Part III Riemannian Geometry
Based on lectures by A. G. Kovalev
Notes taken by Dexter Chua
Lent 2017
These notes are not endorsed by the lecturers, and I have modified them (often
significantly) after lectures. They are nowhere near accurate representations of what
was actually lectured, and in particular, all errors are almost surely mine.
This course is a possible natural sequel of the course Differential Geometry offered in
Michaelmas Term. We shall explore various techniques and results revealing intricate
and subtle relations between Riemannian metrics, curvature and topology. I hope to
cover much of the following:
A closer look at geodesics and curvature. Brief review from the Differential Geometry
course. Geo desic coordinates and Gauss’ lemma. Jacobi fields, completeness and
the Hopf–Rinow theorem. Variations of energy, Bonnet–Myers diameter theorem and
Synge’s theorem.
Hodge theory and Riemannian holonomy. The Hodge star and Laplace–Beltrami
op erator. The Hodge decomposition theorem (with the ‘geometry part’ of the proof).
Bo chner–Weitzenb¨ock formulae. Holonomy groups. Interplays with curvature and de
Rham cohomology.
Ricci curvature. Fundamental groups and Ricci curvature. The Cheeger–Gromoll
splitting theorem.
Pre-requisites
Manifolds, differential forms, vector fields. Basic concepts of Riemannian geometry
(curvature, geodesics etc.) and Lie groups. The course Differential Geometry offered in
Michaelmas Term is the ideal pre-requisite.
Contents
1 Basics of Riemannian manifolds
2 Riemann curvature
3 Geodesics
3.1 Definitions and basic properties
3.2 Jacobi fields
3.3 Further properties of geodesics
3.4 Completeness and the Hopf–Rinow theorem
3.5 Variations of arc length and energy
3.6 Applications
4 Hodge theory on Riemannian manifolds
4.1 Hodge star and operators
4.2 Hodge decomposition theorem
4.3 Divergence
4.4 Introduction to Bochner’s method
5 Riemannian holonomy groups
6 The Cheeger–Gromoll splitting theorem
1 Basics of Riemannian manifolds
Before we do anything, we lay out our conventions. Given a choice of local
coordinates {x
k
}, the coefficients X
k
for a vector field X are defined by
X =
X
k
X
k
x
k
.
In general, for a tensor field X T M
q
T
M
p
, we write
X =
X
X
k
1
...k
q
`
1
...`
p
x
k
1
···
x
k
q
dx
`
1
··· dx
`
p
,
and we often leave out the signs.
For the sake of sanity, we will often use implicit summation convention, i.e.
whenever we write something of the form
X
ijk
Y
i`jk
,
we mean
X
i,j
X
ijk
Y
i`jk
.
We will use upper indices to denote contravariant components, and lower
indices for covariant components, as we have done above. Thus, we always sum
an upper index with a lower index, as this corresponds to applying a covector to
a vector.
We will index the basis elements oppositely, e.g. we write d
x
k
x
k
for a basis element of
T
M
, so that the indices in expressions of the form
A
k
d
x
k
seem to match up. Whenever we do not follow this convention, we will write out
summations explicitly.
We will also adopt the shorthands
k
=
x
k
,
k
=
k
.
With these conventions out of the way, we begin with a very brief summary
of some topics in the Michaelmas Differential Geometry course, starting from
the definition of a Riemannian metric.
Definition
(Riemannian metric)
.
Let
M
be a smooth manifold. A Riemannian
metric
g
on
M
is an inner product on the tangent bundle
T M
varying smoothly
with the fibers. Formally, this is a global section of
T
M T
M
that is fiberwise
symmetric and positive definite.
The pair (M, g) is called a Riemannian manifold.
On every coordinate neighbourhood with coordinates
x
= (
x
1
, ··· , x
n
), we
can write
g =
n
X
i,j=1
g
ij
(x) dx
i
dx
j
,
and we can find the coefficients g
ij
by
g
ij
= g
x
i
,
x
j
and are C
functions.
Example.
The manifold
R
k
has a canonical metric given by the Euclidean
metric. In the usual coordinates, g is given by g
ij
= δ
ij
.
