Abstract
J-trajectories are arc length parameterized curves in almost Hermitian manifold which satisfy the equation \(\nabla _{{\dot{\gamma }}}{\dot{\gamma }}=q J {\dot{\gamma }}\). In this paper J-trajectories in the solvable Lie group \(\mathrm {Sol}_0^4\) are investigated. The first and the second curvature of a non-geodesic J-trajectory in an arbitrary LCK manifold whose anti Lee field has constant length are examined. In particular, the curvatures of non-geodesic J-trajectories in \(\mathrm {Sol}_0^4\) are characterized.
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The second named author is partially supported by JSPS KAKENHI Grant Number 19K03461. The authors would like to thank the referee for her/his careful reading of the manuscript and valuable suggestions for improving this article.
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Appendix A Almost Hamiltonian Systems
Appendix A Almost Hamiltonian Systems
1.1 Almost Symplectic Manifolds
Let N be a manifold. A two-form \(\varPsi \) is said to be an almost symplectic form if it is non-degenerate. A manifold N equipped with an almost symplectic form is called an almost symplectic manifold. It should be remarked that almost symplectic manifolds are even-dimensional. In particular, almost symplectic form is called a symplectic form if it is closed. A manifold N together with a symplectic form \(\varPsi \) is called a symplectic manifold.
On an almost symplectic manifold \((N,\varPsi )\), one can define a vector field \(\xi _H\) uniquely for any prescribed smooth function H on N by
for all vector fields Y on N. The vector field \(\xi _H\) is called the Hamiltonian vector field with Hamiltonian H.
The dynamical system \((N,\xi _H)\) determined by the Hamiltonian vector field \(\xi _H\) will be called an almost Hamiltonian system. Let us consider a local flow \(\{\psi _t\}\) of the Hamiltonian vector field (called a local Hamiltonian flow). Then one can see that the Hamiltonian H is preserved under local Hamiltonian flows, since \(dH(\xi _H)=(\iota _{\xi _H}\varPsi )(\xi _H)=\varPsi (\xi _H,\xi _H)=0\).
An almost Hamiltonian system is called a Hamiltonian system if its almost symplectic form is closed. On a Hamiltonian system \((N,\varPsi ,H)\), Hamiltonian vector field \(\xi _H\) satisfies
Here \(\pounds _{\xi _H}\) denotes the Lie differentiation by \(\xi _H\). Namely \(\xi _H\) is an infinitesimal symmetry of \((N,\varPsi ,H)\). In case \(d\varPsi \not =0\), this property does not hold, in general. By the Cartan’s formula
where \(\iota _{\xi _H}\) denotes the interior product by \(\xi _H\), we deduce that \(\pounds _{\xi _H}\varPsi =0\) holds if and only if
If \(\xi _H\) satisfies this condition, then \(\xi _H\) is said to be a strongly Hamiltonian. Note that Vaisman [41] requires “strongly Hamiltonian” for \(\xi _H\) of almost Hamiltonian systems.
Remark 15
On an almost symplectic manifold \((N,\varPsi )\), a vector field X is said to be locally Hamiltonian if \(d(\iota _{X}\varPsi )=0\). A locally Hamiltonian vector field X admits a local potential H, i.e., locally defined smooth function H so that \(dH=\iota _{X}\varPsi \). Thus X is locally expressed as \(X=\xi _H\).
Lemma 1
([17]) Let Y be a locally Hamiltonian vector field and Z is an infinitesimal symmetry of \(\varPsi \) on an almost symplectic manifold \((N,\varPsi )\) then \([Y,Z]=-\xi _{\varPsi (Y,Z)}\).
The (almost) Poisson bracket \(\{f,h\}\) of functions f and h on N is defined by
The bracket operation \(\{\cdot ,\cdot \}\) satisfies the Jacobi identity if and only if \(d\varPsi =0\).
