4 Contact Hamiltonian Systems for Probability Distribution Functions …
65
When identifying a contact Hamiltonian vector field with a dynamical system, ˙
x
a is
identified with dx
a
/dt for each a.
4.2.3 Legendre Submanifold as Dually Flat Space
As shown in [3], a Legendre submanifold can be related to a dually flat space defined
in information geometry. To state this explicitly, some of notions developed in information geometry are shown below [31, 32].
1. Let (M, g) be a (pseudo) Riemannian manifold, and ∇ a connection on M. If a
connection ∇
∗ satisfies
X [g(Y, Z )] = g(∇ X Y, Z ) + g(Y, ∇
∗
X Z ),
∀ X, Y, Z ∈ S T M
then ∇
∗ is referred to as the dual connection of ∇ with respect to g.
2. On a Riemannian manifold (M, g), if there exists a function such that g =
∇d , then (∇, g) is referred to as a Hessian structure, and the triplet (M, ∇, g)
a Hessian manifold.
3. On a Hessian manifold (M, ∇, g), there exists a coordinate system θ such that
all the connection coefficients vanish everywhere. This θ is referred to as ∇-affine
coordinates. Moreover there exists the ∇
∗ -affine coordinate system denoted by
η, such that
g
∂
∂θ a ,
∂
∂η b
= δ
b
a ,
a, b = 1, . . . , dim M.
This η is referred to as the dual coordinate system of θ with respect to g.
4. On a Riemannian manifold (M, g) with a connection ∇, if the following two
conditions for all X, Y, Z ∈ S T M
1. T
∇
(X, Y ) = 0, where T
∇
(X, Y ) := ∇ X Y − ∇ Y X − [X, Y ],
2. (∇ X g)(Y, Z ) = (∇ Y g)(X, Z ),
are satisfied, then a triplet (M, ∇, g) is referred to as a statistical manifold. The
introduced (1, 2)-tensor field T
∇ is torsion.
5. On a statistical manifold (M, ∇, g), if curvature tensor field R
∇ defined by
R
∇
(X, Y )Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [X,Y ] Z ,
∀ X, Y, Z ∈ S T M
vanishes, then (M, ∇, g) is referred to as a flat statistical manifold. A flat statistical manifold is referred to as a dually flat space, and is locally a Hessian manifold.
In general, given a connection ∇, if T
∇
≡ 0 and R
∇
≡ 0, then the connection ∇
is referred to as being flat.
The following is a relation between contact geometry and information geometry.
65
When identifying a contact Hamiltonian vector field with a dynamical system, ˙
x
a is
identified with dx
a
/dt for each a.
4.2.3 Legendre Submanifold as Dually Flat Space
As shown in [3], a Legendre submanifold can be related to a dually flat space defined
in information geometry. To state this explicitly, some of notions developed in information geometry are shown below [31, 32].
1. Let (M, g) be a (pseudo) Riemannian manifold, and ∇ a connection on M. If a
connection ∇
∗ satisfies
X [g(Y, Z )] = g(∇ X Y, Z ) + g(Y, ∇
∗
X Z ),
∀ X, Y, Z ∈ S T M
then ∇
∗ is referred to as the dual connection of ∇ with respect to g.
2. On a Riemannian manifold (M, g), if there exists a function such that g =
∇d , then (∇, g) is referred to as a Hessian structure, and the triplet (M, ∇, g)
a Hessian manifold.
3. On a Hessian manifold (M, ∇, g), there exists a coordinate system θ such that
all the connection coefficients vanish everywhere. This θ is referred to as ∇-affine
coordinates. Moreover there exists the ∇
∗ -affine coordinate system denoted by
η, such that
g
∂
∂θ a ,
∂
∂η b
= δ
b
a ,
a, b = 1, . . . , dim M.
This η is referred to as the dual coordinate system of θ with respect to g.
4. On a Riemannian manifold (M, g) with a connection ∇, if the following two
conditions for all X, Y, Z ∈ S T M
1. T
∇
(X, Y ) = 0, where T
∇
(X, Y ) := ∇ X Y − ∇ Y X − [X, Y ],
2. (∇ X g)(Y, Z ) = (∇ Y g)(X, Z ),
are satisfied, then a triplet (M, ∇, g) is referred to as a statistical manifold. The
introduced (1, 2)-tensor field T
∇ is torsion.
5. On a statistical manifold (M, ∇, g), if curvature tensor field R
∇ defined by
R
∇
(X, Y )Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [X,Y ] Z ,
∀ X, Y, Z ∈ S T M
vanishes, then (M, ∇, g) is referred to as a flat statistical manifold. A flat statistical manifold is referred to as a dually flat space, and is locally a Hessian manifold.
In general, given a connection ∇, if T
∇
≡ 0 and R
∇
≡ 0, then the connection ∇
is referred to as being flat.
The following is a relation between contact geometry and information geometry.