Inner Product Space

HODGE DECOMPOSITION

1. Some Linear Algebras

In this note, all the basis mentioned are ordered basis and all the vector spaces are real finite dimensional.

1.1. Inner Product Space. Let V be an n-dimensional real vector space. Given any two p-vectors v1∧ · · · ∧ v_p, and w1∧ · · · ∧ w_p in Λ^pV, we set

hv₁∧ · · · ∧ v_p, w₁∧ · · · ∧ w_pi_ΛpV = det (hv_i, w_ji_V)^p_i,j=1.

This function can be extended bilinearly to an inner product on Λ^pV. In other words, if η =P

i1,··· ,ipa_i1···i_pv_i1 ∧ · · · ∧ v_ip and ω =P

j1,··· ,jpb_i1···i_pw_j1 ∧ · · · ∧ w_jp, we define hη, ωi_ΛpV =X

i,j

a_i1···ipb_i1···iphv_i1 ∧ · · · ∧ v_ip, w_j1 ∧ · · · ∧ w_jpi_ΛpV. Lemma 1.1. The bilinear form h·, ·i_Λ^p_V on Λ^pV is an inner product.

Proof. The proof is left to the readers. 

The inner product on Λ^pV defined in Lemma 1.1 is called the inner product induced from V.

1.2. Oriented Vector Space. Let V be an n-dimensional real vector space. Any two ordered basis β = {v₁, · · · , v_n} and γ = {w₁, · · · , w_n} for V can be related by an n × n real invertible matrix A = (a_ij)ⁿ_i,j=1 by

w_j =

n

X

i=1

a_ijv_i.

We denote A by [1]^γ_β. Two ordered basis β, γ is said to be equivalent, denoted by β ∼ γ if det[1]^γ_β > 0.

Lemma 1.2. On the set of all basis for V, the relation ∼ is an equivalence relation.

Proof. The proof will be leave to readers as an exercise.  An orientation of V is a choice of an equivalence class; A finite dimensional real vector space together with an orientation is called an oriented vector space. A basis is said to be a positive basis for V if it belongs to the given orientation. In fact, we can say that V is oriented if V is given a specific ordered basis and a basis is said to be positive if the matrix relating the basis with the given one has positive determinant.

Definition 1.1. An oriented inner product space V is an inner product space together with an orientation.

1

Let V be an n-dimensional inner product space and β = {e1, · · · , en} be an orthonormal basis for V, i.e. he_i, e_ji = δ_ij for 1 ≤ i, j ≤ n. Let V be oriented by β. Denote the dual basis to β by {ψ1, · · · , ψn}. In other words, {ψ₁, · · · , ψn} forms a basis for V^∗, the dual space¹ for V and ψ_i(e_j) = δ_ij.

Theorem 1.1. (Riesz representation theorem) Let ψ be a linear functional on V. (The vector space V needs not be oriented.) Then there is a unique vector ξ_ψ in V such that

ψ(v) = hv, ξ_ψi.

Proof. Let us first proof the uniqueness. If ξ, ξ⁰ are two vectors such that ψ(v) = hv, ξi = hv, ξ⁰i. Then hv, ξ − ξ⁰i = 0 for all v ∈ V. Taking v = ξ − ξ⁰, we find hξ − ξ⁰, ξ − ξ⁰i = 0. By the property of inner product, ξ − ξ⁰ = 0. Hence ξ = ξ⁰.

Now, let us construct such a vector. Notice that if such a vector ξ exists, ψ(ei) = hei, ξi.

We know that any vector v ∈ V, has the Fourier expansion v = Pn

i=1hξ, e_iie_i. ξ must be of the form ξ = Pn

i=1ψ(e_i)e_i. In fact, if we take ξ = Pn

i=1ψ(e_i)e_i, we can verify ξ is the

required vector. 

This lemma also allows us to introduce a natural inner product on V^∗ by setting hψ, ϕi_V^∗= hξ_ψ, ξ_ϕi_V

where ξψ, ξϕ are the unique vectors in V representing ξψ and ξϕ. Let us define a map T : ΛⁿV → R by

T (v₁∧ · · · ∧ v_n) = det(ψ_i(v_j)) and extend it linearly to Λⁿ(V ), i.e.

