Wang, Jenn-Nan Stability for the reconstruction of a Riemannian metric by boundary measurements. (English) Zbl 0969.53018 Inverse Probl. 15, No. 5, 1177-1192 (1999). In this note the author derives a stability estimate for the reconstruction of a Riemannian metric in a bounded domain \(\Omega\subset \mathbb{R}^3\) by a set of boundary measurements related to geodesics’ information from a different point of view. This problem is based on the following inverse problem. Let \(\Omega\subset \mathbb{R}^3\) with smooth boundary \(\Gamma= \partial\Omega\). Assume that \(g(x)\) is a given Riemannian metric in \(\Omega\). The author also assumes that \(\Omega\) is geodesically convex with respect to \(g\). Denote by \(L_g(x,y)\) the length of the geodesic joining \(x\) and \(y\) for \((x,y)\in \Gamma^{2}\).An inverse problem in this setting is whether one can reconstruct the Riemannian metric by the knowledge of \(L(x,y)\) for all \((x,y)\in \Gamma^2\). Based on P. Stefanov and G. Uhlman’s ideas [J. Funct. Anal. 154, 330-358 (1988; Zbl 0915.35066)], the author derives a stability estimate for the inverse problem which is equivalent to the hodograph problem. Reviewer: G.G.Vrănceanu (Bucureşti) Cited in 8 Documents MSC: 53C22 Geodesics in global differential geometry 35R30 Inverse problems for PDEs Keywords:length of a geodesic; Riemannian metric; geodesically convex; length of the geodesic; inverse problem; hodograph problem Citations:Zbl 0915.35066 PDFBibTeX XMLCite \textit{J.-N. Wang}, Inverse Probl. 15, No. 5, 1177--1192 (1999; Zbl 0969.53018) Full Text: DOI