CHAPTER 5 Manifolds

The aim of this chapter is simply to remind you of enough material concerning differentiable manifolds to be able to prove the Hodge theorem (given the deRham theorem) and also to be able to talk freely, later, about symplectic manifolds and such things.

5.1. C^{∞} structures

Let X be a paracompact Hausdorff topological space. A C^{∞} structure on X is a subspace

(5.1) F ⊂ C^{0}(X) = {u : X ⟶ ℝ continuous }

with the following property:

For each \overline{x} ∈ X there exists elements f_{1}, …, f_{n} ∈ F such that for some open neighbourhood Ω ∋ \overline{x}

F : Ω ∋ x ⟼ (f_{1}(x), …, f_{n}(x)) ∈ ℝ^{n} (5.2)

is a homeomorphism onto an open subset of ℝ^{n} and every f ∈ F satisfies

(5.3) f ↾ Ω = g ∘ F for some g ∈ C^{∞}(ℝ^{n}).

The map (5.2) is a coordinate system near \overline{x}. Two C^{∞} structures F_{1} and F_{2} are ‘compatible’ if F_{1} ∪ F_{2} is also a C^{∞} structure. It therefore makes sense to say that:

Definition 5.1. A C^{∞} manifold is a (connected) paracompact Hausdorff topological space with a maximal C^{∞} structure.

The maximal C^{∞} structure is conventionally denoted

C^{∞}(X) ⊂ C^{0}(X).

It is necessarily a ring. If we let C_{c}^{∞}(W) ⊂ C^{∞}(X) denote the subspace of functions which vanish outside a compact subset of W then any local coordinates (5.2) have the property

F^{∗} : C_{c}^{∞}(F (Ω)) ⟷ {u ∈ C^{∞}(X); u = 0 on X\K, K ⊂⊂ Ω}. (5.4)

Futhermore C^{∞}(X) is local:

u : X ⟶ ℝ and ∀ \overline{x} ∈ X ∃ Ω\overline{x} open, Ω\overline{x} ∋ \overline{x},

s.t. u = f\overline{x} = 0 on Ω\overline{x} for some f\overline{x} ∈ C^{∞}(X) ⟹ u ∈ C^{∞}(X).

(5.5)

A map G : X ⟶ Y between C^{∞} manifolds X and Y is C^{∞} if

G^{∗} : C^{∞}(Y) ⟶ C^{∞}(X)

i.e. G ∘ u ∈ C^{∞}(X) for all u ∈ C^{∞}(Y).

5.2. Form bundles

A vector bundle is a triple π : V ⟶ X consisting of two manifolds, X and V, and a C^{∞} map π with each

V_{x} = π^{−1}(x)

having a linear structure such that

F = {u : V ⟶ ℝ, u is linear on each V_{x}}

is a C^{∞} structure on V compatible with C^{∞}(V) (i.e. contained in it, since it is maximal).

The basic example is the cotangent bundle which we defined before. Namely for each \overline{x} ∈ X set

I\overline{x} = \{u ∈ C^{∞}(X); u(\overline{x}) = 0\}

I2 ; \overline{x} = \{u = ∑_{finite} u_{i}u′_{i}; u_{i}, u′_{i} ∈ I\overline{x}\}

T ∗ ; \overline{x}X = I\overline{x}/I2 ; \overline{x}, T^{∗}X = ⋃_{\overline{x}∈X} T ∗ ; \overline{x}X.

So π : T^{∗}X ⟶ X just maps each T ∗ ; \overline{x}X to \overline{x}. We need to give T^{∗}X a C^{∞} structure so that “it” (meaning π : T^{∗}X ⟶ X) becomes a vector bundle. To do so note that the differential of any f ∈ C^{∞}(X)

df : X ⟶ T^{∗}X df(\overline{x}) = [f − f(\overline{x})] ∈ T ∗ ; \overline{x}X

is a section (π ∘ df = Id). Put

F = {u : T^{∗}X ⟶ ℝ; u ∘ df : X ⟶ ℝ is C^{∞} ∀ f ∈ C^{∞}(X)}.

