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<p style='text-indent:20px;'>We prove that a continuous potential <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the perturbed biharmonic operator <inline-formula><tex-math id="M2">\begin{document}$ \Delta_g^2+q $\end{document}</tex-math></inline-formula> on a conformally transversally anisotropic Riemannian manifold of dimension <inline-formula><tex-math id="M3">\begin{document}$ \ge 3 $\end{document}</tex-math></inline-formula> with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [<xref ref-type="bibr" rid="b56">56</xref>]. In particular, our result is applicable and new in the case of smooth bounded domains in the <inline-formula><tex-math id="M4">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional Euclidean space as well as in the case of <inline-formula><tex-math id="M5">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional admissible manifolds.</p>

We show that the knowledge of the Dirichlet–to–Neumann map for a nonlinear magnetic Schrödinger operator on the boundary of a compact complex manifold, equipped with a Kähler metric and admitting sufficiently many global holomorphic functions, determines the nonlinear magnetic and electric potentials uniquely.

We study inverse boundary problems for first order perturbations of the biharmonic operator on a conformally transversally anisotropic Riemannian manifold of dimension n ≥ 3. We show that a continuous first order perturbation can be determined uniquely from the knowledge of the set of the Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.4 YAN complex vector fields, and let q (1) , q(1.4)we have q (1) = q (2) in M. Remark 1.3. Examples of non-simple manifolds M 0 satisfying Assumption 1 include in particular manifolds with strictly convex boundary which are foliated by strictly convex hypersurfaces [39], [41], and manifolds with a hyperbolic trapped set and no conjugate points [20], [21].Remark 1.4. To the best of our knowledge, Theorem 1.2 seems to be the first result where one recovers a vector field uniquely on general CTA manifolds.Remark 1.5. The assumption (1.4) is made for simplicity only and can be removed by performing the boundary determination as done in Appendix A for the vector fields X (1) and X (2) . This can be done by using the approach of [22] combined with its extensions in [32] and [16].

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