Determination of metrics by boundary energy

This paper reformulates a problem of Sharafutdinov [4] and extend the new variant from the single-time context to the multi-time context. Section 1 is dedicated to the single-time case. It starts with well-known facts of describing geodesics as extremals. Then it is formulated and studied the problem of determination of a metric by the boundary energy. The linearization of this problem leads to the ray transform of a tensor field and to moment problem. Section 2 extend the single-time case to the multi-time case. It begins with well-known facts about harmonic maps and continues with determining a pair of metrics from boundary energy. Using the linearization, we extend the idea to multi-ray transform of a distinguished tensor field (moment problem). Mathematics Subject Classification: 53C43, 53C21.