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.