Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

The Dirichlet-to-Neumann (DN) map Λg: C∞(∂M) → C∞(∂M) on a compact Riemannian manifold (M, g) with boundary is defined by Λgh = ∂u/∂ν|∂M, where u is the solution to the Dirichlet problem ∆u = 0, u|∂M = h and ν is the unit normal to the boundary. If gt = g+ tf is a variation of the metric g by a symmetric tensor field f, then Λgt = Λg + tΛ̇f + o(t). We study the question: how do tensor fields f look like for which Λ̇f = 0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: for the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold