MINISTRY OF EDUCATION AND TRAINING
HANOI PEDAGOGICAL UNIVERSITY 2
——————–o0o———————

NGUYEN THI NHU QUYNH

GEODESICS

BACHELOR THESIS

HA NOI, 2019


MINISTRY OF EDUCATION AND TRAINING
HANOI PEDAGOGICAL UNIVERSITY 2
——————–o0o———————

NGUYEN THI NHU QUYNH

GEODESICS

BACHELOR THESIS
Major: Geometry
Supervisor: NGUYEN THAC DUNG

HA NOI, 2019


1

Thesis Assurance

The thesis is completed after self-study and synthetic process with the guidance of
Assoc. Prof. Dr. Nguyen Thac Dung. It is written by following the textbook entitled
”Elementary Differential Geometry” by Andrew Pressley. Most of the contents of this
thesis are taken from the chapter 9: ”Geodesics” in the mentioned textbook.
While the thesis is completed, I consulted some documents which have been mentioned in the bibliography section.
I assure that this thesis is not copied from any other thesis. I certify that these
statements are true. And I will be responsible for their correctness.
Ha Noi, May 5, 2019
Student

Nguyen Thi Nhu Quynh


Bachelor thesis

NGUYEN THI NHU QUYNH

Thesis Acknowledgement

This thesis is conducted at the Department of Mathematics, HANOI PEDAGOGICAL UNIVERSITY 2. The lecturers have imparted valuable knowledge a`n facilitated for me to complete the course and the thesis.
I would like to express my deep respect and gratitude to Dr. Nguyen Thac Dung,
who has direct guidance, help me to complete this thesis. I also want to thank to MS.
Nguyen Thac Dung for his valuable advice and assistance in the course of my degree.
Due to time, capacity and conditions are limited, so the thesis can not avoid errors.
So I am looking forward to receiving valuable comments from teachers and friends.
Ha Noi, May 5, 2019
Student

Nguyen Thi Nhu Quynh


i


Contents

Thesis Assurance

1

Notation

1

Preface

1

1

2

PRELIMINARIES
1.1

Curves in the plane and in the space . . . . . . . . . . .

2

1.2


Curvatures on a smooth surfaces . . . . . . . . . . . . . .

3

1.3

Surfaces and Isometries of surface . . . . . . . . . . . . .

5

2 GEODESICS

9

2.1

Definition and basic properties . . . . . . . . . . . . . . .

9

2.2

Geodesic equations . . . . . . . . . . . . . . . . . . . . .

13

2.3

Geodesics on surfaces of revolution . . . . . . . . . . . .


18

2.4

Geodesics as shortest paths . . . . . . . . . . . . . . . .

27

2.5

Geodesic coordinates . . . . . . . . . . . . . . . . . . . .

34

CONCLUSION

38
38

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Notation
S, S1 , S2 : surface
p,q : point
γ : curve

γ : parametrized curve
γ˙ : vector
γ(t)
˙
: speed
γ¨ : acceleration
kg : geodesic curvature
N: Standard unit normal
θ : the angle between q and p
ϕ :the angle between q and positive x-axis
Tp S : Tangent plane to S at p
Π : plane is perpendicular to the tangent plane of the surface at point
of γ
σ(u, v): surface patch
Γ: Christoffel symbols
E, G, F : coefficient of the first fundamental form.

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Preface
Geodesics are the curves in a surface that make turns just to stay on
the surface and never move sideways. A bug living in the surface and
following such a curve would perceive it to be straight. We will begin
with a definition of geodesics, then present various method for finding
geodesics on surfaces, and later reveal their relationships to shortest

paths.The term geodesic comes from the science of geodesy, which is
concerned with measurements of the earth’s surface.

1


Chapter 1
PRELIMINARIES
1.1

Curves in the plane and in the space

Definition 1.1.1. A parametrized curve in Rn is a smooth map γ :
(α, β) −→ Rn , for some α, β with −∞ ≤ α ≤ β ≤ ∞.
From now on, all parametrized curves will be assumed to be smooth.
˙
Definition 1.1.2. If γ is a parametrized curve, its first derivative γ(t)
is called the tangent vector of γ at the point γ(t).
Definition 1.1.3. If γ : (α, β) −→ Rn is a parametrized curve, its speed
˙
˙
at the point γ(t) is γ(t)
and γ is said to be a unit-speed curve if γ(t)
is a unit vector for all t ∈ (α, β).
Definition 1.1.4. A parametrized curve γ : (α, β) −→ Rn is a reparametrization of a parametrized curve γ : (α, β) −→ Rn if there is a smooth bijection map φ : (α, β) −→ (α, β) ( the reparametrization map) such that
the inverse map φ−1 : (α, β) −→ (α, β) is also smooth and
γ(t) = γ(φ(t)) f or all

2


t ∈ (α, β).


