1Basics of Riemannian manifolds

III Riemannian Geometry

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

instead of d

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

admits

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.

We already know that

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

)

→

Ω

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 ∈ Ω

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

∇

E⊗F

(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.