Differential Geometry
Contents
Chapter 1 - Review of basic notions on smooth manifolds ............................................................ 3
1.1 First Principles ............................................................................................................................ 3
1.2 Tangent Space and Vector Fields ............................................................................................. 4
1.3 Cotangent Space, Vector Bundles and Tensor Fields............................................................ 6
Review of Tensor Product of Vector Spaces............................................................................. 7
Construction of a local frame using a local trivialisation ....................................................... 9
Vector Bundle Construction Lemma (Proof non-examinable) .............................................. 9
1.4 Bundle Maps and isomorphisms ............................................................................................. 9
Construction of new vector bundles from old ....................................................................... 10
Tensor Fields and Differential Forms ...................................................................................... 11
Chapter 2 – Riemannian Manifolds ................................................................................................. 14
2.1 Definition of Riemannian Manifolds..................................................................................... 14
2.2 Expression of the metric in local coordinates....................................................................... 14
2.3 Length and Angle between tangent vectors......................................................................... 15
2.4 Examples of Riemannian metrics........................................................................................... 15
2.5 Existence of a Riemannian Metric.......................................................................................... 16
2.6 Conformal Maps and Isometries............................................................................................ 17
2.7 Extra advantages of having a Riemannian Metric .............................................................. 17
Riemannian Volume Forms ...................................................................................................... 17
Integral of Functions .................................................................................................................. 18
Musical Isomorphisms and gradient of a function ............................................................... 18
Extending the inner product to tensors and tensor fields of
-type ............................ 18
2.8 Bundle Metrics.......................................................................................................................... 19
2.9 Hodge Star Operator and Laplace Operator ........................................................................ 19
Chapter 3: Affine connection and the Levi-Civita Connection ................................................... 21
3.1 Affine Connections .................................................................................................................. 21
3.2 Extension of connections to tensor fields.............................................................................. 22
3.3 Connection along curves and parallel transport ................................................................. 23
3.4 The Levi-Civita Connection .................................................................................................... 24
Chapter 4: Geodesics and Riemannian Distance ........................................................................... 26
4.1 Geodesics ................................................................................................................................... 26
4.2 Exponential Map ...................................................................................................................... 29
4.3 Normal Coordinates ................................................................................................................ 30
4.4 Riemannian Distance ............................................................................................................... 30
4.5 Minimising Properties of Geodesics...................................................................................... 31
Minimising Curves are Geodesics ........................................................................................... 32
Geodesics are Locally Minimising ........................................................................................... 34
Chapter 5: Curvature ......................................................................................................................... 35
5.1 The Riemannian Curvature Tensor ....................................................................................... 35
Local Expressions ....................................................................................................................... 35
Extending
to general tensor fields ....................................................................................... 36
5.2 Sectional Curvature ................................................................................................................. 37
Model Spaces with Constant Curvature: ................................................................................ 38
5.3 Ricci and Scalar Curvature ..................................................................................................... 38
Chapter 6: Jacobi Fields ..................................................................................................................... 40
6.1 Basic Facts about Jacobi Fields ............................................................................................... 40
6.2 Jacobi Fields and Curvature ................................................................................................... 41
6.3 Second Variation Formula ...................................................................................................... 42
Second Variation formula for the energy ............................................................................... 43
Chapter 7: Classical Theorems in Riemannian Geometry ............................................................ 44
7.1 Hopf-Rinow Theorem ............................................................................................................. 44
7.2 Catan-Hadward Theorem ....................................................................................................... 44
7.3 Bonnet-Myers Theorem ........................................................................................................... 45
Chapter 1 - Review of basic notions on smooth manifolds
1.1 First Principles
Hausdorff
A topological space is called Hausdorff if for all
that
and
and
.
Second Countable
A topological space
of .
there exist open sets
such
is second countable if there exists a countable basis for the topology
Locally Euclidean of dimension
A topological space is locally Euclidean of dimension if for every point
, there
exists an open set
and an open set
so that and are homeomorphic.
Definition 1.1
A topological manifold of dimension is a topological space that is Hausdorff, second
countable and locally Euclidean of dimension .
Remark
1. Unless otherwise specified, all manifolds are assumed to be connected.
2. Some people assume “paracompactness” which is used to prove the existence of a
partition of unity.
Definition 1.2
A smooth atlas
of a topological space
a. An open covering { }
}
b. A family {
So that if
is given by
where
open and
⋃
of homeomorphisms onto open subsets
then the map
(
)
Definition 1.3
If
then the diffeomorphism
map.
(
(
)
) is a diffeomorphism.
(
) is known as the transition
Definition 1.4
A smooth structure on a Hausdorff topological space is an equivalence class of atlases, with
two atlases and being equivalent if, for
and (
with
)
then the transition map
open sets of
(
)
(
) is a diffeomorphism (as a map between
).
It is easy to check that this is an equivalence relation, since the inverse of a diffeomorphism
and composition of two diffeomorphisms are diffeomorphisms.
Definition 1.5
A smooth manifold
a smooth structure.
of dimension
is a topological manifold of dimension
together with
Definition 1.6
Let
be two manifolds of dimension
respectively. A map
is called smooth
at
if there exist charts
with
and
with
and the composition
is smooth (as a map between open sets of
). is called smooth if it is smooth at every
.
In particular:


A smooth map
A smooth map
.
is a smooth function.
from an open interval
is called a smooth curve in
Definition 1.7
A map
is called a diffeomorphism if it is smooth, bijective and its inverse
is also smooth.
Definition 1.8
A map is called an immersion (respectively submersion) if is injective (respectively
surjective) and its differential is injective (respectively surjective).
Definition 1.9
A map is called an embedding if
Definition 1.10
If
is an immersion then
Definition 1.11
If
is an embedding, then
submanifold) of ; equivalently
is an immersion and homeomorphic onto its image.
is an immersed submanifold of .
is an embedded submanifold (or a regular
is a submanifold of using the subspace topology.
1.2 Tangent Space and Vector Fields
Notation

Let

{
Given a point
{
{
} and let
}
, denote
}
Definition 1.12
a. The tangent vector to the curve
given by the formula:
at
|
is the map
(
)
(
)
b. A tangent vector at
with
. That is,
Remark
A tangent vector at
Leibniz property:
is the tangent vector at
of some curve
is known as a linear functional defined on
Definition 1.13
The tangent space at
, deonted by
which satisfies the
is the set of all tangent vectors at .
Remark
1.
admits a structure of a real vector space.
2. Given a chart
with
and coordinates
|
| form a basis of
the coordinate vectors
. Note that there is no standard choice of basis as
the coordinates are dependent upon which chart you are using.
In other words, any tangent vector
Proposition (Change of variables formula)
Given two coordinate systems
and
∑
∑
Then
Proof
Pick
{
for all
{
} and apply
∑
|
and a vector field
then if
can be written as
|
∑
|
}.
to the function
. Then
(
)
∑
| (
)
(
)
∑
| (
)
∑
∑
Hence
Einstein Summation convention
The vector
∑
| in Einstein summation convention is written instead as
| .
