Rigidity for metrics with the same lengths of geodesics (Q1128267)

The authors prove that it is possible to recover a Riemannian metric in a bounded domain \(\Omega\in\mathbb{R}^3\) up to an isometry \(\psi\) which is the identity on the boundary, by knowing the lengths of geodesics joining points on the boundary. The metric is considered to be close to the Euclidean metric \(e\). Consider two Riemannian metrics \(g_1\) and \(g_2\) with \(\overline\Omega\) strictly convex with respect to both and \(d_{g_1}(x, y)= d_{g_2}(x, y)\), \(\forall x,y\in\Gamma= \partial\Omega\), where \(d_g(x,y)\) denotes the geodesic distance between \(x\) and \(y\). The authors prove that there exists an \(\varepsilon> 0\) such that, if \(g_m- e\in C^{12}_{(0)}(\Omega)\), \(\|g_m- e\|_{C^{12}(\overline\Omega)}< \varepsilon\), \(m= 1,2\), then there exists a \(C^{11}\) diffeomorphism \(\psi+\overline\Omega\to \overline\Omega\) such that \(\psi|_\Gamma= Id\) and \(\psi^*g_1= g_2\), where \(C^k_{(0)}(\Omega)\) denotes the set of all \(f\in C^k(\overline\Omega)\) such that \(\partial^\alpha f=0\) on \(\Gamma\) for \(|\alpha|\leq k\). The proof of this result uses the Fourier transform of \(m(x)\), where \(\{m_{ij}\}= \{g^{ij}_1\}- \{g^{ij}_2\}\).