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NGUYEN THI NHU QUYNH

Curvatures on a smooth surfaces

Definition 1.2.1. If γ is a unit-speed curve with parameter t, its curvature k(t) at the point γ(t) is defined to be γ¨ (t) (∆t)2 .
Definition 1.2.2. If γ is a unit-speed curve on an oriented surface S,
then γ˙ is a unit vector and is a tangent vector to S. Hence, γ˙ is perpendicular to the unit normal N of S, so γ,
˙ N and N ×γ˙ are perpendicular
unit vectors. Since γ is unit-speed, γ¨ is perpendicular to γ,
˙ and hence is
a linear combination of N and N×γ:
˙
γ¨ = kn N + kg N × γ˙
The scalars kn and kg are called the normal curvature and the geodesic
curvature of γ, respectively.
Definition 1.2.3. γ is a normal section of the surface if γ is the intersection of the surface with the a plane Π that is perpendicular to the
tangent plane of the surface at every point of γ.
Corollary 1.2.4. The curvature k, normal curvature kn and geodesic
curvature kg of a normal section of a surface are related by
kn = ±k,

kg = 0.


Proposition 1.2.5. (Gauss Equations)
Let σ(u, v) be a surface patch with first and second fundamental forms
Edu2 + 2F dudv + Gdv 2 and Ldu2 + 2M dudv + N dv 2

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where Then,
σuu = Γ111 σu + Γ211 σv + LN,
σuv = Γ112 σu + Γ212 σv + M N,
σvv = Γ122 σu + Γ222 σv + N N,
where
GEu − 2F Fu + F Ev
,
2(EG − F 2 )
GEv − F Gu
,
Γ112 =
2(EG − F 2 )
2GFv − GGu + F Gv
Γ122 =
,
2(EG − F 2 )

2EFu − EEv + F Eu
,

2(EG − F 2 )
EGu − 2F Ev
,
Γ212 =
2(EG − F 2 )
EGv − 2F Fv + F Gu
Γ222 =
.
2(EG − F 2 )

Γ111 =

Γ211 =

The six Γ coefficients in these formulas are called Christoffel symbols.
Note that they depend only on the first fundamental form of σ.
Proposition 1.2.6. Let γ(t) = σ(u(t), v(t)) be a curve on a surface
patch σ, and let v(t) = α(t)σu + β(t)σv be a tangent vector field along γ,
where α and β are smooth function of t. Then, v is parallel along γ if
and only if the following equations are satisfied:
α˙ + (Γ111 u˙ + Γ112 v)α
˙ + (Γ112 u˙ + Γ122 v)β
˙ =0
β˙ + (Γ211 u˙ + Γ212 v)α
˙ + (Γ212 u˙ + Γ222 v)β
˙ =0
These equations involve only the first fundamental form of σ.

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1.3

NGUYEN THI NHU QUYNH

Surfaces and Isometries of surface

Surface is a subset of R3 that looks like a piece of R2 in the vicinity
of any given point such as the surface of the Earth, although actually
nearly spherical, appears to be a flat plane to an observer on the surface
who sees only to the horizon. In other words, we have the following
definition.
Definition 1.3.1. A subset S of R3 is a surface if, for every point p ∈ S,
there is an open set U in R2 and an open set W in R3 containing p such
that S ∩ W is homeomorphic to U . A subset of a surface S of the form
S ∩ W , where W is an open subset of R3 , is called an open subset of
S. A homeomorphism σ : U −→ S ∩ W is called a surface patch or
parametrization of the open subset S ∩ W of S. A collection of such
surface patches whose images cover the whole of S is called an atlas of
S.
Definition 1.3.2. A ruled surface is a surface that is a union of straight
lines, called the rulings (or sometimes the generators) of the surface.