The Differential
Given
and
and
, choose a curve
with
and
(this is possible due to the theorem about existence of solutions of linear first
order ODEs). Then consider the map
mapping
.
This is a linear map between two vector spaces and it is independent of the choice of .
Definition 1.14
The linear map
defined above is called the derivative or differential of
image
is called the push forward of at
.
Remark
Sometimes
is denoted as
at , while the
.
Definition 1.15
The tangent bundle of a smooth dimensional manifold
of the tangent spaces at
. That is:
, denoted
is the disjoint union
∐
The elements of
will be represented by either
Remark:
admits a smooth structure which makes
or the pair
.
itself a smooth manifold of dimension
Definition 1.16
Given a smooth manifold , a vector field is a map
and is called smooth if it is smooth as a map from to
mapping
.
,
.
Notation
{
for
is an -vector space: for
,
define
,
Definition 1.17
For
, the Lie bracket [
Observe that for
Proof
Take
}
and
mapping
]
and
.
, then the Lie bracket satisfies the Jacobi identity
] ] [[
] ] [[
] ]
[[
then
[[
] ]
[[
]
[
[[
]
]
] ]
[
]
[
]
1.3 Cotangent Space, Vector Bundles and Tensor Fields
Definition 1.18
Let be a smooth -manifold, and
. We define cotangent space at , denoted by
to be the dual space of the tangent space at :
(
)
{
Elements of
are called cotangent vectors or tangent convectors at .
Remark:
}.
→
is called
and referred to
, so it is a cotangent vector at .
1. For
smooth, the composition
as the differential of . Note that
2. For a chart (
) of
the dual basis of {
Definition 1.19
The disjoint union
, then {
and
| }
}
. In fact {
is a basis of
} is
.
∐
is called the cotangent bundle of
Remark
is a smooth manifold of dimension
.
Review of Tensor Product of Vector Spaces
Definition 1.20
The tensor product of finitely many vector spaces
-
{
Definition 1.21
The elements in the tensor product
.
is defined on the set
}.
⏟
⏟
are called
-tensors
or -contravariant, -covariant tensors.
Remark
is a vector space.
Definition 1.22
̃
Given
̃)
(
̃
̃
then the tensor product
̃
̃
̃ is an element of
̃
̃
̃
given by
̃
̃
Remark
The Tensor Product is bilinear and associative however it is in general not commutative; that
is
in general.
Definition 1.23
is called reducible if it can be written in the form
,
for
.
Definition 1.24
Choose two indices
where
for
. For any reducible tensor
̂
̂
. Let
Then
. We extend this linearly to get a linear map
tensor-contraction.
Example
Let be a vector space with basis { } and dual basis {
then
∑
(
)
∑
∑
}. Let
.
.
which is called
∑
Remark
From the example above we deduce that
Definition 1.25
An antisymmetric (or alternating
of all -forms on is denoted
is the trace operation
-tensor)
is called a -form on , and the space
{
}.
Definition 1.26
A (smooth real) vector bundle of rank , denoted
is a smooth manifold of
dimension
(the total space) a smooth manifold of dimension (the base manifold)
and a smooth surjective map
(projection map) with the following properties:
1. There exists an open cover { }
2. For any point
,
(
of
)
diagram (in this case
3. Whenever
takes the form
is smooth. For each
(
)
The pairs
and diffeomorphisms
{ }
⏟
and we get a commutative
is projection onto the first component).
(
the diffeomorphism
(
, the set
),
)
(
)
where
is called the fibre over . Observe that
{ }
are called local trivialisations. The maps
are called
transition maps.
Examples:
1. The trivial bundle (
) where is projection onto the first component.
2. The tangent bundle
3. The infinite Möbius band:
Take an infinite strip, and identify the two edges with each other, giving one a twist.
The infinite Möbius band is a non-orientable manifold and a non-trivial vector
bundle over
with fibre .
Definition 1.27
A smooth section of a vector bundle
is a map
so that
for all
. is called a smooth section if it is smooth as a map from
{
}
Denote
Example
Take
and then
that is
to .
Suppose a rank vector bundle admits sections which are everywhere linearly
independent, then we call such a collection of sections a frame (or global frame or frame
field). When these sections are only locally defined, we call the collection a local frame.
Remark
A Global frame does not always exist, but a local frame always exists (as a manifold is
locally Euclidean).
Construction of a local frame using a local trivialisation
} be the standard basis of
Let {
when within an open set
given by
for
and
.
, a local frame is
Examples
Let
Let
and let (
) be a chart. A local frame is given by {
and let (
}|
) be a chart. A local frame is given by {
}|
.
Vector Bundle Construction Lemma (Proof non-examinable)
Lemma 1.28
Let be an -dimensional smooth manifold and { }
an open cover of . Suppose for
each
that there exists a smooth map
satisfying the following
condition:
Then there exists a smooth real vector bundle
of rank with local trivialisations
. The transition maps are given by
; the change of basis matrix
between the coordinate system induced by and the coordinate system induced from .
Idea of Proof
Consider (∐
)
, where
(
that (∐
)
)
is defined as follows: For
for all transition matrices
define
,
then
. Then we prove
is a vector bundle. The Details are in [G-H-L].
Example
is a vector bundle. We use the Lemma.
∐
Define
and let
. Let
coordinate system of
and
and (
) be two charts for
which is the transition map in
and
. For
between the
. Being a linear map this is smooth and clearly for all
(
)
exists a smooth real vector bundle
which is simply
. Hence by the lemma there
of rank
with local trivialisation
1.4 Bundle Maps and isomorphisms
Definition 1.29
Suppose
and ( ̃ ̃ ̃) are two vector bundles. A smooth map
smooth bundle map from
to ( ̃ ̃ ̃) if
̃ is called a
̃ such that the following diagram commutes
1. there exists a smooth map
That is, ̃(
2.
induces a linear map from
)
to ̃
(
) for all
for any
Definition 1.30
Two vector bundles
and ( ̃ ̃ ̃) are called smoothly isomorphic if there exists a
smooth bundle map between them which is a diffeomorphism.
Remarks
1. If two vector bundles are isomorphic they are viewed as “the same”
2. A vector bundle which is isomorphic to the trivial bundle (
) is called trivial.
Claim
A vector bundle is trivial if and only if it admits a global frame.
Example
is a trivial vector bundle. By joining the two ends there are two possibilities either
the result is
if there is no twisting, while if there is twisting we get the infinite Mobius
band.
Construction of new vector bundles from old
Dual Bundle
∐
Take a vector bundle
where
∐
. Replace
with its dual
and consider
Let
by as in Definition 1.26. Then the transition maps for the dial bundle
denoted (
)
((
) )
(
(
)
) . Observe that (
(
Example
Take
the tangent bundle over
and thus
.
(
)
(
) (
)
) . Then we apply Lemma 1.28.
. The cotangent bundle
Tensor Product of Vector Bundles
Suppose
is a vector bundle of rank
the same base manifold
)
are
then define
and ( ̃
̃
∐
is the dual bundle of
̃) is a vector bundle of rank over
̃
This is well defined because
and ̃ are vector spaces. Let be an open cover of
̃
̃ be the local trivialisations and transition maps to and ̃ respectively.
Then the transition maps and local trivialisations for
̃ are given by
̃ ̃
̃
̃
Subbundle
A vector bundle ( ̃ ̃ ̃) is called a subbundle of a vector bundle
, the fibre ̃ is a linear subspace of .
Example
The space of alternating -tensors on ,
∐
is a sub bundle of ⨂
⋀
⋀
if for each point
⏟
Pullback Bundle
Let
be a smooth map and
be a vector bundle over . Then the pullback
∐
bundle
is a vector bundle over .
In conclusion, if




⏟
⋀
is a vector bundle then the following are all vector bundles:
⏟
(which is a subbundle of ⨂
for a smooth map
Tensor Fields and Differential Forms
Definition 1.31
Define the
-tensor bundle (or tensor bundle of
vector bundle
⏟
⏟
An
-type) of a manifold
to be the
-tensor field (or an -contravariant, covariant tensor field) is a section of
Denote
{
} we adopt the convention that
, the space of all smooth functions over .
Using local coordinates ( ) we can write an
Here
-tensor field
is a local frame for
Note that Tensor Fields are
as
.
-linear
Definition 1.32
Let
be a smooth map between two smooth manifolds and
covariant tensor field. We define a -covariant tensor field
over
(⏟
In this case,
is called the pullback of
by .
)
(
by
) be a
.
Proposition 1.33 (Properties of Pullback)
Suppose
is a smooth map and
a smooth map for
(
. Then
manifolds and
a.
b.
c.
d. If
)
(
) and
smooth
. In particular,
and ( ) are local coordinates in a chart containing the point
(
)
(
) (
)
then
(
)
Proof
The proofs are all exercises.
Definition 1.34
A differential -form on a smooth manifold
(which is a subbundle of ⨂
⋀
⋀
is a smooth section of the vector bundle
)
Notation
We denote the set of all differential -forms on
convention that
. Note that a
Locally we write a differential -form as
by
(⋀ ). We adopt the
tensor field is a 1-form.
. In this case
is a local frame for ⋀
Exterior derivative
The exterior derivative is a map
and if is a
function and a
vector field on
which is -linear such that
then
.
Integration of differential forms
is well defined only if is orientable,
∫
has compact support and is a differential -form on
and has a partition of unity and
.
Theorem (Partition of unity) (Proof non-examinable)
Let be a smooth manifold which is countable at infinity (that is a countable union of
compact sets). Then there exists a partition of unity for . The partition of unity statement is
].
in [
Remarks
1. All the “natural” manifolds we are considering will be countable at infinity. In
particular, all compact manifolds are countable at infinity and
⋃ {
}
| |
2. In some textbooks, they assume manifolds are paracompact in order to get a partition
of unity.
Remarks about tensor fields
1. Tensor contractions
for tensors extend to tensor fields.
2. One can consider symmetric and antisymmetric tensor fields (smooth anti-symmetric
3. If
the set of
Let
Then
where
̃
̃
⏟
)
-tensor fields then write
̃
̃
̃
for coordinate systems { } {̃}
̃
⏟
Chapter 2 – Riemannian Manifolds
2.1 Definition of Riemannian Manifolds
Definition 2.1
An inner product (or scalar product) on a vector space
is:
⟩ ⟨
⟩ for all
1. Symmetric: ⟨
⟩
⟨
⟩
⟨
2. Bilinear: ⟨
and
⟩
3. Positive-definite: ⟨
for all
is a function ⟨
⟩
⟩
⟩
⟩ and ⟨
⟨
that
⟨
⟩ for all
.
Definition 2.2
A Riemannian metric on a smooth manifold is a map that assigns every point
an
⟨ ⟩ on
inner product
, which depends smoothly on ; that is for all
⟨
⟩ is smooth.
the function
mapping
Equivalently, a Riemannian metric is a symmetric, positive definite
-tensor field on .
Definition 2.3
A pair
of a manifold
manifold.
equipped with a Riemannian metric
is called a Riemannian
Remark
If the Positive Definite condition in Definition 2.1 is replaced with
4. Non-degenerate: ⟨
Then
⟩
for all
implies that
is called a semi or pseudo-Riemannian metric.
2.2 Expression of the metric in local coordinates
and a chart (
Given a Riemannian manifold
⟨
mapping
,(
For each

Symmetric:

Positive Definite:
(
Definition 2.4
These functions
is an
(
matrix that is:
)
for any
) is an invertible matrix)
are called the local representations of the Riemannian metric
respect to the coordinates (
Recall that
| ⟩
|
)
) consider the functions
).
so using the local frame {
} for
we write
with
Remark
Using the symmetric part of two 1 forms,
we can also write
2.3 Length and Angle between tangent vectors
Definition 2.5
Suppose
is a Riemannian manifold and
to be | |
tangent vector
between
(
Two vectors
and
√⟨
⟩ (Recall
⟨
) by
Definition 2.6
Given a smooth curve
formula
[
]
( |
⟩) and the angle
.
or equivalently if ⟨
for all and ⟨
⟩
, the length of between
∫ |
Where
⟨
⟩
| || |
are orthogonal if
are called orthonormal if | |
. We define the length (or norm) of a
⟩
for all
and
. Vectors
.
is defined by the
|
)
2.4 Examples of Riemannian metrics
1. Euclidean Metric (canonical metric)
on
2. Induced Metric
Let
be a Riemannian manifold and
smooth manifold (that is is a smooth map and
metric on is defined
an immersion where is a
is injective). Then the induced
(
)
to be an immersion in order to get the positive definite
Note that we require
condition on
.
3. Induced Metric
on
| from the Euclidean space
The induced metric on
sometimes denoted
by the inclusion
is called the standard (or round) metric on
(clearly is an immersion).
Consider stereographic projection
and denote the inverse map
. Then
Gives the Riemannian metric for .
4. Product metric
If
are two Riemannian manifolds then the product
admits a
Riemannian metric
called the product metric defined by
Where
for
. We use the fact that
5. Warped Product
Suppose
are two Riemannian manifolds. Then
is the warped product of
(or denoted
) where
smooth positive function.
is a
Example
{ } can be written as
In polar coordinates,
the Euclidean metric on
. It can be considered the warped product of
where
.
6. Conformal Metric
Let
be a Riemannian manifold and
be a smooth positive function.
Then
is also a Riemannian manifold. This change of metric preserves the
angle between two vectors
(Check)
is said to be conformal to
. We can define an equivalence relation
if is conformal to . Denote the equivalence classes by [ ]. In particular when is
constant the two Riemannian metrics are said to be homothetic.
Remarks:
is a Riemannian Manifold and { } a local frame for
and { } is its
⟨
⟩. If { } is an orthonormal basis
dual frame then
where
1. Suppose
then { } is the dual frame implies that
∑
.
2.
2.5 Existence of a Riemannian Metric
Theorem 2.7
On any smooth manifold
there exists at least one Riemannian Metric.
Proof
Let
be an atlas of such that { } is locally finite. For each
we define on
the Riemannian metric
(recall that
by definition is a
diffeomorphism onto its image hence an immersion.)
