3Geodesics
III Riemannian Geometry
3.1 Definitions and basic properties
We will eventually want to talk about geodesics. However, the setup we need to
write down the definition of geodesics can be done in a much more general way,
and we will do that.
The general setting is that we have a vector bundle π : E → M.
Definition
(Lift)
.
Let
π
:
E → M
be a vector bundle with typical fiber
V
.
Consider a curve
γ
: (
−ε, ε
)
→ M
. A lift of
γ
is a map
γ
E
: (
−ε, ε
)
→ E
if
π ◦γ
E
= γ, i.e. the following diagram commutes:
E
(−ε, ε) M
π
γ
γ
E
.
For
p ∈ M
, we write
E
p
=
π
−1
(
{p}
)
∼
=
V
for the fiber above
p
. We can think
of
E
p
as the space of some “information” at
p
. For example, if
E
=
T M
, then the
“information” is a tangent vector at
p
. In physics, the manifold
M
might represent
our universe, and a point in
E
p
might be the value of the electromagnetic field
at p.
Thus, given a path
γ
in
M
, a lift corresponds to providing that piece of
“information” at each point along the curve. For example, if
E
=
T M
, then we
can canonically produce a lift of
γ
, given by taking the derivative of
γ
at each
point.
Locally, suppose we are in some coordinate neighbourhood
U ⊆ M
such that
E is trivial on U . After picking a trivialization, we can write our lift as
γ
E
(t) = (γ(t), a(t))
for some function a : (−ε, ε) → V .
One thing we would want to do with such lifts is to differentiate them, and
see how it changes along the curve. When we have a section of
E
on the whole
of
M
(or even just an open neighbourhood), rather than just a lift along a
curve, the connection provides exactly the information needed to do so. It is not
immediately obvious that the connection also allows us to differentiate curves
along paths, but it does.
Proposition.
Let
γ
: (
−ε, ε
)
→ M
be a curve. Then there is a uniquely
determined operation
∇
dt
from the space of all lifts of
γ
to itself, satisfying the
following conditions:
(i) For any c, d ∈ R and lifts ˜γ
E
, γ
E
of γ, we have.
∇
dt
(cγ
E
+ d˜γ
E
) = c
∇γ
E
dt
+ d
∇˜γ
E
dt
(ii) For any lift γ
E
of γ and function f : (−ε, ε) → R, we have
∇
dt
(fγ
E
) =
df
dt
+ f
∇γ
E
dt
.
(iii)
If there is a local section
s
of
E
and a local vector field
V
on
M
such that
γ
E
(t) = s(γ(t)), ˙γ(t) = V (γ(t)),
then we have
∇γ
E
dt
= (∇
V
s) ◦ γ.
Locally, this is given by
∇γ
E
dt
i
= ˙a
i
+ Γ
i
jk
a
j
˙x
k
.
The proof is straightforward — one just checks that the local formula works,
and the three properties force the operation to be locally given by that formula.
Definition
(Covariant derivative)
.
The uniquely defined operation in the propo-
sition above is called the covariant derivative.
In some sense, lifts that have vanishing covariant derivative are “constant”
along the map.
Definition
(Horizontal lift)
.
Let
∇
be a connection on
E
with Γ
i
jk
(
x
) the
coefficients in a local trivialization. We say a lift γ
E
is horizontal if
∇γ
E
dt
= 0.
Since this is a linear first-order ODE, we know that for a fixed
γ
, given any
initial a(0) ∈ E
γ(0)
, there is a unique way to obtain a horizontal lift.
Definition
(Parallel transport)
.
Let
γ
: [0
,
1]
→ M
be a curve in
M
. Given any
a
0
∈ E
γ(0)
, the unique horizontal lift of
γ
with
γ
E
(0) = (
γ
(0)
, a
0
) is called the
parallel transport of
a
0
along
γ
(0). We sometimes also call
γ
E
(1) the parallel
transport.
Of course, we want to use this general theory to talk about the case where
M
is a Riemannian manifold,
E
=
T M
and
∇
is the Levi-Civita connection of
g
. In this case, each curve
γ
(
t
) has a canonical lift independent of the metric or
connection given simply by taking the derivative ˙γ(t).
Definition
(Geodesic)
.
A curve
γ
(
t
) on a Riemannian manifold (
M, g
) is called
a geodesic curve if its canonical lift is horizontal with respect to the Levi-Civita
connection. In other words, we need
∇˙γ
dt
= 0.
In local coordinates, we write this condition as
¨x
i
+ Γ
i
jk
˙x
j
˙x
k
= 0.
