´E POINCAR D U A L I T Y A N G L E SO NR I E MA N N I A N A N I FO L D S WI T HB O U N D A R Y

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POINCAR´

E DUALITY ANGLES ON RIEMANNIAN

MANIFOLDS WITH BOUNDARY

Clayton Shonkwiler

A Dissertation

in

Mathematics

Presented to the Faculties of the University of Pennsylvania in Partial

Fulfillment of the Requirements for the Degree of Doctor of Philosophy

2009

Dennis DeTurck

Supervisor of Dissertation

Herman Gluck

Supervisor of Dissertation

Tony Pantev

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Acknowledgments

I am incredibly lucky to have two wonderful advisors in Dennis DeTurck and Herman Gluck. They are amazing mathematicians, teachers, friends and role models and their guidance, support, insight and good humor has been indispensable. Thank you.

Thank you also to Paul Melvin and Rafal Komendarczyk, who have been knowledgeable and friendly sources of advice, patient teachers and tireless collaborators.

All of my fellow graduate students deserve thanks for making Penn’s math department such a great place, but Chris Jankowski, Jen Hom, Paul Rowe, John Olsen, Andrew Obus, David Fithian, Michael McDuffee, Wil Brady, Kendra Nelsen and Naomi Utgoff merit special mention as friends, inspirations, occasional commiserators and, in several cases, extremely patient officemates.

Lisa Traynor and Josh Sabloff run the best seminar in the area and have made me, at least one day a week, a morning person.

Shea Vela-Vick, who gave me a ride to that seminar each week, fits so many of the above categories that he defines his own category. Thank you, Shea, for being a friend and an inspiration. Janet Burns, Monica Pallanti, Paula Scarborough and Robin Toney made life easier in many ways that I noticed and probably many more that I didn’t. Thanks for your patience, reminders and smiles.

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ABSTRACT

POINCAR ´E DUALITY ANGLES ON RIEMANNIAN MANIFOLDS WITH BOUNDARY

Clayton Shonkwiler

Dennis DeTurck and Herman Gluck, Advisors

On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces correspond-ing to cohomology comcorrespond-ing from the interior and boundary of the manifold. The principal angles between these interior subspaces are all acute and are calledPoincaré duality angles. This disserta-tion determines the Poincaré duality angles of a collection of interesting manifolds with boundary derived from complex projective spaces and from Grassmannians, providing evidence that the Poincaré duality angles measure, in some sense, how “close” a manifold is to being closed.

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List of Figures

1.1 The concrete realizations of the absolute and relative cohomology groups . . . 2

2.1 Hp_N(M) andHp_D(M) . . . 18

3.1 S2n−1(cost)×S1(sint) sitting over the hypersurface at distancetfrom CPn−1 . . 28

4.1 Elements of the subGrassmanniansG1Rn+1 andG2Rn+1. . . 38

4.2 The Stiefel bundleV2Rn+1→G2Rn+1 . . . 41

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Chapter 1

Introduction

Consider a closed, smooth, oriented Riemannian manifold Mn. For any pwith 0 ≤ p≤n, the Hodge Decomposition Theorem [Hod34, Kod49] says that the pth cohomology group Hp₍_M_;

R)

is isomorphic to the space of closed and co-closed differential p-forms onM. Thus, the space of such forms (called harmonic fieldsby Kodaira) is a concrete realization of the cohomology group

Hp₍_M_;

R) inside the space Ωp(M) of allp-forms onM.

SinceM is closed,∂M =∅, soHp₍_M_;

R) =Hp(M, ∂M;R) and thus the concrete realizations

of Hp(M;R) and Hp(M, ∂M;R) coincide. This turns out not to be true for manifolds with

non-empty boundary.

When Mn is a compact, smooth, oriented Riemannian manifold with non-empty boundary

∂M, the relevant version of the Hodge Decomposition Theorem was proved by Morrey [Mor56] and Friedrichs [Fri55]. It gives concrete realizations of bothHp₍_M_;

R) andHp(M, ∂M;R) inside

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determine how close the manifold M is to being closed.

The relative positions of these spaces was recently described more completely by DeTurck and Gluck [DG04]. They noted that Hp(M;R) and Hp(M, ∂M;R) each has a portion consisting of

those cohomology classes coming from the boundary ofM and another portion consisting of those classes coming from the “interior” of M. DeTurck and Gluck showed that these interior and boundary portions manifest themselves as orthogonal subspaces inside the concrete realizations of Hp₍_M_;

R) and Hp(M, ∂M;R). This leads to a refinement of the Hodge–Morrey–Friedrichs

decomposition which says, in part, that

(i). the concrete realizations of Hp₍_M_;

R) andHp(M, ∂M;R) meet only at the origin,

(ii). the boundary subspace of each is orthogonal to all of the other, and

(iii). the principal angles between the interior subspaces are all acute.

This behavior is depicted in Figure 1.1.

H_p

(_M ;

R₎

bound ar

y

bound ary

H

p(M

, ∂M

;R)

interior in

terior

Figure 1.1: The concrete realizations of the absolute and relative cohomology groups

The principal angles between the interior subspaces of the concrete realizations of Hp₍_M_;

R)

and Hp₍_{M, ∂M}_;

R) are invariants of a Riemannian manifold with boundary and were christened

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interior and since the concrete realizations of the absolute and relative cohomology groups coincide on such a manifold, it seems reasonable to guess that the Poincar´e duality angles go to zero as a manifold closed up.

One aim of this dissertation is to provide evidence for this hypothesis by determining the Poincar´e duality angles for some interesting manifolds which are intuitively close to being closed. For example, consider the complex projective space _CPn with its usual Fubini-Study metric and consider the manifold

Mr:=CPn−Br(q)

obtained by removing a ball of radius r centered at the point q ∈_CPn_{. If the Poincar´}_{e duality} angles do measure how close a manifold is to being closed, the Poincar´e duality angles of Mr should be small whenris near zero.

Theorem 1. For 1 ≤ k ≤n−1 there is a non-trivial Poincar´e duality angle θ2kr between the

concrete realizations of H2k(Mr;R) andH2k(Mr, ∂Mr;R) which is given by

cosθ_r2k= 1−sin 2n_r

q

(1 + sin2nr)2₊(n−2k)2 k(n−k) sin

2n_r

.

Indeed, asr→0, the Poincar´e duality anglesθ2k

r →0. Moreover, asrapproaches its maximum value ofπ/2, theθ2k_r →π/2.

Theorem 1 immediately generalizes to other non-trivial D2-bundles over CPn−1 with an

ap-propriate metric, so this asymptotic behavior of the Poincar´e duality angles is not dependent on being able to cap off the manifold with a ball.

Removing a ball around some point in a closed manifold is just a special case of removing a tubular neighborhood of a submanifold. With this in mind, consider G2Rn+2, the Grassmannian

of oriented 2-planes in Rn+2, and define

Nr:=G2Rn+2−νr G1Rn+1,

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Theorem 2. For 1 ≤ k ≤ n−1 there is exactly one Poincar´e duality angle θ2k

r between the

concrete realizations of H2k₍_N_r;

R)andH2k(Nr, ∂Nr;R)given by

cosθr2k=

1−sinnr

q

(1 + sinnr)2₊(n−2k)2 k(n−k) sin

n_r.

Again, theθr2k→0 asr→0 andθ2kr →π/2 asrapproaches its maximum value of π/2. Whenn= 2, the GrassmannianG2R4is isometric to the productS2 1/

√

2

×S2 ₁_/√₂ and the subGrassmannianG1R3corresponds to the anti-diagonal 2-sphere∆ (cf. [GW83]). Hence, thee the Riemannian manifold with boundary S2 ₁_/√₂

×S2 ₁_/√₂

−νr

e

∆has a single Poincar´e duality angleθ2

r in dimension 2 given by

cosθ2_r=1−sin 2_r

1 + sin2r.

Theorems 1 and 2 suggest the following conjecture:

Conjecture 3. Let Mm be a closed, smooth, oriented Riemannian manifold and let Nn be a closed submanifold of codimension ≥2. Define the compact Riemannian manifold

Mr:=M −νr(N),

whereνr(N)is the open tubular neighborhood of radiusraboutN (restrictingrto be small enough that ∂Mr is smooth). Then, ifθrk is a Poincar´e duality angle ofMr in dimensionk,

θkr =O(rm−n)

forr near zero.

The Poincar´e duality angles seem to be interesting invariants even in isolation, but the other aim of this dissertation is to show that they are related to the Dirichlet-to-Neumann operator for differential forms, which arises in certain inverse problems of independent interest.

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map for functions which arises in the problem of Electrical Impedance Tomography (EIT). The EIT problem was first posed by Calder´on [Cal80] in the context of geoprospecting, but is also of considerable interest in medical imaging.