I.e.
(
) for all
, for all
.
Let { } with
be a partition of unity with respect to the open cover { } . Then
∑
define
is a Riemannian metric on . Note that the sum is well defined because
{ } is locally finite.
The three requirements for a Riemannian metric are easy to check:
Note that is positive definite: for all
,
there exists
so that
Then
and
because is positive definite so
.
.
2.6 Conformal Maps and Isometries
Definition
A smooth map
map with conformal factor
between two Riemannian manifolds is called a conformal
if
.
Note
A conformal map preserves angles; that is
.
(
) for all
and
Example:
. We consider stereographic projection.
. As stereographic projection is
a diffeomorphism its inverse
is a conformal map. It follows from an exercise
sheet that
is a conformal map with conformal factor
.
Definition 2.9
A Riemannian Manifold
is locally flat if for every point
diffeomorphism
between an open neighbourhoods of
Definition 2.10
Given two Riemannian Manifolds
diffeomorphism
such that
Remark
In particular an isometry
a Riemannian manifold form a group.
Example:
{ }
there exists a conformal
and
of
.
and
, they are called isometric if there is a
. Such a diffeomorphism is called an isometry.
is called an isometry of
. All Isometries on
is isometric to the warped product
where
.
Definition 2.11
are called locally isometric if for every point
there is an isometry
from an open neighbourhood of in and an open neighbourhood of
.
Definition 2.12
A Riemannian metric
Definition 2.13
Suppose
on
is called a flat metric if
is an immersion. Then
is locally isometric to
is isometric if
.
2.7 Extra advantages of having a Riemannian Metric
Riemannian Volume Forms
Lemma 2.14
On any Oriented Riemannian -manifold
there is a unique -form
where
is an orthonormal basis for some
,
satisfying
.
in
.
is called the Riemannian Volume Form. In local coordinates ( )
This unique -form
then
√
Recall that
(
)
. See Sheet 3 for details.
Integral of Functions
Definition 2.15
Let
be an oriented Riemannian -manifold with its Riemannian volume form
is a compactly supported smooth function on then
is a new -form which is
compactly supported. We can define the integral of over as
∫
. If
∫
Musical Isomorphisms and gradient of a function
Let
be a Riemannian manifold. Then
. The isomorphisms between
are called the musical isomorphisms.
⟨
⟩
⟨
⟩ is an
For
, consider the map
. The map
isomorphism for Riemannian manifolds called “flat”
mapping
(
) ⟨
and its inverse
Together
⟩
called “sharp”
.
are known as the musical isomorphisms.
They extend to vector fields and -forms to give a linear map
for the local expressions of and see below:
In local coordinates ( ) write
Write
so
to denote the
then
and
.
th component of the inverse matrix (
) . Then write
.
Let
. The gradient of
is defined as
For the Properties of the gradient of a function, see exercise sheet 3.
Take
. Then
(
)
.
Remark
The musical isomorphisms can be defined for a vector space
product ⟨ ⟩. Then (flat)
and (sharp)
Extending the inner product to tensors and tensor fields of
Suppose is a vector space with an inner product ⟨ ⟩.
1. Define the inner product of two
tensors
equipped with an inner
-type
as ⟨
⟩
⟨⏟ ⏟⟩
and
2. We extend the inner product to reducible
⟨
⟨
⟩
⟨
3. We extend the inner product to general
Let
⟨
be a Riemannian Manifold and
-tensors
⟩
⟩
⟨
⟩
⟨
tensors by linearity.
⟨ ⟩. Given
⟩
then ⟨
⟩
⟩
Let
be a vector bundle and
∐
2.8 Bundle Metrics
Recall from linear algebra:
On a vector space , a bilinear form
can be considered as an element
. Given a vector bundle
a bundle metric is a map that assigns each fibre
inner product ⟨ ⟩ which depends smoothly on
.
Definition 2.16
A bundle metric on the vector bundle
symmetric and positive definite.
Notation
Given
is an element of
or ⟨
we write their inner product as
an
which is
⟩
Example:
Any vector bundle admits a bundle metric (use partition of unity).
Remark
Given a Vector bundle
with a bundle metric we can define an isomorphism
. W can extend to any
tensor products of and .
2.9 Hodge Star Operator and Laplace Operator
Recall Linear Algebra:
Let be a vector space with an inner product ⟨
exterior product ⋀
{
{
⟨
. Consider the -fold
} these are the -vectors while ⋀
} is the set of -forms. We have an inner product on ⋀
⟩
⟩) and extend it linearly to
(⟨
⋀ . If
orthonormal basis for , then {
of ⋀
⟩ and
are an
} is an orthonormal basis
.
Definition
} and { ̃
Two ordered bases {
̃} are said to have the same orientation if
is a positive multiple of ̃
̃ and have negative orientation if the multiple
is negative.
is said to be oriented if an ordered basis is distinguished as positive.
Suppose
is oriented. We define the Hodge Star Operator ⋀
for
⋀
} is a positively oriented basis of .
where {
by
Exercise
This operation is independent of basis but dependent on the orientation.
Let
be a Riemannian Manifold which is orientable and has dimension . Since is
oriented, select an orientation for all tangent spaces
. Also on all cotangent spaces
.
induces an inner product on each
, thus we obtain the Hodge Star Operator
Which extends to a bundle map
⋀(
)
⋀(
(⋀
)
(⋀
)
), i.e.
.
Remarks



where
For
set
for
Laplace Operator
Define
harmonic if
define the codifferential
. Sometimes books write
.
by
.
. Let
by mapping
instead of
. It is convention so
. Then
is called
Chapter 3: Affine connection and the Levi-Civita Connection
3.1 Affine Connections
Motivation
An affine connection is a kind of directional derivative of vector fields on a manifold.
Imagine a vector field on
(that is a map
). Take a point
and choose a
tangent vector
.
Denote by
the directional derivative of
at
in the direction . Write
. Then
We shall generalise to the case that
and
.
We could define
in local coordinates, however in order to obtain a definition of
that
is independent of the local coordinates we need some additional structure on ; an affine
connection.
Definition 3.1
An affine/linear connection on a manifold
satisfying:
is a map
C1. For each fixed
then the map
for all
and
C2. For each
then the map
for all
and
.
C3.
for all
⏟
mapping
is
-linear. That is
.
is -linear. That is
Example
The canonical connection on the Euclidean space
(
is
)
Remarks
1. More generally we can define a connection on a vector bundle
as the map
satisfying the conditions similar to C1-C3 in Definition 3.1.
An affine connection on is then a connection on
.
2.
is called the directional derivative on in the direction of .
3. An affine connection is not a tensor field because
is not
-linear in .
4.
is also denoted by .
Lemma 3.2 (The local Expression)
Let be an affine connection on a manifold and let
local frame { } on
for
by
and
(
Where the functions
)
on
frame { } are defined by
(
)
( (
)
be expressed in terms of a
. Then
)
are called the Christoffel Symbols of
∑
(
)
with respect to the local
Remark
depends only on the value of
at
and the values of
in a neighbourhood of . See
Exercise Sheet 4 for further discussions.
for local coordinates ( ).
