## ABSTRACT

Chapter 11

Integrable Geodesic Flows

on Two-dimensional Surfaces

11.1. STATEMENT OF THE PROBLEM

Let M

n

be a smooth Riemannian manifold with a Riemannian metric g

ij

(x). Recall

that geodesics of the given metric are dened as smooth parameterized curves

(t) = (x

(t); : : : ; x

n

(t))

that are solutions to the system of dierential equations

r

_

_ = 0 ;

where _ =

d

dt

is the velocity vector of the curve , and r is the covariant derivation

operator related to the symmetric connection associated with the metric g

ij

. In local

coordinates, these equations can be rewritten in the form

d

x

i

dt

+

X

i

jk

dx

j

dt

dx

k

dt

= 0 ;

where

i

jk

(x) are smooth functions called the Christoel symbols of the connec-

tion r and dened by the following explicit formulas:

i

jk

(x) =

X

g

is

@g

sj

@x

k

+

@g

ks

@x

j

@g

kj

@x

s

:

The geodesics can be interpreted as trajectories of a single mass point that moves

on the manifold without any external action, i.e., by inertia. Indeed, the equation

CH. 11.