Does every manifold admit a metric? Recall
Theorem
(Whitney embedding theorem)
.
Every smooth manifold
M
an embedding into
R
k
for some
k
. In other words,
M
is diffeomorphic to a
submanifold of R
k
. In fact, we can pick k such that k 2 dim M.
Using such an embedding, we can induce a Riemannian metric on
M
by
restricting the inner product from Euclidean space, since we have inclusions
T
p
M T
p
R
k
=
R
k
.
More generally,
Lemma.
Let (
N, h
) be a Riemannian manifold, and
F
:
M N
is an immersion,
then the pullback g = F
h defines a metric on M.
The condition of immersion is required for the pullback to be non-degenerate.
In Differential Geometry, if we do not have metrics, then we tend to consider
diffeomorphic spaces as being the same. With metrics, the natural notion of
isomorphism is
Definition
(Isometry)
.
Let (
M, g
) and (
N, h
) be Riemannian manifolds. We
say
f
:
M N
is an isometry if it is a diffeomorphism and
f
h
=
g
. In other
words, for any p M and u, v T
p
M, we need
h
(df)
p
u, (df )
p
v
= g(u, v).
Example.
Let
G
be a Lie group. Then for any
x
, we have translation maps
L
x
, R
x
: G G given by
L
x
(y) = xy
R
x
(y) = yx
These maps are in fact diffeomorphisms of G.
G
admits a Riemannian metric, but we might want
to ask something stronger does there exist a left-invariant metric? In other
words, is there a metric such that each L
x
is an isometry?
Recall the following definition:
Definition
(Left-invariant vector field)
.
Let
G
be a Lie group, and
X
a vector
field. Then X is left invariant if for any x G, we have d(L
x
)X = X.
We had a rather general technique for producing left-invariant vector fields.
Given a Lie group
G
, we can define the Lie algebra
g
=
T
e
G
. Then we can
produce left-invariant vector fields by picking some X
e
g, and then setting
X
a
= d(L
a
)X
e
.
The resulting vector field is indeed smooth, as shown in the differential geometry
course.
Similarly, to construct a left-invariant metric, we can just pick a metric at
the identity and the propagating it around using left-translation. More explicitly,
given any inner product on h·, ·i on T
e
G, we can define g by
g(u, v) = h(dL
x
1
)
x
u, (dL
x
1
)
x
vi
for all
x G
and
u, v T
x
G
. The argument for smoothness is similar to that
for vector fields.
Of course, everything works when we replace “left” with “right”. A Rie-
mannian metric is said to be bi-invariant if it is both left- and right-invariant.
These are harder to find, but it is a fact that every compact Lie group admits a
bi-invariant metric. The basic idea of the proof is to start from a left-invariant
metric, then integrate the metric along right translations of all group elements.
Here compactness is necessary for the result to be finite.
We will later see that we cannot drop the compactness condition. There are
non-compact Lie groups that do not admit bi-invariant metrics, such as
SL
(2
, R
).
Recall that in order to differentiate vectors, or even tensors on a manifold,
we needed a connection on the tangent bundle. There is a natural choice for the
connection when we are given a Riemannian metric.
Definition
(Levi-Civita connection)
.
Let (
M, g
) be a Riemannian manifold.
The Levi-Civita connection is the unique connection
: Ω
0
M
(
T M
)
1
M
(
T M
)
on M satisfying
(i) Compatibility with metric:
Zg(X, Y ) = g(
Z
X, Y ) + g(X,
Z
Y ),
(ii) Symmetry/torsion-free:
X
Y
Y
X = [X, Y ].
Definition
(Christoffel symbols)
.
In local coordaintes, the Christoffel symbols
are defined by
j
x
k
= Γ
i
jk
x
i
.
With a bit more imagination on what the symbols mean, we can write the
first property as
d(g(X, Y )) = g(X, Y ) + g(X, Y ),
while the second property can be expressed in coordinate representation by
Γ
i
jk
= Γ
i
kj
.
The connection was defined on
T M
, but in fact, the connection allows us to
differentiate many more things, and not just tangent vectors.