Figure 1.1:

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where δ is a vector in the direction of the line passing through γ. And
point p lies on one of the given straight lines, which intersects γ at q.
By the definition of ruled surface, we can show that, a ruled surface
has a parametrization as follows
σ(u, v) = γ(u) + vδ(u)
where δ(u) is a non-zero vector in the direction of the line passing
through γ(u).

And σ : U −→ R3 is a smooth map, where U =

(u, v) ∈ R2 |α < u < β . Moreover, denoting d/dt by a dot, σu = γ˙ +
˙ σv = δ.
v δ,
Definition 1.3.3. If S1 and S2 are surfaces, a smooth map f : S1 −→ S2
is called a local isometry if it takeks any curve in S1 to a curve of the
same length in S2 . If a local isometry f : S1 −→ S2 exists, we say that
S1 and S2 are locally isometric.
Definition 1.3.4. A sphere is the set of point of R3 that are a fixed
distance ( the radius of the sphere) from a fixed point (its center). For
example, the sphere of radius 1 and center the origin, called the unit
sphere, is
S 2 = (x, y, z) ∈ R3 | x2 + y 2 + z 2 = 1.
The parametrization of S 2 is that given by latitude θ and longitude
ϕ. If p is a point of sphere, the line through p parallels to the z -axis
intersects the xy-plane at a point q. Then θ is the angle between q and
p and ϕ is the angle between q and the positive x -axis. The circles on

the sphere corresponding to a constant value of θ are called parallels;
those corresponding to a constant value of ϕ are called meridians. The
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Figure 1.2:

latitude-longitude parametrization of S 2 is
p = (cosθcosϕ, cosθsinϕ, sinθ).
Definition 1.3.5. A surface of revolution is the surface obtained by
rotating a plane curve, called the profile curve, around a straight line in
the plane. The circles obtained by rotating a fixed point on the profile
curve around the axis of rotation are called the parallels of the surface,
and the curves on the surface obtained by rotating the profile curve
through a fixed angle are called its meridians.

Figure 1.3:

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Let us take the axis of rotation to be the z-axis and the plane to be

the xz-plane. Any point p of the surface is obtained by rotating some
point q of the profile curve through an angle v (say) around the z-axis.
If
γ(u) = (f (u), 0, g(u))
is a parametrization of the profile curve containing q, p is of the form
σ(u, v) = (f (u)cosv.f (u)sinv, g(u)).
Proposition 1.3.6. Let p and q be distinct points of S 2 . If p = −q, the
shortest length joining p and q. If p = −q , any great semicircle joining
p and q is a shortest curve joining these two points.
Theorem 1.3.7. (Inverse Function Theorem) Let f : U −→ Rn be a
smooth map defined on an open subset U of Rn (n ≥ 1). Assume that,
at some point x0 ∈ U , the Jacobian matrix J(f ) is invertible. Then,
there is an open subset V of Rn and a smooth map g : V −→ Rn such
that
(i) y0 = f (x) ∈ V
(ii) g(y0 ) = x0
(iii) g(V ) ⊆ U
(iv) g(V ) is an open subset of Rn
(v) f (g(y)) = y for all y ∈ V
In particular, g : V −→ g(V ) and f : g(V ) −→ V are inverse bijections.

8


Chapter 2
GEODESICS
2.1

Definition and basic properties


If we drive along a ”straight” road, we do not have to turn the wheel of
our car to the right of left. However,the road is not, in fact, a straight
line as the surface of the earth is, to a good approximation, a sphere
and there can be no straight line on the surface of a sphere. If the road
is represented by a curve γ, its acceleration γ¨ will be non-zero, but we
perceive the curve as being straight because the tangential component
of γ¨ is zero, in other words because γ¨ is perpendicular to the surface.
Definition 2.1.1. A curve γ(t) on a surface S is called a geodesic if
at every point γ(t) the acceleration γ¨ is zero or perpendicular to the
tangent plane of the surface at the point γ(t), i.e., parallel to its unit
normal, for all values of the parameter t.
Equivalently, γ is a geodesic if and only if its tangent vector γ is parallel
along γ.
Proposition 2.1.2. A geodesic γ(t) on a surface S has constant speed.
Proof. We have
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d
γ˙
dt