Take
Example
By the Motivation Section, the canonical connection on
(
)
locally looks like
. The Christoffel Symbols of the canonical connection with respect to the
coordinate fame
are all identically 0.
Proposition 3.3 (Existence of an affine connection)
Every smooth manifold admits at least one affine connection.
Proof
Non-examinable. The proof uses a partition of unity.
3.2 Extension of connections to tensor fields
We extend
⏟
to
⏟
Lemma 3.4
Given an affine connection on , then there is a unique map
mapping
to
which coincides with the given connection on
satisfying conditions C1-C3 and additionally:
C4. On
then is given by
C5. satisfies the following product rule with respect to the tensor products:
(
C6.
commutes with all tensor contractions:
tensor contraction
(
)
)
̂
)
Remarks:
we have (
1. For any
(
)
(
(
∑
)
(
)
)
Therefore we have proved that
2. Similarly, for
we have
)
∑
(
Definition 3.5
Let be an affine connection on
(
)
(
)
)
. For
)
where represents any
̂
(
(
we define
by
The
tensor field
is called the total covariant derivative.
Definition 3.6
A tensor field is called parallel if
Remark
More generally for
we can define
inductively by
3.3 Connection along curves and parallel transport
Lemma 3.7
For
depends only on the value of
such that
and
at
and the values of
along any curve
Proof
Exercise Sheet 4.
Definition 3.8
Let
be a curve in
for all
.
Denote by
. A map
is said to be a vector field along
{
if
}
Definition 3.9
Let
be a curve in . For any
choose a neighbourhood
frame { }
for
. Then we can write any
as
We define the covariant derivative of along as
( )
(Recall that
(
of
(
and a local
) for near .
)
)
Remark:
1. The above definition is well defined and independent of the choice of local frame .
To check take ̃
2.
satisfies the following properties -linear but not
-linear)
for all
. Moreover for all
we have
̃ for all
3. Suppose there exists a smooth vector field ̃ on such that
.
̃
Then
Definition 3.10
is called parallel along
curve is parallel along
geodesic.
that is
if
. If the velocity vector field
then the
( ) of a
is said to be a self-parallel curve or
Proposition 3.11
Given an affine connection on a curve
and a tangent vector
the curve there exists a unique parallel vector field
along such that
Proof
Take a local frame { } for
around
to { }. We write
(
(
Suppose
. Let
) for near
at a point
.
on
be the Christoffel Symbols with respect
.
) is a solution then
∑ [(
{
)
(
(
)
(
)]
(
)
)
So
(
{
)
We find the terms of the solution
Definition 3.12
Let
be a curve in
and
(
)
by solving this system of linear ODEs.
be a point on the curve. Then the map
Where
and
is the unique extension of to a parallel vector field along the
curve . The map is called the parallel transport from
to
.
3.4 The Levi-Civita Connection
Definition 3.13
Let be an affine connection on a Riemannian Manifold
. is called compatible with
if for every curve
and every pair of parallel vector fields
along then
is
constant.
Remark
1.
is compatible with if and only if
. For more equivalent definitions of
compatibility see Exercise Sheet 4. One such definition is
(
)
2. For a vector bundle
with a bundle metric (that is restricted to a fibre
is just on and depends smoothly on .) Then for a connection on
we
have
is compatible with if and only if (
for all
)
and
.
Since
mapping
is
-linear in , can be
thought of as a map
More generally we can extend the definition so that for
(⏟
⏟
)
(
⏟
and
⏟
)
is compatible with
the case in part 1.
if and only if
. Observe that when
we return to
Definition 3.14
An affine connection on a manifold is called symmetric or torsion free if
[
] for all
.
[
]. Then is
Define
mapping
-linear and antisymmetric. is called the torsion tensor of . See Exercise Sheet 4 for
details.
Theorem 3.15
Given a Riemannian Manifold
there exists a unique affine connection on that is
symmetric and compatible with . Moreover it is determined by the formula . This
desired connection is known as the Levi-Civita Connection.:
⟨
⟩
{ ⟨
⟩
⟨
⟩
⟨
⟩
⟨
[
]⟩
⟨ [
]⟩
⟨ [
]⟩}
Idea of Proof:
Assume exists with the desired properties. Using compatibility of with we get
⟨
⟩
⟨
⟩ ⟨
⟩
[
] so replacing this in the expression above
Since is symmetric we have
gives
]
]⟩ ⟨
⟨
⟩
⟨
⟩ ⟨ [
⟩
⟨
⟩ ⟨ [
⟩
Cycling
By computing
⟨
⟩ ⟨
gives two similar formulae:
⟨
⟩
⟨
⟩ ⟨ [
⟨
⟩
⟨
⟩ ⟨ [
]⟩
]⟩
⟨
⟨
⟩
⟩
we obtain
⟩
]⟩ ⟨
]⟩ ⟨
⟨
⟩ ⟨ [
⟩
⟨
⟩ ⟨ [
⟩
⟨
⟩
]⟩ ⟨
⟨ [
⟩
Now using symmetry of the metric, we get the desired formula
this implies uniqueness
of the Levi-Civita Connection.
Moreover
determines an affine connection. It is simple to see that this connection
satisfies conditions C1-C3 in Definition 3.1.
⟩
⟨
Remarks:
1. We can find the local expressions of the Christoffel Symbols of the Levi-Civita
connection with respect to the local coordinates ( ). See Sheet 4 for details. The
result is that
If
and
⟨
⟩. Then
(
2.
is an isometry.
)
Chapter 4: Geodesics and Riemannian Distance
Assume throughout this chapter that we have a Riemannian Manifold
Civita connection .
with a Levi-
4.1 Geodesics
A Geodesic is a curve
identically zero.
which has zero acceleration; that is the derivative of
is
Definition 4.1
A curve
from an open interval
is called a geodesic if
for all
That is
is parallel along . Moreover, the restriction of to some closed interval
[
]
is called a geodesic segment or arc.
Theorem 4.2 (Local Existence and Uniqueness of Geodesics)
Given
and
there exists
and a geodesic
and
.
such that
. Moreover, any two geodesics agree on their common domain.
Proof
Choose local coordinate chart (
) of . Without loss of generality we may assume that
corresponds to the origin. Let
be the Christoffel Symbols of the Levi-Civita Connection on
with respect to the local frame
such that
and
|
(
; that is
. Suppose
is a geodesic
(
. Therefore we can write
)
). Write
then
{
{
( )
By the Definition of Geodesic, and the definition of Covariant Derivative:
(( )
(
))
( )
( )
(
)
Now we treat the two components of the sums separately: to emphasis this we change the
index of the first component from to and expand the second component involving the
connection:
(
)
(
)
( )
(
)
(
)
(
)
( )
(
)
(
)
( )
(
)
[(
)
( )
Therefore we get the initial value problem:
(
)
]
(
)
(
)
(
(
)
(
)
)
(
{
)
( )
(
)
(
From ODE Theory, there exists
has a unique solution for
(
)
)
such that the system of second order linear equations
.
Now we can repeat the entire argument for any chart (
) for . By repeating as
necessary we can extend the domain of the geodesic . By the local uniqueness, it follows
that any two geodesics coincide on their common domains.