The Dirichlet-to-Neumann map Λ is a map Ωp(∂M)→Ωn−p−1(∂M) and can be used to define the Hilbert transform T =dΛ−1_{. The connection to the Poincar´}_{e duality angles is given by the} following theorem:

Theorem 4. If θp₁, . . . , θ_kp are the principal angles between the concrete realizations of Hp(M;R)

andHp₍_{M, ∂M}_;

R)(i.e. the Poincar´e duality angles in dimension p), then the quantities

(−1)pn+p+ncos2θp_i

are the non-zero eigenvalues of a suitable restriction of T2.

Belishev and Sharafutdinov showed that the cohomology groups of M can be completely de-termined from the boundary data (∂M,Λ) and, in fact, that this data determines the long exact sequence of the pair (M, ∂M). Theorem 4 shows that the data (∂M,Λ) not only determines the interior and boundary subspaces of the cohomology groups, but detects their relative positions as subspaces of differential forms onM.

At the end of their paper, Belishev and Sharafutdinov posed the following question:

Can the multiplicative structure of cohomologies be recovered from our data (∂M,Λ)? Till now, the authors cannot answer the question.

The mixed cup product

∪:Hp(M;R)×Hq(M, ∂M;R)→Hp+q(M, ∂M;R)

can be reconstructed from the Dirichlet-to-Neumann map when the relative class comes from the boundary subspace, giving a partial answer to Belishev and Sharafutdinov’s question:

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When the manifold M occurs as a region in Euclidean space, all relative cohomology classes come from the boundary subspace, so Theorem 5 has the following immediate corollary:

Corollary 6. IfMn is a compact region inRn, the boundary data(∂M,Λ)completely determines

the mixed cup product on M.

The expression for the reconstruction of the mixed cup product given in the proof of Theorem 5 makes sense even when the relative class does not come from the boundary subspace, suggesting that the data (∂M,Λ) may determine the full mixed cup product. It remains an interesting question whether the Dirichlet-to-Neumann data determines the absolute or relative cup products onM.

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Chapter 2

Preliminaries

2.1 Poincar´

e duality angles

DeTurck and Gluck’s Poincar´e duality angles arise from a refinement of the classical Hodge– Morrey–Friedrichs decomposition for Riemannian manifolds with boundary. This section reviews the Hodge decomposition theorem for closed manifolds and the Hodge–Morrey–Friedrichs decom-position for manifolds with boundary in building up to the definition of the Poincar´e duality angles and the proof of DeTurck and Gluck’s main theorem.

2.1.1 The Hodge decomposition for closed manifolds

LetMn_{be a closed, oriented, smooth Riemannian manifold of dimension}_n_{. For each}_p_{between 0} and nlet Ωp(M) be the space of smooth differentialp-forms onM and let Ω(M) =Ln_i=0Ωi(M) be the algebra of all differential forms onM.

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space of exact p-forms, thepth de Rham cohomology group is defined as the quotient

H_dRp (M) :=Cp_/Ep_.

De Rham’s theorem [dR31] states that

H_dRp (M)∼=Hp(M;R),

the pth singular cohomology group with real coefficients. If Ω(M) is equipped with an inner product, it is natural to expect that the cohomology group Hp(M;R) will be realized as the

orthogonal complement of the space of exactp-forms inside the space of closedp-forms onM. It is the Riemannian metric which gives an inner product on Ω(M).

The metric allows the Hodge star

?: Ωp(M)→Ωn−p(M) to be defined for eachp. In turn, the co-differential

δ= (−1)n(p+1)+1? d ?: Ωp(M)→Ωp−1 and the Hodge Laplacian

∆ =dδ+δd

are defined using the exterior derivative and the Hodge star. Then theL2_{inner product on Ω}p₍_M_{) is defined as}

hα, βiL2 := Z

M

α∧? β

for α, β ∈Ωp₍_M_{). This inner product is extended to all of Ω(}_M_{) be declaring that, for} _i ₆₌_j_, Ωi(M) is orthogonal to Ωj(M).

Consider the following subspaces of Ωp(M):

Ep₍_M_{) :=}_{ω_∈_Ωp₍_M_{) :}_ω₌_dη_{for some} _η_∈_Ωp−1_}

cEp₍_M_{) :=}_{ω_∈_Ωp₍_M_{) :}_ω₌_δξ_{for some}_ξ_∈_Ωp+1₍_M₎_}

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the spaces of exact p-forms, co-exact p-forms and harmonic p-fields, respectively. On closed manifolds the space Hp₍_M_{) of harmonic} _p_{-fields coincides with the space}

b

Hp₍_M_{) :=}_{ω_∈_Ωp₍_M_{) : ∆}_ω_{= 0}_}

of harmonicp-forms, but this is special to closed manifolds and fails on manifolds with boundary. With the above notation in place, the Hodge Decomposition Theorem can now be stated:

Hodge Decomposition Theorem. LetMn_{be a closed, oriented, smooth Riemannian manifold.}

For each integer p such that 0 ≤ p≤ n, the spaceΩp₍_M₎ _{of smooth} _{p-forms on} _M _{admits the}

L2-orthogonal decomposition

Ωp(M) =cEp₍_M₎_{⊕ H}p₍_M₎_{⊕ E}p₍_M₎_. _(2.1.1)

Moreover, the spaceHp₍_M₎_{of harmonic}_{p-fields is finite-dimensional and}

Hp₍_M₎_∼

=Hp(M;R).

The Hodge Decomposition Theorem has its historical roots in the work of Helmholtz [Hel58]; the modern version was developed through the work of Hodge [Hod34, Hod41], Weyl [Wey40] and Kodaira [Kod49]. A complete proof is given in Warner’s book [War83].

The fact thatHp₍_M₎∼₌_Hp₍_M_;

R) follows from the decomposition (2.1.1), which implies that

the space of closed p-forms Cp₍_M_{) =} _Hp₍_M₎_{⊕ E}p₍_M_{). In other words, the space of harmonic}

p-fieldsHp₍_M_{) is the orthogonal complement of}_Ep₍_M_{) inside the space of closed}_p_{-forms. Then,} as expected from the de Rham theorem,

Hp(M;R)∼=H_dRp (M) =Cp₍_M₎_/Ep₍_M₎_∼

=Hp₍_M₎_.

2.1.2 The Hodge–Morrey–Friedrichs decomposition

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The space Ωp₍_M_{) and the maps} _d_, _?_, _δ _{and ∆ are defined just as in Section 2.1.1. Let}_d ∂, ?∂,

δ∂ and ∆∂ denote the exterior derivative, Hodge star, co-derivative and Laplacian on the closed Riemannian manifold∂M.

One consequence of the Hodge theorem for closed manifolds is that the pth real cohomology group Hp₍_M_;

R) can be realized as the subspace of harmonicp-fieldsHp(M) inside the space of

p-forms. When the boundary is non-empty the spaceHp₍_M_{) is infinite-dimensional and so is much} too big to represent the cohomology. Also, on a manifold with boundary there are twodifferent

pth cohomology groups: the absolute cohomology group Hp₍_M_;

R) and the relative cohomology

group Hp₍_{M, ∂M}_;

R).

That being said, the de Rham theorem stated earlier for closed manifolds holds equally well for the absolute cohomology groups on a manifold with boundary: Hp(M;R) is isomorphic to

the quotient of the space of closed p-forms by the space of exact p-forms. Thus, the concrete realization of the absolute cohomology group Hp₍_M_;

R) ought to be the orthogonal complement

ofEp₍_M_{) inside the space of closed}_p_-forms.

For the relative version, consider those smoothp-formsω whose pullback i∗ω to ∂M is zero. Define

Ωp(M, ∂M) :={ω∈Ωp(M) :i∗ω= 0},

the space ofrelative p-formsonM. Let

Cp(M, ∂M) :={ω∈Ωp(M, ∂M) :dω= 0}

Ep₍_{M, ∂M}_{) :=}_{ω_∈_Ωp₍_{M, ∂M}_{) :}_ω₌_dη_{for some} _η_∈_Ωp−1₍_{M, ∂M}₎_}

denote the subspaces of closed relative p-forms and relatively exact p-forms, respectively. The

relative de Rham cohomology groupscan then be defined as

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Duff [Duf52] proved the relative version of de Rham’s theorem, namely that

Hp(M, ∂M;R)∼=H_dRp (M, ∂M).

Thus, the concrete realization of the relative cohomology group Hp(M, ∂M;R) ought to be the

orthogonal complement of the subspace of relatively exactp-forms inside the space of closed relative

p-forms.