2

=


d
(γ.
˙ γ)
˙ = 2¨
γ .γ.
˙
dt

Since γ is geodesic, γ¨ is perpendicular to the tangent plane which contains γ.
˙ Hence, γ¨ .γ˙ = 0. Subsequently, d γ˙ 2 /dt = 0. Therefore, the
speed γ˙ is constant. The proof is complete.
Proposition 2.1.3. A unit-speed curve on a surface is a geodesic if and
only if its geodesic curvature is zero everywhere.
Proof. Let γ be a unit-speed geodesic in a patch σ of the surface S , and
N be the standard unit normal of σ, so that the geodesic curvature is
kg = γ¨ .(N × γ)
˙

(2.1)

If γ¨ is zero, then the above implies that kg = 0. Otherwise, by definition
γ¨ is parallel to N, it is obviously perpendicular to N × γ˙ , so by equation
2.1 kg = 0.
Conversely, suppose that kg = 0. Then γ¨ is perpendicular to N × γ.
˙
But then, since γ,
˙ N and N × γ˙ are perpendicular unit vectors in R3 ,
and since γ¨ is perpendicular to γ,
˙ it follows that γ¨ is parallel to N. The
proof is complete.

Proposition 2.1.4. Any (part of a) straight line on a surface is a
geodesic.
Proof. This is obvious, for any straight line has a (constant speed) parametrization of the form
γ(t) = a + bt,

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where a and b are constant vectors, and clearly γ¨ = 0. The proof is
complete.
Example 2.1.5. All straight lines in the plane are geodesics, as are the
rulings of any ruled surface, such as those of a ( generalized) cylinder or
a ( generalized) cone, or the straight lines on a hyperboloid of one sheet.

Figure 2.1:

Proposition 2.1.6. Any normal section of a surface is a geodesic
Proof. A normal section is a curve produced by slicing the surface with a
plane that contains the surface normal at every point of the curve. The
commonest example of a normal section is a section formed by a plane
of symmetry. So, any intersection with a plane of symmetry is always a
geodesic. The proof is complete.
Example 2.1.7. A great circle on a sphere is its intersection with a
plane Π passing through the center O of the sphere. Every point p on
the great circle defines a vector with Op which is perpendicular to the
tangent plane at p. Thus, the great circle is a geodesic.

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Figure 2.2:

Example 2.1.8. The intersection of a generalized cylinder with a plane
Π perpendicular to the rulings of the cylinder is a geodesic. Clearly, the
unit normal at such an intersection point is perpendicular to the rulings,
and hence contained in Π. So, Π perpendicular to the tangent plane at
the point.

Figure 2.3:

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NGUYEN THI NHU QUYNH

Geodesic equations

To determine all the geodesics on a given surface, we need to solve differential equations stated in following theorem.
Theorem 2.2.1. A curve γ on a surface S is a geodesic if and only if,

for any part γ(t) = σ(u(t), v(t)) of γ contained in a surface patch σ of
S, the following two equation are satisfied:
1
d
(E u˙ + F v)
˙ = (Eu u˙ 2 + 2Fu u˙ v˙ + Gu v˙ 2 ),
dt
2
d
1
(F u˙ + Gv)
˙ = (Ev u˙ 2 + 2Fv u˙ v˙ + Gv v˙ 2 ),
dt
2

(2.2)

where Edu2 + 2F dudv + Gdv 2 is the first fundamental form of σ.
The different equations (2.2) are called the geodesic equations.
Proof. The tangent plane is spanned by σu and σv . By the definition the
curve γ is a geodesic if and only if γ¨ is perpendicular to σu and σv so
γ¨ .σu = γ¨ .σv = 0. Since γ˙ = uσ
˙ u + vσ
˙ v , γ¨ .σu = 0 becomes
d
(uσ
˙ u + vσ
˙ v ) .σu = 0.
dt
We rewrite the left hand side of the above equation:

d
d
dσu
(uσ
˙ u + vσ
˙ v ) .σu = ((uσ
˙ u + vσ
˙ v ).σu ) − (uσ
˙ u + vσ
˙ v ).
dt
dt
dt
d
= (E u˙ + F v)
˙ − (uσ
˙ u + vσ
˙ v ).(uσ
˙ uu + vσ
˙ uv )
dt
d
= (E u˙ + F v)
˙ − (u˙ 2 (σu .σuu ) + u˙ v(σ
˙ u .σuv + σv .σuu ) + v˙ 2 (σv .σuv )).
dt
(2.3)
Now,
Eu = (σu .σu ).u = σuu .σu + σu .σuu = 2σu .σuu ,
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1
1
so σu .σuu = Eu . Similarly σv .σuv = Gu . Finally,
2
2
σu .σuv + σv .σuu = (σu .σv )u = Fu .
Substituting these values into (2.3), we obtain
d
d
1
(uσ
˙ u + vσ
˙ v ) .σu = (E u˙ + F v)
˙ − (Eu u˙ 2 + 2Fu u˙ v˙ + Gu v˙ 2 ).
dt
dt
2
This establishes the first differential equation. Similarly, other equations
can be established from
d
(uσ
˙ u + vσ
˙ v ) .σv = 0.
dt


