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The complete Dirichlet-to-Neumann map for differential forms

The Dirichlet-to-Neumann map for differential forms on a Riemannian manifold with boundary is a generalization of the classical Dirichlet-to-Neumann map which arises in the problem of Electrical Impedance Tomography. We synthesize the two different approaches to defining this operator by giving an invariant definition of the complete Dirichlet-to-Neumann map for differential forms in terms of two linear operators {\Phi} and {\Psi}. The pair ({\Phi}, {\Psi}) is equivalent to Joshi and Lionheart's operator {\Pi} and determines Belishev and Sharafutdinov's operator {\Lambda}. We show that the Betti numbers of the manifold are determined by {\Phi} and that {\Psi} determines a chain complex whose homologies are explicitly related to the cohomology groups of the manifold.


Introduction
We consider the problem of recovering the topology of a compact, oriented, smooth Riemannian manifold (M, g) with boundary from the Dirichletto-Neumann map for differential forms. The classical Dirichlet-to-Neumann map for functions was first defined by Calderón [Cal80], and has been shown to recover surfaces up to conformal equivalence [LU01,Bel03] and realanalytic manifolds of dimension ≥ 3 up to isometry [LTU03].
The classical Dirichlet-to-Neumann map was generalized to an operator on differential forms independently by Joshi and Lionheart [JL05] and Belishev and Sharafutdinov [BS08]. Joshi and Lionheart called their operator Π and showed that the data (∂M, Π) determines the C ∞ -jet of the Riemannian metric at the boundary. Krupchyk, Lassas, and Uhlmann have recently extended this result to show that (∂M, Π) determines a real-analytic manifold up to isometry [KLU10].
On the other hand, Belishev and Sharafutdinov called their Dirichletto-Neumann map Λ and showed that (∂M, Λ) determines the cohomology groups of the manifold M . Shonkwiler [Sho09] demonstrated a connection between Λ and invariants called Poincaré duality angles and showed that the cup product structure of the manifold M can be partially recovered from (∂M, Λ).
The operators Π and Λ are similar, but do not appear to be equivalent. One of the advantages of Belishev and Sharafutdinov's Λ, especially for the task of recovering topological data, is that it is defined invariantly. In this paper we provide an invariant definition of Joshi and Lionheart's operator Π, which we give in terms of two auxiliary operators Φ : Ω k (∂M ) → Ω n−k−1 (∂M ) and Ψ : Ω k (∂M ) → Ω k−1 (∂M ).
We can easily show that Λ is determined by Φ and Ψ, so it makes sense to regard Π as the "complete" Dirichlet-to-Neumann operator on differential forms.
Belishev and Sharafutdinov's proof that the Betti numbers of M can be recovered from the data (∂M, Λ) was somewhat circuitous, as it involved determining the dimension of the image of the operator G = Λ ± d ∂ Λ −1 d ∂ .
In contrast, it is straightforward to recover the Betti numbers of M from Φ. The operator Ψ turns out to be a chain map and the homology of the chain complex (Ω * (∂M ), Ψ) is given in terms of a mixture of absolute and relative cohomology groups of M .
This, in turn, implies that the space of k-forms on ∂M contains an "echo" (detected by Π) of the (k + 1)st relative cohomology group of M .
Corollary 3. The space Ω k (∂M ) of k-forms on ∂M contains a subspace isomorphic to H k+1 (M, ∂M ; R) which is distinguished by the Dirichlet-to-Neumann operator Π. Specifically, When n = 2 and k = 0, Theorem 1 and Corollary 3 imply that all the cohomology groups of a surface are contained in Ω 0 (∂M ).
Corollary 4. All of the cohomology groups of a surface M with boundary can be realized inside the space of smooth functions on ∂M , where they can be recovered by the Dirichlet-to-Neumann operator Π.
Since Ψ is a chain map, it is natural to try to define associated cochain maps and compute their cohomologies. In this spirit, we define Ψ = ±⋆ ∂ Ψ⋆ ∂ and show that it is the adjoint of Ψ. Not surprisingly, Finally, we define another cochain map Θ with the same cohomology as Ψ. It turns out that Θ = ±d ∂ Φ 2 , so the cohomology of Ψ (and hence the homology of Ψ) is completely determined by the operator Φ. With this in mind, restating Corollary 3 in terms of Φ and specializing to the case k = 0 yields the following: Corollary 5. A copy of the cohomology group H n−1 (M ; R) is distinguished by the operator Φ inside Ω 0 (∂M ), the space of smooth functions on ∂M . Specifically, The above results all suggest that the operator Π (and, in particular, Φ) encodes more information about the topology of M than does the operator Λ. Thus far nobody has been able to use Λ to recover the cohomology ring structure on M , but perhaps this will be easier to recover from the operator Π. Another interesting question relates to the linearized inverse problem of recovering the metric: can the results of [Sha09] be strengthened if the data Λ are replaced with the richer data (Φ, Ψ)?
