Argument shift method and sectional operators: applications to differential geometry

This text does not contain any new results, it is just an attempt to present, in a systematic way, one construction which makes it possible to use some ideas and notions well-known in the theory of integrable systems on Lie algebras to a rather different area of mathematics related to the study of projectively equivalent Riemannian and pseudo-Riemannian metrics. The main observation can be formulated, yet without going into details, as follows: The curvature tensors of projectively equivalent metrics coincide with the Hamiltonians of multi-dimensional rigid bodies. Such a relationship seems to be quite interesting and may apparently have further applications in differential geometry. The wish to talk about this relation itself (rather than some new results) was one of motivations for this paper.