The parallels between the above calculation of the Poincar´e duality angles forNrto the calculation in the CPn story suggests that a similar strategy may work for other Grassmannians.
If GkRn+1 is the Grassmannian of oriented k-planes in Rn+1, then there are two obvious
subGrassmannians,
where a (k−1)-plane inRn specifies the k-plane in Rn+1 which contains it and the xn+1-axis.
These subGrassmannians sit at distance π/2 from each other inGkRn+1 and each is the other’s
focal locus. Using the same ideas as in Section 4.1, GkRn+1 is the cohomogeneity one manifold
corresponding to the group diagram
SO(n) SO(k)×SO(n−k) SO(k−1)×SO(n−k+ 1) SO(k−1)×SO(n−k) A principal orbit SO(n) SO(k−1)×SO(n−k) forms a bundle overGkRn= SO(k)SO(n)×SO(n−k) with fiber
SO(k)×SO(n−k)
SO(k−1)×SO(n−k) =
SO(k)
SO(k−1) =S k−1.
Each principal orbit is also a bundle over Gk−1Rn= SO(k−1)SO(n)×SO(n−k+1) with fiber
SO(k−1)×SO(n−k+ 1)
SO(k−1)×SO(n−k) =
SO(n−k+ 1)
SO(n−k) =S n−k. The fiber over GkRn is scaled by sint and the fiber overGk−1Rn is scaled by cost.
This then gives a fairly complete geometric picture of the manifold with boundary
Nr:=GkRn+1−νr(Gk−1Rn).
Emulating the case whenk= 2, the prototype for a closed and co-closedpk-form onNrwould then be
ω=f(t)(dα)p+g(t)α∧(dα)p−1∧τ
where αrestricts to anSO(n)-invariant (k−1)-form on each principal orbit.
However, there may not always be Poincar´e duality angles in the expected dimensions. For example, whenk= 3 andn= 5, the manifold
hasnoPoincar´e duality angles.
To see this, note thatG3R6is a 3(6−3) = 9-dimensional manifold, soNris also a 9-manifold. Since Nr is a D3-bundle over the six-dimensional submanifold G3R5 ' G2R5, it has the same
absolute cohomology as G2R5. Hence, the cohomology ofNr occurs in dimensions 0, 2, 4 and 6 since the closed manifoldG2R5has 1-dimensional real cohomology groups in dimensions 0, 2, 4 and
6. Using Poincar´e–Lefschetz duality, this means thatNrhas one-dimensional relative cohomology groups in dimensions 3, 5, 7 and 9. Therefore,
Hi(Nr;R) and Hi(Nr, ∂Nr;R)
cannot both be non-zero for any i, so there are no Poincar´e duality angles.
SinceG3R5 is homeomorphic toG2R5, the same holds even ifNris defined instead as
Nr:=G3R6−νr G3R5.
This suggests that the next interesting case will be the Grassmannian G4R8, in which the
subGrassmannians G3R7 and G4R7 are the focal loci of each other. Since G3R7 and G4R7 are
homeomorphic, the Poincar´e duality angles should be the same regardless of which is removed to get a manifold with boundary. It seems that finding harmonic representatives of the first Pontryagin form on G4R8 will be key in determining the Poincar´e duality angles of the manifold
Chapter 5
Connections with the
Dirichlet-to-Neumann operator
The goal of this chapter is to elucidate the connection between the Poincar´e duality angles and the Dirichlet-to-Neumann map for differential forms, then exploit that to find a partial reconstruction of the mixed cup product from boundary data. Throughout this chapter, Mn is a compact, oriented, smooth Riemannian manifold with non-empty boundary∂M.
5.1
The Dirichlet-to-Neumann map and Poincar´e duality
angles
Supposeθp1, . . . , θkpare the Poincar´e duality angles ofM in dimensionp; i.e. θp1, . . . , θkpare the prin- cipal angles between the interior subspaces E∂H
p
N(M) and cEH p
D(M). Recall from Section 2.1.3 that, if projD :HpN(M)→ HpD(M) is the orthogonal projection onto the space of Dirichlet fields and projN : HpD(M) → HpN(M) is the orthogonal projection onto the space of Neumann fields,
then the cos2θp
i are the non-zero eigenvalues of the composition
projN◦projD:HpN(M)→ HpN(M).
It is this interpretation of the Poincar´e duality angles which yields the connection to the Dirichlet-to-Neumann map.