For any vector fields X and Y on an almost symplectic manifold \((N,\varPsi )\), by Cartan’s formula, we have
In particular, if X and Y are locally Hamiltonian, we have
As a result, for strongly Hamiltonian vector fields \(\xi _f\) and \(\xi _h\),
Namely we have
1.2 The Canonical Two-Form
Let M be an m-manifold. Denote by \(T^{*}M\) its cotangent bundle. Take a local coordinate system \((x^1,x^2,\ldots ,x^m)\) of M, then it induces a fiber coordinates \((p_1,p_2,\ldots ,p_m)\). Then \(\vartheta =p_{j}\,dx^j\) is globally defined one-form on \(T^{*}M\) and called the canonical one-form (also called the Liouville form). The two-form \(\varPsi =-d\vartheta =dx^{j}\wedge dp_{j}\) is a symplectic form on \(T^{*}M\). The symplectic form \(\varPsi \) is called the canonical two-form on \(T^{*}M\). For a prescribed Hamiltonian H, the Hamiltonian vector field \(\xi _H\) is locally expressed as
Take a curve \({\bar{\gamma }}(t)\) in \(T^{*}M\). Then \({\bar{\gamma }}(t)=(x^i(t);p_i(t))\) is an integral curve of \(\xi _H\) if and only if it satisfies the Hamilton equation:
1.3 Geodesic Flows
Let us consider a Riemannian m-manifold (M, g), then its cotangent bundle \(T^{*}M\) is identified with the tangent bundle TM via the metric g. The so-called musical isomorphism
is a vector bundle isomorphism. We denote by \(\pi \) the natural projection of TM onto M. Take a local coordinate system \((x^1,x^2,\dots ,x^m)\) with fiber coordinates \((u^1,u^2,\dots ,u^m)\). Then the pull-backed one-form \(\flat ^{*}\vartheta \) is expressed as \(\flat ^{*}\vartheta =g_{ij}u^{j}dx^{i}\). Then the pull-backed two-form \(\varPhi :=\flat ^{*}\varPsi \) is computed as
The pull-backed two-form \(\varPhi \) is a symplectic form on TM. Let us consider the kinetic energy on TM:
Then we get a Hamiltonian system \((TM,\varPhi , E)\). The Hamiltonian vector field on TM with Hamiltonian E is called the geodesic spray and has local expression:
The integral curves of \(\xi _E\) are solutions to the system
One can see that for every trajectory \({\bar{\gamma }}(t)\) of the Hamiltonian system \((TM,\varPhi ,E)\), the projected curve \(\gamma (t)=\pi (\gamma (t))\) in M is a geodesic.
1.4 Magnetic Trajectories
Let us consider a two-form F on a Riemannian manifold (M, g). Express F as
Then we introduce an endomorphism field \(\phi \) by
and represent it locally as
then
Equivalently we have
We deform the symplectic form \(\varPhi \) of TM as
for some constant q. Then \(\varPhi _F\) has local expression
The deformed two-form \(\varPhi _F\) is still non-degenerate. When F is a magnetic field, then \(\varPhi _F\) is a symplectic form on TM (and hence \(\varPhi \) itself is a magnetic field).
Let us consider an almost Hamiltonian system \((TM,\varPhi _F,E)\). The Hamiltonian vector field is given by
where \(\mathrm {v}\{\phi (u)\}\) is a vertical vector field on TM globally defined by (cf. [35])
The Hamilton equation is
This is a second order ODE
This system has coordinate-free expression
Proposition 8
Let (M, g) be a Riemannian manifold with a magnetic field F. Then the magnetic trajectory equation is the Hamilton equation of the Hamiltonian system \((TM,\varPhi _F,E)\).
Example 5
Let (M, J, g) be an almost Hermitian manifold with fundamental two-form \(\Omega \). Then the J-trajectory equation
is the Hamilton equation of the almost Hamiltonian system \((TM,\varPhi _{-\Omega },E)\).