Remark
The solutions to the geodesic equation are smooth as functions depending on the initial
conditions. It follows from the local existence and uniqueness (Theorem 4.2) that for any
point
and
there is a unique maximal geodesic.
That is, a geodesic defined on some open interval containing 0 and cannot be extended to
any larger domain as a geodesic.
Examples:
1. Geodesics in
,
. In this case the Christoffel Symbols
a vector
equation (
)
to
. Given
. In the statement of Theorem 4.2, we need to solve the
for all
and thus
and
and ( )
is the Geodesic.
2. Geodesics on
We use polar coordinates
Project
for all
for all . Therefore
. We obtain the formulae:
in the plane spanned by
and . Then
. By associating
to
and
. Also (
That is
so
and
to
then (
)
(
.
(
)
)
)
(
)
. Recall the formula for the Christoffel
Symbols:
(
)
So
(
)
(
)
(
)
(
)
Recall the Geodesic equations:
(
Let
) be a geodesic then (
)
( )
(
)
(
)
.
(
Then if
) is a geodesic on
then
(
)
We will find a special solution. When
and
we see that it is a
solution to the differential equations above, and this solution corresponds to a
portion of a great circle passing through and . By uniqueness of geodesics and
using rotations, the geodesics on
are portions of great circles.
3. Geodesics on the Upper Half Hyperbolic Plane
{
Let
} and a metric
method above by computing
and
. Then using the same
and then
we can determine all geodesics.
They are of the forms of semicircles and vertical lines as shown below:
See Exercise Sheet 5 for details.
4. Geodesics on the Poincaré Disc
Let
{
} and
(
)
then as before we can show that all geodesics are of the form of arcs normal to the
tangent of the disc:
4.2 Exponential Map
This is map from
to
. First we need a Stronger Version of Theorem 4.2:
Theorem 4.2’
For
there exists a neighbourhood of and
with | |
then there exists a unique geodesic
so that for all
with
mapping (
. Moreover the map
)
and
and
is smooth.
Definition 4.4
Let
be the set of all vectors so that
is defined on an interval containing [
Then we define the exponential map at by
mapping
where
is the unique geodesic as in Theorem 4.2’.
∐
Take
and define the exponential map by
mapping
].
.
Example:
We take
{ } is surjective and a
. In fact
diffeomorphism.
Lemma 4.5 (Homogeneity of Geodesics)
Given
and
we have
whenever either side is defined.
Proof
Proof not examinable.
Proposition 4.6 (Properties of the Exponential Map)
1.
is an open subset of
containing all zero-vectors and hence
is a smooth map
between smooth manifolds and . (Any open subset of a manifold is itself a
manifold).
2. For any
the maximal geodesic
is given by
for
all such that both sides are well defined.
3.
and (
)
for all
Proof
Non-examinable.
Remarks:
2. Means that
3. As (
maps the radial lines in
)
then
to radial geodesics in
passing through .
is locally a diffeomorphism.
4.3 Normal Coordinates
Lemma 4.7 (Normal Neighbourhood Theorem)
Given
there exists a neighbourhood of in
and another neighbourhood
of
such that the exponential map
is a diffeomorphism.
Such is called a normal neighbourhood of .In particular, there exists
such that
is a diffeomorphism where
(
)
{
| |
}.
Terminology: For
ball and
Given
with as in Lemma 4.7, we call
a closed geodesic ball. Also,
(
choose an orthonormal basis { } of
mapping
an open geodesic
) is called a geodesic sphere.
. This gives an isomorphism
.
Let be a normal neighbourhood of , we define the chart map
The coordinates associated to the chart
are called normal coordinates centred at .
They map
Then
maps
.
It is extremely useful to take normal coordinates centred at
which implies that
.
for all
, since then
.
4.4 Riemannian Distance
Given a Riemannian manifold
and
we can find a metric
metric space whose induced topology is identical to the given topology on
so
is a
.
Definition 4.8
]
A map [
is called a smooth curve if
is a smooth map (in the usual
[
]
sense) and
is continuous, and all derivatives of extend continuously to (with
respect to a chart containing
). Similarly for .
For some
̃ [
where ̃
. ̃
is defined for all
and
,
Definition 4.9
A continuous map
subdivision
[
]
is called a piecewise smooth curve if there exists a finite
such that the restrictions |[
] are smooth curves
for each .
Remark
1. The length of a piecewise smooth curve is defined as
2. We say joins the points
and
∑
( |[
])
.
Given any two points
consider the set of all piecewise smooth curves joining
[ ]
{piecewise smooth curves [
]
}
namely
Remark
For a connected manifold
connected.
,
Definition 4.10
We define a map
Distance between
and
as every connected smooth manifold is path
by
.
is called the Riemannian
and .
Lemma 4.11
Any Riemannian Manifold equipped with the Riemannian Distance is a metric space
whose induced topology agrees with the topology of the underlying set
Proof
See Exercise Sheet 6 Question 4.
Remark
For
sufficiently small, a geodesic ball
{
}
(
) coincides with the metric ball
4.5 Minimising Properties of Geodesics
The statement “geodesics are shortest paths” is true under certain conditions.
Minimising Curves are Geodesics
Definition 4.12
]
A piecewise smooth curve [
is called a minimising curve if
[
] ; that is any other piecewise smooth curve joining
An equivalent condition on
(
is
and
)
Definition 4.13
]
Let [
be a smooth curve. A smooth variation of is a smooth map
[
].
mapping
such that
for all
Consider, for
[
the curve
lengths. We will give a formula for
|
speed parameterisation; that is |
|
]
mapping
[
]
and then their
under the assumption
has unit
for all (with respect to the metric).
Definition 4.14
The velocity vector of the curve
mapping
denoted by
for all
.
at
gives (as varies) a vector field along the given curve
called the variational vector field of .
Theorem 4.15 (First Variation Formula of Length)
]
Suppose [
is a smooth curve with unit speed parametrisation; that is |
[
]
Take
is a smooth variation of with variational vector field
above. Then
⟨
|
Remark
More generally, for
with |
|
|
⟩|
∫ ⟨
⟩
for all , we have a similar formula:
⟨
|
⟩|
|
∫ ⟨
⟩
|
|
Idea of Proof of Theorem 4.15
|
Now,
|
∫ |
|
∫
|
|
|
|
.
as
|
|
|
⟨
|
⟩
|
⏞
|
|
⟨
⟩
⟨ ⟩
⟨
|
⟩
⟨
⏞
By the Symmetry Lemma in Exercise Sheet 4,
⟩|
. Therefore
⟨ ⟩
|
|
⟨
|
⟩|
⏞
⟨
⟨
⟩|
∫ ⟨
|
|
⟩|
⟨
⟩|
Therefore
∫
|
⟨
⟨
Theorem 4.16
If a smooth curve [
then it is a geodesic.