Then F = C^{∞}(T^{∗}X) is a maximal C^{∞} structure on T^{∗}X and

F_{lin} = \{u : T^{∗}X ⟶ ℝ, linear on each T ∗ ; \overline{x}X; u ∈ F\}

is therefore compatible with it. Clearly df is C^{∞}.

Any (functorial) operation on finite dimensional vector spaces can be easily seen to generate new vectors bundles from old. Thus duality, tensor product, exterior powers all lead to new vector bundles:

T_{x}X = (T_{x}^{∗}X)^{∗} , T X = ⋃_{x∈X} T_{x}X

is the tangent bundle

Λ_{x}^{k}X = {u : T_{x}X × ⋯ × T_{x}X^{k factors} ⟶ ℝ; u is multilinear and antisymmetric }

leads to the k-form bundle

Λ^{k}X = ⋃_{x∈X} Λ_{x}^{k}X, Λ^{1}X ≃ T^{∗}X

where equivalence means there exists (in this case a natural) C^{∞} diffeomorphism mapping fibres to fibres linearly (and in this case projecting to the identity on X).

A similar construction leads to the density bundles

Ω_{x}^{α}X = {u : T_{x}X ∧ ⋯ ∧ T_{x}X^{n=dim X, textfactors} ⟶ ℝ; absolutely homogeneous of degree α}

that is

u(tv_{1} ∧ … v_{n}) = |t|^{α}u(v, ∧ ⋯ ∧ v_{n}).

These are important because of integration. In general if π : V ⟶ X is a vector bundle then

C^{∞}(X; V) = {u : X ⟶ V; π ∘ u = Id}

is the space of sections. It has a natural linear structure. Suppose W ⊂ X is a coordinate neighbourhood and u ∈ C^{∞}(X; Ω), Ω = Ω^{1}X, has compact support in W. Then the coordinate map gives an identification

Ω_{x}^{∗}X ⟷ Ω_{F (x)}^{∗}ℝ^{n} ∀ α

and

∫ u = ∫_{ℝ^{n}} g_{u}(x), u = g_{u}(x)|dx|

is defined independent of coordiantes. That is the integral

∫ : C_{c}^{∞}(X; Ω) ⟶ ℝ

is well-defined.

5.3. Pseudodifferential operators

Let X be a C^{∞} manifold, and let C_{c}^{∞}(X) ⊂ C^{∞}(X) be the space of C^{∞} functions of compact support. Then, for any m ∈ ℝ, Ψ^{m}(X) is the space of linear operators

A : C_{c}^{∞}(X) ⟶ C^{∞}(X) (5.6)

with the following properties. First, if ϕ, ψ ∈ C^{∞}(X) have disjoint supports then ∃ K ∈ C^{∞}(X × X; Ω_{R})

such that ∀ u ∈ C_{c}^{∞}(X) ϕAψu = ∫_{X} K(x, y)u(y), (5.7)

and secondly if F : W ⟶ ℝ^{n} is a coordinate system in X and ψ ∈ C_{c}^{∞}(X) has support in W then

∃ B ∈ Ψ_{∞}^{m}(ℝ^{n}), supp(B) ⊂ W × W s.t.
ψAψu ↾ W = F^{∗}(B((F^{−1})^{∗}(ψu))) ∀ u ∈ C_{c}^{∞}(X).

This is a pretty horrible definition, since it requires us to check every coordinate system, at least in principle. In practice the coordinate-invariance we proved earlier means that this is not necessary and also that there are plenty of examples!

Any open cover of a C^{∞} manifold has a partition of unity subordinate to it, i.e. if A_{r} ⊂ X are open sets for r ∈ R and

X = ⋃_{r∈R} A_{r}

there exists ϕ_{i} ∈ C_{c}^{∞}(X), all non-negative with locally finite support:

∀ i supp(ϕ_{i}) ∩ supp(ϕ_{j}) ≠ ∅ for a finite set of indices j,

where each supp(ϕ_{i}) ⊂ A_{r} for some r = r(i) and

∑_{i} ϕ_{i}(x) = 1 ∀ x.