Now let (M, J, g) be a LCK manifold. We choose \(F=-\varOmega \). Then the perturbed two-form \(\varPhi ^J:=\varPhi _{-\varOmega }\) is given by
and hence
Thus the Hamiltonian vector field
is strongly Hamiltonian if and only if
for all vector fields \({\bar{X}}\) and \({\bar{Y}}\) on TM. Here we note that any point \((p;v)\in TM\), we have
1.5 The Almost Hamiltonian Systems Derived from J-Trajectories on \(\mathrm {Sol}^4_0\)
Set
on \(\mathrm {Sol}_0^4\), then the fiber coordinates are given by
The Christoffel symbols of \(\mathrm {Sol}_0^4\) are
Thus the geodesic spray is given by
where
The vertical vector field \(\mathrm {v}\{J(u)\}\) is given by
Thus we obtain
The symplectic form \(\varPhi \) of \(T\mathrm {Sol}_0^4\) is computed as
Next, we have
Hence we have
In the original notation we have
The Hamiltonian vector field \(\xi _f^J\) is given by the following Proposition.
Proposition 9
The Hamiltonian vector field \(\xi ^J_f\) corresponding to a function f is given by
Example 6
The dual one-forms of the right invariant vector fields \(e_1^R\), \(e_2^R\), \(e_3^R\), \(e_4^R\) are given by:
These one-forms are regarded as right invariant functions
on \(T\mathrm {Sol}_0^4\). The Hamiltonian vector fields \(\xi _1:=\xi ^J_{f_1}\), \(\xi _2:=\xi ^J_{f_2}\), \(\xi _3:=\xi ^J_{f_3}\), \(\xi _4:=\xi ^J_{f_4}\) are given by
Now let us investigate the strong Hamiltonian property of \((T\mathrm {Sol}_0^4,\varPhi ^J,E)\). Since \(\varPhi \) is closed, we have
One can see that
This shows that \(\xi ^J_E\) is not strongly Hamiltonian.
For any smooth functions f and h on \(T\mathrm {Sol}_0^4\), we introduce the (almost) Poisson bracket by
Since \(d\varPhi ^J\not =0\), this bracket does not satisfy Jacobi identity.
1.6 Tangent Group
Let G be a Lie group, then its tangent bundle is a Lie group with multiplication:
The resulting Lie group TG is called the tangent group of G. The tangent group is identified with the semi-direct product \(G\ltimes {\mathfrak {g}}\) via the left Maurer–Cartan form as:
Note that under this identification the Lie algebra \({\mathfrak {g}}=T_{e}G\) is regarded as the Lie algebra of all left invariant smooth vector fields of G. The multiplication law is transformed as
Remark 16
If we identify TG with \(G\times {\mathfrak {g}}\) via the right Maurer–Cartan form, the multiplication law becomes
The Lie algebra \({\mathfrak {g}}\) is regarded as the Lie algebra of right invariant smooth vector fields of G.
In the case of \(\mathrm {Sol}_0^4\), the tangent group \(T\mathrm {Sol}_0^4=\mathrm {Sol}_0^4\ltimes \mathfrak {sol}_0^4\) is \({\mathbb {R}}^8\) with multiplication:
The abelian subgroup \(G(1,2,3)\subset T\mathrm {Sol}_0^4\) given in Example 3 acts on \(T\mathrm {Sol}_0^4\) by right as
Note that the left action is just a translation
Let us consider an arc length parametrized J-trajectory \(\gamma (s)=(x(s),y(s),z(s),t(s))\) in \(\mathrm {Sol}_0^4\). Then its velocity vector field \({\dot{\gamma }}(s)\) is a smooth curve in \(T\mathrm {Sol}_0^4\) with parametrization
Let us represent \(\gamma (s)^{-1}\) as \(\gamma (s)^{-1}=({\underline{x}}(s),{\underline{y}}(s), {\underline{z}}(s),{\underline{t}}(s))\). We define a curve \({\underline{\gamma }}(s)\) of G(1, 2, 3) by
Then we get
We know that
are right invariant functions on \(\mathrm {Sol}_0^4\). This result implies that to solve the ODE system of J-trajectories, one needs to obtain t-coordinate t(s) first.
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Erjavec, Z., Inoguchi, Ji. J-Trajectories in 4-Dimensional Solvable Lie Group \(\mathrm {Sol}_0^4\). Math Phys Anal Geom 25, 8 (2022). https://doi.org/10.1007/s11040-022-09418-5
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DOI: https://doi.org/10.1007/s11040-022-09418-5