This time, we obtain a second-order ODE. So a geodesic is uniquely specified
by the initial conditions
p
=
x
(0) and
a
=
˙x
(0). We will denote the resulting
geodesic as γ
p
(t, a), where t is the time coordinate as usual.
Since we have a non-linear ODE, existence is no longer guaranteed on all
time, but just for some interval (
−ε, ε
). Of course, we still have uniqueness of
solutions.
We now want to prove things about geodesics. To do so, we will need to apply
some properties of the covariant derivative we just defined. Since we are lazy,
we would like to reuse results we already know about the covariant derivative
for vector fields. The trick is to notice that locally, we can always extend
˙γ
to a
vector field.
Indeed, we work in some coordinate chart around
γ
(0), and we wlog assume
˙γ(0) =
∂
∂x
1
.
By the inverse function theorem, we note that
x
1
(
t
) is invertible near 0, and we
can write
t
=
t
(
x
1
) for small
x
1
. Then in this neighbourhood of 0, we can view
x
k
as a function of x
1
instead of t. Then we can define the vector field
˙γ(x
1
, ··· , x
k
) = ˙γ(x
1
, x
2
(x
1
), ··· , x
k
(x
1
)).
By construction, this agrees with ˙γ along the curve.
Using this notation, the geodesic equation can be written as
∇
˙γ
˙γ
γ(t)
= 0,
where the
∇
now refers to the covariant derivative of vector fields, i.e. the
connection itself.
γ
Using this, a lot of the desired properties of geodesics immediately follow from
well-known properties of the covariant derivative. For example,
Proposition. If γ is a geodesic, then |˙γ(t)|
g
is constant.
Proof.
We use the extension
˙γ
around
p
=
γ
(0), and stop writing the underlines.
Then we have
˙γ(g( ˙γ, ˙γ)) = g(∇
˙γ
˙γ, ˙γ) + g( ˙γ, ∇
˙γ
˙γ) = 0,
which is valid at each q = γ(t) on the curve. But at each q, we have
˙γ(g( ˙γ, ˙γ)) = ˙x
k
∂
∂x
k
g( ˙γ, ˙γ) =
d
dt
|˙γ(t)|
2
g
by the chain rule. So we are done.
At this point, it might be healthy to look at some examples of geodesics.
Example.
In
R
n
with the Euclidean metric, we have Γ
i
jk
= 0. So the geodesic
equation is
¨x
k
= 0.
So the geodesics are just straight lines.
Example.
On a sphere
S
n
with the usual metric induced by the standard
embedding S
n
→ R
n+1
. Then the geodesics are great circles.
To see this, we may wlog
p
=
e
0
and
a
=
e
1
, for a standard basis
{e
i
}
of
R
n+1
. We can look at the map
ϕ : (x
0
, ··· , x
n
) 7→ (x
0
, x
1
, −x
2
, ··· , −x
n
),
and it is clearly an isometry of the sphere. Therefore it preserves the Riemannian
metric, and hence sends geodesics to geodesics. Since it also preserves
p
and
a
,
we know
ϕ
(
γ
) =
γ
by uniqueness. So it must be contained in the great circle
lying on the plane spanned by e
0
and e
1
.
Lemma.
Let
p ∈ M
, and
a ∈ T
p
M
. As before, let
γ
p
(
t, a
) be the geodesic with
γ(0) = p and ˙γ(0) = p. Then
γ
p
(λt, a) = γ
p
(t, λa),
and in particular is a geodesic.
Proof. We apply the chain rule to get
d
dt
γ(λt, a) = λ ˙γ(λt, a)
d
2
dt
2
γ(λt, a) = λ
2
¨γ(λt, a).
So
γ
(
λt, a
) satisfies the geodesic equations, and have initial velocity
λa
. Then
we are done by uniqueness of ODE solutions.
Thus, instead of considering
γ
p
(
t, a
) for arbitrary
t
and
a
, we can just fix
t
= 1, and look at the different values of
γ
p
(1
, a
). By ODE theorems, we know
this depends smoothly on
a
, and is defined on some open neighbourhood of
0 ∈ T
p
M.
Definition
(Exponential map)
.
Let (
M, g
) be a Riemannian manifold, and
p ∈ M. We define exp
p
by
exp
p
(a) = γ(1, a) ∈ M
for a ∈ T
p
M whenever this is defined.
We know this function has domain at least some open ball around 0
∈ T
p
M
,
and is smooth. Also, by construction, we have exp
p
(0) = p.
In fact, the exponential map gives us a chart around
p
locally, known as
geodesic local coordinates. To do so, it suffices to note the following rather trivial
proposition.