The definition of the relative p-forms introduces the boundary condition i∗ω = 0, which is called theDirichlet boundary condition. A formωsatisfying the Dirichlet boundary condition can be thought of as “normal” to the boundary, since, for any x∈∂M andV1, . . . , Vp∈TxM,

ω(V1, . . . , Vp)6= 0

only if one of the Vi has a non-trivial component in the direction of the inward-pointing unit normal vector N. Forms satisfying the Dirichlet boundary condition are just the relative forms and the relatively exact forms are just those which are exact and whose primitives satisfy the Dirichlet boundary condition.

Forx∈∂M, letπ:TxM →Tx∂M be the orthogonal projection. Then it is natural to think of a formω∈Ωp₍_M_{) as “tangent” to the boundary if}

ω(V1, . . . , Vp) =ω(πV1, . . . , πVp)

for any V1, . . . , Vp ∈TxM. Equivalently, a formω is tangent to the boundary if the contraction

ιNω of the unit inward-pointing normal vectorN intoω is zero. Therefore, ifωis tangent to the

boundary,

0 =?∂ιNω=i∗? ω.

The boundary condition i∗? ω = 0 is called the Neumann boundary condition. If ω ∈ Ωp₍_M₎ satisfies the Neumann condition, then ? ω∈Ωn−p(M) satisfies the Dirichlet condition. Likewise,

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LetcEp₍_M_),_Hp₍_M_{) and}_Ep₍_M_{) be the spaces of co-exact}_p_{-forms, harmonic}_p_{-fields and exact}

p-forms, respectively. Throughout what follows, juxtaposition of letters denotes intersections; for example,

cEHp₍_M_{) :=}_cEp₍_M₎_{∩ H}p₍_M₎ _and _EHp₍_M_{) :=}_Ep₍_M₎_{∩ H}p₍_M₎_.

Unlike in the case of a closed manifold, these spaces are non-trivial on a compact manifold with boundary.

SubscriptsN andD indicate Neumann and Dirichlet boundary conditions, respectively:

cE_Np(M) :={ω∈Ωp(M) :ω=δξfor some ξ∈Ωp+1(M) wherei∗? ξ= 0} H_Np(M) :={ω∈Ωp(M) :dω= 0, δω= 0, i∗? ω= 0}

Hp_D(M) :={ω∈Ωp(M) :dω= 0, δω= 0, i∗ω= 0}

E_Dp(M) :={ω∈Ωp(M) :ω=dη for someη∈Ωp−1(M) wherei∗η= 0}.

A key subtlety is that the boundary conditions apply to the primitiveof ω in the definitions of

cE_Np(M) and E_Dp(M), whereas they apply to the formω itself in the definitions of Hp_N(M) and

Hp_D(M).

Of course, ifω∈ E_Dp(M), thenω=dη for some Dirichlet formη∈Ωp−1₍_M_{) and}

i∗dη=d∂i∗η= 0,

so elements ofE_Dp(M) do themselves satisfy the Dirichlet boundary condition. In particular, this means that E_Dp(M) is precisely the space of relatively exact p-forms. Likewise, ifω ∈ cE_Np(M), thenω=δξ for some Neumann formξ∈Ωp+1(M) and

i∗? δξ= (−1)p+1i∗d ? ξ= (−1)p+1d∂i∗? ξ= 0, so forms incE_Np(M) satisfy the Neumann boundary condition.

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Hodge–Morrey–Friedrichs Decomposition Theorem. LetM be a compact, oriented, smooth Riemannian manifold with non-empty boundary ∂M. Then the space Ωp₍_M₎_{can be decomposed}

as

Ωp(M) =cE_Np(M)⊕ Hp_N(M)⊕ EHp₍_M₎_{⊕ E}p

D(M) (2.1.2)

=cE_Np(M)⊕cEHp(M)⊕ Hp_D(M)⊕ E_Dp(M), (2.1.3)

where the direct sums are L2-orthogonal. Moreover,

Hp(M;_R)∼=Hp_N(M)

Hp(M, ∂M;R)∼=HpD(M) Morrey proved that

Ωp(M) =cE_Np(M)⊕ Hp₍_M₎_{⊕ E}p

D(M) (2.1.4)

and Friedrichs gave the two decompositions of the harmonic fields:

Hp₍_M_{) =}_Hp

N(M)⊕ EH

p₍_M₎ _(2.1.5)

=cEHp₍_M₎_{⊕ H}p

D(M); (2.1.6)

both were influenced by the work of Duff and Spencer [DS52]. The orthogonality of the components follows immediately from Green’s Formula:

Green’s Formula. Let α∈Ωp−1(M)andβ ∈Ωp(M). Then hdα, βiL2− hα, δβi_L2 =

Z ∂M

i∗α∧i∗? β.

As in the closed case, the isomorphisms ofHp_N(M) andHp_D(M) with the absolute and relative cohomology groups follow immediately from the decompositions (2.1.2) and (2.1.3) along with the de Rham and Duff theorems. In particular, (2.1.2) shows that Hp_N(M) is the orthogonal complement of the exactp-forms inside the space of closedp-forms, so it is the concrete realization of Hp₍_M_;

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relatively exact p-forms E_Dp(M) inside the space of closed relative p-forms (i.e. those satisfying the Dirichlet boundary condition), so it is the concrete realization of Hp₍_{M, ∂M}_;

R).

A complete proof of the Hodge–Morrey–Friedrichs decomposition is given in Chapter 2 of Schwarz’s book [Sch95].

2.1.3 Poincar´

e duality angles

The Hodge–Morrey–Friedrichs Decomposition Theorem shows that there are concrete realizations

Hp_N(M) and Hp_D(M) of the absolute and relative pth cohomology groups inside the space of

p-forms. In fact,

Hp_N(M)∩ Hp_D(M) ={0},

so these concrete realizations meet only at the origin. This follows from the strong unique con-tinuation theorem of Aronszajn, Krzywicki and Szarski [AKS62] (cf. [Sch95, Theorem 3.4.4] for details). But Hp_N(M) and H_Dp(M) are not orthogonal in general and so cannot both appear in the same orthogonal decomposition of Ωp₍_M_{). As DeTurck and Gluck note, the best that} can be done is the following five-term decomposition, which is an immediate consequence of the Hodge–Morrey–Friedrichs decomposition:

Theorem 2.1.1. Let M be a compact, oriented, smooth Riemannian manifold with non-empty boundary. Then the space Ωp₍_M₎_{of smooth} _{p-forms on}_M _{has the direct-sum decomposition}

Ωp(M) =cE_Np(M)⊕ EcEp₍_M₎_⊕₍_Hp

N(M) +H p

D(M))⊕ E p D(M).

In the statement of Theorem 2.1.1, the symbol⊕indicates an orthogonal direct sum, whereas the symbol + just indicates a direct sum.

Proof of Theorem 2.1.1. Since the spaces Hp_N(M) and H_Dp(M) meet only at the origin, the sum

Hp_N(M) +Hp_D(M) is direct. Using the Hodge–Morrey–Friedrichs decomposition, the orthogonal complement of Hp_N(M) inside Hp₍_M_{) is}_EHp₍_M_{), while the orthogonal complement of} _Hp

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insideHp₍_M_{) is}_cEHp₍_M_{). Hence, the orthogonal complement of}_Hp

N(M)+H p

D(M) insideHp(M) is

EHp₍_M₎_∩_cEHp₍_M_{) =}_EcEp₍_M₎_.

The rest of the proposed decomposition of Ωp(M) is the same as in the Hodge–Morrey–Friedrichs Decomposition Theorem, so this completes the proof of Theorem 2.1.1.

DeTurck and Gluck’s key insight was that the non-orthogonality ofH_Np(M) andHp_D(M) has to do with the fact that some of the cohomology ofM comes from the “interior” ofM and some comes from the boundary.

In absolute cohomology, the interior subspace is very easy to identify. Consider the map

i∗:Hp(M;R)→Hp(∂M;R). The kernel ofi∗ certainly deserves to be called the interior portion

ofHp(M;R), but it is not clear what the boundary portion should be. SinceHp(M;R)∼=HpN(M), the interior portion of the absolute cohomology is identifiable as the subspace of the harmonic Neumann fields which pull back to zero in the cohomology of the boundary; i.e.

E∂Hp_N(M) :={ω∈ Hp_N(M) :i∗ω=dϕfor some ϕ∈Ωp−1(∂M)}.

On the other hand, it is the boundary subspace which is easy to identify in relative cohomology. Letj:M = (M,∅)→(M, ∂M) be the inclusion and consider the long exact de Rham cohomology sequence of the pair (M, ∂M)

· · · H_dRp (∂M) i∗

H_dRp (M) j∗

H_dRp (M, ∂M) d

H_dRp−1(∂M) · · · . (2.1.7) The mapd:H_dRp (∂M)→H_dRp (M, ∂M) takes a closed (p−1)-formϕon∂M, extends it arbitrarily to a (p−1)-formϕ_eonM, then definesd[ϕ] := [dϕ_e]. The formdϕ_eis certainly exact, but is not in general relatively exact. Then the image ofH_dRp−1(∂M) insideH_dRp (M, ∂M) is the natural portion to interpret as coming from the boundary.

