle can move without accelerating tangentially. For this reason,they are also recognized as kinetic energy minimizers.
46
Schematic of oscillation of a geodesic on an oblate ellipsoid.
47
2.3 Manifolds in higher dimensions: volume
element, geodesics
In theory, differential forms on manifolds in higher dimensional Euclideanspaces are useful to extrapolate results from lower dimensions. We illustratethis in the following analysis of the ellipsoid M(n) for n ≥ 3. Note that axialsymmetry and uniqueness of the outward unit normal at each point on themanifold are key in obtaining results in computations and proofs. A hyper-surface in Rn+1 is an n-dimensional submanifold of the space, giving us thatM(n) is a hypersurface in Rn+1. We now proceed to examine hypersufaces’volume forms and their geodesics, which fundamentally involve the elementof arclength.
2.3.1 Higher dimensional volume forms
Theorem 2.3.1. Let M be a hypersurface in Rn+1 and
f : U ⊆ Rn −→ Mu = (u1, u2, ..., un) 7−→ f(u) = p
be a differentiable parametrization of M then the volume of M can be obtainedfrom the formula
vol(M) =
∫U
√det[gij]
where gij is from the Riemannian structure;
gij = gji = 〈 ∂f∂ui
,∂f
∂uj〉 for 1 ≤ i, j ≤ n,
of the Riemannian metric of M defined by
g =n∑
i,j=1
gijdui ⊗ duj.
Proof. The aim is to get∫MdV as an equal integral having U as the domain
of integration using the pullback f ∗.∫M
dV =
∫U
f ∗(dV )
48
Define ω ∈∧n(TpM) by
ω(v1, · · · , vn) = det
n(p)v1...vn
where n(p) is the outward unit normal to M at p. If (v1, · · · , vn) form anorthonormal basis for TpM with [v1, · · · , vn] = µp then ω(v1, · · · , vn) = 1meaning that ω is the volume element dV. We may define
ω(v1, · · · , vn) := 〈(v1 × · · · × vn), n(p)〉 = ±‖v1 × · · · × vn‖
since the unit vector n(p) is perpendicular to each of the vectors v1, · · · , vn.The operation above is a multiple cross product which can be executed withn vectors in Rn+1 to produce a mutually orthogonal vector. As we will seebriefly, this operation induces an inner product on Rn when the norm is ap-plied to it.
For each u ∈U , we have
f ∗(dV )((e1)u, · · · , (en)u) = dV (f∗((e1)u), · · · , f∗((en)u))
= dV
(∂f
∂u1
, · · · , ∂f∂un
)= ω
(∂f
∂u1
, · · · , ∂f∂un
)
Let (v1, · · · , vn) be an orthonormal basis for TpM , then we have
ω(v1, · · · , vn) = ±1. We can then express each of the vectors∂f
∂uias a linear
combination of the basis vectors. Let us set
∂f
∂u1...∂f
∂un
= [aij]
v1...vn
so that
∂f
∂ui=∑n
k=1aikvk
and 〈 ∂f∂ui
,∂f
∂uj〉 =
∑nk,l=1aikajl〈vk, vl〉 =
∑nk=1aikajk
49
But from theorem 1.2.1, we have ω
(∂f
∂u1
, · · · , ∂f∂un
)= ±det[aij] which
implies that
(ω(
∂f
∂u1
, · · · , ∂f∂un
)
)2
= (det[aij])2
= (det[aij])(det[aji])
= det([aij][aji])
= det[ζij]
where ζij =∑n
k=1aikajk which we have already seen to equal
〈 ∂f∂ui
,∂f
∂uj〉 := gij.
Hence,∣∣∣∣ω( ∂f
∂u1
, · · · , ∂f∂un
)∣∣∣∣ =
∥∥∥∥ ∂f∂u1
× · · · × ∂f
∂un
∥∥∥∥ =√det[gij]
and this gives us
vol(M) = |∫
M
dV |
= |∫
U
f ∗dV ((e1), · · · , (en))|du1 ∧ du2 ∧ · · · ∧ dun
=
∫U
√det[gij]du1 ∧ du2 ∧ · · · ∧ dun
Of course, the system of coordinates in U must cover the manifold ex-actly once to obtain the correct volume, meaning that the parametrizationf : U →M should be a bijection, except possibly for a negligible subset of M.
Remark - Observe that we may express gu as gu(v, w) = vT [gij]w forv, w ∈ Rn
u which is bilinear, symmetric and positive definite and so is aninner product on the vector space Rn
u. Concretely, we have gp as the usualinner product on TpM pulled back to Rn
u by the parametrization f since wehave vT [gij]w = 〈Df(v), Df(w)〉. The matrix [gij] is also referred to as the
50
matrix of the first fundamental form.
