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Although the main results of the paper are obtained in the two-dimensional case, the problems under consideration can be posed in an arbitrary dimension. Let (M, g) be a compact Riemannian manifold with boundary. The boundary rigidity problem consists of determining the Riemannian metric g on M , up to isometries which are the identity on the boundary, by measuring the geodesic distance, dg, between boundary points. It is easy to see that the answer to this question is no in general. We can find metrics g such that there exists x0 ∈ M satisfying dist(x0, ∂M) > supx,y∈∂Mdg(x, y). Then we can change the metric g in a neighborhood of x0 without changing the boundary distance function. A compact Riemannian manifold (M, g) with boundary is called boundary rigid if the metric g is uniquely determined by dg, up to isometries which are the identity on the boundary. There are a few classes of metrics which are known to be boundary rigid (see [13] for references). In this paper we consider the linearized problem. Let g be a one-parameter family of Riemannian metrics on M such that dgτ = dg0 , −e < τ < e. For a geodesic γ : [a, b] → M of the metric g, with γ(a), γ(b) ∈ ∂M it is easy to see that

On Deformation Boundary Rigidity and Spectral Rigidity of Riemannian Surfaces with No Focal Points