Proposition. We have
(d exp
p
)
0
= id
T
p
M
,
where we identify T
0
(T
p
M)
∼
=
T
p
M in the natural way.
All this is saying is if you go in the direction of
a ∈ T
p
M
, then you go in the
direction of a.
Proof.
(d exp
p
)
0
(v) =
d
dt
exp
p
(tv) =
d
dt
γ(1, tv) =
d
dt
γ(t, v) = v.
Corollary. exp
p
maps an open ball
B
(0
, δ
)
⊆ T
p
M
to
U ⊆ M
diffeomorphically
for some δ > 0.
Proof. By the inverse mapping theorem.
This tells us the inverse of the exponential map gives us a chart of
M
around
p. These coordinates are often known as geodesic local coordinates.
In these coordinates, the geodesics from p have the very simple form
γ(t, a) = ta
for all a ∈ T
p
M and t sufficiently small that this makes sense.
Corollary.
For any point
p ∈ M
, there exists a local coordinate chart around
p
such that
– The coordinates of p are (0, ··· , 0).
– In local coordinates, the metric at p is g
ij
(p) = δ
ij
.
– We have Γ
i
jk
(p) = 0 .
Coordinates satisfying these properties are known as normal coordinates.
Proof.
The geodesic local coordinates satisfies these property, after identifying
T
p
M
isometrically with (
R
n
, eucl
). For the last property, we note that the
geodesic equations are given by
¨x
i
+ Γ
i
jk
˙x
k
˙x
j
= 0.
But geodesics through the origin are given by straight lines. So we must have
Γ
i
jk
= 0.
Such coordinates will be useful later on for explicit calculations, since when-
ever we want to verify a coordinate-independent equation (which is essentially
all equations we care about), we can check it at each point, and then use normal
coordinates at that point to simplify calculations.
We again identify (T
p
N, g(p))
∼
=
(R
n
, eucl), and then we have a map
(r, v) ∈ (0, δ) ×S
n−1
7→ exp
p
(rv) ∈ M
n
.
This chart is known as geodesic polar coordinates. For each fixed
r
, the image of
this map is called a geodesic sphere of geodesic radius
r
, written Σ
r
. This is an
embedded submanifold of M.
Note that in geodesic local coordinates, the metric at 0
∈ T
p
N
is given by
the Euclidean metric. However, the metric at other points can be complicated.
Fortunately, Gauss’ lemma says it is not too complicated.
Theorem
(Gauss’ lemma)
.
The geodesic spheres are perpendicular to their
radii. More precisely,
γ
p
(
t, a
) meets every Σ
r
orthogonally, whenever this makes
sense. Thus we can write the metric in geodesic polars as
g = dr
2
+ h(r, v),
where for each r, we have
h(r, v) = g|
Σ
r
.
In matrix form, we have
g =
1 0 ··· 0
0
.
.
. h
0
The proof is not hard, but it involves a few subtle points.
Proof. We work in geodesic coordinates. It is clear that g(∂
r
, ∂
r
) = 1.
Consider an arbitrary vector field
X
=
X
(
v
) on
S
n−1
. This induces a vector
field on some neighbourhood B(0, δ) ⊆ T
p
M by
˜
X(rv) = X(v).
Pick a direction
v ∈ T
p
M
, and consider the unit speed geodesic
γ
in the direction
of v. We define
G(r) = g(
˜
X(rv), ˙γ(r)) = g(
˜
X, ˙γ(r)).
We begin by noticing that
∇
∂
r
˜
X − ∇
˜
X
∂
r
= [∂
r
,
˜
X] = 0.
Also, we have
d
dr
G(r) = g(∇
˙γ
˜
X, ˙γ) + g(
˜
X, ∇
˙γ
˙γ).
We know the second term vanishes, since
γ
is a geodesic. Noting that
˙γ
=
∂
∂r
,
we know the first term is equal to
g(∇
˜
X
∂
r
, ∂
r
) =
1
2
g(∇
˜
X
∂
r
, ∂
r
) + g(∂
r
, ∇
˜
X
∂
r
)
=
1
2
˜
X(g(∂
r
, ∂
r
)) = 0,
since we know that g(∂
r
, ∂
r
) = 1 constantly.
Thus, we know
G
(
r
) is constant. But
G
(0) = 0 since the metric at 0 is the
Euclidean metric. So G vanishes everywhere, and so ∂
r
is perpendicular to Σ
g
.
Corollary. Let a, w ∈ T
p
M. Then
g((d exp
p
)
a
a, (d exp
p
)
a
w) = g(a, w)
whenever a lives in the domain of the geodesic local neighbourhood.