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

AbstractFor a compact n-dimensional Riemannian manifold (M,g) with boundary i:∂M⊂M, the Dirichlet-to-Neumann (DN) map Λg:Ωk(∂M)→Ωn−k−1(∂M) is defined on exterior differential forms by Λgφ=i∗(⋆dω), where ω solves the boundary value problem Δω=0, i∗ω=φ, i∗δω=0. For a symmetric second rank tensor field h on M, let Λ˙h=dΛg+th/dt|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 Ωk-rigid if there is no other h satisfying Λ˙h=0. We prove that the Ωk-rigidity is equivalent to the density of the range of some bilinear form on the space Hexk+1(M) of exact harmonic fields