T (av1∧ · · · ∧ v_n+ bw1∧ · · · ∧ w_n) = aT (v1∧ · · · ∧ v_n) + bT (w1∧ · · · ∧ w_n) for any a, b ∈ R and v1∧ · · · ∧ v_n, and w₁∧ · · · ∧ w_n in ΛⁿV.

Lemma 1.3. Let V be as above. T is an linear isomorphism of vector spaces.

Proof. As long as we can show that T is surjecitve, we know that T must be an isomorphism by dimension reason. Note that

T (e1∧ · · · ∧ e_n) = det(ψi(ej))ⁿ_i,j=1 = det In= 1.

Hence T (ae1∧ · · · ∧ e_n) = a for all a ∈ R and thus T is surjective.  We can also easily see that from definition

v₁∧ · · · ∧ v_n= T (v₁∧ · · · ∧ v_n)(e₁∧ · · · ∧ e_n) for any v1∧ · · · ∧ v_n∈ ΛⁿV. By definition,

T = ψ1∧ · · · ∧ ψ_n.

Let η be a p-vector in Λ^pV. We define an operator T_η : Λ^n−pV → R by T_η(ω) = T (η ∧ ω).

by the Riesz representation, we can choose a unique n − p-vector said ∗η in Λ^n−pV so that Tη(ω) = hω, ∗ηiΛ^pV.

It is equivalent to say that ∗η is the unque n − p vector in Λ^n−pV such that η ∧ ω = hω, ∗ηi_Λ^p_V(e₁∧ · · · ∧ e_n),

1The dual space of a vector space V is the space of all linear functionals on V.

for all ω ∈ Λ^n−pV.

Lemma 1.4. The function ∗ : Λ^pV → Λ^n−pV is a linear operator.

Proof. The proof is a corollary of the Riesz representation theorem.  Now, let us compute ∗1. By definition, T1(e1∧ · · · ∧ e_n) = he1 ∧ · · · ∧ e_n, ∗1i = 1. Since he₁ ∧ · · · ∧ e_n, e₁ ∧ · · · ∧ e_ni = 1, by uniqueness, we obtain ∗1 = e₁ ∧ · · · ∧ e_n. Similarly,

∗(e₁∧ · · · ∧ e_n) = 1.

Lemma 1.5. Let A be an n × n real matrix and w_i =Pn

i=1a_ije_i for 1 ≤ i ≤ n. Then w₁∧ · · · ∧ w_n= det(A)(e₁∧ · · · ∧ e_n).

Proof. We know w1∧ · · · ∧ w_n= a(e1∧ · · · ∧ e_n), for a = T (w1∧ · · · ∧ w_n). By definition, T (w1∧ · · · ∧ w_n) = (ψ1∧ · · · ∧ ψ_n)(w1, · · · , wn) = det(ψi(wj))ⁿ_i,j=1= det A.

 Since ∗ is a linear operator from Λ^pV → Λ^n−pV, we only need to compute ∗(ei1∧ · · · ∧ e_ip) for all 1 ≤ i₁ < · · · < i_p ≤ n.

Given an ordered set of indices I = {1 ≤ i₁ < · · · < i_p≤ n}, denote eI = ei1 ∧ · · · ∧ e_ip.

Let I = {1 ≤ i1 < · · · < ip ≤ n} and J = {1 ≤ j₁ < · · · < jq ≤ n} be two sets of indices.

If I ∩ J 6= ∅, e_I∧ e_J = 0. If I ∪ J = {1, 2, · · · , n}, then e_I∧ e_J = ±e₁∧ · · · ∧ e_n. In other words, ∗eI = ±eJ and ∗eJ = ±eI. In fact,

∗e_I =

(eJ if the ordered set I ∪ J is an even permutation of {1, · · · , n},

−e_J if the ordered set I ∪ J is an odd permutation of {1, · · · , n}.

As it turns out, we can prove

Proposition 1.1. ∗∗ = (−1)^p(n−p) : Λ^pV → Λ^pV.

The proof is based on the following identity:

eI∧ e_J = (−1)^p(n−p)eJ∧ e_I. Proposition 1.2. For any ω, η ∈ Λ^p(V ), we have

η ∧ ∗ω = hω, ηi(e₁∧ · · · ∧ e_n) = ω ∧ ∗η.

Proof. If we can show that η ∧ ∗ω = hω, ηi(e1∧ · · · ∧ e_n), then by the symmetry of inner product, ω ∧ ∗η = η ∧ ∗ω. So then we only need to verify this identity on basis.