2. The operators Φ and Ψ Throughout this paper, (M, g) will be a smooth, compact, oriented Riemannian manifold of dimension n ≥ 2 with nonempty boundary. The term "smooth" is used as a synonym for "C ∞ -smooth". Let i : ∂M ֒→ M be the identical embedding and let Ω(M ) = n k=0 Ω k (M ) be the graded algebra of smooth differential forms on M . We use the standard operators d, δ, ∆, and ⋆ on Ω(M ), as well as their analogues d ∂ , δ ∂ , ∆ ∂ , and ⋆ ∂ on Ω(∂M ).
Joshi When applied to forms, the meaning of the normal derivative ∂/∂ν needs to be specified. Instead, we prefer to give an equivalent definition of Π in invariant terms. To do so, note that the restriction ω| ∂M is determined by two boundary forms, i * ω and i * ⋆ ω. Likewise, the data ∂ω/∂ν| ∂M are equivalent to the two boundary forms i * ⋆ dω and i * δω. Hence, we will define the operator where ω ∈ Ω k (M ) is the solution to the boundary value problem Since Π sends pairs of forms to pairs of forms, it is somewhat cumbersome to work with in practice. Instead of using it directly, we find a pair of operators (Φ, Ψ) which is equivalent to Π. Define the linear operators Here ω ∈ Ω k (M ) is the solution to the boundary value problem Now it is straightforward to express Π in terms of Φ and Ψ. We write Π as the matrix Then, comparing (1) and (3), From (1) and (2), the operators Π 12 and Π 22 are given by where ε solves the boundary value problem If ε ∈ Ω k (M ) is the solution to this boundary value problem for ψ ∈ Ω n−k (∂M ), then the form ω = ⋆ε solves the problem Comparing this to (4), we see that (5) Φψ = i * ⋆ dω and Ψψ = i * δω. Since (1) and (5) imply that Therefore, the operator Π can be expressed in terms of Φ and Ψ as Belishev and Sharafutdinov's version of the Dirichlet-to-Neumann map is the operator given by where ω ∈ Ω k (M ) is a solution to the boundary value problem We can now express the operator Λ in terms of Φ and Ψ. Given ϕ ∈ Ω k (∂M ), let ω ∈ Ω k (M ) solve the boundary value problem (7) and set ψ = i * ⋆ ω. Then ω solves the boundary value problem (2), so we have that With the help of (6) we can rewrite this equation as the system Eliminating ψ from the system yields the expression The fact that the operator ΨΦ −1 Ψ is well-defined follows from Corollary 4.3, stated below. We take this opportunity to record some useful relations involving Φ and Ψ: Lemma 2.1. The operators Φ and Ψ satisfy the following relations: Proof. Given ϕ ∈ Ω k (∂M ), let ω ∈ Ω k (M ) solve the boundary value problem (4). Then Letting ξ = δω, we certainly have ∆ξ = 0. Pulling ξ and ⋆ξ back to the boundary yields Therefore, ξ solves the boundary value problem and so (14) ΦΨϕ = i * ⋆ dξ and Ψ 2 ϕ = i * δξ.
Finally, (12) is proved along the same lines. From (16) we have Again making use of the fact that δdω = −dδω, this implies that In turn, we can use (13) and (15) to rewrite the above formula as Comparing with (17), this produces the desired relation (12).
Remark 2.2. The key properties of the operator Λ are expressed by the equalities Λd ∂ = 0, d ∂ Λ = 0, and Λ 2 = 0. It is straightforward to check that these equalities follow from (8) In contrast, it is much more straightforward to recover the Betti numbers of M from the operator Φ.
In particular, Φ is a Fredholm operator with index zero.
Proof. If ϕ ∈ Ω k (∂M ) such that Φ k ϕ = 0, then the boundary value problem is solvable. Using Green's formula, The right side of this equation equals zero since ω solves the boundary value problem (20). Hence, ω is a harmonic Neumann field since i * ⋆ ω = 0, and so ϕ = i * ω ∈ i * H k N (M ). The converse statement is immediate: if ϕ = i * ω for ω ∈ H k N (M ), then ω solves the boundary value problem (20) and hence ϕ ∈ ker Φ k .
On the other hand, a form ψ ∈ Ω n−k−1 (∂M ) is in the image of Φ k if and only if the boundary value problem is solvable. The defining condition (19) of (i * H k N (M )) ⊥ is precisely the necessary and sufficient condition for the solvability of this boundary value problem [Sch95, Corollary 3.4.8].
