5Smooth embedded surfaces (in ℝ3)
IB Geometry
5.2 Geodesics
We now come to the important idea of a geodesic. We will first define these for
Riemannian metrics, and then generalize it to general embedded surfaces.
Definition (Geodesic). Let
V ⊆ R
2
u,v
be open, and
E
d
u
2
+ 2
F
d
u
d
v
+
G
d
v
2
be a Riemannian metric on V . We let
γ = (γ
1
, γ
2
) : [a, b] → V
be a smooth curve. We say
γ
is a geodesic with respect to the Riemannian metric
if it satisfies
d
dt
(E ˙γ
1
+ F ˙γ
2
) =
1
2
(E
u
˙γ
2
1
+ 2F
u
˙γ
1
˙γ
2
+ G
u
˙γ
2
2
)
d
dt
(F ˙γ
1
+ G ˙γ
2
) =
1
2
(E
v
˙γ
2
1
+ 2F
v
˙γ
1
˙γ
2
+ G
v
˙γ
2
2
)
for all t ∈ [a, b]. These equations are known as the geodesic ODEs.
What exactly do these equations mean? We will soon show that these are
curves that minimize (more precisely, are stationary points of) energy. To do so,
we need to come up with a way of describing what it means for
γ
to minimize
energy among all possible curves.
Definition (Proper variation). Let
γ
: [
a, b
]
→ V
be a smooth curve, and let
γ(a) = p and γ(b) = q. A proper variation of γ is a C
∞
map
h : [a, b] ×(−ε, ε) ⊆ R
2
→ V
such that
h(t, 0) = γ(t) for all t ∈ [a, b],
and
h(a, τ) = p, h(b, τ) = q for all |τ| < ε,
and that
γ
τ
= h( ·, τ) : [a, b] → V
is a C
∞
curve for all fixed τ ∈ (−ε, ε).
Proposition. A smooth curve
γ
satisfies the geodesic ODEs if and only if
γ
is
a stationary point of the energy function for all proper variation, i.e. if we define
the function
E(τ ) = energy(γ
τ
) : (−ε, ε) → R,
then
dE
dτ
τ=0
= 0.
Proof. We let γ(t) = (u(t), v(t)). Then we have
energy(γ) =
Z
b
a
(E(u, v) ˙u
2
+ 2F (u, v) ˙u ˙v + G(u, v) ˙v
2
) dt =
Z
b
a
I(u, v, ˙u, ˙v) dt.
We consider this as a function of four variables
u, ˙u, v, ˙v
, which are not necessarily
related to one another. From the calculus of variations, we know
γ
is stationary
if and only if
d
dt
∂I
∂ ˙u
=
∂I
∂u
,
d
dt
∂I
∂ ˙v
=
∂I
∂v
.
The first equation gives us
d
dt
(2(E ˙u + F ˙v)) = E
u
˙u
2
+ 2F
u
˙u ˙v + G
u
˙v
2
,
which is exactly the geodesic ODE. Similarly, the second equation gives the other
geodesic ODE. So done.
Since the definition of a geodesic involves the derivative only, which is a local
property, we can easily generalize the definition to arbitrary embedded surfaces.
Definition (Geodesic on smooth embedded surface). Let
S ⊆ R
3
be an embed-
ded surface. Let Γ : [
a, b
]
→ S
be a smooth curve in
S
, and suppose there is a
parametrization
σ
:
V → U ⊆ S
such that
im
Γ
⊆ U
. We let
θ
=
σ
−1
be the
corresponding chart.
We define a new curve in V by
γ = θ ◦ Γ : [a, b] → V.
Then we say Γ is a geodesic on
S
if and only if
γ
is a geodesic with respect to
the induced Riemannian metric.
For a general Γ : [
a, b
]
→ V
, we say Γ is a geodesic if for each point
t
0
∈
[
a, b
],
there is a neighbourhood
˜
V
of
t
0
such that
im
Γ
|
˜
V
lies in the domain of some
chart, and Γ|
˜
V
is a geodesic in the previous sense.
Corollary. If a curve Γ minimizes the energy among all curves from
P
= Γ(
a
)
to Q = Γ(b), then Γ is a geodesic.
Proof.
For any
a
1
, a
2
such that
a ≤ a
1
≤ b
1
≤ b
, we let Γ
1
= Γ
|
[a
1
,b
1
]
. Then Γ
1
also minimizes the energy between
a
1
and
b
1
for all curves between Γ(
a
1
) and
Γ(b
1
).
If we picked
a
1
, b
1
such that Γ([
a
1
, b
1
])
⊆ U
for some parametrized neigh-
bourhood
U
, then Γ
1
is a geodesic by the previous proposition. Since the
parametrized neighbourhoods cover
S
, at each point
t
0
∈
[
a, b
], we can find
a
1
, b
1
such that Γ([a
1
, b
1
]) ⊆ U . So done.
This is good, but we can do better. To do so, we need a lemma.
Lemma. Let
V ⊆ R
2
be an open set with a Riemannian metric, and let
P, Q ∈ V
.
Consider
C
∞
curves
γ
: [
a, b
]
→ V
such that
γ
(0) =
P, γ
(1) =
Q
. Then such a
γ
will minimize the energy (and therefore is a geodesic) if and only if
γ
minimizes
the length and has constant speed.
This means being a geodesic is almost the same as minimizing length. It’s
just that to be a geodesic, we have to parametrize it carefully.
Proof.