We know that {eI} forms an orthonormal basis for Λ^pV. If I 6= I⁰ with #I = #I⁰ = p, he_I, e_I⁰i = 0. In this case, I ∩({1, · · · , n}\I⁰) is nonempty, e_I∧∗e_I⁰ = 0. Hence the statement holds when ω = eI and η = eI⁰ with I 6= I⁰. Now let us compute eI ∧ ∗e_I. By definition,

∗e_I = e_J when I ∪ J is an even permutation of {1, · · · , n}. Then e_I∧ e_J = e1∧ · · · ∧ e_n. On the other hand, he_I, e_Ii = 1. We find

eI∧ ∗e_I = e1∧ · · · ∧ e_n= heI, eIi(e₁∧ · · · ∧ e_n).

 Lemma 1.6. Let {v₁, · · · , v_n} be a positive basis for V.

∗1 = 1

q

det (hvi, vji)ⁿ_i,j=1

v₁∧ · · · ∧ v_n.

Proof. Let A = (aij)ⁿ_i,j=1 be the matrix relating {e1, · · · , en} and {v₁, · · · , vn}, i.e. v_j = Pn

i=1aijei. Then we know

v₁∧ · · · ∧ v_n= (det A)e₁∧ · · · ∧ e_n. By the orthonormality of {e₁, · · · , e_n}, we obtain

hv_j, v_ki =X

i,k

a_ija_lkhe_i, e_ki =X

i,k

a_ija_lkδ_ik =

n

X

i=1

a_ija_ik.

If we denote gjk = hvj, vki, and G = (g_jk)ⁿ_j,k=1, then G = A^∗A. By the property of determi- nant, det G = (det A)². Then we find

v₁∧ · · · ∧ v_n= (det G)^1/2e₁∧ · · · ∧ e_n.

We finish the proof using the fact ∗1 = e1∧ · · · ∧ e_n.  Let V1, · · · , Vm be inner product spaces and V =Lm

i=1Vi be the vector space direct sum of V₁, · · · , V_m. On V, we introduce a form

(1.1) h(v₁, · · · , vn), (w1, · · · , wn)iV =

n

X

i=1

hv_i, wii_Vi.

One can easily verify that (1.1) introduces an inner product on V. The vector space V together with the inner product (1.1) is called the direct sum of inner product spaces V₁, · · · , V_m.

Using this construction, we can introduce a natural inner product on Λ^∗V =L

i≥0ΛⁱV, where Λ⁰V = R induced from the inner product on V.

2. On Riemannian Manifold

Let (M, g) be a compact oriented connected smooth manifold of dimension n. Then for each x ∈ M, the inner product g(x) on T_xM induces an inner product on T_x^∗M. Locally, if g has the expression

g(x) =

n

X

i,j=1

g_ij(x)dxⁱ⊗ dx^j, then

g⁻¹(x) =

n

X

i,j=1

g^ij(x) ∂

∂xⁱ ⊗ ∂

∂x^j

is the inner product on T_x^∗M induced from g. Here (g^ij(x)) is the inverse matrix of (g_ij(x)).

Hence for each x, we obtain the Hodge ∗ operator ∗x : Λ^pT_x^∗ → Λ^n−pT_x^∗M. Since M is oriented, we obtain a global operator

∗ : Ω^p(M ) → Ω^n−p(M ).

We call dµ = ∗1 the volume form on M. Using Lemma 1.6, locally, dµ =

q

det(gij(x))dx¹∧ · · · ∧ dxⁿ. For each α, β ∈ Ω^p(M ), we can introduce an inner product:

hα, βi_p = Z

M

hα(x), β(x)i_Λ^p_Tx^∗_Mdµ = Z

M

α ∧ ∗β.

The L²-completion of Ω^p(M ) is denoted by L²(Ω^pM ). Similarly, we can introduce an inner product h·, ·i on Ω^∗(M ) =L

p≥0Ω^p(M ) whose L²-completion is denoted by L²(Ω^∗M ).