The proof is complete.
The geodesic equations are non-linear differential equations, and are
usually difficult or impossible to solve explicitly.
Example 2.2.2. We determine the geodesics on the unit sphere S 2 by
solving the geodesic equations. For the usual parametrization by latitude
θ and longitude ϕ,
σ(θ, ϕ) = (cosθcosϕ, cosθsinϕ, sinθ),
we found the first fundamental form is
dθ2 + cos2 θdϕ2 .
We might as well restrict ourselves to unit-speed curves γ(t) = σ(θ(t), ϕ(t)),
so that

θ˙2 + ϕ˙ 2 cos2 θ = 1,
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and if γ is a geodesic the second equation in (2.2) gives
d
(ϕcos
˙ 2 θ) = 0,
dt
ϕcos
˙ 2 θ = Ω,

so that


where Ω is a constant. If Ω = 0, then ϕ˙ = 0 and so ϕ is constant and γ
is part of a meridian.We assume that ϕ˙ = 0 from now on.
The unit-speed condition gives
Ω2
θ˙2 = 1 −
,
cos2 θ
so along the geodesic we have
dθ
dϕ

2

=

θ˙2
= cos2 θ(Ω−2 cos2 θ − 1),
2
ϕ˙

and hence
dθ
,
cosθ Ω−2 cos2 θ − 1
√

±(ϕ − ϕ0 ) =

where ϕ0 is a constant. The integral can be evaluated by making the

substitution u = tanθ. This gives
±(ϕ − ϕ0 ) =

√

du
= sin−1
Ω−2 − 1 − u2

√

u
Ω−2 − 1

,

and hence
tanθ = ±

Ω−2 − 1 sin(ϕ − ϕ0 ).

This implies that the coordinates x = cosθ cosϕ, y = cosθ sinϕ and
z = sinθ of γ(t) satisfy the equation z = ax + by,
√
√
where a = ∓ Ω−2 − 1 sinϕ0 , and b = ± Ω−2 − 1 cosϕ0 . This shows
that γ is contained in the intersection of S 2 with a plane passing through
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the origin. Hence, in all cases, γ is part of great circle.
The geodesic equations can be expressed in a different, but equivalent,
form which is sometimes more useful than that in Theorem 2.2.1.
Proposition 2.2.3. A curve γ on a surface S is a geodesic if and only
if, for any part γ(t) = σ(u(t), v(t)) of γ contained in a surface patch σ
of S, the following two equation are satisfied:
u¨ + Γ111 u˙ 2 + 2Γ112 u˙ v˙ + Γ122 v˙ 2 = 0
v¨ + Γ211 u˙ 2 + 2Γ212 u˙ v˙ + Γ222 v˙ 2 = 0.
Proof. γ is a geodesic if and only if γ˙ is parallel along γ. Since γ˙ =
uσ
˙ u + vσ
˙ v , the equations in the statement of the proposition follow from
previous proposition.The proof is complete.
Proposition 2.2.4. Let p be a point of a surface S, and let t be a unit
tangent vector to S at p. Then, there exists a unique unit-speed geodesic
γ on S which passed through p and has tangent vector t there.
In short, there is a unique geodesic through any given point of a surface in any given tangent direction.
Proof. The geodesic equations are of the form
u¨ = f (u, v, u,
˙ v),
˙

v¨ = g(u, v, u,
˙ v),
˙


(2.4)

where f and g are smooth functions of four variables u,v,u,
˙ v.
˙ For any
given constant a,b,c and d, and any value t0 of t, there is a solution of
above equation such that
u(t0 ) = a, v(t0 ) = b, u(t
˙ 0 ) = c, v(t
˙ 0 ) = d,
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and such that u(t), v(t) are defined and smooth for all t satisfying |t −
t0 | < , where is some positive number. Moreover, any two solutions of
equation(2.4) satisfying (2.5) agree for all values of t such that |t − t0 | <
, where

is some positive number ≤ .