Specifically, the Hilbert transform is closely related to the projections projD and projN, as illustrated by the following propositions:
Proposition 5.1.1. If ω∈ HpN(M)andprojDω=η ∈ HpD(M)is the orthogonal projection of ω ontoHpD(M), then
T i∗ω= (−1)np+1i∗? η.
Proposition 5.1.2. If λ∈ HpD(M) andprojNλ=σ∈ H p
N(M) is the orthogonal projection of λ
ontoHpN(M), then
T i∗? λ= (−1)n+p+1i∗σ.
Proposition 5.1.2 is proved by applying the Hodge star and then invoking Proposition 5.1.1.
Proof of Proposition 5.1.1. Using the first Friedrichs decomposition (2.1.5),
ω=δξ+η∈cEHp(M)⊕ Hp
D(M). (5.1.1)
Sinceω satisfies the Neumann boundary condition,
0 =i∗? ω=i∗?(δξ+η) =i∗? δξ+i∗? η,
so
i∗? η=−i∗? δξ. (5.1.2)
On the other hand, sinceη satisfies the Dirichlet boundary condition,
The formξ can be chosen such that
∆ξ= 0 and dξ= 0
(cf. [Sch95, (4.11)] or [BS08, Section 2]), which means that? ξsolves the boundary value problem
∆ε= 0, i∗ε=i∗? ξ, and i∗δε= 0.
Therefore,
(−1)np+n+2Λi∗? ξ= (−1)np+n+2i∗? d ? ξ=i∗ω.
Then, using the definition of the Hilbert transform T =d∂Λ−1and (5.1.2),
T i∗ω= (−1)np+n+2d∂Λ−1Λi∗? ξ= (−1)np+nd∂i∗? ξ = (−1)np+2i∗? δξ
= (−1)np+1i∗? η,
as desired.
Taken together, Propositions 5.1.1 and 5.1.2 can be seen as a clarification of Belishev and Sharafutdinov’s Lemma 7.1, which says that T maps i∗HpN(M) toi∗HnN−p(M).
Consider the restrictionTe of the Hilbert transformT toi∗H p
N(M). SinceTeis closely related to the orthogonal projections projD and projN, it should come as no surprise that Te2 is closely related to the composition projN ◦projD, the eigenvalues of which are the cos2θ
i. Indeed, the connection between the Dirichlet-to-Neumann map and the Poincar´e duality angles is given by: Theorem 4. If θ1p, . . . , θpk are the principal angles between E∂H
p
N(M) andcE∂H p
D(M) (i.e. the
Poincar´e duality angles in dimension p), then the quantities
(−1)pn+p+ncos2θpi
Proof. If δα∈cEHpN(M), the boundary subspace of HpN(M), then, by Theorem 2.1.3, the form
δαis orthogonal to HpD(M). By Proposition 5.1.1, then,T ie∗δα= 0, so i∗cEH p
N(M) is contained in the kernel of Te2.
Combining Propositions 5.1.1 and 5.1.2, the eigenforms ofTe2 are precisely the eigenforms of
projN ◦projD. Moreover, if ωi ∈ E∂H p
N(M) is an eigenform of projN ◦projD corresponding to a non-zero eigenvalue, then ωi is the Neumann field realizing the Poincar´e duality angleθ
p
i. Hence
projN ◦projD(ωi) = cos2θipωi.
Therefore,
e
T2i∗ω=T2i∗ωi= (−1)pn+p+ncos2θpii
∗ω
i,
so the (−1)pn+p+ncos2θipare the non-zero eigenvalues ofTe2.
Note that the domain ofTe2 is imGn−p−1, which is determined by the Dirichlet-to-Neumann map. Thus, Theorem 4 implies that the Dirichlet-to-Neumann operator not only determines the cohomology groups of M, as shown by Belishev and Sharafutdinov, but determines the interior and boundary cohomology.
Corollary 5.1.3. The boundary data(∂M,Λ)distinguishes the interior and boundary cohomology of M.
Proof. By Theorem 4, the pullback i∗cEHpN(M) of the boundary subspace is precisely kernel of the operatorTe2, while the pullbacki∗E∂H
p
N(M) of the interior subspace is the image ofTe2. Since harmonic Neumann fields are uniquely determined by their pullbacks to the boundary,
i∗cEHpN(M)=∼cEHpN(M) and i∗E∂HpN(M)∼=E∂HpN(M),
so the interior and boundary absolute cohomology groups are determined by the data (∂M,Λ). Since ? cEHpN(M) = EHDn−p(M) and ?E∂HNp(M) = cE∂HDn−p(M) for each p, the interior and boundary relative cohomology groups are also determined by the data (∂M,Λ).