Firstly, the connection
induces a unique covariant derivative on
T
M
, also
denoted , defined uniquely by the relation
Xhα, Y i = h∇
X
α, Y i + hα,
X
Y i
for any X, Y Vect(M) and α
1
(M).
To extend this to a connection
on tensor bundles
T
q,p
(
T M
)
q
(T
M)
p
for any p, q 0, we note the following general construction:
In general, suppose we have vector bundles
E
and
F
, and
s
1
Γ(
E
) and
s
2
Γ(
F
). If we have connections
E
and
F
on
E
and
F
respectively, then
we can define
EF
(s
1
s
2
) = (
E
s
1
) s
2
+ s
1
(
F
s
2
).
Since we already have a connection on
T M
and
T
M
, this allows us to extend
the connection to all tensor bundles.
Given this machinery, recall that the Riemannian metric is formally a section
g
Γ(
T
M T
M
). Then the compatibility with the metric can be written in
the following even more compact form:
g = 0.
2 Riemann curvature
With all those definitions out of the way, we now start by studying the notion
of curvature. The definition of the curvature tensor might not seem intuitive
at first, but motivation was somewhat given in the III Differential Geometry
course, and we will not repeat that.
Definition
(Curvature)
.
Let (
M, g
) be a Riemannian manifold with Levi-Civita
connection . The curvature 2-form is the section
R = −∇ Γ(
V
2
T
M T
M T M ) Γ(T
1,3
M).
This can be thought of as a 2-form with values in
T
M T M
=
End
(
T M
).
Given any X, Y Vect(M), we have
R(X, Y ) Γ(End T M ).
The following formula is a straightforward, and also crucial computation:
Proposition.
R(X, Y ) =
[X,Y ]
[
X
,
Y
].
In local coordinates, we can write
R =
R
i
j,k`
dx
k
dx
`
i,j=1,...,dim M
2
M
(End(T M )).
Then we have
R(X, Y )
i
j
= R
i
j,k`
X
k
Y
`
.
The comma between j and k is purely for artistic reasons.
It is often slightly convenient to consider a different form of the Riemann
curvature tensor. Instead of having a tensor of type (1, 3), we have one of type
(0, 4) by
R(X, Y, Z, T ) = g(R(X, Y )Z, T )
for X, Y, Z, T T
p
M. In local coordinates, we write this as
R
ij,k`
= g
iq
R
q
j,k`
.
The first thing we want to prove is that
R
ij,k`
enjoys some symmetries we might
not expect:
Proposition.
(i)
R
ij,k`
= R
ij,`k
= R
ji,k`
.
(ii) The first Bianchi identity:
R
i
j,k`
+ R
i
k,`j
+ R
i
`,jk
= 0.
(iii)
R
ij,k`
= R
k`,ij
.
Note that the first Bianchi identity can also be written for the (0
,
4) tensor as
R
ij,k`
+ R
ik,`j
+ R
i`,jk
= 0.
Proof.
(i)
The first equality is obvious as coefficients of a 2-form. For the second
equality, we begin with the compatibility of the connection with the metric:
g
ij
x
k
= g(
k
i
,
j
) + g(
i
,
k
j
).
We take a partial derivative, say with respect to x
`
, to obtain
2
g
ij
x
`
x
k
= g(
`
k
i
,
j
)+g(
k
i
,
`
j
)+g(
`
i
,
k
j
)+g(
i
,
`
k
j
).
Then we know
0 =
2
g
x
`
x
k
2
g
x
k
x
`
= g([
`
,
k
]
i
,
j
) + g(
i
, [
`
,
k
]
j
).
But we know
R(
k
,
`
) =
[
k
,∂
`
]
[
k
,
`
] = [
k
,
`
].
Writing R
k`
= R(
k
,
`
), we have
0 = g(R
k`
i
,
j
) + g(
i
, R
k`
j
) = R
ji,k`
+ R
ij,k`
.
So we are done.
(ii) Recall
R
i
j,k`
= (R
k`
j
)
i
= ([
`
,
k
]
j
)
i
.