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long exact sequence

· · · Hp(∂M) i∗

Hp_N(M) j∗

Hp_D(M) d

Hp−1₍_∂M₎

· · ·

and the boundary subspace of Hp_D(M) consists of those Dirichlet fields which are exact: the subspace EHp_D(M).

The Hodge star takes Hn_D−p(M) to H_Np(M) and takes its boundary subspace EHn_D−p(M) to

cEHp_N(M); regard this as the boundary subspace ofHp_N(M). Likewise, the interior subspace of

Hp_D(M) is identified as the image under the Hodge star of the interior subspace E∂H n−p N (M) of

Hn_N−p(M), namely the subspace

cE∂Hp_D(M) :={ω∈ Hp_D(M) :i∗? ω=dψfor someψ∈Ωn−p−1(∂M)}.

DeTurck and Gluck’s refinement of the Hodge–Morrey–Friedrichs decomposition says that in both Hp_N(M) and Hp_D(M) the boundary subspace is the orthogonal complement of the interior subspace.

Theorem 2.1.2 (DeTurck–Gluck). The spaces Hp_N(M) and Hp_D(M) admit the L2_-orthogonal

decompositions into interior and boundary subspaces

Hp_N(M) =cEH_Np(M)⊕ E∂HpN(M) (2.1.8)

Hp_D(M) =EHp_D(M)⊕cE∂Hp_D(M). (2.1.9)

Proof. It is sufficient to prove (2.1.8), since substituting n−pforp in (2.1.8) and applying the Hodge star gives (2.1.9).

The fact that the two terms on the right hand side of (2.1.8) are orthogonal follows from Green’s formula. If δξ∈cEHp_N(M) and ω ∈ E∂H

p

N(M), then i∗ω=dϕfor someϕ∈Ω

p−1₍_∂M₎ and

hω, δξiL2 =hdω, ξi_L2− Z

∂M

i∗ω∧i∗? ξ=−

Z ∂M

dϕ∧i∗? ξ

sinceω is closed. Extendϕarbitrarily to some formϕ_e∈Ωp−1₍_M_{); then}

hω, δξiL2 =− Z

∂M

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Running Green’s formula again,

hω, δξiL2 =hd e

ϕ, δξiL2 =h e

ϕ, δδξiL2+ Z

∂M

i∗ϕ_e∧i∗? δξ= 0 sinceδξis a Neumann form and hencei∗_{? δξ}_{= 0. Thus, the spaces}_cEHp

N(M) andE∂H p

N(M) are orthogonal.

The proof that the sum ofcEHp_N(M) andE∂H p

N(M) is equal toH p

N(M) proceeds in two steps. Step 1: Letcp₁, . . . , cp

g be absolutep-cycles which form a homology basis for i∗Hp(∂M;R)⊂

Hp(M;R). The goal is to show that there is a uniquep-form incEHpN(M) with preassigned periods

C1, . . . , Cg on thesep-cycles.

Extend the given homology basis fori∗Hp(M;R) to a homology basis

cp₁, . . . , cp_g, cp_g+1, . . . , cp_s

for all ofHp(M;R), wheres=bp(M) is thepth Betti number ofM. Let

cn₁−p, . . . , cn_g−p, cn_g+1−p, . . . , cn_s−p (2.1.10) be a Poincar´e dual basis forHn−p(M, ∂M;R), wherecn1−p, . . . , c

n−p

g are relative (n−p)-cycles and

cn_g+1−p, . . . , cns−p are absolute (n−p)-cycles. SinceHn_D−p(M)∼=Hn−p₍_{M, ∂M}_;

R), there is a unique (n−p)-formηe∈ H n−p

D (M) with preas-signed periods

F1, . . . , Fg, Fg+1, . . . , Fs

on the basis (2.1.10). The form η_eis exact (i.e. in the boundary subspaceEHn_D−p(M)) precisely when

Fg+1=. . .=Fs= 0.

Since the focus is on the boundary subspace, start with _eη∈ EHn_D−p(M) having periods

F1, . . . , Fg, Fg+1=. . .=Fs= 0.

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To prove that there is an element of cEHp_N(M) having arbitrary preassigned periods on

cp₁, . . . cp

g, it suffices to show that (F1, . . . , Fg)7→(C1, . . . , Cg) is an isomorphism.

Suppose some set ofF-values gives all zeroC-values, meaning thati∗ηis zero in the cohomology of∂M. In other words, the formi∗η is exact, meaning thatη ∈ E∂H

p

N(M), the interior subspace ofHp_N(M). SinceE∂H

p

N(M) is orthogonal tocEH p

N(M), this implies thatη= 0, soeη=±? η= 0 and hence the periodsFi ofeη must have been zero.

Therefore, the map (F1, . . . , Fg)7→(C1, . . . , Cg) is an isomorphism, completing Step 1. Step 2: Let ω ∈ Hp_N(M) and let C1, . . . , Cg be the periods of ω on the above p-cycles

cp₁, . . . , cp

g. Let α∈cEH p

N(M) be the unique form guaranteed by Step 1 having the same periods on this homology basis.

Thenβ =ω−αhas zero periods on the p-cyclescp₁, . . . , cpg; since β is a closed form onM, it certainly has zero period on each p-cycle of ∂M which bounds in M. Hence,β has zero periods on allp-cycles of∂M, meaning thati∗β is exact, soβ∈ E∂H

p N(M). Therefore, ω =α+β ∈ cEHp_N(M) +E∂H

p

N(M), so H p

N(M) is indeed the sum of these two subspaces, as claimed in (2.1.8). This completes the proof of the theorem.

Theorem 2.1.2 allows the details of Figure 1.1 to be filled in, as shown in Figure 2.1.

H

_p N

₍

M

)

c EHp N₍ M ) EH p D (M )

H

p D

(

M

)

E∂ H p N (M ) c_E ∂ H p D (_M )

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With the interior and boundary subspaces given explicitly, DeTurck and Gluck’s main theorem can now be stated:

Theorem 2.1.3(DeTurck–Gluck). LetMn _{be a compact, oriented, smooth Riemannian manifold}

with nonempty boundary ∂M. Then within the spaceΩp₍_M₎_of _{p-forms on}_M_,

(i). The concrete realizationsHp_N(M)andHp_D(M)of the absolute and relative cohomology groups Hp₍_M_;

R)andHp(M, ∂M;R) meet only at the origin.

(ii). The boundary subspacecEH_Np(M)ofHp_N(M)is orthogonal to all ofHp_D(M)and the boundary subspaceEHp_D(M)of Hp_D(M) is orthogonal to all ofHp_N(M).

(iii). No larger subspace of Hp_N(M) is orthogonal to all of Hp_D(M) and no larger subspace of H_Dp(M)is orthogonal to all ofHp_N(M).

(iv). The principal angles between the interior subspaces E∂HpN(M) of H p

N(M) and cE∂HpD(M)

ofHp_D(M)are all acute.

The fact thatE∂H p

N(M) andcE∂H p

D(M) have the same dimension is a straightforward conse-quence of Poincar´e–Lefschetz duality. The principal angles between these subspaces are invariants of the Riemannian structure onM and are called thePoincar´e duality angles.

Proof. (i). As was already mentioned, the fact that Hp_N(M)∩ Hp_D(M) ={0} follows from the strong unique continuation theorem of Aronszajn, Krzywicki and Szarski.

(ii). Ifδξ∈cEHp_N(M) andη∈ Hp_D(M), then, using Green’s formula

hη, δξiL2 =hdη, ξi_L2− Z

∂M

i∗η∧i∗? ξ = 0

since η is closed and satisfies the Dirichlet boundary condition. Therefore, the boundary subspacecEHp_N(M) is orthogonal to all ofHp_D(M).

Likewise, ifdγ∈ EHp_D(M) andω∈ H_Np(M), then

hdγ, ωiL2 =hγ, δωi_L2+ Z

∂M

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sinceω is co-closed and satisfies the Neumann boundary condition. Therefore,EHp_D(M) is orthogonal to all ofHp_N(M).

(iii). This result follows from the Friedrichs decompositions (2.1.5) and (2.1.6), which said that

Hp(M) =Hp_N(M)⊕ EHp(M)

=cEHp₍_M₎_{⊕ H}p D(M).

If a formω∈ Hp_N(M) is orthogonal to all of Hp_D(M), thenω ∈cEHp₍_M_{) and, therefore,} _ω is in the boundary subspacecEHp_N(M) ofHp_N(M).

