ABSTRACT
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
molecules.