So we have
R
i
j,k`
+ R
i
k,`j
+ R
i
`,jk
= [(
`
k
j
k
`
j
) + (
j
`
k
`
j
k
) + (
k
j
`
j
k
`
)]
i
.
We claim that
`
k
j
`
j
k
= 0.
Indeed, by definition, we have
(
k
j
)
q
= Γ
q
kj
= Γ
q
jk
= (
j
k
)
q
.
The other terms cancel similarly, and we get 0 as promised.
(iii) Consider the following octahedron:
R
ik,`j
= R
ki,j`
R
i`,jk
= R
`i,kj
R
j`,ki
= R
`j,ik
R
jk,i`
= R
kj,`i
R
ij,k`
= R
ji,`k
R
k`,ij
= R
`k,ji
The equalities on each vertex is given by (i). By the first Bianchi identity,
for each greyed triangle, the sum of the three vertices is zero.
Now looking at the upper half of the octahedron, adding the two greyed
triangles shows us the sum of the vertices in the horizontal square is
(
2)
R
ij,k`
. Looking at the bottom half, we find that the sum of the
vertices in the horizontal square is (2)R
k`,ij
. So we must have
R
ij,k`
= R
k`,ij
.
What exactly are the properties of the Levi-Civita connection that make
these equality works? The first equality of (i) did not require anything. The
second equality of (i) required the compatibility with the metric, and (ii) required
the symmetric property. The last one required both properties.
Note that we can express the last property as saying
R
ij,k`
is a symmetric
bilinear form on
V
2
T
p
M.
Sectional curvature
The full curvature tensor is rather scary. So it is convenient to obtain some
simpler quantities from it. Recall that if we had tangent vectors
X, Y
, then we
can form
|X Y | =
p
g(X, X)g(Y, Y ) g(X, Y )
2
,
which is the area of the parallelogram spanned by X and Y . We now define
K(X, Y ) =
R(X, Y, X, Y )
|X Y |
2
.
Note that this is invariant under (non-zero) scaling of
X
or
Y
, and is symmetric
in
X
and
Y
. Finally, it is also invariant under the transformation (
X, Y
)
7→
(X + λY, Y ).
But it is an easy linear algebra fact that these transformations generate all
isomorphism from a two-dimensional vector space to itself. So
K
(
X, Y
) depends
only on the 2-plane spanned by X, Y . So we have in fact defined a function on
the Grassmannian of 2-planes,
K
:
Gr
(2
, T
p
M
)
R
. This is called the sectional
curvature (of g).
It turns out the sectional curvature determines the Riemann curvature tensor
completely!
Lemma.
Let
V
be a real vector space of dimension
2. Suppose
R
0
, R
00
:
V
4
R
are both linear in each factor, and satisfies the symmetries we found
for the Riemann curvature tensor. We define
K
0
, K
00
:
Gr
(2
, V
)
R
as in the
sectional curvature. If K
0
= K
00
, then R
0
= R
00
.
This is really just linear algebra.
Proof. For any X, Y, Z V , we know
R
0
(X + Z, Y, X + Z, Y ) = R
00
(X + Z, Y, X + Z, Y ).
Using linearity of
R
0
and
R
00
, and cancelling equal terms on both sides, we find
R
0
(Z, Y, X, Y ) + R
0
(X, Y, Z, Y ) = R
00
(Z, Y, X, Y ) + R
00
(X, Y, Z, Y ).
Now using the symmetry property of R
0
and R
00
, this implies
R
0
(X, Y, Z, Y ) = R
00
(X, Y, Z, Y ).
Similarly, we replace Y with Y + T , and then we get
R
0
(X, Y, Z, T ) + R
0
(X, T, Z, Y ) = R
00
(X, Y, Z, Y ) + R
00
”(X, T, Z, Y ).
We then rearrange and use the symmetries to get
R
0
(X, Y, Z, T ) R
00
(X, Y, Z, T ) = R
0
(Y, Z, X, T ) R
00
(Y, Z, X, T ).
We notice this equation says
R
0
(
X, Y, Z, T
)
R
00
(
X, Y, Z, T
) is invariant under
the cyclic permutation
X Y Z X
. So by the first Bianchi identity, we
have
3(R
0
(X, Y, Z, T ) R
00
(X, Y, Z, T )) = 0.