Proof. A form ψ ∈ (i * H k N (M )) ⊥ belongs to the range of Φ, so the equation Φϕ = ψ is solvable. If Φϕ 1 = Φϕ 2 , then the form ϕ 1 − ϕ 2 ∈ ker Φ is closed, The apparent similarity between the operator d ∂ Φ −1 and the Hilbert transform T = d ∂ Λ −1 defined by Belishev and Sharafutdinov is no accident, as the following proposition demonstrates. Thus, the connection to the Poincaré duality angles of M [Sho09, Theorem 4] comes directly from the definition of Φ (and hence Π) without using Λ as an intermediary.
where the term on the right-hand side is understood to be the restriction of d ∂ Φ −1 to im Λ = i * H k (M ).

The homology of the chain complex (Ω * (∂M ), Ψ)
We saw in Lemma 2.1 that Ψ 2 = 0, so it is natural to ask: what is the homology of the chain complex (Ω * (∂M ), Ψ)?
In other words, the homology groups of (Ω * (∂M ), Ψ) contain the absolute cohomology groups of M in the same dimension and echoes of the relative cohomology groups of M in one higher dimension. This behavior is similar to that exhibited by the cohomology of harmonic forms studied by Cappell, DeTurck, Gluck, and Miller [CDGM06].
Since H k (M ; R) ≃ ker Φ k (by Theorem 1) and since it will turn out that im Ψ k+1 completely misses ker Φ k , we can see the echo of the (k+1)st relative cohomology group of M inside the space of k-forms on ∂M .
Corollary 3. The space Ω k (∂M ) of k-forms on ∂M contains a space isomorphic to H k+1 (M, ∂M ; R) which is distinguished by the Dirichlet-to-Neumann operator Π. Specifically,   The operator G k is the restriction to Ω k (∂M ) of the operator G defined in (18).

Equations (21) and (22) imply that
Since δdζ is co-exact and since the space of co-exact k-forms is precisely the orthogonal complement of the space of k-forms satisfying a Dirichlet boundary condition, (23) implies that δdζ = 0. Hence, dζ is co-closedbut E k D (M ) is precisely the orthogonal complement of the space of co-closed k-forms, so it follows that dζ = 0. Therefore, ω = δξ + κ is co-closed. Since both ω and δξ ∈ cE k N (M ) satisfy a Neumann boundary condition, κ must be a harmonic Neumann field. Moreover, since both ω and κ are harmonic, it follows that δξ is harmonic. Hence, . Conversely, forms in this space are clearly in the kernel of Ψ.
In (24) the sum of spaces is not, a priori, direct, but directness of the sum follows immediately from the fact that harmonic forms are uniquely determined by their boundary values [Sch95, Theorem 3.4.10].
The term i * H k N (M ) = ker Φ k in (24) is exactly the space described in (i), so the lemma will follow from showing that i * (cE k N (M ) ∩ ker ∆) is the direct sum of the spaces described in (ii) and (iii).
Conversely, if ϕ ∈ ker G k ∩ i * (C k (M )) ⊥ , then ϕ = i * δξ for some δξ ∈ cE k N (M ) which has a conjugate form. This implies that dδξ is both exact and co-exact, and it is straightforward to check that ξ can be chosen to be harmonic. Case 2: If ξ is not harmonic, then it belongs to the space Lemma 3], and so i * N k is the space given in (iii).
The directness of the sum ker G k ∩ i * (C k (M )) ⊥ + i * N k again follows from the fact that harmonic forms are uniquely determined by their boundary values.
We can now determine the image of Ψ k+1 .
The first equality above is due to the fact that ⋆ is an isometry and the second follows because H n−k N (M ) is orthogonal to the space of exact forms on M .
Putting (25) and (26) together shows that for any λ N ∈ H n−k N (M ), so Lemma 3.1 implies that ϕ ∈ im Φ n−k , as desired.
5.1. The operator Ψ. The fact that Ψ 2 = 0 is immediate: Let Ψ k be the restriction of Ψ to Ω k (∂M ). Since ⋆ ∂ is an isomorphism, ker Ψ k ≃ ker Ψ n−k−1 and im Ψ k−1 ≃ im Ψ n−k , and so Thus, we can use Theorem 2 to determine the cohomology groups of Ψ. The obvious guess, suggested by experience with Λ and by the duality given in (27), is that Ψ is the adjoint of Ψ.
Finally, combining (12) with (33) yields This last expression makes it clear that Θ is a cochain map: Notice that (Ω * (∂M ), Θ) has the same cohomology as (Ω * (∂M ), Ψ). We omit the proof of Proposition 5.3, which is somewhat long and technical, though not particularly difficult. Two perhaps surprising consequences are: (i) Since Θ has the same cohomology as Ψ, the homology of Ψ can be completely recovered from that of Θ. However, by (34), Θ = d ∂ Ψd ∂ , so pre-and post-composing Ψ by d ∂ does not change the (co)homology. (ii) By (32) and (33), Hence, the homology of Ψ is completely determined by the operator Φ, and the results of Corollaries 3 and 4 depend only on Φ. In that spirit, the following is a restatement of the k = 0 case of Corollary 3.