Suppose we need to construct a geodesic passing through a point p on
a surface patch σ(u, v) of S, say p = σ(a, b), and that t = cσu +dσv ,where
a,b,c and d are scalars and the derivatives are valuated at u=a, v=b. A
unit-speed curve γ(t) = σ(u(t), v(t)) passes through p at t = t0 if and

only if u(t0 ) = a, v(t0 ) = b, and has tangent vector t there if and only if
cσu + dσv = t = γ(t
˙ 0 ) = u(t
˙ 0 )σu + v(t
˙ 0 )σv ,
i.e., u(t
˙ 0 ) = c, v(t
˙ 0 ) = d. Thus, finding a ( unit-speed) geodesic γ passing
through p at t = t0 and having tangent vector t is equivalent to solving
the geodesic equations subject to the initial conditions (2.5). But we
have said above that this problem has a unique solution.The proof is
complete.
Example 2.2.5. All straight lines in a plane are geodesics. Since there
is a straight line in the plane through any given point of the plane in
any given direction parallel to the plane, it follows from Proposition 2.2.4
that there are no other geodesics.
Corollary 2.2.6. Any local isometry between two surfaces takes the
geodesics of one surface to the geodesics of the other.
Proof. Let S1 and S2 be the two surface, let f : S1 −→ S2 be the local
isometry and let γ1 be a geodesic on S1 , p be a point on γ1 and σ(u, v)
be a surface patch S1 with p in its image. Then, the part of γ1 lying in
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the patch σ is of the form γ1 (t) = σ(u(t), v(t) with a < t < b, say, where
the smooth functions u and v satisfy the geodesic equations (2.2), with

E, F and G being the coefficients of the first fundamental form of σ. We
have fo σ is a patch of S2 with the same first fundamental form as σ.
Hence, γ2 (t) = f (σ(u(t), v(t))), with a < t < b, is a geodesic on S2 . This
implies that γ¨2 is perpendicular to S2 at f (p). As this is true for all p,
γ2 is a geodesic on S2 . The proof is complete.
Example 2.2.7. Let S1 be the infinite strip in the xy-plane given by 0 <
x < 2π, and S2 be the circular cylinder x2 + y 2 = 1 with the ruling x=1,
y=0 removed. Then S1 is covered by a single patch σ1 (u, v) = (u, v, 0),
and S2 by the patch σ2 (u, v) = (sinu, cosu, v), where 0 < x < 2π for both
cases. Consider the map f : (u, v, 0) −→ (sinu, cosu, v) from S1 and S2 .
We can verify that both patches have the same first fundamental form
du2 + dv 2 . Therefore f is an isometry, and S1 and S2 are isometric.

Figure 2.4:

2.3

Geodesics on surfaces of revolution

It turns out that, although the geodesic equations for a surface of revolution cannot usually be solved explicitly, they can be used to get a good
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qualitative understanding of the geodesics on such a surface.
We parametrize the surface of the revolution in the usual way
σ(u, v) = (f (u)cosv, f (u)sinv, g(u)),

2

2

df
dg
where we assume that f > 0 and
+
= 1 ( we used a
du
du
dot there to denote d/du, but now a dot is reserved for d/dt, where t
is parameter along a geodesic).The first fundamental form of σ is du2 +
f (u)2 dv 2 . Referring to Equation (2.2),
u¨ = f (u)

d
(f (u)2 v)
˙ = 0.
dt

df 2
v˙ ,
du

(2.6)

We might as well consider unit-speed geodesics, so that
u˙ 2 + f (u)2 v˙ 2 = 1.


(2.7)

From this, we make the following easy deductions:
Proposition 2.3.1. On the surface of revolution
σ(u, v) = (f (u)cosv, f (u)sinv, g(u)),
(i) Every meridian is a geodesic.
(ii) A parallel u = u0 is a geodesic if and only if df /du = 0 when u = u0 ,
i.e., u0 is a stationary point of f.
Proof. On a meridian, we have v= constant so the the second equation
in (2.6) is obviously satisfied. Equation u˙ 2 + f (u)2 v˙ 2 = 1 gives u˙ = ±1,
so u˙ is constant and the first equation in (2.6) is also satisfied.
For (ii) , if u = u0 is constant, then by equation u˙ 2 + f (u)2 v˙ 2 = 1,
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