So we must have R
0
= R
00
.
Corollary.
Let (
M, g
) be a manifold such that for all
p
, the function
K
p
:
Gr(2, T
p
M) R is a constant map. Let
R
0
p
(X, Y, Z, T ) = g
p
(X, Z)g
p
(Y, T ) g
p
(X, T )g
p
(Y, Z).
Then
R
p
= K
p
R
0
p
.
Here
K
p
is just a real number, since it is constant. Moreover,
K
p
is a smooth
function of p.
Equivalently, in local coordinates, if the metric at a point is
δ
ij
, then we have
R
ij,ij
= R
ij,ji
= K
p
,
and all other entries all zero.
Of course, the converse also holds.
Proof.
We apply the previous lemma as follows: we define
R
0
=
K
p
R
0
p
and
R
00
=
R
p
. It is a straightforward inspection to see that this
R
0
symmetry properties of
R
p
, and that they define the same sectional curvature.
So R
00
= R
0
. We know K
p
is smooth in p as both g and R are smooth.
We can further show that if
dim M >
2, then
K
p
is in fact independent of
p
under the hypothesis of this function, and the proof requires a second Bianchi
identity. This can be found on the first example sheet.
Other curvatures
There are other quantities we can extract out of the curvature, which will later
be useful.
Definition (Ricci curvature). The Ricci curvature of g at p M is
Ric
p
(X, Y ) = tr(v 7→ R
p
(X, v)Y ).
In terms of coordinates, we have
Ric
ij
= R
q
i,jq
= g
pq
R
pi,jq
,
where g
pq
denotes the inverse of g.
This
Ric
is a symmetric bilinear form on
T
p
M
. This can be determined by
Ric(X) =
1
n 1
Ric
p
(X, X).
The coefficient
1
n1
is just a convention.
There are still two indices we can contract, and we can define
Definition
(Scalar curvature)
.
The scalar curvature of
g
is the trace of
Ric
respect to g. Explicitly, this is defined by
s = g
ij
Ric
ij
= g
ij
R
q
i,jq
= R
qi
iq
.
Sometimes a convention is to define the scalar curvature as
s
n(n1)
In the case of a constant sectional curvature tensor, we have
Ric
p
= (n 1)K
p
g
p
,
and
s(p) = n(n 1)K
p
.
Low dimensions
If
n
= 2, i.e. we have surfaces, then the Riemannian metric
g
is also known as
the first fundamental form, and it is usually written as
g = E du
2
+ 2F du dv + G dv
2
.
Up to the symmetries, the only non-zero component of the curvature tensor is
R
12,12
, and using the definition of the scalar curvature, we find
R
12,12
=
1
2
s(EG F
2
).
Thus
s/
2 is also the sectional curvature (there can only be one plane in the
tangent space, so the sectional curvature is just a number). One can further
check that
s
2
= K =
LN M
2
EG F
2
,
the Gaussian curvature. Thus, the full curvature tensor is determined by the
Gaussian curvature. Also,
R
12,21
is the determinant of the second fundamental
form.
If n = 3, one can check that R(g) is determined by the Ricci curvature.
3 Geodesics
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
(
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.
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
n1
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
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
n1
. 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.
3.2 Jacobi fields
Fix a Riemannian manifold
M
. Let’s imagine that we have a “manifold” of all
smooth curves on
M
. Then this “manifold” has a “tangent space”. Morally,
given a curve
γ
, a “tangent vector” at
γ
in the space of curve should correspond
to providing a tangent vector (in M ) at each point along γ:
Since we are interested in the geodesics only, we consider the “submanifold” of
geodesics curves. What are the corresponding “tangent vectors” living in this
“submanifold”?
In rather more concrete terms, suppose
f
s
(
t
) =
f
(
t, s
) is a family of geodesics
in
M
indexed by
s
(
ε, ε
). What do we know about
f
s
s=0
, a vector field
along f
0
?