Likewise, if η∈ Hp_D(M) is orthogonal to all ofH_Np(M), thenη ∈ EHp₍_M_{) and therefore}_η is in the boundary subspaceEHp_D(M).

(iv). By (i) and (iii), the principal angles between the interior subspaces can be neither 0 norπ/2, so they must all be acute.

Suppose thatM is a Riemannian manifold with boundary and thatθp₁, . . . , θ_kpare the Poincar´e duality angles in dimension p, i.e. θ₁p, . . . , θp_k are the principal angles between E∂H

p

N(M) and

cE∂H p

D(M). Then, ifprojD:H p

N(M)→ H p

D(M) is the orthogonal projection,projDcEH p

N(M) = 0 andproj_DE∂H

p

N(M) =cE∂H p

D(M). Since the cosines of the principal angles between twok-planes are the singular values of the orthogonal projection from one to the other, the cosθp_i are the non-zero singular values ofproj_D. Likewise, ifproj_N :H_Dp(M)→ Hp_N(M) is the orthogonal projection, the cosθp_i are also the non-zero singular values ofproj_N.

Thus, for 1≤i≤k, the quantities cos2θ_ip are the non-zero eigenvalues of the compositions

proj_N ◦proj_D and proj_D◦proj_N.

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2.2 The Dirichlet-to-Neumann map

2.2.1 The classical Dirichlet-to-Neumann map and the problem of

Elec-trical Impedance Tomography

The Neumann map for differential forms is a generalization of the classical Dirichlet-to-Neumann operator for functions. The classical Dirichlet-to-Dirichlet-to-Neumann operator arises in connection with the problem of Electrical Impedance Tomography (EIT), which was originally posed by Calder´on [Cal80] in the context of geoprospecting but which is also of interest in medical imaging (cf. [Hol05] for an overview of medical applications).

The problem of EIT is to determine the conductivity inside an open subset Ω⊂_R3 _{(or, more} generally, Ω ⊂ _Rn_{) by creating voltage potentials on the boundary and measuring the induced} current flux through the boundary. At low frequencies the electrical potentialuon Ω is governed by the Laplace equation

∇ ·γ∇u= 0, (2.2.1)

whereγ= (γij) is the positive-definite matrix giving the conductivity at pointsx∈Ω (cf. [CIN99, Appendix 1] for a derivation of (2.2.1) from Maxwell’s equations). The current flux through the boundary is the normal component of the current density at the boundary; denoting this by j,

j=−γ∂u ∂ν,

where ν is the unit inward-pointing normal vector. If f = u|∂Ω is the electrical potential on the boundary, then the problem of EIT is to determine the conductivity γ on all of Ω from the voltage-to-current map

f 7→ −γ∂u ∂ν.

Define the classical Dirichlet-to-Neumann operator Λcl:C∞(∂Ω)→C∞(∂Ω) as the map

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where ∆u= 0 on Ω ⊂ Rn and u|∂Ω = f. In dimension ≥ 3, Lee and Uhlmann [LU89] showed that the problem of EIT is equivalent to determining an associated Riemannian metricgfrom the Dirichlet-to-Neumann map Λcl, where

gij = detγk` 1/(n−2)

γij−1.

This restatement of the problem of EIT in terms of the inverse problem of determining the Rie-mannian metric g from the Dirichlet-to-Neumann map Λcl makes sense on an arbitrary compact Riemannian manifoldM with non-empty boundary, regardless of whether or notM is a region in Euclidean space, so this is the preferred mathematical formulation.

A prototypical theorem for this inverse problem is the following:

Theorem 2.2.1 (Lee–Uhlmann [LU89]). If Mn is a simply-connected, compact, real-analytic, geodesically convex Riemannian manifold with boundary and n ≥ 3, then (∂M,Λcl) determines

the Riemannian metric on M up to isometry. If “real-analytic” is replaced with “smooth”, then

(∂M,Λcl)determines theC∞-jet of the metric at the boundary of M.

Generalizations of the above theorem are given by Lassas–Uhlmann [LU01] and Lassas–Taylor– Uhlmann [LTU03].

In dimension two, the problem of EIT and the problem of determining the Riemannian metric from Λcl are distinct. In this context the EIT problem for isotropic conductivities was solved by Nachman [Nac96]; Sylvester [Syl90] showed that the anisotropic case reduces to the isotropic case. Determining the metric from Λclis too much to ask for in dimension two, but Lassas and Uhlmann showed that the conformal class of a surface with boundary is determined by Λcl.

2.2.2 The Dirichlet-to-Neumann map for differential forms

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equivalent, but Belishev and Sharafutdinov’s notation and definitions are used throughout this dissertation.

LetMn be a compact, oriented, smooth Riemannian manifold with non-empty boundary∂M. Define the Dirichlet-to-Neumann map Λp : Ωp(∂M) → Ωn−p−1(∂M) for any 0 ≤ p≤ n−1 as follows.

Ifϕ∈Ωp₍_∂M_{) is a smooth}_p_{-form on the boundary, then the boundary value problem}

∆ω= 0, i∗ω=ϕ and i∗δω= 0 (2.2.2) can be solved (cf. [Sch95, Lemma 3.4.7]). The solution ω ∈Ωp₍_M_{) is unique up to the addition} of an arbitrary harmonic Dirichlet field λ∈ Hp_D(M). Define

Λpϕ:=i∗? dω.

Then Λpϕis independent of the choice ofωsince takingdωeliminates the ambiguity in the choice ofω. Define

Λ := n−1 M i=0

Λi.

Whenϕ is a function (i.e. ϕ∈Ω0₍_∂M_{)), suppose} _u_∈_Ω0₍_M_{) is a harmonic function which} restricts toϕon the boundary. Sinceδu= 0,usolves the boundary value problem (2.2.2). Hence,

Λ0ϕ=i∗? u=

∂u

∂ν dvol∂M = (Λclϕ) dvol∂M,

so Λ is indeed a generalization of the classical Dirichlet-to-Neumann map.

In the spirit of Theorem 2.2.1, Joshi and Lionheart showed that the Dirichlet-to-Neumann map for differential forms recovers information about the metric onM:

Theorem 2.2.2 (Joshi–Lionheart). For any p such that 0 ≤ p ≤ n−1, the data (∂M,Λp)

determines theC∞-jet of the Riemannian metric at the boundary ofM.

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Lemma 2.2.3 (Belishev–Sharafutdinov). If ϕ∈ Ωp₍_∂M₎ _and _ω _∈ _Ωp₍_M₎ _{solves the boundary}

value problem (2.2.2), then dω ∈ Hp+1₍_M₎ _and _δω _{= 0}_{. Hence,} _(2.2.2) _{is equivalent to the}

boundary value problem

∆ω= 0, i∗ω=ϕ and δω= 0. (2.2.3) Lemma 2.2.4 (Belishev–Sharafutdinov). For any 0≤p≤n−1, the kernel of Λp coincides with

the image of Λn−p−1. Moreover, a form ϕ ∈ Ωp belongs to ker Λp = im Λn−p−1 if and only if

ϕ=i∗ω for some harmonic field ω∈ Hp₍_M₎_{. In other words,}

i∗Hp₍_M_{) = ker Λp}_{= im Λn}

−p−1.

Knowledge of the kernel of the Dirichlet-to-Neumann map yields lower bounds on the Betti numbers bp(M) ofM andbp(∂M) of∂M:

Theorem 2.2.5 (Belishev–Sharafutdinov). The kernel ker Λp of the Dirichlet-to-Neumann map Λp contains the spaceEp(∂M)of exactp-forms on∂M and

dim [ker Λp/Ep₍_∂M_)]_≤_min_{b_p(_M₎_{, b}_p(_∂M₎_}_.

A refinement of this theorem, including a direct-sum decomposition of ker Λp, is given in Proposition 5.2.3.

Two other operators come to attention in Belishev and Sharafutdinov’s story. The first is the

Hilbert transformT, defined as T :=d∂Λ−1. The Hilbert transform is obviously not well-defined on all forms on ∂M, but is well-defined on i∗Hp₍_M_{) = im Λ}

n−p−1 for any 0≤p≤n−1. The analogy between the mapTand the classical Hilbert transform from complex analysis is explained in Belishev and Sharafutdinov’s Section 5.

The other interesting operatorGp: Ωp(∂M)→Ωn−p−1(∂M) is defined as

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Belishev and Sharafutdinov’s main theorem shows that knowledge of Λ (and thus ofG) yields knowledge of the cohomology ofM:

Theorem 2.2.6 (Belishev–Sharafutdinov). For any0≤p≤n−1,

imGn−p−1=i∗H p N(M).

