4Hodge theory on Riemannian manifolds

III Riemannian Geometry

4.1 Hodge star and operators

Throughout this chapter, we will assume our manifolds are oriented, and write

n

of the dimension. We will write

ε ∈

Ω

n

(

M

) for a non-vanishing form defining

the orientation.

Given a coordinate patch

U ⊆ M

, we can use Gram-Schmidt to obtain a

positively-oriented orthonormal frame

e

1

, ··· , e

n

. This allows us to dualize and

obtain a basis ω

1

, ··· , ω

n

∈ Ω

1

(M), defined by

ω

i

(e

i

) = δ

ij

.

Since these are linearly independent, we can multiply all of them together to

obtain a non-zero n-form

ω

1

∧ ··· ∧ ω

n

= aε,

for some

a ∈ C

∞

(

U

),

a >

0. We can do this for any coordinate patches, and

the resulting

n

-form agrees on intersections. Indeed, given any other choice

ω

0

1

, ··· , ω

0

n

, they must be related to the original

ω

1

, ··· , ω

n

by an element

Φ ∈ SO(n). Then by linear algebra, we have

ω

0

1

∧ ··· ∧ ω

0

n

= det(Φ) ω

1

∧ ··· ∧ ω

n

= ω

1

∧ ··· ∧ ω

n

.

So we can patch all these forms together to get a global

n

-form

ω

g

∈

Ω

n

(

M

)

that gives the same orientation. This is a canonical such

n

-form, depending only

on

g

and the orientation chosen. This is called the (Riemannian) volume form

of (M, g).

Recall that the

ω

i

are orthonormal with respect to the natural dual inner

product on

T

∗

M

. In general,

g

induces an inner product on

V

p

T

∗

M

for all

p

= 0

,

1

, ··· , n

, which is still denoted

g

. One way to describe this is to give an

orthonormal basis on each fiber, given by

{ω

i

1

∧ ··· ∧ ω

i

p

: 1 ≤ i

1

< ··· < i

p

≤ n}.

From this point of view, the volume form becomes a form of “unit length”.

We now come to the central definition of Hodge theory.

Definition

(Hodge star)

.

The Hodge star operator on (

M

n

, g

) is the linear map

? :

V

p

(T

∗

x

M) →

V

n−p

(T

∗

x

M)

satisfying the property that for all α, β ∈

V

p

(T

∗

x

M), we have

α ∧ ?β = hα, βi

g

ω

g

.

Since g is non-degenerate, it follows that this is well-defined.

How do we actually compute this? Since we have vector spaces, it is natural

to consider what happens in a basis.

Proposition.

Suppose

ω

1

, ··· , ω

n

is an orthonormal basis of

T

∗

x

M

. Then we

claim that

?(ω

1

∧ ··· ∧ ω

p

) = ω

p+1

∧ ··· ∧ ω

n

.

We can check this by checking all basis vectors of

V

p

M

, and the result drops

out immediately. Since we can always relabel the numbers, this already tells us

how to compute the Hodge star of all other basis elements.

We can apply the Hodge star twice, which gives us a linear endomorphism

?? :

V

p

T

∗

x

M →

V

p

T

∗

x

M. From the above, it follows that

Proposition.

The double Hodge star

??

:

V

p

(

T

∗

x

M

)

→

V

p

(

T

∗

x

M

) is equal to

(−1)

p(n−p)

.

In particular,

?1 = ω

g

, ?ω

g

= 1.

Using the Hodge star, we can define a differential operator:

Definition

(Co-differential (

δ

))

.

We define

δ

: Ω

p

(

M

)

→

Ω

p−1

(

M

) for 0

≤ p ≤

dim M by

δ =

(

(−1)

n(p+1)+1

? d? p 6= 0

0 p = 0

.

This is (sometimes) called the co-differential.

The funny powers of (−1) are chosen so that our future results work well.

We further define

Definition (Laplace–Beltrami operator ∆). The Laplace–Beltrami operator is

∆ = dδ + δd : Ω

p

(M) → Ω

p

(M).

This is also known as the (Hodge) Laplacian.

We quickly note that

Proposition.

?∆ = ∆ ? .