The matrix [hij] for 1 ≤ i, j ≤ n is the second fundamental form where
hij = 〈 ∂2f
∂ui∂uj, ψ〉
and ψ is the outward unit normal to the hypersurface.
The matrix of the shape operator of M is given by [gij]−1[hij] and we
obtain the following quantities from it:
i.) The principal curvatures λ1, · · · , λn, which are the eigenvalues of theshape operator;
ii.) The Gauss - Kronecker curvature λ1 · · ·λn, which is the product ofthe n eigenvalues of the shape operator and also equals its determinant;
iii.) The mean curvature 1n(λ1 + · · · + λn), which is the mean of the n
eigenvalues of the shape operator and also equals its trace divided by n.
We also have a matrix of the third fundamental form of M given by[〈 ∂ψ∂ui
,∂ψ
∂uj〉]ij
,
where ψ is the outward unit normal to the hypersurface.
The next theorem gives a means of evaluating volumes of n-dimensionalsubmanifolds of Rn from relevant transformations or maps.
Theorem 2.3.2.Let S and P be differentiable n-dimensional submanifolds of Rn withP = f(S) for a smooth map
f : S −→ Rn
(u1, u2, ..., un) 7−→ (x1, x2, ..., xn)
Then the volume of P is given by∫S
df 1 ∧ df 2 ∧ · · · ∧ dfn =
∫S
J(f)du1 ∧ du2 ∧ · · · ∧ dun.
51
Proof. We have by definition ∫P
ω =
∫S
f ∗ω
where ω is the volume element of P. In this case, the volume element for S,P and Rn is the determinant given by ω = det = dx1 ∧ dx2 ∧ · · · ∧ dxn on therange and det = du1 ∧ du2 ∧ · · · ∧ dun on the domain. Hence the volume ofP equals∫
S
f ∗(dx1 ∧ dx2 ∧ · · · ∧ dxn)
=
∫S
f ∗dx1 ∧ f ∗dx2 ∧ · · · ∧ f ∗dxn
=
∫S
df 1 ∧ df 2 ∧ · · · ∧ dfn
These equalities follow from the properties of the pullback operator listedin section 1.2.2. Also, we derive the volume formula as follows.∫
S
f ∗ω(e1, · · · , en)du1 ∧ du2 ∧ · · · ∧ dun
=
∫S
ω(f)(f∗(e1), · · · , f∗(en))du1 ∧ du2 ∧ · · · ∧ dun
=
∫S
det(f∗(e1), · · · , f∗(en))du1 ∧ du2 ∧ · · · ∧ dun
=
∫S
det(∂f
∂u1
, · · · , ∂f∂un
)du1 ∧ du2 ∧ · · · ∧ dun
=
∫S
J(f)du1 ∧ du2 ∧ · · · ∧ dun (J(f) is the Jacobian determinant)
2.3.2 Higher dimensional geodesics
The calculus of variations in higher dimensions is key to studying minimalgeodesics within the hypersuface M , which in any case are locally length-minimizing curves. M as a subset of Rn+1 is parametrized by n variables sowe set as a parametrization for M
f : V ⊆ Rn −→M ⊂ Rn+1 ; (u1, u2, · · · , un) 7−→ (x1, x2, · · · , xn+1)
The arclength function is given by
I =
∫ds =
∫ √(dx1)2 + (dx2)2 + · · ·+ (dxn+1)2
And
dxi =n∑j=1
∂xi∂uj
duj
52
which gives us that
(dxi)2 =
n∑j=1
(∂xi∂uj
)2
(duj)2 + 2
n∑k,l=1k 6=l
∂xi∂uk
∂xi∂ul
dukdul
We then have
I =
∫ √√√√√ n∑j=1
[n+1∑i=1
(∂xi∂uj
)2
(duj)2
]+ 2
n∑k,l=1k 6=l
[n+1∑i=1
∂xi∂uk
∂xi∂ul
dukdul
]
The expression under the integral is the element of arclength for M while thecoefficients therein constitute the Riemannian structure. More specifically,
the coefficient of duiduj is 〈 ∂f∂ui
,∂f
∂uj〉. We may then write,
I =
∫ √∑n
i,j=1gijduiduj.
