Chapter 12

Liouville Classication

of Integrable Geodesic Flows

on Two-Dimensional Surfaces

In this chapter, we discuss the results by E. N. Selivanova [311], V. V. Kalash-

nikov (Jr.) [175], Nguyen Tien Zung, L. S. Polyakova [263], [264], V. S. Matveev [224]

devoted to the topology of Liouville foliations of integrable geodesic ows on

two-dimensional surfaces. We begin with the simplest case of global Liouville

metrics on the two-dimensional torus, where the structure of the Liouville fo-

liation, on the one hand, is the most natural and, on the other hand, serves

a good model for the description of all other cases. As we shall see shortly,

integrable geodesic ows on a two-dimensional surface are similar in many

respects. However, each class of such geodesic ows is distinguished among

the others by some specic properties of Liouville foliations. Following the general

idea of our book, we shall formulate the nal answer in terms of marked