Lemma 2.1. Let d^∗: Ω^p(M ) → Ω^p−1(M ) be the operator defined by d^∗= (−1)^n(p+1)+1∗d∗, where d : Ω^p−1(M ) → Ω^p is the de Rham differential operator. Then

(2.1) hdα, βi = hα, d^∗βi

for any α ∈ Ω^p−1(M ) and β ∈ Ω^p(M ), i.e. d^∗ is the (formal) adjoint of d.

One can check that (d^∗)² = 0.

Definition 2.1. The Laplace operator on Ω^p(M ) is defined by

∆ = (d + d^∗)² = d^∗d + dd^∗.

A p-form ω ∈ Ω^p(M ) is said to be harmonic if ∆ω = 0. The space of harmonic p-forms is denoted by H^p.

Corollary 2.1. ∆ is (formally) self-adjoint:

h∆α, βi = hα, ∆βi for any α, β ∈ Ω^p(M ).

One can easily verify that

h∆α, αi = kdαk²+ kd^∗αk². It follows from this identity that

Corollary 2.2. A p-form α is harmonic if and only if dα = 0 and d^∗α = 0.

A p-form α is said to be co-closed if d^∗α = 0. This corollary tells us that a p-form is harmonic if and only if it is closed and co-closed.

If f ∈ C^∞(M ) = Ω⁰(M ) is harmonic function, then df = 0. In this case, f is a locally constant function on M. Since M is connected, f is a constant. We conclude that:

Corollary 2.3. Every harmonic function on M is a constant.

One can verify that ∆ : Ω^p(M ) → Ω^p(M ) is an elliptic operator. By the theory of P.D.E,

∆ is a Fredholm operator, i.e. ker ∆ and coker ∆ are finite dimensional vector spaces. Since

∆ is self-adjoint, coker ∆ = ker ∆. This implies that we have an orthogonal direct sum decomposition:

Ω^p(M ) = H^p⊕ Im ∆.

This gives us the following orthogonal direct sum decomposion:

Ω^p(M ) = H^p⊕ dΩ^p−1(M ) ⊕ d^∗Ω^p+1(M ).

As a result,

ker d|_Ωp(M )= H^p⊕ dΩ^p−1(M ).

The de Rham cohomology group is the quotient space H_dR^p (M ) = ker d|_Ωp(M )/dΩ^p−1(M ).

We obtain the following isometric isomorphism:

H_dR^p (M ) ∼= H^p.

Theorem 2.1. Let (M, g) be a compact oriented connected Riemannian manifold. Then dim_RH_dR^p (M ) < ∞.

Let M be as above. We define a bilinear map:

B : H_dR^p (M ) × H_dR^n−p(M ) → R by B([ω], [η]) = R

Mω ∧ η, where ω and η are representatives of [ω] and [η] respectively.

Then B is a well-defined map. Notice that for a nonzero closed p-form α, we have hα, αi =

Z

M

α ∧ ∗α > 0.

This shows that B([α], [∗α]) > 0. In other words, B is a non degenerate² bilinear form.

Theorem 2.2. Let M be as above. The vector space H_dR^p (M ) is isomorphic to H_dR^n−p(M )^∗ via B.

Proof. The theorem follows from the following lemma. 

Lemma 2.2. Let V, W be finite dimensional vector spaces and B : V ×W → R be a bilinear form. If B is non degenerate, then V can be identified with W^∗ and W^∗ can be identified with V.

Proof. Let Φ : V → W^∗ be the linear map defined by Φ(v) = B(v, ·). Since B is bilinear, Φ(v) ∈ W^∗. Since B is non degenerate, ker Φ = {0}. Similarly, Ψ : W → V^∗ defined by w 7→ B(·, w) is linear and ker Φ = {0}. By the rank nullity lemma and dim V = dim V^∗, dim W = dim W^∗, we find dim V = dim W. Again, by the rank nullity lemma, Φ : V → W^∗ and Ψ : W → V^∗ are linear isomorphism.

 This implies that dim H_dR^p (M ) = dim H_dR^n−p(M ).

Corollary 2.4. Let M be a compact connected oriented smooth manifold of dimension n and bi(M ) be its i-th Betti number, i.e. bi(M ) = dim H_dRⁱ (M ). Then

(1) b_n(M ) = 1,

(2) bp(M ) = bn−p(M ) for 0 ≤ p ≤ n.

2Let V, W be vector spaces. A bilinear form B : V × W → R is non degenerate if for each v 6= 0 in V, there is w ∈ W so that B(v, w) 6= 0.