We begin by considering such families that fix the starting point
f
(0
, s
), and
then derive some properties of
f
s
in these special cases. We will then define a
Jacobi field to be any vector field along a curve that satisfies these properties.
We will then prove that these are exactly the variations of geodesics.
Suppose
f
(
t, s
) is a family of geodesics such that
f
(0
, s
) =
p
for all
s
. Then
in geodesics local coordinates, it must look like this:
For a fixed p, such a family is uniquely determined by a function
a(s) : (ε, ε) T
p
M
such that
f(t, s) = exp
p
(ta(s)).
The initial conditions of this variation can be given by a(0) = a and
˙a(0) = w T
a
(T
p
M)
=
T
p
M.
We would like to know the “variation field” of
γ
(
t
) =
f
(
t,
0) =
γ
p
(
t, a
) this
induces. In other words, we want to find
f
s
(
t,
0). This is not hard. It is just
given by
(d exp
p
)
ta
0
(tw) =
f
s
(t, 0),
As before, to prove something about
f
, we want to make good use of the
properties of
. Locally, we extend the vectors
f
s
and
f
t
to vector fields
t
and
s
. Then in this set up, we have
˙γ =
f
t
=
t
.
Note that in
f
t
, we are differentiating
f
with respect to
t
, whereas the
t
on
the far right is just a formal expressions.
By the geodesic equation, we have
0 =
dt
˙γ =
t
t
.
Therefore, using the definition of the curvature tensor R, we obtain
0 =
s
t
t
=
t
s
t
R(
s
,
t
)
t
=
t
s
t
+ R(
t
,
s
)
t
We let this act on the function f. So we get
0 =
dt
ds
f
t
+ R(
t
,
s
)
f
t
.
We write
J(t) =
f
s
(t, 0),
which is a vector field along the geodesic γ. Using the fact that
ds
f
t
=
dt
f
s
,
we find that J must satisfy the ordinary differential equation
2
dt
2
J + R( ˙γ, J) ˙γ = 0.
This is a linear second-order ordinary differential equation.
Definition
(Jacobi field)
.
Let
γ
: [0
, L
]
M
be a geodesic. A Jacobi field is a
vector field J along γ that is a solution of the Jacobi equation on [0, L]
2
dt
2
J + R( ˙γ, J) ˙γ = 0. ()
We now embark on a rather technical journey to prove results about Jacobi
fields. Observe that ˙γ(t) and t ˙γ(t) both satisfy this equation, rather trivially.
Theorem.
Let
γ
: [0
, L
]
N
be a geodesic in a Riemannian manifold (
M, g
).
Then
(i) For any u, v T
γ(0)
M, there is a unique Jacobi field J along Γ with
J(0) = u,
J
dt
(0) = v.
If
J(0) = 0,
J
dt
(0) = k ˙γ(0),
then
J
(
t
) =
kt ˙γ
(
t
). Moreover, if both
J
(0)
,
J
dt
(0) are orthogonal to
˙γ
(0),
then J(t) is perpendicular to ˙γ(t) for all [0, L].
In particular, the vector space of all Jacobi fields along
γ
have dimension
2n, where n = dim M .
The subspace of those Jacobi fields pointwise perpendicular to
˙γ
(
t
) has
dimensional 2(n 1).
(ii) J
(
t
) is independent of the parametrization of
˙γ
(
t
). Explicitly, if
˜γ
(
t
) =
˜γ(λt), then
˜
J with the same initial conditions as J is given by
˜
J(˜γ(t)) = J(γ(λt)).
This is the kind of theorem whose statement is longer than the proof.
Proof.
(i)
Pick an orthonormal basis
e
1
, ··· , e
n
of
T
p
M
, where
p
=
γ
(0). Then
parallel transports
{X
i
(
t
)
}
via the Levi-Civita connection preserves the
inner product.
We take e
1
to be parallel to ˙γ(0). By definition, we have
X
i
(0) = e
i
,
X
i
dt
= 0.
Now we can write
J =
n
X
i=1
y
i
X
i
.
Then taking g(X
i
, ·) of () , we find that
¨y
i
+
n
X
j=2
R( ˙γ, X
j
, ˙γ, X
i
)y
j
= 0.