In fact one can do slightly better than this, for a covering by coordinate neighbourhoods.

Lemma 5.1. There exists a partition of unity on X (a C^{∞} manifold) ϕ_{i} s.t. for every i

⋃_{j}{supp(ϕ_{j}); supp(ϕ_{j}) ∩ supp(ϕ_{i}) ≠ 0}

is contained in a coordinate neighbourhood!

Using such a partition of unity we see that every element of Ψ^{m}(X) can be written in the form

A = ∑_{i} ∑_{j} ϕ_{j}F^{∗}K_{A}ϕ_{i}

where the terms have smooth kernels if the supports of ψ_{i} and ψ_{j} do not meet, or else are pseudodifferential operators in any local coordinates in a A patch containing both supports. Below we shall use this to prove:

Theorem 5.1. Let X be a compact C^{∞} manifold then the pseudodifferential operators Ψ^{∗}(X) form a symbol-filtered ring.

If X is a C^{∞} manifold we have defined the space Ψ^{m}(X) as consisting of those linear operators

A : C_{c}^{∞}(X) ⟶ C^{∞}(X) (5.8)

which are given locally by pseudodifferential operators of order m on X, in a precise sense. Let us recast this definition in terms of the Schwartz kernel theorem. Over the product, X^{2}, consider the right density bundle, Ω_{R} = π_{R}^{∗}Ω. Here we use the pull-back operation on vector bundles:

Theorem 5.2. If W ⟶ Y is a C^{∞} vector bundle and F : X ⟶ Y is a C^{∞} map then F^{∗}W ⟶ X is a well-defined C^{∞} vector bundle over X with total space

F^{∗}W = ⋃_{x∈X} W_{F (x)}; (5.9)

if ϕ ∈ C^{∞}(Y; W) then F^{∗}ϕ is a section of F^{∗}W and C^{∞}(X; F^{∗}W) is spanned by C^{∞}(X) · F^{∗}C^{∞}(Y; W).

Distributional sections of any C^{∞} vector bundle can be defined in two equivalent ways:

(5.10) “Algebraically” C^{−∞}(X; W) = C^{−∞}(X) ⨂_{C^{∞}(X)} C^{∞}(X; W)

or

(5.11) “Analytically” C^{−∞}(X; W) = [C_{c}^{∞}(X; Ω ⊗ W′)]

where W′ is the dual bundle and Ω the density bundle over X. In order to use (5.11) we need to define a topology on C_{c}^{∞}(X; U) for any vector bundle U over X. One can do this by reference to local coordinates.

If W ⟶ X is a vector bundle the spaces S^{m}(W) of symbols on W is welldefined for each m ∈ ℝ.

Proposition 5.1. A pseudodifferential operator A ∈ Ψ^{m}(X) can be written in terms of its kernel

K_{A} ∈ C^{∞}(X^{2}; Ω_{R})

where

K_{A} is C^{∞} in X^{2}\Δ (5.12)

and if X_{i} ⊂ X is a coordinate partition, ρ_{i} ∈ C_{c}^{∞}(X) has support in X_{i} then in terms of the same coordinates x_{i} = F_{i}^{∗}(x_{i}) and y_{i} = F^{∗}(x_{i}) in the two factors of X

ρ_{i}(x)ρ(y)K_{A} = K_{i}(x, y)|dy|, K_{i} ∈ Ψ^{m}(ℝ^{n}). (5.13)

Suppose ρ_{i}^{2} is a partition of unity of X, subordinate to a coordinate covering. For each i the symbol of K_{i} in (5.13) is an equivalence class on ℝ^{n} × (ℝ^{n}\0), with support in supp(ρ_{i}) ∗ (ℝ^{2}\0). Set

σ_{m}(τ) = ∑_{{i,π(τ)∈supp(ρ_{i})}} a_{i}(x^{(i)}, ξ^{(i)}) (5.14)

where

τ = F_{i}^{∗}(∑_{j} ξ_{j}^{(i)} · dx_{j}) ξ^{(i)} · dx ∈ T_{x^{(i)}}^{∗}ℝ^{n}, x^{(i)} = F_{i}(π(τ))

and the a_{i} are representations of the symbols of the K_{i}. This defines a function on T^{∗}X\0, in fact the equivalence class

σ_{m}(A) ∈ S^{m−[1]}(T^{∗}X) (5.15)

is well-defined.