]
⟩|
⟩|
⟩|
∫ ⟨
|
∫ ⟨
|
⟩
⟩
is minimising and it has constant speed parameterisation
Idea of Proof of Theorem 4.16
| |
| for all . If it is zero, then
Assume |
is constant so by definition a
geodesic. If it is not zero, then we apply the first variation formula:
Take a smooth variation [
so
we have
]
such that
and
. By the general first variation formula of length,
|
Since
|
is minimising, (
therefore for any
)
|
∫ ⟨
⟩
for all
hence
|
, we have
|
|
∫ ⟨
⟩
Such a
was arbitrary (from the arbitrary choice of smooth variation) and the inner
[
]; that is
product ⟨ ⟩ is positive definite. Therefore
for all
is a
geodesic.
Given any curve
speed curve.
[
]
mapping
we can find a diffeomorphism to a unit
Geodesics are Locally Minimising
Consider the Riemannian Manifold
. Given two points
we can find two
geodesics joining and ,
where
and
. and are geodesics
because they are a portion of great circles. In fact is minimising, but is not minimising.
Therefore a geodesic need not be globally minimising.
Theorem 4.17 (Geodesics are Locally minimising)
Let
, a normal neighbourhood of and
be a geodesic ball centred at . Then
every geodesic segment [ ]
with
is minimising.
That is,
for all piecewise smooth curves [ ]
with
and
. Moreover, this inequality is strict unless is a monotone reparameterisation of
.
Proof
Non Examinable!
A useful technical Lemma (used in the Proof of 4.17) is the following:
Lemma 4.18 (Gauss Lemma)
Let
and
such that
⟨(
(
is well-defined. Let
)
(
⟩
)
⟨
). Then
⟩
Proof
Non-Examinable.
Remark
In particular if ⟨
⟩
then ⟨(
)
(
)
⟩
. This means that the radial
geodesics are orthogonal to the geodesics spheres. (See Exercise Sheet 6).
Chapter 5: Curvature
In this chapter
is always a Riemannian manifold with a Riemannian Metric
Levi-Civita Connection .
and a
5.1 The Riemannian Curvature Tensor
Given
then in general
. So
is not commutative in
general. The Riemannian curvature tensor measures the extent to which the covariant
derivative is not commutative.
Definition 5.1
Let be a manifold with an affine connection . The curvature tensor of
mapping
where
[
is the map
]
Observe that this map is
-linear in
. The curvature tensor of the Levi-Civita
Connection is called the Riemannian Curvature Tensor.
Remark
1.
is a tensor field of type
that is
.
2. The value of
at a point
depends only on the values of
3.
is antisymmetric in and .
4. Some people would define
[
] .
.
Given a Riemannian manifold then the curvature tensor of its Levi-Civita Connection is
called the Riemannian Curvature Tensor.
Example
Consider
and [
then
]
for all
Therefore
Using the metric
denoted again by
for any
(
so
; recall that
)
hence
for
(
)
.
⟨ ⟩ we define the Riemannian Curvature Tensor of type
(or
) as:
⏟
Local Expressions
Let ( ) be local coordinates. Then
⟨
for all
⟩
which is
And it is possible to find formulas for
By definition,
properties of :
and
.
is antisymmetric in
. In fact there are more symmetric
Proposition 5.2 (Algebraic Properties of the Riemannian Curvature Tensor)
1.
2.
3. First Bianchi Identity:
4. Second Bianchi Identity:
[Fix the last two entries and cycle the first three]
Proof
1. By Compatibility of the Metric with ,
⟨
⟩ ⟨
⟩ ⟨
⟩
⟨
⟩
⟨
⟩ ⟨
Similarly:
⟨
⟩
⟨
⟩ ⟨
By computing
we have:
[
]⟨
⟩ ⟨
On the other hand
[
]⟨
⟩ ⟨ [
⟩
]
⟨
⟩
⟨
⟩
⟨
⟩
⟨
⟩
⟩
⟨
⟩
⟨
⟩
⟩
[
]
⟩
Then
⟨
⟩
[
]
⟨
⟩ ⟨
⟩
2. We Apply the First Bianchi Identity Four Times:
⟨
[
]
⟩
Then by using antisymmetric properties in part 1, on the sum of these identities we
achieve the desired result
3. See Exercise Sheet 7 for a proof of the First and Second Bianchi Identities.
Extending to general tensor fields
Recall the affine connection on can be extended to general tensor fields:
So we can extend the Levi-Civita Connection and the Riemannian Curvature Tensor. We can
consider
mapping
[
]
Properties (Proofs are on the Example Sheet)
a.
b.
c. For any tensor contraction ,
5.2 Sectional Curvature
Definition 5.3
Given a point
basis {
} of
and a two-dimensional subspace
of
and define the sectional curvature of
⟨
⟩
at
we take an orthonormal
as (
associated to
)
Remarks
) does not depend on choice of basis of the orthonormal basis of
} { ̃ ̃} of .
̃ ̃ ̃ ̃ for orthonormal bases {
} is any basis of
b. If {
(not necessarily orthonormal) then
⟨
⟩
( )
⟨
⟩⟨
⟩ ⟨
⟩
a.
(
then (
If
details.
) is the Gauss Curvature of
. That is
at . See exercise Sheet 8 for
The Riemannian Curvature tensor determines all sectional curvature. The converse is also
true; the sectional curvature completely determines the Riemannian Curvature Tensor.
Proposition 5.4
Assume that
satisfying the properties a, b, c in proposition 5.2. Then if
then
; that is
are two
-linear maps
for all
for all
Proof
See GHL.
Proposition 5.5
Given a point
1.
2.
(
)
then the following properties are equivalent:
for all 2-planes
⟨
⟩
⟨
⟩ for all
.
Definition 5.6
A Riemannian Manifold is called a constant curvature Riemannian Manifold if ( )
for
all two-planes
for all
.
A Riemannian manifold whose Riemannian curvature tensor vanishes (equivalently, if all
sectional curvature vanishes) is called flat.
Remark
is flat
Riemannian metric
is locally isometric to some Euclidean Space
is flat.
The
Example
is flat and so
.
is flat.
Model Spaces with Constant Curvature:
1. The Euclidean Space
has constant curvature 0
2. The unit sphere
has constant curvature 1
3. The Hyperbolic Space has constant curvature -1
a. Poincare half space model:
{
}
In the case
see Exercise Sheet 2.
b. Poincare ball model
{
In the case
}
[
see Exercise Sheet 2.
]
Additional Exercise:
Compute the Sectional Curvature of these model spaces.
5.3 Ricci and Scalar Curvature
Definition 5.7
The Ricci Curvature Tensor is the map
where for each point
,
mapping
{
Where we extend
(
)
}
to a vector field ̃ then use the Riemann Curvature Tensor:
(
̃
)
Remarks
1.
(
) is an endomorphism of
2.
is
-linear in and
∑
3. If
is an orthonormal basis for
then
where
. Therefore
is symmetric in and .
4. The above definition of
is independent of the choice of orthonormal basis:
∑
∑
5. Let { } be a local frame with dual {
then
(In Exercise Sheet 7, take
̃
̃
}. Locally we can write
and
(
where
)
.