Now, we illustrate computations using the specific manifold M(n). Let usconsider the parametrization for M(n) given by
f0 : [−π2, π
2]n−1× [0, 2π] −→M(n) ; (u1, u2, · · · , un) 7−→ (x1, x2, · · · , xn+1);
f 10 := x1 = a
n∏j=1
cosuj
f i0 := xi = an−i+1∏j=1
cosuj sinun−i+2, if 2 ≤ i ≤ n
fn+10 := xn+1 = b sinu1
The parametrization given above covers the manifold exactly once and ityields the following system of partial derivatives.
∂xi∂uj
=
0 for i ≥ n− j + 3. Otherwise,
−a sinujn∏
k=1,k 6=jcosuk for i = 1
−a sinuj(n−i+1∏k=1,k 6=j
cosuk) sinun−i+2 for 2 ≤ i ≤ n− j + 1
an−i+2∏k=1
cosuk for i = n− j + 2 6= n+ 1
b cosu1 for i = n+ 1
53
When j < l < n, then
〈∂f0
∂uj,∂f0
∂ul〉 =
n+1∑i=1
∂xi∂uj
∂xi∂ul
= a2 sinuj sinul cosuj cosul(n∏
k=1,k 6=j,lcos2uk+sin2un
n−1∏k=1,k 6=j,l
cos2uk+sin2un−1
n−2∏k=1,k 6=j,l
cos2uk+
. . .+ sin2ul+1
l∏k=1,k 6=j,l
cos2uk)− a2 sinuj sinul cosuj cosul(l−1∏
k=1,k 6=j,lcos2uk)
= a2 sinuj sinul cosuj cosul(l−1∏
k=1,k 6=j,lcos2uk)(
n∏k=l+1
cos2uk+sin2unn−1∏k=l+1
cos2uk+
sin2un−1
n−2∏k=l+1
cos2uk + . . .+ sin2ul+1 − 1) = 0.
We can similarly check that 〈∂f0
∂ui,∂f0
∂un〉 = 0 for 1 ≤ i ≤ n − 1, hence
confirming that
(∂f0
∂ui
)1≤i≤n
forms an orthogonal basis for TpM(n). Using
the parametrization f0, we conveniently have an elimination of all terms gijfrom the expression ds whenever i 6= j. Our arclength function then becomes
I =
∫ √√√√ n∑i=1
gii(dui)2 =
∫ √√√√√gjj +n∑i=1i 6=j
gii
(duiduj
)2
duj
Geodesics are the curves on M(n) for which their principal unit normalseverywhere coincide with the outward unit normal to the manifold. Let theproblem posed be to find a geodesic path within the manifold M(n) given astarting point and direction, in other words, vp ∈ TpM(n), then the geodesic isunique and we may proceed by use of Euler-Lagrange differential equations.In executing this process, we must make pairwise selections of the ui’s tovary. By repeatedly varying a certain one along with one of the others ata time, we create a system of n − 1 differential equations in n unknowns.Such a system would yield the parametric equation of a curve in Rn+1 whenrestricted to M(n), if it could be reduced. Observing that
gjj =
a2sin2u1 + b2cos2u1 for j = 1
(a2)j−1∏k=1
cos2uk for 2 ≤ j ≤ n
we see that none of the terms from the Riemannian structure depend onun so we will begin by varying this term while deriving the Euler-Lagrange
54
equations.
I =
∫ √g11 +
∑n
j=2gjj
(dujdu1
)2
du1 and we get the following as an Euler-
Lagrange equation when minimizing I.
∂L1
∂un− d
du1
(∂L1
∂u′n
)= 0, where
L1 =
√g11 +
∑nj=2gjj
(dujdu1
)2
and u′n =dundu1
∂L1
∂u′n=
gnnu′n
L1
=(a2)
n−1∏k=1
cos2ukdundu1√
g11 +∑n
j=2gjj
(dujdu1
)2
Since∂L1
∂un= 0 , then
∂L1
∂u′nis a constant. Therefore,
(a2)n−1∏k=1
cos2ukdundu1
= γ
√g11 +
∑nj=2gjj
(dujdu1
)2
. . . eq(2.2)
for a real constant γ.
Likewise, we obtain similar geodesic differential equations by repeatedlyvarying un and the other ui’s one at a time. We can preview each equation toobserve the appearance of the geodesic solution in submanifolds of M(n). Inthe equation just obtained, when we set each ui to 0 except u1 and un, we viewthe appearance of the geodesic curve in (R×R×0n−2×R)∩M(n), adheringto the axial orientation specified in the parametrization f0. The co-ordinates
in this submanifold are (a cosu1 cosun, a cosu1 sinun,
n−2︷ ︸︸ ︷0, 0, · · · , 0, b sinu1), which
is clearly isometric to M(2). Our equation then becomes
a2cos2u1
(dundu1
)= γ
√a2sin2u1 + b2cos2u1 + a2cos2u1
(dundu1
)2
=⇒ dundu1
= γ
√a2sin2u1 + b2cos2u1
a2cos2u1(a2cos2u1 − γ2)
The associated solution here has identical properties to what we have forgeodesics of M(2). (This becomes clear when we apply the parametrization
55
Φ from section 1 of this chapter in the geodesic equation (2.1) derived after-wards in section 2.)
