Linearized inverse problem for the Dirichlet-to-Neumann map on differential forms

Abstract

For a compact n-dimensional Riemannian manifold ( M , g ) with boundary i : ∂ M ⊂ M , the Dirichlet-to-Neumann (DN) map Λ g : Ω k ( ∂ M ) → Ω n − k − 1 ( ∂ M ) is defined on exterior differential forms by Λ g φ = i ∗ ( ⋆ d ω ) , where ω solves the boundary value problem Δ ω = 0 , i ∗ ω = φ , i ∗ δ ω = 0 . For a symmetric second rank tensor field h on M, let Λ ˙ h = d Λ g + t h / d t | t = 0 be the Gateaux derivative of the DN map in the direction h. We study the question: for a given ( M , g ) , how large is the subspace of tensor fields h satisfying Λ ˙ h = 0 ? Potential tensor fields belong to the subspace since the DN map is invariant under isomeries fixing the boundary. For a manifold of an even dimension n, the DN map on ( n / 2 − 1 ) -forms is conformally invariant, therefore spherical tensor fields belong to the subspace in the case of k = n / 2 − 1 . The manifold is said to be Ω k -rigid if there is no other h satisfying Λ ˙ h = 0 . We prove that the Ω k -rigidity is equivalent to the density of the range of some bilinear form on the space H e x k + 1 ( M ) of exact harmonic fields.