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Although one expects the association between minima and positive curvatures
on the one side and negative curvatures along some directions and saddles on the
other side to be essentially true for most choices of the metric, a particular choice
of g among the many possible ones must be made in order to perform explicit
calculations. The most immediate choice would probably be that of considering
as our manifold M the N-dimensional surface z = V (q
1
;:::;q
N
) itself, i.e., the
graph of the potential energy V as a function of the N coordinates q
1
;:::;q
N
of
the conguration space, and to dene g as the metric induced on that surface
by its immersion in R
N+1
. Although perfectly reasonable, this choice has two
drawbacks: (i) the explicit expressions for the curvatures in terms of derivatives of
V are rather complicated and (ii) the link between the properties of the dynamics
and the geometry is not very precise, i.e., one cannot prove that the geometry
completely determines the dynamics and its stability. For these reasons in Refs. [25]
a particular choice of (M;g), referred to as the Eisenhart metric [26], such that the
link between geometry and dynamics is more clear was proposed. The properties
of this metric have been previously thoroughly reviewed in [27, 28] so that we refer
the reader to the latter references for the details. Let us only recall that given
a potential energy V (q
1
;:::;q
N
) this (pseudo-Riemannian) metric is dened on a
the conguration space with two extra dimensions, MR
2
, with local coordinates
(q
0
;q
1
;:::;q
N
;q
N+1
), and its arc-length is
ds
2
=
i;j
dq
i
dq
j
2V (q)(dq
0
)
2
+ 2dq
0
dq
N+1
:
(6.13)
The metric tensor will be referred to as g
E
and its components are
0
@
1
A
2V (q)
0 0
1
0
1 0
0
.
.
.
.
.
.
.
g
E
=
(6.14)
0
0 1
0
1
0 0
0
as can be derived by Eq. (6.13).
The geodesics of this metric are the natural motions of a Hamiltonian system
with standard kinetic energy and potential energy V (see [27]). The nonvanishing
components of the curvature tensor are
R
0i0j
= @
i
@
j
V ;
(6.15)
it can then be shown that the Ricci curvature (6.11) in the direction of motion, i.e.,
in the direction of the velocity vector v of the geodesic, is given by
K
R
(v) =4V ;
(6.16)
where4V is the Laplacian of the potential V , and that the scalar curvatureR
identically vanishes. We note that K
R
(v) is nothing but a scalar measure of the
average curvature \felt" by the system during its evolution; we will refer to it