Consider the spacial case of (

M, g

) = (

R

n

, eucl

), and

p

= 0. Then a straight-

forward calculation shows that

∆f = −

∂

2

f

∂x

2

1

− ··· −

∂

2

f

∂x

2

n

for each

f ∈ C

∞

(

R

n

) = Ω

0

(

R

n

). This is just the usual Laplacian, except there

is a negative sign. This is there for a good reason, but we shall not go into that.

More generally, metric

g

=

g

ij

d

x

i

d

x

j

on

R

n

(or alternatively a coordinate

patch on any Riemannian manifold), we have

ω

g

=

p

|g| dx

1

∧ ··· ∧ dx

n

,

where |g| is the determinant of g. Then we have

∆

g

f = −

1

p

|g|

∂

j

(

p

|g|g

ij

∂

i

f) = −g

ij

∂

i

∂

j

f + lower order terms.

How can we think about this co-differential δ? One way to understand it is

that it is the “adjoint” to d.

Proposition. δ

is the formal adjoint of d. Explicitly, for any compactly sup-

ported α ∈ Ω

p−1

and β ∈ Ω

p

, then

Z

M

hdα, βi

g

ω

g

=

Z

M

hα, δβi

g

ω

g

.

We just say it is a formal adjoint, rather than a genuine adjoint, because

there is no obvious Banach space structure on Ω

p

(

M

), and we don’t want to go

into that. However, we can still define

Definition

(

L

2

inner product)

.

For

ξ, η ∈

Ω

p

(

M

), we define the

L

2

inner

product by

hhξ, ηii

g

=

Z

M

hξ, ηi

g

ω

g

,

where ξ, η ∈ Ω

p

(M).

Note that this may not be well-defined if the space is not compact.

Under this notation, we can write the proposition as

hhdα, βii

g

= hhα, δβii

g

.

Thus, we also say δ is the L

2

adjoint.

To prove this, we need to recall Stokes’ theorem. Since we don’t care about

manifolds with boundary in this course, we just have

Z

M

dω = 0

for all forms ω.

Proof. We have

0 =

Z

M

d(α ∧ ?β)

=

Z

M

dα ∧ ?β +

Z

M

(−1)

p−1

α ∧ d ? β

=

Z

M

hdα, βi

g

ω

g

+ (−1)

p−1

(−1)

(n−p+1)(p−1)

Z

M

α ∧ ? ? d ? β

=

Z

M

hdα, βi

g

ω

g

+ (−1)

(n−p)(p−1)

Z

M

α ∧ ? ? d ? β

=

Z

M

hdα, βi

g

ω

g

−

Z

M

α ∧ ?δβ

=

Z

M

hdα, βi

g

ω

g

−

Z

M

hα, δβi

g

ω

g

.

This result explains the funny signs we gave δ.

Corollary. ∆ is formally self-adjoint.

Similar to what we did in, say, IB Methods, we can define

Definition

(Harmonic forms)

.

A harmonic form is a

p

-form

ω

such that ∆

ω

= 0.

We write

H

p

= {α ∈ Ω

p

(M) : ∆α = 0}.

We have a further corollary of the proposition.

Corollary. Let M be compact. Then

∆α = 0 ⇔ dα = 0 and δα = 0.

We say α is closed and co-closed .

Proof. ⇐ is clear. For ⇒, suppose ∆α = 0. Then we have

0 = hhα, ∆αii = hhα, dδα + δdαii = kδαk

2

g

+ kdαk

2

g

.

Since the L

2

norm is non-degenerate, it follows that δα = dα = 0.

In particular, in degree 0, co-closed is automatic. Then for all

f ∈ C

∞

(

M

),

we have

∆f = 0 ⇔ df = 0.

In other words, harmonic functions on a compact manifold must be constant.

This is a good way to demonstrate that the compactness hypothesis is required,

as there are many non-trivial harmonic functions on R

n

, e.g. x.

Some of these things simplify if we know about the parity of our manifold. If

dim M = n = 2m, then ?? = (−1)

p

, and

δ = − ? d?

whenever

p 6

= 0. In particular, this applies to complex manifolds, say

C

n

∼

=

R

2n

,

with the Hermitian metric. This is to be continued in sheet 3.