Determining anisotropic real-analytic metric from boundary electromagnetic information

For a compact, connected, oriented Riemannian $3$-manifold $(M, g)$ with smooth boundary $\partial M$, we explicitly give a local representation and a full symbol expression for the electromagnetic Dirichlet-to-Neumann map by factorizing Maxwell's equations and using an isometric transform. We prove that one can reconstruct a compact, connected, real-analytic Riemannian $3$-manifold $M$ with boundary from the set of tangential electric fields and tangential magnetic fields, given on a non-empty open subset $\Gamma$ of the boundary, of all electric and magnetic fields with tangential electric data supported in $\Gamma$. We note that for this result we need no assumption on the topology of the manifold other than compactness and connectedness, nor do we need a priori knowledge of all of $\partial M$. In addition, as a by-product of the explicit symbol expression of $\Lambda_{g,\Gamma}$, we show that for a given smooth Riemannian metric $g$, the electromagnetic Dirichlet-to-Neumann map $\Lambda_{g,\Gamma}$ uniquely determines all order tangential and normal derivatives of electromagnetic parameters $\mu$ and $\sigma$ on $\Gamma$. Therefore, $\mu$ and $\sigma$ are completely determined in $M$ by $\Lambda_{g,\Gamma}$ if these two parameter functions and metric $g$ are all real analytic in $M$ up to $\Gamma$.