Recall the Cauchy-Schwartz inequality for continuous functions
f, g ∈
C[0, 1], which says
Z
1
0
f(x)g(x) dx
2
≤
Z
1
0
f(x)
2
dx
Z
1
0
g(x)
2
dx
,
with equality iff
g
=
λf
for some
λ ∈ R
, or
f
= 0, i.e.
g
and
f
are linearly
dependent.
We now put f = 1 and g = k˙γk. Then Cauchy-Schwartz says
(length γ)
2
≤ energy(γ),
with equality if and only if ˙γ is constant.
From this, we see that a curve of minimal energy must have constant speed.
Then it follows that minimizing energy is the same as minimizing length if we
move at constant speed.
Is the converse true? Are all geodesics length minimizing? The answer is
“almost”. We have to be careful with our conditions in order for it to be true.
Proposition. A curve Γ is a geodesic iff and only if it minimizes the energy
locally, and this happens if it minimizes the length locally and has constant
speed.
Here minimizing a quantity locally means for every
t ∈
[
a, b
], there is some
ε > 0 such that Γ|
[t−ε,t+ε]
minimizes the quantity.
We will not prove this. Local minimization is the best we can hope for, since
the definition of a geodesic involves differentiation, and derivatives are local
properties.
Proposition. In fact, the geodesic ODEs imply kΓ
0
(t)k is constant.
We will also not prove this, but in the special case of the hyperbolic plane,
we can check this directly. This is an exercise on the third example sheet.
A natural question to ask is that if we pick a point
P
and a tangent direction
a, can we find a geodesic through P whose tangent vector at P is a?
In the geodesic equations, if we expand out the derivative, we can write the
equation as
E F
F G
¨γ
1
¨γ
2
= something.
Since the Riemannian metric is positive definite, we can invert the matrix and
get an equation of the form
¨γ
1
¨γ
2
= H(γ
1
, γ
2
, ˙γ
1
,
˙
γ
2
)
for some function
H
. From the general theory of ODE’s in IB Analysis II, subject
to some sensible conditions, given any
P
= (
u
0
, v
0
)
∈ V
and a = (
p
0
, q
0
)
∈ R
2
,
there is a unique geodesic curve
γ
(
t
) defined for
|t| < ε
with
γ
(0) =
P
and
˙γ
(0) = a. In other words, we can choose a point, and a direction, and then there
is a geodesic going that way.
Note that we need the restriction that
γ
is defined only for
|t| < ε
since we
might run off to the boundary in finite time. So we need not be able to define it
for all t ∈ R.
How is this result useful? We can use the uniqueness part to find geodesics.
We can try to find some family of curves
C
that are length-minimizing. To prove
that we have found all of them, we can show that given any point
P ∈ V
and
direction a, there is some curve in C through P with direction a.
Example. Consider the sphere
S
2
. Recall that arcs of great circles are length-
minimizing, at least locally. So these are indeed geodesics. Are these all? We
know for any
P ∈ S
2
and any tangent direction, there exists a unique great
circle through
P
in this direction. So there cannot be any other geodesics on
S
2
,
by uniqueness.
Similarly, we find that hyperbolic line are precisely all the geodesics on a
hyperbolic plane.
We have defined these geodesics as solutions of certain ODEs. It is possible
to show that the solutions of these ODEs depend
C
∞
-smoothly on the initial
conditions. We shall use this to construct around each point
P ∈ S
in a surface
geodesic polar coordinates. The idea is that to specify a point near
P
, we can
just say “go in direction
θ
, and then move along the corresponding geodesic for
time r”.
We can make this (slightly) more precise, and provide a quick sketch of how
we can do this formally. We let
ψ
:
U → V
be some chart with
P ∈ U ⊆ S
. We
wlog
ψ
(
P
) = 0
∈ V ⊆ R
2
. We denote by
θ
the polar angle (coordinate), defined
on V \ {0}.
θ
Then for any given
θ
, there is a unique geodesic
γ
θ
: (
−ε, ε
)
→ V
such that
γ
θ
(0) = 0, and ˙γ
θ
(0) is the unit vector in the θ direction.
We define
σ(r, θ) = γ
θ
(r)
whenever this is defined. It is possible to check that
σ
is
C
∞
-smooth. While
we would like to say that
σ
gives us a parametrization, this is not exactly true,
since we cannot define
θ
continuously. Instead, for each
θ
0
, we define the region
W
θ
0
= {(r, θ) : 0 < r < ε, θ
0
< θ < θ
0
+ 2π} ⊆ R
2
.
Writing V
0
for the image of W
θ
0
under σ, the composition
W
θ
0
V
0
U
0
⊆ S
σ
ψ
−1
is a valid parametrization. Thus σ
−1
◦ ψ is a valid chart.
The image (
r, θ
) of this chart are the geodesic polar coordinates. We have
the following lemma:
Lemma (Gauss’ lemma). The geodesic circles
{r
=
r
0
} ⊆ W
are orthogonal to
their radii, i.e. to
γ
θ
, and the Riemannian metric (first fundamental form) on
W
is
dr
2
+ G(r, θ) dθ
2
.
This is why we like geodesic polar coordinates. Using these, we can put the
Riemannian metric into a very simple form.
Of course, this is just a sketch of what really happens, and there are many
holes to fill in. For more details, go to IID Differential Geometry.
Definition (Atlas). An atlas is a collection of charts covering the whole surface.
The collection of all geodesic polars about all points give us an example.
Other interesting atlases are left as an exercise on example sheet 3.