Then the claims of the theorem follow from the standard existence and
uniqueness of solutions of differential equations.
In particular, for the orthogonality part, we know that
J
(0) and
J
dt
(0)
being perpendicular to
˙γ
is equivalent to
y
1
(0) =
˙y
1
(0) = 0, and then
Jacobi’s equation gives
¨y
1
(t) = 0.
(ii) This follows from uniqueness.
Our discussion of Jacobi fields so far has been rather theoretical. Now that
we have an explicit equation for the Jacobi field, we can actually produce some
of them. We will look at the case where we have constant sectional curvature.
Example.
Suppose the sectional curvature is constantly
K R
, for
dim M
3.
We wlog |˙γ| = 1. We let J along γ be a Jacobi field, normal to ˙γ.
Then for any vector field T along γ, we have
hR( ˙γ, J) ˙γ, T i = K(g( ˙γ, ˙γ)g(J, T ) g( ˙γ, J)g( ˙γ, T )) = Kg(J, T ).
Since this is true for all T , we know
R( ˙γ, J) ˙γ = KJ.
Then the Jacobi equation becomes
2
dt
2
J + KJ = 0.
So we can immediately write down a collection of solutions
J(t) =
sin(t
K)
K
X
i
(t) K > 0
tX
i
(t) K = 0
sinh(t
K)
K
X
i
(t) K < 0
.
for i = 2, ··· , n, and this has initial conditions
J(0) = 0,
J
dt
(0) = e
i
.
Note that these Jacobi fields vanishes at 0.
We can now deliver our promise, proving that Jacobi fields are precisely the
variations of geodesics.
Proposition.
Let
γ
: [
a, b
]
M
be a geodesic, and
f
(
t, s
) a variation of
γ(t) = f(t, 0) such that f(t, s) = γ
s
(t) is a geodesic for all |s| small. Then
J(t) =
f
s
is a Jacobi field along ˙γ.
Conversely, every Jacobi field along
γ
can be obtained this way for an
appropriate function f.
Proof.
The first part is just the exact computation as we had at the beginning of
the section, but for the benefit of the reader, we will reproduce the proof again.
2
J
dt
=
t
t
f
s
=
t
s
f
t
=
s
t
f
t
R(
t
,
s
) ˙γ
s
.
We notice that the first term vanishes, because
t
f
t
= 0 by definition of geodesic.
So we find
2
J
dt
= R( ˙γ, J) ˙γ,
which is the Jacobi equation.
The converse requires a bit more work. We will write
J
0
(0) for the covariant
derivative of
J
along
γ
. Given a Jacobi field
J
along a geodesic
γ
(
t
) for
t
[0
, L
],
we let ˜γ be another geodesic such that
˜γ(0) = γ(0),
˙
˜γ(0) = J(0).
We take parallel vector fields X
0
, X
1
along ˜γ such that
X
0
(0) = ˙γ(0), X
1
(0) = J
0
(0).
We put X(s) = X
0
(s) + sX
1
(s). We put
f(t, s) = exp
˜γ(s)
(tX(s)).
In local coordinates, for each fixed s, we find
f(t, s) = ˜γ(s) + tX(s) + O(t
2
)
as t 0. Then we define
γ
s
(t) = f(t, s)
whenever this makes sense. This depends smoothly on
s
, and the previous
arguments say we get a Jacobi field
ˆ
J(t) =
f
s
(t, 0)
We now want to check that
ˆ
J
=
J
. Then we are done. To do so, we have to
check the initial conditions. We have
ˆ
J(0) =
f
s
(0, 0) =
d˜γ
ds
(0) = J(0),
and also
ˆ
J
0
(0) =
dt
f
s
(0, 0) =
ds
f
t
(0, 0) =
X
ds
(0) = X
1
(0) = J
0
(0).
So we have
ˆ
J = J.
Corollary. Every Jacobi field J along a geodesic γ with J(0) = 0 is given by
J(t) = (d exp
p
)
t ˙γ(0)
(tJ
0
(0))
for all t [0, L].