Proposition 5.2. The principal symbol map in (5.15), defined as in (5.14), gives a short exact sequence:

0 ↪ Ψ^{m−1}(X) ↪ Ψ^{m}(X) ⟶^{σ_{m}} S^{m−[1]}(T^{∗}X) ⟶ 0. (5.16)

Proof. First we need to check that σ_{m}(A) is indeed well-defined. This involves checking what happens under a change of coordinate covering and a change of partition of unity subordinate to it. First, under a change of partition of unity, subordinate to a fixed covering note that

ρ′_{j}(x)ρ′_{j}(y)K_{A} = ∑_{i} ρ′_{j}(x)ρ_{i}^{2}(x)ρ′_{j}(y)K_{A}

≡ ∑_{i} ρ′_{j}(x)ρ_{i}(x)ρ_{i}(y)ρ′_{j}(y)K_{A}

≡ (ρ′_{j})^{2} ∑_{i} ρ_{i}(x)K_{A}ρ_{j}(y).

where equality is modulo Ψ^{m−1}, since [ϕ, Ψ^{m}] ⊂ Ψ^{m−1} for any C^{∞} function ϕ.

It follows from (5.16) that the principal symbols, defined by (5.14), for the two partitions of unity agree.

For a change of coordinate covering it suffices to use the transformation law for the principal symbol under a diffeomorphism of ℝ^{n} and the freedom, just established, to choose the partition of unity to be subordinates to both coordinate coverings. Thus σ_{m} is well-defined.

Certainly if A ∈ Ψ^{m−1}(X) then σ_{m}(N) ≡ 0. Moreover if A ∈ Ψ^{m}(X) and σ_{m}(A) ≡ 0 then all the operators ρ_{i}(x)Kρ_{i}(y) are actually of order m − 1. Since

A ≡ ∑_{i} ∑_{i} ρ′_{i}ρ_{j}(X)Aρ′_{i}ρ_{j} mod Ψ^{m−1}(X) (5.17)

for any two partitions of units ρ_{i}^{′2}, (ρ′_{i})^{2} we can choose the ρ′_{i} to each have support in a region where ρ_{j} ≠ 0 for some j. Then (5.17) shows that A ∈ Ψ^{m−1}(X).

Thus it only remains to show that the map σ_{m} is surjective. If a ∈ S^{m}(T^{∗}X) choose A_{i} ∈ Ψ_{∞}^{m}(ℝ^{n}) by

σ_{L}(A_{i}) = ρ_{i}(x)(F^{∗})^{−1}a_{i}ρ_{i}(y) ∈ S_{∞}^{m}(ℝ^{n} × ℝ^{n})

and set

A = ∑_{i} F_{i}^{∗}A_{i}G_{i}^{∗} G_{i} = F_{i}^{−1}. (5.18)

Then, from (5.14) σ_{m}(A) ≡ a by invariance of the principal symbol.

The other basic properties of the calculus are easily established. For example

σ_{m+m′}(A · B) = σ_{m}(A) · σ_{m}(B) (5.19)

if A ∈ Ψ^{m}(X), B ∈ Ψ^{m′}(X), X compact. Similarly note that

AB = ∑_{i,j} ρ_{i}^{2}Aρ_{j}^{2}B = ∑_{i,j} ρ_{i}Aρ_{i} · ρ_{j}Bρ_{i} mod Ψ^{m′_{m}−1}

which gives (5.19).

