Geodesic flows of Riemannian metrics on manifolds are one of the classical objects in geometry. A particular place among them is occupied by integrable geodesic flows. We consider them in the context of the general theory of integrable Hamiltonian systems, and in particular, from the viewpoint of a new topological classification theory, which was recently developed for integrable Hamiltonian systems with two degrees of freedom. As a result, we will see that such a new approach is very useful for a deeper understanding of the topology and geometry of integrable geodesic flows. The main object to be studied in our paper is the class of integrable geodesic flows on two-dimensional surfaces. There are many such flows on surfaces of small genus, in particular, on the sphere and torus. On the contrary, on surfaces of genus 9 > 1, no such flows exist in the analytical case. One of the most important and interesting problems consists in the classification of integrable flows up to different equivalence relations such as (1) an isometry, (2) the Liouville equivalence, (3) the trajectory equivalence (smooth and continuous), and (4) the geodesic equivalence. In recent years, a new technique was developed, which gives, in particular, a possibility to classify integrable geodesic flows up to these kinds of equivalences. This technique is presented in our paper, together with various applications. The first part of our book, namely, Chaps.
Philosophers of science have produced a variety of definitions for the notion of one sentence, theory or hypothesis being closer to the truth, more verisimilar, or more truthlike than another one. The definitions put forward by philosophers presuppose at least implicitly that the subject matter with which the compared sentences, theories or hypotheses are concerned has been specified,! and the property of closeness to the truth, verisimilitude or truth likeness appearing in such definitions should be understood as closeness to informative truth about that subject matter. This monograph is concerned with a special case of the problem of defining verisimilitude, a case in which this subject matter is of a rather restricted kind. Below, I shall suppose that there is a finite number of interrelated quantities which are used for characterizing the state of some system. Scientists might arrive at different hypotheses concerning the values of such quantities in a variety of ways. There might be various theories that give different predictions (whose informativeness might differ , too) on which combinations of the values of these quantities are possible. Scientists might also have measured all or some of the quantities in question with some accuracy. Finally, they might also have combined these two methods of forming hypotheses on their values by first measuring some of the quantities and then deducing the values of some others from the combination of a theory and the measurement results.