2010/07/06 by Katsiaryna Krupchyk, Krupchyk, Katsiaryna, Matti Lassas +4
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #math.AP #msc:35R30 #msc:58A10 #msc:58J10 #msc:58J32
paper · pdf · doi:10.48550/arxiv.1007.0979
arxiv created 2010/07/06 · arxiv updated 2010/07/07
Consider a real-analytic orientable connected complete Riemannian manifold M with boundary of dimension n≥ 2 and let k be an integer 1≤ k≤ n. In the case when M is compact of dimension n≥ 3, we show that the manifold and the metric on it can be reconstructed, up to an isometry, from the set of the Cauchy data for harmonic k-forms, given on an open subset of the boundary. This extends a result of [13] when k=0. In the two-dimensional case, the same conclusion is obtained when considering the set of the Cauchy data for harmonic 1-forms. Under additional assumptions on the curvature of the manifold, we carry out the same program when M is complete non-compact. In the case n≥ 3, this generalizes the results of [12] when k=0. In the two-dimensional case, we are able to reconstruct the manifold from the set of the Cauchy data for harmonic 1-forms.