In § 4.4 we used the symbol calculus to construct a left and right parametrix for an elliptic element of Ψ^{m}(X), where X is compact, i.e. an element B ∈ Ψ^{−m}(X), such that

AB − Id, BA − Id ∈ Ψ^{−∞}(X). (5.20)

As a consequence of this construction note that:

Proposition 5.3. If A ∈ Ψ^{m}(X) is elliptic, and X is compact, then

A : C^{∞}(X) ⟶ C^{∞}(X) (5.21)

is Fredholm, i.e. has finite dimensional null space and closed range with finite dimensional complement. If ν is a non-vanishing C^{∞} measure on X and a generalized inverse of A is defined by

Gu = f if u ∈ Ran(A), Af = u, f ⊥_{ν} Nul(A)
Gu = 0 if u ⊥_{ν} Ran(A) (5.22)

then G ∈ Ψ^{−m}(X) satisfies

\begin{aligned}GA & = Id −π_{N} \\ AG & = Id −π_{R} \end{aligned} (5.23)

where π_{N} and π_{R} are ν-orthogonal projections onto the null space of A and the ν-orthocomplement of the range of A respectively.

Proof. The main point to note is that E ∈ Ψ^{−∞}(X) is smoothing,

E : C^{−∞}(X) ⟶ C^{∞}(X) ∀ E ∈ Ψ^{−∞}(X). (5.24)

Such a map is compact on L^{2}(X), i.e. maps bounded sets into precompact sets by the theorem of Ascoli and Arzela. The second thing to recall is that a Hilbert space with a compact unit ball is finite dimensional. Then

Nul(A) = {u ∈ C^{∞}(X); Au = 0} = {u ∈ L^{2}(X); Au = 0} (5.25) since, from (5.25) Au = 0 ⟹ (BA − Id)u = −Eu, E ∈ Ψ^{∞}(X), so Au = 0, u ∈ C^{−∞}(X) ⟹ u ∈ C^{∞}(X). Then

Nul(A) = {u ∈ L^{2}(X); Au = 0 ∫ |u|^{2}ν = 1} ⊂ L^{2}(X)

is compact since it is closed (A is continuous) and so Nul(A) = E(Nul(A)) is precompact. Thus Nul(A) is finite dimensional.

Next let us show that Ran(A) is closed. Suppose f_{j} = Au_{j} ⟶ f in C^{∞}(X), u_{j} ∈ C^{∞}(X). By what we have just shown we can assume that u_{j} ⊥_{ν} Nul(A). Now if B is the parametrix

u_{j} = Bf_{j} + Eu_{j}, E ∈ Ψ^{−∞}(X).

Suppose, along some subsequence, ∥u_{j}∥_{ν} ⟶ ∞. Then

\frac{u_{j}}{∥u_{j}∥_{ν}} = B (\frac{f_{j}}{∥u_{j}∥_{ν}} ) + E (\frac{u_{j}}{∥u_{j}∥_{ν}} )

shows that \frac{u_{j}}{∥u_{j}∥_{ν}} lies in a precompact subset of L^{2}, \frac{u_{j}}{∥u_{j}∥_{ν}} ⟶ u. This is a contradiction, since Au = 0 but ∥u∥ = 1 and u ⊥_{ν} Nul(A). Thus the norm sequence ∥u_{j}∥ is bounded and therefore the sequence has a weakly convergent subsequence, which we can relabel as u_{j}. The parametrix shows that u = Bf_{j} + Eu_{j} is strongly convergent with limit u, which satisfies Au = f.

Finally we have to show that Ran(A) has a finite dimensional complement. If π_{R} is orthogonal projection off Ran(A) then from the second part of (5.20) f = π_{R}E′f for some smoothing operator E. This shows that the orthocomplement has compact unit ball, hence is finite dimensional.

Notice that it follows that the two projections in (5.23) are both smoothing operators of finite rank.

5.4. Pseudodifferential operators on vector bundles

We have just shown that any elliptic pseudodifferential operator, A ∈ Ψ^{m}(X) on a compact manifold X has a generalized inverse B ∈ Ψ^{−m}(X) where

\begin{aligned}BA & = Id −π_{N} \\ AB & = Id −π_{R} \end{aligned} (5.26)

where π_{N} and π_{R} are the orthogonal projections onto the null space of A and the orthocomplement of the range of A with respect to a prescribed C^{∞} positive density ν, both are elements of Ψ^{−∞}(X) and have finite rank. To use this theorem in geometric situations we need first to make the “trivial” extension to operators on sections of vector bundles.

