4 Contact Hamiltonian Systems for Probability Distribution Functions …
61
4.2 Underlying Geometry
In this section mathematical definitions and facts commonly used in the later sections
are summarized. In this paper manifolds are assumed smooth and connected. In
addition tensor fields are assumed smooth and real. The set of vector fields on a
manifold M is denoted by S T M, that of p-forms denoted by S
p
M, the interior
product with X ∈ S T M is denoted by ı X , and the Lie derivative along X ∈ S T M
denoted by L X .
4.2.1 Para-Contact Metric Manifolds
In this subsection definitions and some existing statements are summarized, and the
notations used in this paper are fixed (see [20, 29]). Note that there are several ways
to define (para-) contact metric manifold.
Let M be a (2m + 1)-dimensional manifold (m ≥ 1). An almost para-contact
structure on M is a triplet (φ, ξ, λ), where ξ is a vector field, λ a one-form, φ :
S T M → S T M a (1, 1)-tensor field such that
(i) : φ
2
= Id − λ ⊗ ξ, (ii) : λ(ξ ) = 1, and (iii) : ker(λ) = Im(φ) = D
+
+ D
−
,
where ker(λ) := {X ∈ S T M | λ(X ) = 0 }, Im(φ) := {φ(X, −) ∈ S T M| X ∈
S T M}, D
± are eigen-spaces of D := ker(λ) whose eigenvalues are ±1, and Id
is the identity operator. A pseudo Riemannian metric tensor field g satisfying
g(φ X, φY ) = − g(X, Y ) + λ(X ) λ(Y ),
∀X, Y ∈ S T M
is referred to as a metric tensor compatible with an almost para-contact structure. It is verified for non-compact manifolds that any almost para-contact structure
admits a metric tensor field compatible with an almost para-contact structure. Then
(M, φ, ξ, λ, g) is referred to as an almost para-contact metric manifold.
On almost para-contact metric manifolds, one can show that
λ φ = 0, φ ξ = 0, λ(X ) = g(X, ξ), g(ξ, ξ ) = 1, and g(φ X, Y ) + g(X, φY ) = 0,
for ∀X, Y ∈ S T M. If g of an almost para-contact metric manifold satisfies
g(X, φY ) =
1
2
dλ(X, Y ),
∀X, Y ∈ S T M
(4.3)
then, (M, φ, ξ, λ, g) is referred to as a para-contact metric manifold, where
the convention of the numerical factor dλ(X, Y ) = X λ(Y ) − Y λ(X ) − λ([X, Y ]),
([X, Y ] := XY − Y X) has been adapted. This convention corresponds to
61
4.2 Underlying Geometry
In this section mathematical definitions and facts commonly used in the later sections
are summarized. In this paper manifolds are assumed smooth and connected. In
addition tensor fields are assumed smooth and real. The set of vector fields on a
manifold M is denoted by S T M, that of p-forms denoted by S
p
M, the interior
product with X ∈ S T M is denoted by ı X , and the Lie derivative along X ∈ S T M
denoted by L X .
4.2.1 Para-Contact Metric Manifolds
In this subsection definitions and some existing statements are summarized, and the
notations used in this paper are fixed (see [20, 29]). Note that there are several ways
to define (para-) contact metric manifold.
Let M be a (2m + 1)-dimensional manifold (m ≥ 1). An almost para-contact
structure on M is a triplet (φ, ξ, λ), where ξ is a vector field, λ a one-form, φ :
S T M → S T M a (1, 1)-tensor field such that
(i) : φ
2
= Id − λ ⊗ ξ, (ii) : λ(ξ ) = 1, and (iii) : ker(λ) = Im(φ) = D
+
+ D
−
,
where ker(λ) := {X ∈ S T M | λ(X ) = 0 }, Im(φ) := {φ(X, −) ∈ S T M| X ∈
S T M}, D
± are eigen-spaces of D := ker(λ) whose eigenvalues are ±1, and Id
is the identity operator. A pseudo Riemannian metric tensor field g satisfying
g(φ X, φY ) = − g(X, Y ) + λ(X ) λ(Y ),
∀X, Y ∈ S T M
is referred to as a metric tensor compatible with an almost para-contact structure. It is verified for non-compact manifolds that any almost para-contact structure
admits a metric tensor field compatible with an almost para-contact structure. Then
(M, φ, ξ, λ, g) is referred to as an almost para-contact metric manifold.
On almost para-contact metric manifolds, one can show that
λ φ = 0, φ ξ = 0, λ(X ) = g(X, ξ), g(ξ, ξ ) = 1, and g(φ X, Y ) + g(X, φY ) = 0,
for ∀X, Y ∈ S T M. If g of an almost para-contact metric manifold satisfies
g(X, φY ) =
1
2
dλ(X, Y ),
∀X, Y ∈ S T M
(4.3)
then, (M, φ, ξ, λ, g) is referred to as a para-contact metric manifold, where
the convention of the numerical factor dλ(X, Y ) = X λ(Y ) − Y λ(X ) − λ([X, Y ]),
([X, Y ] := XY − Y X) has been adapted. This convention corresponds to