The boundary rigidity problem consists in determining a Riemannian metric of a Riemannian manifolds with boundary by measuring the boundary distance function (lengths of geodesics) joining boundary points. This problem arises in differential geometry, as well as in geophysics in an attempt to determine the inner structure of the Earth by measuring the travel times of seismic waves.
Calderon’s inverse boundary problem consists in determining the electrical conductivity inside a body by making voltage and current measurements at the boundary. This inverse problem is also called Electrical Impedance Tomography (EIT). The boundary information is encoded in the Dirichlet-to-Neumann (DN) map and the inverse problem is to determine the coefficients of the conductivity equation (an elliptic partial differential equation) knowing the DN map.
A connection between these two inverse problems has led to a solution of the boundary rigidity problem in two dimensions for simple Riemannian metrics.