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