Since harmonic Neumann fields are uniquely determined by their pullbacks to the boundary, this

means that imGn−p−1 ∼= Hp_N(M) ∼= Hp(M;R). In other words, the boundary data (∂M,Λ)

completely determines the absolute cohomology groups of M.

By Poincar´e–Lefschetz duality,Hp₍_M_;

R)∼=Hn−p₍_{M, ∂M}_;

R), so the above theorem

immedi-ately implies that (∂M,Λ) also determines the relative cohomology groups ofM.

A key feature of Theorem 2.2.6 is that the cohomology groupsHp(M;R) andHp(M, ∂M;R)

are not just determined abstractly by (∂M,Λ), but can be realized as particular subspaces of differential forms on ∂M. The content of Theorem 4 is that these shadows of Hp₍_M_;

R) and

Hp₍_{M, ∂M}_;

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Chapter 3

Poincar´

e duality angles on

complex projective space

The simplest example of a compact manifold with boundary which is intuitively close to being closed is one which is obtained from a closed manifold by removing a small ball. Therefore, determining the Poincar´e duality angles for an interesting class of such manifolds should provide insight into whether the Poincar´e duality angles measure how close a manifold is to being closed. To that end, forn≥2 consider the complex projective spaceCPn with its usual Fubini-Study

metric and letx∈CPn. For 0< r < π/2, define the one-parameter family of compact Riemannian

manifolds with boundary

Mr:=CPn−Br(x),

where Br(x) is the open ball of radiusrcentered atx. The cohomology groups ofMr are

Hi(Mr;_R)∼=H2n−i(Mr, ∂Mr;R) =

        

R fori= 2k, 0≤k≤n−1

0 otherwise.

(3.0.1)

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Theorem 1. For 1 ≤ k ≤n−1 there is a non-trivial Poincar´e duality angle θ2k

r between the

concrete realizations of H2k₍_M_r;

R) andH2k(Mr, ∂Mr;R) which is given by

cosθ_r2k= 1−sin 2n_r

q

(1 + sin2nr)2₊(n−2k)2 k(n−k) sin

2n_r

. (3.0.2)

For small r, cosθ2k

r = 1 +O(r2n) and 2n= dimCP

n _{= codim}_{x}_{, so the Poincar´}_{e duality} angles not only go to zero as r →0, but seem to detect the codimension of the point removed. Also, for r near its maximum value ofπ/2, cosθ2k

r =O(r2) and 2 = codim CP n−1

, which is the base manifold onto whichMr collapses asr→π/2.

In order to compute these angles it is necessary to get a detailed geometric picture ofMr.

3.1 The geometric situation

The cut locus of the point x ∈ CPn is a copy of CPn−1 sitting at constant distance π/2 from

x. Thus, thinking of xand this copy of_CPn−1 as sitting at opposite “ends” of _CPn, the space in between the two is foliated by copies ofS2n−1 _{given by exponentiating the concentric spheres} aroundxinTxCPn.

This follows simply from the definition of the cut locus, but can also be seen as follows. There is a Hopf fibrationH :S2n+1→CPn given by identifying points on the same complex line. The

preimage H−1(x) is a copy ofS1, the preimage H−1(CPn−1) is a copy ofS2n−1 and S2n+1 can

be viewed as the join of these copies of S1_and_S2n−1_{. Hence, the region between these spheres is} foliated by copies of S1_×_S2n−1_{; metrically, the leaves are given by}

S2n−1(cost)×S1(sint),

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πcost S1(sint)

S2n−1_(cos_t₎

πsintcost

Hopf fiber

y −y

Figure 3.1: S2n−1_(cos_t₎_×_S1_(sin_t_{) sitting over the hypersurface at distance}_t_from

CPn−1

Metrically,H S2n−1(cost)×S1(sint)is a round sphere which has been scaled by a factor of cost and whose Hopf fibers have been scaled by an additional factor of sint. This is illustrated in Figure 3.1, where the “big diagonal” is a Hopf fiber in S2n+1 _{and the “little diagonal” is a} Hopf circle in the horizontal S2n−1_(cos_t_{). The distance between antipodal points} _y _and _−y _on the horizontal Hopf circle isπcost. However, the minimum distance from−yto the Hopf fiber in

S2n−1_(cos_t₎_×_S1_(sin_t_{) containing} _y _is_π_sin_t_cos_t_{, so in the quotient the Hopf direction is scaled} by an additional factor of sint.

Therefore, the hypersurface at distancet fromCPn−1is the Berger sphere

S2n−1(cost)sint

which is obtained from the unit sphere by scaling the entire sphere by cost and the Hopf fibers by an additional factor of sint.

Since the manifold Mr is just CPn−Br(x), it consists of all the spheres S2n−1(cost)sintfor 0< t≤π/2−ralong with_CPn−1with its usual metric (which, with an abuse of notation, could be identified with S2n−1_{(cos 0)sin 0). Hence,} _M

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same absolute cohomology asCPn−1since the fibers are contractible. This justifies the statement

of the cohomology groups ofMrgiven in (3.0.1).

Moreover, since∂Mris homeomorphic toS2n−1, none of the cohomology in dimensions strictly between 0 and 2ncomes from the boundary, meaning that there are non-trivial Poincar´e duality angles between the concrete realizations ofH2k₍_M_;

R) andH2k(M, ∂M;R) for all 1≤k≤n−1.

3.2 Finding harmonic fields

The goal of this section is to find harmonic 2k-fields satisfying Neumann and Dirichlet boundary conditions for each 1≤k≤n−1. Finding such harmonic fields and measuring the angles between them is sufficient to determine the Poincar´e duality angles sinceH2k₍_M_;

R) and H2k(M, ∂M;R)

are both 1-dimensional with no boundary subspace for each suchk.

These harmonic fields must be isometry-invariant; otherwise they could be averaged over the action of the isometry group to get isometry-invariant forms representing the same cohomology class. But this averaging does not affect whether the form is closed or co-closed, so the harmonic fields must have been isometry-invariant to start with.

Any isometry ofMrmust map the hypersurfaceS2n−1(cost)sintat constant distanceπ/2−r−t from the boundary to itself and must preserve the Hopf fibers on this hypersurface since they are scaled differently from the other directions. Equivalently, any isometry of Mr extends to an isometry ofCPnwhich fixes the pointx, so the isometry group ofMris just the isotropy subgroup ofCPn(the identity component of which isSU(n)). In either formulation, the Hopf direction and

the tdirection are invariant directions and the (2n−2)-plane distribution orthogonal to both is also invariant.

With this in mind, letht:S2n−1→S2n−1(cost)sintbe the obvious diffeomorphism of the unit sphere with the hypersurface at constant distance t from_CPn−1 and letH :S2n−1 _→

CPn−1 be

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push-forward by ht of the unit vector field in the Hopf direction onS2n−1. Define αto be the 1-form dual toA, letτ=dtbe the 1-form dual to _∂t∂ and defineη to be the 2-form which restricts on each S2n−1(cost)sint to

(H◦ht)∗η_CPn−1,

where η_CPn−1 is the standard symplectic form on _CPn−1. Then η is the symplectic form on the (2n−2)-plane distribution orthogonal to bothAand _∂t∂.

Away fromCPn−1the manifoldMris topologically a productS2n−1×I, so exterior derivatives can be computed as in S2n−1. Thus

dα=−2η

andτ andη are closed.

The goal is to use the above information to construct closed and co-closed 2k-forms ωN and

ωDsatisfying Neumann and Dirichlet conditions, respectively. Since such harmonic fields must be isometry-invariant and thus map S2n−1_(cos_t_)sin

t to itself and preserve the Aand _∂t∂ directions, they take the form

ωN :=fN(t)ηk+gN(t)α∧ηk−1∧τ (3.2.1)

ωD:=fD(t)ηk+gD(t)α∧ηk−1∧τ. The requirement thatωN be closed means that

0 =dωN =d(fN(t)ηk) +d(gN(t)α∧ηk−1∧τ) =f_N0 (t)ηk∧τ−2gN(t)ηk∧τ, meaning that 0 =f_N0 (t)−2gN(t). Hence,ωN is closed if and only if

gN(t) = 1 2f

0

N(t). (3.2.2)

Likewise, ωD is closed if and only ifgD(t) = 1₂fD0 (t).

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is dual to the Hopf direction onS2n−1_(cos_t_)sin_{t, which is scaled by a factor of sin}_t_cos_t_{from the} Hopf direction on the unit sphere. Therefore, the unit vector field in the direction of A is given by _sin_tcos1 _tAand the dual 1-form is

sintcost α.