As usual there are two ways (at least) to approach this extension; the high road and the low road. The “low” road is to go back to the definition of Ψ^{m}(X) and to generalize to Ψ^{m}(X; V, W). This just requires to take the definition, following (5.6), but using a covering with respect to which the bundles V, W are both locally trivial. The local coordinate representatives of the pseudodifferential operator are then matrices of pseudodifferntial opertors. The symbol mapping becomes

Ψ^{m}(X; V, W) ⟶ S^{m−[1]} (T^{∗}X; Hom(V, W)) (5.27) where Hom(V, W) ≃ V ⊗ W′ is the bundle of homomorphisms from V to W and the symbol space consists of symbolic sections of the lift of this bundle to T^{∗}X. We leave the detailed description and proof of these results to the enthusiasts.

So what is the “high” road. This involves only a little sheaf-theoretic thought. Namely we want to define the space Ψ^{m}(X; V, W) using Ψ^{m}(X) by:

Ψ^{m}(X; V, W) = Ψ^{m}(X) ⨂_{C^{∞}(X^{2})} C^{∞}(X^{2}; V ⊠ W′). (5.28)

To make sense of this we first note that Ψ^{m}(X) is a C^{∞}(X^{2})-module as is the space C^{∞}(X^{2}; V ⊠ W′) where V ⊠ W′ is the “exterior” product:

(V ⊠ W′)_{(x,y)} = V_{x} ⊗ W′_{y}. (5.29)

The tensor product in (5.28) means that

(5.30) A ∈ Ψ^{m}(X; V, W) is of the form A = ∑_{i} A_{i} · G_{i}

where A_{i} ∈ Ψ^{m}(X), G_{i} ∈ C^{∞}(X^{2}; V ⊠ W′) and equality is fixed by the relation

ϕA · G − A · ϕG ≡ 0. (5.31)

Now what we really need to note is:

Proposition 5.4. For any compact C^{∞} manifold Y and any vector bundle U over Y

C^{−∞}(Y; U) ≡ C^{−∞}(Y) ⨂_{C^{∞}(Y)} C^{∞}(Y; U). (5.32)

Proof. C^{−∞}(Y; U) = (C^{∞}(Y; Ω ⊗ U′))′ is the definition. Clearly we have a mapping

C^{−∞}(Y) ⨂_{C^{∞}(Y)} C^{∞}(Y; U) ∋ ∑_{i} A_{i} · g_{i} ⟶ C^{−∞}(Y; U)

given by

∑_{i} u_{i} · g_{i}(ψ) = ∑_{i} u_{i}(g_{i} · ψ) (5.33)

since g_{i}ψ ∈ C^{∞}(Y; Ω) and linearity shows that the map descends to the tensor product. To prove that the map is an isomorphism we construct an inverse. Since Y is compact we can find a finite number of sections g_{i} ∈ C^{∞}(Y; U) such that any u ∈ C^{∞}(Y; U) can be written

u = ∑_{i} h_{i}g_{i} h_{i} ∈ C^{∞}(Y).

By reference to local coordinates the same is true of distributional sections with

h_{i} = u · q_{i} q_{i} ∈ C^{∞}(Y; U′).

This gives a left and right inverse.

Theorem 5.3. The calculus extends to operators sections of vector bundles over any compact C^{∞} manifolds.

5.9. COMPLEX POWERS 87

5.5. Laplacian on forms Since this is more differential geometry than differential analysis I will be brief.

We already defined the exterior derivative d : C^{∞}(X) ⟶ C^{∞}(X; T^{∗}X) (5.34) as part of the definition of T^{∗}X, i.e.

df(\overline{x}) = [f(x − f(\overline{x})] ∈ T ∗ ; \overline{x}X ∀ \overline{x} ∈ X.

The importance of the form bundles is tha they give a resolution of (5.34).

[Actually I didn’t mean to be this brief, but was interupted while writing the lecture!]

5.6. Hodge theorem 5.7. Heat kernel 5.8. Resolvent 5.9. Complex powers

88 5. MANIFOLDS