Now, we vary un and uk; 1 6= k 6= n following the above process.
I =
∫ √gkk +
∑n
j=1j 6=k
gjj
(dujduk
)2
duk
∂Lk∂un− d
duk
(∂Lk∂u′n
)= 0, where
Lk =
√gkk +
∑nj=1j 6=k
gjj(dujduk
)2
and u′n =dunduk
=⇒ ∂Lk∂u′n
=gnnu
′n
Lk= γ (γ is a real constant).
Hence, we have
gnndunduk
= γ
√gkk +
∑nj=1j 6=k
gjj
(dujduk
)2
. . . eq(2.3)
Setting each ui to 0 except uk and un, we view the appearance of thegeodesic curve in (R×R×0n−k−1×R×0k−1)∩M(n), adhering to the axialorientation specified in the parametrization f0. The co-ordinates in this sub-
manifold are (a cosuk cosun, a cosuk sinun,
n−k−1︷ ︸︸ ︷0, 0, · · · , 0, a sinuk,
k−1︷ ︸︸ ︷0, 0, · · · , 0) which
is isometric to aS2. Our equation then becomes
a2cos2uk
(dunduk
)= γ
√a2 + a2cos2uk
(dunduk
)2
⇒ dunduk
= γ
√1
cos2uk(a2cos2uk − γ2)
⇒ un = γ
∫ √1
cos2uk(a2cos2uk − γ2)duk
=
∫duk
cosuk
√( aγ)2cos2uk − 1
= Arctan
sinuk√( aγ)2cos2uk − 1
+ δ (δ ∈ R is constant)
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= Arcsin
Tanuk√( aγ)2 − 1
+ δ
⇒ sin(un − δ) =Tanuk√( aγ)2 − 1
⇒ sinun cos δ − cosun sin δ =1
cosuk
sinuk√( aγ)2 − 1
⇒ a cosuk sinun cos δ − a cosuk cosun sin δ =a sinuk√( aγ)2 − 1
⇒ x2 cos δ − x1 sin δ =xn−k+2√( aγ)2 − 1
This is the equation of a plane passing through the origin so its restrictionto the submanifold (R × R × 0n−k−1 × R × 0k−1) ∩M(n) appears as agreat circle of the sphere.
Notice that we get equation (2.3) from equation (2.2) simply by multi-
plying the left and right sides of equation (2.2) bydu1
duk. This means that the
best we can get from the system of equations we have created is a sectionalsolution of the curve. To obtain the general solution, we must set the ar-clength function in a single parameter. For instance,
I =
∫L(u1, u2(u1), · · · , un−1(u1), un(u1), u′2, · · ·u′n)du1,
where u′i =duidu1
. By the canonical Euler - Lagrange equations, we have
∂L
∂ui− d
du1
(∂L
∂u′i
)= 0 for 2 ≤ i ≤ n.
(In problems where we have not provided explicit analytical solutions, theoutlines are laid for alternative numerical approximation techniques in suchinstances. These can be implemented with appropriate computerized soft-ware such as MAPLE, MATLAB and MATHEMATICA.)
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Bibliography
[1] CALCULUS ON MANIFOLDS - Michael Spivak,copyright 1965, W.A. Benjamin Inc, New York, USA.
[2] GLOBAL ANALYSIS: DIFFERENTIAL FORMS IN ANALYSIS,GEOMETRY AND PHYSICS - Ilka Agricola, Thomas Friedrich,copyright 2002 by the American Mathematical Society.
[3] GEODESICS ON AN ELLIPSOID - PITTMAN’S METHODRod Daekin,http://rmit.academia.edu/RodDeakin/Papers/137514/Geodesics on an ellipsoid -Pittmans method.
[4] DIFFERENTIAL GEOMETRY : CURVES - SURFACES -MANIFOLDS - Wolfgang Kuhnel,copyright 2003 by the American Mathematical Society.
[5] CALCULUS OF VARIATIONS - Jurgen Jost and Xiangqing Li-Jost,copyright 1998 by the Cambridge University Press.
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