The 2-form η is dual to a 2-plane distribution tangent to S2n−1_(cos_t₎

sint but orthogonal to A. Such directions are scaled by a factor of cost. If those directions are normalized, the dual 2-form must be cos2_{t η}_{. Finally,} ∂

∂t is a unit vector field, soτ is already normalized. The volume form onMris, therefore,

1

(n−1)!α∧η

n−1_∧_τ, so the relevant computations of the Hodge star are

?

₁

(k−1)!sintcos

2k−1_{t α}

∧ηk−1∧τ

= 1

(n−k)!cos

2n−2k_{t η}n−k

?

1

k!cos 2k_{t η}k

= 1

(n−k−1)!sintcos

2n−2k−1_{t α}_∧_ηn−k−1_∧_τ. Combining this with (3.2.1) yields

? ωN =

(k−1)! (n−k−1)!

ksintcos2n−4k−1t fN(t)α∧ηn−k−1∧τ+ 1

n−k

cos2n−4k+1_t sint gN(t)η

n−k

,

(3.2.3) so

d ? ωN =

(k−1)! (n−k−1)!

−2ksintcos2n−4k−1t fN(t)

− 1 n−k

(2n−4k+ 1) cos2n−4kt+cos

2n−4k+2_t sin2t

gN(t)−

cos2n−4k+1t

sint g

0

N(t)

ηn−k∧τ.

Using (3.2.2) and simplifying, this implies thatωN being co-closed is equivalent tofN(t) satisfying the ODE

0 =f_N00(t)−((2n−4k+ 1) tant+ cott)f_N0 (t)−4k(n−k) tan2t fN(t). (3.2.4) Solutions of this ODE take the form

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yielding

gN(t) = 1 2f

0

N(t) =−kC1sintcos2k−1t+ (n−k)C2

sint

cos2n−2k+1_t. The boundary ofMr occurs att=π/2−r, so, using (3.2.3),

i∗? ωN =

(k−1)! (n−k)!

cos2n−4k+1(π/2−r)

sin(π/2−r) gN(π/2−r)η n−k_. Therefore,ωN satisfies the Neumann boundary conditioni∗? ωN = 0 if and only if

0 =gN(π/2−r) =−kC1sin(π/2−r) cos2k−1(π/2−r) + (n−k)C2

sin(π/2−r) cos2n−2k+1₍_π/₂₋_r₎ =−kC1cosrsin2k−1r+ (n−k)C2

cosr

sin2n−2k+1r,

meaning that

C2=

k

n−kC1sin

2n

r.

RenamingC1 asCN, this implies that

fN(t) =CN

cos2kt+ k

n−ksin

2n

r 1

cos2n−2k_t

(3.2.5)

gN(t) =CN

−ksintcos2k−1t+ksin2nr sint

cos2n−2k+1_t

.

SinceωDmust be closed and co-closed,fDandgDalso satisfy the equations (3.2.2) and (3.2.4) and so take the same basic form asfN andgN. The Dirichlet boundary conditioni∗ωD= 0 implies that fD(π/2−r) = 0, so

fD(t) =CD

cos2kt−sin2nr 1

cos2n−2k_t

(3.2.6)

gD(t) =CD

−ksintcos2k−1t−(n−k) sin2nr sint

cos2n−2k+1_t

for some constantCD.

3.3 Normalizing the forms

The angleθk betweenωN andωD is given by

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Therefore, the constantsCN andCD should be chosen such that

kωNkL2 = 1 =kωDkL2.

Using (3.2.1) and (3.2.3),

ωN∧? ωN =

(k−1)! (n−k−1)!

ksintcos2n−4k+1t fN(t)2+ 1

n−k

cos2n−4k+1_t sint gN(t)

2

α∧ηn−1∧τ.

Thus,

hωN, ωNiL2 = Z

Mr

ωN ∧? ωN

= Z

Mr

(k−1)! (n−k−1)!

n−k

2

α∧ηn−1∧τ. Mris a product away from CPn−1 (which has measure zero), so Fubini’s Theorem implies that

hωN, ωNiL2 =

Z π/2−r 0

" Z

S2n−1_(cos_t)sin_t

α∧ηn−1

#

(k−1)! (n−k−1)!

ksintcos2n−4k+1tfN(t)2 + 1

n−k

2

dt.

(3.3.1) Since 1

(n−1)!sintcos

2n−1_{t α}_∧_ηn−1 _{is the volume form on}_S2n−1_(cos_t₎

sintand since volS2n−1(cost)sint= sintcos2n−1tvolS2n−1(1),

the expression (3.3.1) reduces to

hωN, ωNiL2 =Q

Z π/2−r 0

n−k

cos2n−4k+1t

sint gN(t)

2

dt,

where

Q:= (n−1)!(k−1)! (n−k−1)! volS

2n−1₍₁₎ _(3.3.2)

Using the expressions forfN(t) andgN(t) given in (3.2.5) a straightforward computation yields

hωN, ωNiL2 =C_N2Q

Z π/2−r 0

_kn

n−ksintcos

2n−1_t₊ k2n (n−k)2sin

4n_r sint cos2n+1_t

dt

=CN2Q

k

2(n−k)

1 + 2k−n

n−k sin

2n

r− k n−ksin

4n

r

=C_N2Q k

2(n−k)

1 + k

n−ksin

2n_r

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Therefore, the condition kωNkL2 = 1 means thatCN should be chosen as

CN = v u u t

2(n−k)! volS2n−1_{(1) (}_n₋_1)!_k_!_{1 +} k

n−ksin

2n_r ₁₋_sin2n_r. (3.3.3)

An entirely analogous calculation shows thatkωDkL2 = 1 when

CD= v u u t

2(n−k)! volS2n−1_{(1) (}_n₋_{1)! (}_k₋_1)! k

n−k+ sin

2n_r ₁₋_sin2n_r. (3.3.4)

3.4 Computing the Poincar´

e duality angle

Now that the formsωN andωD are completely pinned down, computing the angle between them is straightforward:

cosθ_r2k=hωN, ωDiL2

sinceωN andωD have unit norm.

Using (3.2.1) and the modification of (3.2.3) appropriate for? ωD,

ωN∧? ωD=

(k−1)! (n−k−1)!

ksintcos2n−4k−1t fN(t)fD(t) + 1

n−k

cos2n−4k+1t

sint gN(t)gD(t)

α∧ηn−1∧τ.

Therefore, using the same trick to integrate out S2n−1_(cos_t₎

sintas in (3.3.1) and the ensuing few lines,

cosθk =Q

Z π/2−r 0

ksintcos2n−4k−1t fN(t)fD(t) + 1

n−k

cos2n−4k+1_t

sint gN(t)gD(t)

dt

The definitions offN(t), fD(t), gN(t) andgD(t) then yield

cosθk =CNCDQ

Z π/2−r 0

kn

n−ksintcos

2n−1_t₋ kn

n−ksin

4n_r sint cos2n+1_t

dt

=CNCDQ

k

2(n−k) 1−sin 2n_r2

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The expressions forQ,CN andCD given in (3.3.2), (3.3.3) and (3.3.4) give

CNCDQ

k

2(n−k) =

q k n−k (1−sin2nr)

r

1 +_n₋k_ksin2nr _n₋k_k + sin2nr

= 1

(1−sin2nr) q

1 + sin2nr2

+(n_k(n−₋2k)_k)2sin2nr .

This allows (3.4.1) to be simplified as

cosθk=

1−sin2nr

q

1 + sin2nr2

+(n_k(n−2k)₋_k)2sin2nr

, (3.4.2)

completing the proof of Theorem 1.

3.5 Generalization to other bundles

The manifoldMr constructed above is topologically aD2-bundle over CPn−1 with Euler class 1.

Other non-trivial D2_{-bundles over}

CPn−1 have the same rational cohomology, so they also have

a single Poincar´e duality angle between the concrete realizations of the absolute and relative cohomology groups.

TheD2_{-bundle with Euler class}_m_over

CPn−1has boundaryL(m,1), the lens space which is

the quotient ofS2n−1by the action of Z/mZgiven by

e2πi/m·(z0, . . . , zn) =

e2πi/mz0, . . . , e2πi/mzn

.

The lens spaceL(m,1) fibers overCPn−1; whenn= 2 this corresponds to the non-singular Seifert

fibration of the three-dimensional lens spaceL(m,1) overS2_.

LetL(m,1)t be the quotient of S2n−1(cost)sint by this action of Z/mZ. Then L(m,1)t is a

“Berger lens space” obtained from the standard lens spaceL(m,1) by scaling the whole space by costand the circle fibers over _CPn−1 by an additional factor of sint.

