This text is an introduction to Calderón’s inverse conductivity problem on Rie-mannian manifolds. This problem arises as a model for electrical imaging in anisotropic media, and it is one of the most basic inverse problems in a geometric setting. The problem is still largely open, but we will discuss recent developments based on complex geometrical optics and the geodesic X-ray transform in the case where one restricts to a fixed conformal class of conductivities.

This work is based on lectures for courses given at the University of Helsinki in 2010 and at Universidad Autónoma de Madrid in 2011. It has therefore the feeling of a set of lecture notes for a graduate course on the topic, together with exercises and also some problems which are open at the time of writing this. The main focus is on manifolds of dimension three and higher, where one has to rely on real variable methods instead of using complex analysis. The text can be considered as an introduction to geometric inverse problems, but also as an introduction to the use of real analysis methods in the setting of Riemannian manifolds.