This is just a reiteration of the fact that if we pull back to the geodesic local
coordinates, then the variation must look like this:
But this corollary is stronger, in the sense that it holds even if we get out of the
geodesic local coordinates (i.e. when exp
p
no longer gives a chart).
Proof.
Write
˙γ
(0) =
a
, and
J
0
(0) =
w
. By above, we can construct the variation
by
f(t, s) = exp
p
(t(a + sw)).
Then
(d exp
p
)
t(a+sw)
(tw) =
f
s
(t, s),
which is just an application of the chain rule. Putting
s
= 0 gives the result.
It can be shown that in the situation of the corollary, if
a w
, and
|a|
=
|w| = 1, then
|J(t)| = t
1
3!
K(σ)t
3
+ o(t
3
)
as t 0, where σ is the plane spanned by a and w.
3.3 Further properties of geodesics
We can now use Jacobi fields to prove interesting things. We now revisit the
Gauss lemma, and deduce a stronger version.
Lemma (Gauss’ lemma). Let a, w T
p
M, and
γ = γ
p
(t, a) = exp
p
(ta)
a geodesic. Then
g
γ(t)
((d exp
p
)
ta
a, (d exp
p
)
ta
w) = g
γ(0)
(a, w).
In particular,
γ
is orthogonal to
exp
p
{v T
p
M
:
|v|
=
r}
. Note that the latter
need not be a submanifold.
This is an improvement of the previous version, which required us to live in
the geodesic local coordinates.
Proof. We fix any r > 0, and consider the Jacobi field J satisfying
J(0) = 0, J
0
(0) =
w
r
.
Then by the corollary, we know the Jacobi field is
J(t) = (d exp
p
)
ta
tw
r
.
We may write
w
r
= λa + u,
with
a u
. Then since Jacobi fields depend linearly on initial conditions, we
write
J(t) = λt ˙γ(t) + J
n
(t)
for a Jacobi field J
n
a normal vector field along γ. So we have
g(J(r), ˙γ(r)) = λr|˙γ(r)|
2
= g(w, a).
But we also have
g(w, a) = g(λar + u, a) = λr|a|
2
= λr|˙γ(0)|
2
= λr|˙γ(r)|
2
.
Now we use the fact that
J(r) = (d exp
p
)
ra
w
and
˙γ(r) = (d exp
p
)
ra
a,
and we are done.
Corollary
(Local minimizing of length)
.
Let
a T
p
M
. We define
ϕ
(
t
) =
ta
,
and ψ(t) a piecewise C
1
curve in T
p
M for t [0, 1] such that
ψ(0) = 0, ψ(1) = a.
Then
length(exp
p
ψ) length(exp
p
ϕ) = |a|.
It is important to interpret this corollary precisely. It only applies to curves
with the same end point in
T
p
M
. If we have two curves in
T
p
M
whose end
points have the same image in
M
, then the result need not hold (the torus would
be a counterexample).
Proof.
We may of course assume that
ψ
never hits 0 again after
t
= 0. We write
ψ(t) = ρ(t)u(t),
where ρ(t) 0 and |u(t)| = 1. Then
ψ
0
= ρ
0
u + ρu
0
.
Then using the extended Gauss lemma, and the general fact that if
u
(
t
) is a unit
vector for all t, then u · u
0
=
1
2
(u · u)
0
= 0, we have
d
dx
(exp
p
ψ)(t)
2
=
(d exp
p
)
ψ(t)
ψ
0
(t)
2
= ρ
0
(t)
2
+ 2g(ρ
0
(t)u(t), ρ(t)u
0
(t)) + ρ(t)
2
|(d exp
p
)
ψ(t)
u
0
(t)|
2
= ρ
0
(t)
2
+ ρ(t)
2
|(d exp
p
)
ψ(t)
u
0
(t)|
2
,
Thus we have
length(exp
p
ψ)
Z
1
0
ρ
0
(t) dt = ρ(1) ρ(0) = |a|.
Notation. We write Ω(p, q) for the set of all piecewise C
1
curves from p to q.
We now wish to define a metric on M, in the sense of metric spaces.
Definition
(Distance)
.
Suppose
M
is connected, which is the same as