Define the 2n-dimensional Riemannian manifold Mr,m modeled on Mr by replacing every appearance of S2n−1_(cos_t₎

sint in the definition ofMr byL(m,1)t. ThenMr,m is topologically a

D2_{-bundle over}

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All of the computations in Sections 3.2–3.4 follow through verbatim except that each appear-ance of volS2n−1_{(1) must be replaced by} 1

mvolS

2n−1_{(1). The}_m_{’s cancel in the end, so it follows} that

Theorem 3.5.1. For 1 ≤ k ≤ n−1, the Poincar´e duality angle θ2kr,m between the concrete

realizations ofH2k₍_M

r,m;R)andH2k(Mr,m, ∂Mr,m;R)is given by

cosθ_r,m2k = (1−sin 2n_r₎ q

1 + sin2nr2

+(n_k(n−2k)₋_k)2sin2nr .

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Chapter 4

Poincar´

e duality angles on

Grassmannians

The example of_CPn−Br(x) is a special case of a more general phenomenon in a couple of different ways. _CPn is a simple example of a Grassmannian (namely, of complex lines in _Cn+1_{); it would} be interesting to determine the Poincar´e duality angles of other Grassmannians. Also, removing a ball centered at some point in a closed manifold to get a manifold with boundary is the simplest way of removing a tubular neighborhood of a submanifold from a closed manifold.

With that in mind, consider G2Rn+2, the Grassmannian of oriented 2-planes in Rn+2. The

usual metric on G2Rn+2 =SO(n+ 2)/SO(n) is the Riemannian submersion metric induced by

the bi-invariant metrics on SO(n+ 2) andSO(n). For any mthe bi-invariant metric on SO(m) is inherited from the Euclidean metric on the space of m×mmatrices and has the unfortunate feature that the standard basis vectors Eij for so(m) have length

√

2. For the purposes of this chapter it is more convenient to reduce all linear dimensions in all the orthogonal groups by√2 and then assign any associated homogeneous spaces, such asG2Rn+2, the resulting Riemannian

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en+2

v

w

v∧w∈G2Rn+1

v∧e

n+2

∈G1R

n+1

Figure 4.1: Elements of the subGrassmanniansG1Rn+1 andG2Rn+1

The Grassmannian G2Rn+2 contains two obvious subGrassmannians G2Rn+1 and G1Rn+1:

oriented 2-planes in Rn+1 are certainly also oriented 2-planes in Rn+2, while an oriented line in Rn+1 determines the 2-plane in Rn+2 containing that line and the xn+2-axis, oriented from the

line to the xn+2-axis (see Figure 4.1). This is entirely analogous to the situation inCPn. Letting

x= (0 : 0 :. . .: 0 : 1), the relevant submanifolds there were_CPn−1, the space of complex lines in

Cn, and {(0 : 0 :. . .: 0 : 1)}, which can be identified with the space of complex 0-planes inCn.

Paralleling the_CPn story, define the one-parameter family of manifolds

Nr:=G2Rn+2−νr G1Rn+1,

whereνr G1Rn+1is the open tubular neighborhood of radiusraroundG1Rn+1. Since an oriented

line in_Rn+1 _{is determined by a unit vector in}

Rn+1, the manifoldG1Rn+1is just the unit sphere

Sn_{. Just as}

CPn−1was the cut locus of{p},G2Rn+1is the focal locus ofG1Rn+1(and vice versa),

so νr G1Rn+1is topologically the tangent bundle of G1Rn+1'Sn. Hence, the boundary

∂νr(G1Rn+1)

=∂Nr

is homeomorphic to the unit tangent bundle U Sn_.

Since G1Rn+1 is the focal locus of G2Rn+1, the manifold Nr is a disk bundle over G2Rn+1.

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The fibers overG2Rn+1are contractible, soNrhas the same absolute cohomology asG2Rn+1:

Hi(Nr;R) =

                  

R fori= 2k, 0≤k≤n−1, k6= n−21

R2 ifn−1 even andi=n−1

0 otherwise.

SinceHi₍_N

r;R)∼=H2n−i(Nr, ∂Nr;R) by Poincar´e–Lefschetz duality, replacingiby 2n−iin the

above gives the relative cohomology of Nr. Thus, the absolute and relative cohomology groups are both non-trivial in dimension 2kfor 1≤k≤n−1. Whennis even,∂Nr'U Sn has the same rational cohomology asS2n−1, so all of this cohomology is interior. Whennis odd,U Sn has the same rational cohomology asSn×Sn−1, so all of this cohomology is interior exceptHn−1(Nr;R)

andHn−1(M, ∂M;R) could potentially have a 1-dimensional boundary subspace. This turns out

to be the case and the goal of this chapter is to prove Theorem 2:

Theorem 2. For 1 ≤ k ≤ n−1 there is exactly one Poincar´e duality angle θ2k

r between the

concrete realizations of H2k₍_N_r;

R)andH2k(Nr, ∂Nr;R)given by

cosθ_r2k= 1−sin n_r

q

(1 + sinnr)2₊(n−2k)2 k(n−k) sin

n

r .

As in theCPnstory, whenris small cosθ2kr = 1+O(rn) andn= codimG1Rn+1, so the Poincar´e

duality angles not only go to zero as r →0, but they apparently detect the codimension of the manifold removed. Also, for r close to its maximum value ofπ/2, the quantity cosθ2kr =O(r2) and 2 = codimG2Rn+1, which is the base manifold onto whichNr collapses asr→π/2.

4.1 The geometry of

N

r

The Grassmannian G2Rn+2 can be viewed as a cohomogeneity one manifold as follows. As a

homogeneous space,

G2Rn+2=

SO(n+ 2)

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where theSO(2) acts by rotating thex1x2-plane and theSO(n) rotates the orthogonaln-plane. The group SO(n+ 1) embeds into SO(n+ 2) as the subgroup fixing the xn+2-axis. This embedding induces an action of SO(n+ 1) on G2Rn+2. This induced action of SO(n+ 1) is

transitive on the copies of G2Rn+1 andG1Rn+1 sitting inside ofG2Rn+2, and of course

G2Rn+1=

SO(n+ 1)

SO(2)×SO(n−1), G1R

n+1₌_Sn₌ SO(n+ 1)

SO(n) .

The action ofSO(n+ 1) preserves the principal angle a 2-plane makes with thexn+2-axis, but is transitive on each collection of 2-planes making the same such angle, so the principal orbits of this action are those 2-planes making an acute angle θ with thexn+2-axis. The principal orbit corresponding to the angle θ forms the hypersurface at constant distance θ from G1Rn+1 and

distance π/2−θfrom G2Rn+1.

The set of oriented 2-planes making an acute angleθwith the xn+2-axis is homeomorphic to the Stiefel manifoldV2Rn+1 of oriented 2-frames inRn+1. This can be seen as follows: if P is an

oriented 2-plane making a principal angleθwith thexn+2-axis, letvbe the unit vector inP which realizes this angle. Letwbe the orthogonal unit vector inP such that (v, w) gives the orientation ofP. Thenwis orthogonal to thexn+2-axis, sow∈Rn+1. Sinceθ6= 0, the orthogonal projection

projv ofvto Rn+1 has length sinθ6= 0, so

_proj_v sinθ, w

gives an oriented 2-frame inRn+1.

The upshot is that G2Rn+2 is the cohomogeneity one manifold corresponding to the group

diagram

SO(n+ 1)

SO(2)×SO(n−1) SO(n)

SO(n−1) with principal orbits

SO(n+ 1)

SO(n−1) =V2R

´E POINCAR D U A L I T Y A N G L E SO NR I E MA N N I A N A N I FO L D S WI T HB O U N D A R Y

POINCAR´

E DUALITY ANGLES ON RIEMANNIAN

MANIFOLDS WITH BOUNDARY

Clayton Shonkwiler

A Dissertation

in

Mathematics

Presented to the Faculties of the University of Pennsylvania in Partial

Fulfillment of the Requirements for the Degree of Doctor of Philosophy

2009

Dennis DeTurck

Supervisor of Dissertation

Herman Gluck

Supervisor of Dissertation

Tony Pantev

Acknowledgments

Contents

List of Figures

Chapter 1

Introduction

Chapter 2

Preliminaries

2.1

Poincar´

e duality angles

2.1.1

The Hodge decomposition for closed manifolds

2.1.2

The Hodge–Morrey–Friedrichs decomposition

2.1.3

Poincar´

e duality angles

H

(

M

)

H

(

M

)

2.2

The Dirichlet-to-Neumann map

2.2.1

The classical Dirichlet-to-Neumann map and the problem of

Elec-trical Impedance Tomography

2.2.2

The Dirichlet-to-Neumann map for differential forms

Chapter 3

Poincar´

e duality angles on

complex projective space

3.1

The geometric situation

3.2

Finding harmonic fields

3.3

Normalizing the forms

3.4

Computing the Poincar´

e duality angle

3.5

Generalization to other bundles

Chapter 4

Poincar´

e duality angles on

Grassmannians

4.1

The geometry of

N

₍