66
S. Goto and H. Hino
Proposition 1 (A contact manifold and a strictly convex function induce a dually
flat space, [3, 19]). Let (C, λ) be a (2m + 1)-dimensional contact manifold, (x, y, z)
a set of coordinates such that λ = dz −
m
a=1 y a dx
a with x = {x
1
, . . . , x
m
} and
y = {y 1 , . . . , y m }, and a strictly convex function depending only on x. Then,
((C, λ), , ) induces an m-dimensional dually flat space.
4.2.4 Legendre Submanifold as Attractor
As shown in [3] and [21], a class of relaxation processes, initial states approach to
the equilibrium state as time develops, can be formulated as contact Hamiltonian
vector fields on contact manifolds. This statement on a class of contact Hamiltonian
vector fields can be summarized as follows.
Proposition 2 (Legendre submanifold as an attractor, [3]). Let (C, λ) be a (2m + 1)dimensional contact manifold with λ being a contact form, (x, y, z) its coordinates
so that λ = dz −
m
a=1 y a dx
a , and a function depending only on x. Then, one
has the following.
1. The contact Hamiltonian vector field associated with the contact Hamiltonian
h : C → R with
h (x, y, z) := (x) − z,
(4.8)
gives
d
dt
x
a
= 0,
d
dt
y a =
∂ ∂
∂ x a − y a ,
d
dt
z = (x) − z,
(4.9)
where a = 1, . . . , m.
2. The Legendre submanifold generated by , given by (4.7), is an invariant manifold for the contact Hamiltonian vector field.
3. Every point on C \ A approaches to A along an integral curve as time
develops. Equivalently A is an attractor in C :
lim
t→∞
( x(t), y(t), z(t) ) ∈ A .
(4.10)
4. Let {x(0), y(0), z(0)} be a point on C \ A . Then for any t ∈ R, it follows that
h (x(t), y(t), z(t)) = exp(− t) h (x(0), y(0), z(0)).
(4.11)
Points on the integral curve of the contact Hamiltonian vector field in Proposition 2
can be characterized by the Mrugala metric tensor field G. To state this, one defines
the length of a curve γ : R → M, ( t → γ (t) ), associated with a vector field ˙
γ :=
(d/dt)γ on a (pseudo) Riemannian manifold (M, g) as
1
1 In [27, 28], the sign of the length of curves is not correct.
S. Goto and H. Hino
Proposition 1 (A contact manifold and a strictly convex function induce a dually
flat space, [3, 19]). Let (C, λ) be a (2m + 1)-dimensional contact manifold, (x, y, z)
a set of coordinates such that λ = dz −
m
a=1 y a dx
a with x = {x
1
, . . . , x
m
} and
y = {y 1 , . . . , y m }, and a strictly convex function depending only on x. Then,
((C, λ), , ) induces an m-dimensional dually flat space.
4.2.4 Legendre Submanifold as Attractor
As shown in [3] and [21], a class of relaxation processes, initial states approach to
the equilibrium state as time develops, can be formulated as contact Hamiltonian
vector fields on contact manifolds. This statement on a class of contact Hamiltonian
vector fields can be summarized as follows.
Proposition 2 (Legendre submanifold as an attractor, [3]). Let (C, λ) be a (2m + 1)dimensional contact manifold with λ being a contact form, (x, y, z) its coordinates
so that λ = dz −
m
a=1 y a dx
a , and a function depending only on x. Then, one
has the following.
1. The contact Hamiltonian vector field associated with the contact Hamiltonian
h : C → R with
h (x, y, z) := (x) − z,
(4.8)
gives
d
dt
x
a
= 0,
d
dt
y a =
∂ ∂
∂ x a − y a ,
d
dt
z = (x) − z,
(4.9)
where a = 1, . . . , m.
2. The Legendre submanifold generated by , given by (4.7), is an invariant manifold for the contact Hamiltonian vector field.
3. Every point on C \ A approaches to A along an integral curve as time
develops. Equivalently A is an attractor in C :
lim
t→∞
( x(t), y(t), z(t) ) ∈ A .
(4.10)
4. Let {x(0), y(0), z(0)} be a point on C \ A . Then for any t ∈ R, it follows that
h (x(t), y(t), z(t)) = exp(− t) h (x(0), y(0), z(0)).
(4.11)
Points on the integral curve of the contact Hamiltonian vector field in Proposition 2
can be characterized by the Mrugala metric tensor field G. To state this, one defines
the length of a curve γ : R → M, ( t → γ (t) ), associated with a vector field ˙
γ :=
(d/dt)γ on a (pseudo) Riemannian manifold (M, g) as
1
1 In [27, 28], the sign of the length of curves is not correct.