Definition 5.8
The Ricci Curvature of a Riemannian Manifold
is the quadratic form associated to the
Ricci Curvature Tensor; that is the map
mapping
Remarks
1. For a unit tangent vector
{
} and compute
that is | |
, we choose an orthonormal basis
∑
⏟
{
2. As the Ricci Curvature Tensor
Ricci Curvature.
is symmetric it is completely determined by the
we define an endomorphism ̃
Given a point in
⟩
given by ⟨̃
Definition 5.9
The Scalar Curvature of a Riemannian Manifold
{̃
}
Remark
If { } is a basis of
If
̃
then
then
(
}
mapping
is the function
(
̃
mapping
)
)
Definition 5.10
A Riemannian Manifold
such that
⏟
is called an Einstein Manifold if there exists a constant
Namely,
for all
and for all
.
Remark
1. If
has constant curvature and
then
is an Einstein Manifold
with
.
2. For
the Riemannian Curvature Tensor is entirely given by the scalar curvature.
3. For
the Riemannian Curvature Tensor is entirely given by the Ricci Curvature.
In particular, if
then
is Einstein
has constant curvature.
Chapter 6: Jacobi Fields
Jacobi Fields lead to a geometric interpretation of curvature.
6.1 Basic Facts about Jacobi Fields
Definition 6.1
]
Let [
be a geodesic on
]
smooth map [
[
1.
for all
2. For fixed , the curve
and
. A variation of
such that
on
through geodesics is a
]
is a geodesic.
|
Consider the variational vector field of , that is
. Since
is smooth,
is smooth.
Theorem 6.2
Let be a geodesic and a variation of through geodesics. If
is the variational
vector field of then it satisfies the following equation known as the Jacobi Equation:
(
)
Definition 6.3
]
Let [
be a geodesic. A vector field along
[
]
the Jacobi equation for all
is said to be a Jacobi Field if it satisfies
Proposition 6.4 (Existence and Uniqueness of Jacobi Fields)
]
Let [
be a geodesic and
. For any pair
Jacobi Field along
there exists a unique
satisfying the initial condition
Remarks
1. Let
be a geodesic on a Riemannian Manifold
{
Then
is a -dimensional linear subspace of
linearly independent Jacobi Fields along .
Note that by using a parallel basis {
system of linear ODEs.
} along
,
. Denote by
}
. Therefore there exist
then the Jacobi Equation is just a
2. Notice that
and
are linearly independent Jacobi Fields along . In fact any
⟩
can be written uniquely as
where ⟨
Examples
1.
.
so
where
and
are constant vector fields along
.
2. Riemannian Manifold of Constant Curvature:
⟩
Let be a unit speed geodesic,
a Jacobi Field normal to (that is ⟨
and
. Then
where
is a parallel vector field along , and
.
Proposition 6.5
A vector field along a geodesic
through geodesics.
Corollary 6.6
Let
,
Jacobi Field
is a Jacobi Field if and only if comes from a variation of
be a geodesic and
. Then the
(
)
and
is given by the formula:
and
along such that
(
)
6.2 Jacobi Fields and Curvature
We shall relate the rate of spreading of geodesics that start from the same point
the curvature at .
Proposition 6.7
Let
be a unit speed geodesic on
⟨
⟩
. Consider
with
and
|
| about
is given by |
with
and take
and
(
) with | |
. Then the Taylor Expansion of
|
where
Corollary 6.8
With the same conditions as above,
|
|
̃
̃
Remark
The geodesics
spread apart less than the rays in
if
.
Given
we can consider it as either a point, a vector or a constant vector field on
is a geodesic in
is a Jacobi Field along . Using the exponential map gives
and
.
.
Now we give a geometrical interpretation of sectional curvature, which measures a
deviation of a Riemannian Manifold from Euclidean Space
Corollary 6.9
In a -plane
of
, consider a circle of radius centred at the origin
and denote by
the length of , which is the image of the above circle under the exponential map
(
Then
).
Remark
Suppose ( ) are normal coordinates centred at
|
.
. Then
| see Exercise Sheet 8 to use the Jacobi Field to compute the Taylor
Expansion.
Therefore
⏟
√
(
|
⏟
|
)
6.3 Second Variation Formula
|
We now compute
Theorem 6.10 (Second Variation Formula)
]
Let [
be a unit speed geodesic, [
its variational vector field. Then
|
Where
|
∫ (|
⟨⏟
]
a smooth variation of
)
⟨
|
and
⟩|
⟩
Definition 6.11
The index form of the geodesic is a symmetric bilinear map defined on all smooth vector
fields along which vanish at the end points, given by
.
∫ ⟨
Proposition 6.12
For
with
for all
with
and ⟨
⟩
⟩
, it is a Jacobi Field if and only if
Remark
Second Variation formula for the energy
For a smooth curve
[
]
∫ |
energy of the curve as
|
We can also define
∫ |
we have
|
∫ ⟨
for the length. We define the
. Then we can find
∫ |
|
|
⟩
|
⟨
|
⟩|
Chapter 7: Classical Theorems in Riemannian Geometry
7.1 Hopf-Rinow Theorem
Definition 7.1
A Riemannian Manifold
is called geodesically complete at
if every geodesic
passing through is defined on the whole real line; that is the exponential map
is
defined over the whole tangent space
.
is called geodesically complete if it is
geodesically complete at any point
, that is
is defined on all of
.
Examples




is geodesically complete
Take
then
is not geodesically complete.
(
) the Poincare disc is geodesically complete
{ }
is not geodesically complete.
Lemma 7.2
If
is connected, let
. Suppose is geodesically complete at . Then for any
there exists a minimising geodesic joining and .
Theorem 7.3 (Hopf-Rinow)
Let
be a connected Riemannian Manifold, and let be the Riemannian distance
function on with respect to . Then the four following conditions are equivalent:
1.
is complete as a metric space. (Any Cauchy Sequence converges)
2.
is geodesically complete.
3.
is geodesically complete at a point
.
4. The closed and bounded subsets of are compact.
Moreover, any of the above conditions implies that for any two points
minimising geodesic joining and .
there exists a
Proof
Non-examinable.
Remark
A Riemannian Manifold if called complete if any of the four conditions are satisfied.
7.2 Catan-Hadward Theorem
Theorem 7.4 (Catan-Hadward)
Let be a complete Riemannian manifold of constant sectional curvature . Suppose
then is diffeomorphic to
. More precisely, for any point
is a diffeomorphism.
Theorem 7.5 (Classification of Riemannian Manifolds of constant sectional curvature)
Let be a complete Riemannian Manifold of constant sectional curvature . Suppose
. The
is isometric to the space form where:
√
{
}
{(
|⏟ |
| |
)
7.3 Bonnet-Myers Theorem
Definition 7.6
The Diameter of a Riemannian Manifold
is
{
}.
Theorem 7.7 (Bonnet-Myers)
Suppose
and
is a complete and connected Riemannian Manifold of dimension
for which
Then
is compact and has diameter at most
.
Remarks
1.
2. This theorem is sharp for a round sphere
of radius . That is
.
(
)
3. The hypothesis cannot be relaxed to
. Indeed, the paraboloid
{
} equipped with the induced metric satisfies
is not compact, though it is complete.
and
but it