Periodic \(J\)-trajectories on \(\mathbb{R} \times \mathbb{S}^3\)
If \((M,g)\) is a Riemannian manifold, then the magnetic curves on \(M\) represent, in Physics, trajectories of charged particles moving under the action of a magnetic field. A magnetic field on \(M\) is defined by a closed 2-form \(\Omega\), and equivalently, a skew-symmetric \((1,1)\)-tensor field \(\varphi\), called the Lorentz force on \(M\), can be defined by \(g(\varphi X,Y)=\Omega(X,Y)\). A magnetic trajectory of this dynamical system \((M,g,\varphi)\) is a solution of the Lorentz equation \((*)\) \(\nabla_{\dot\gamma}\dot\gamma=q\,\varphi\dot\gamma\), where \(q\neq 0\) is known as the strength of the magnetic field. If \(\{T=\dot\gamma,\nu_1,\dots,\nu_r\}\) is an orthonormal frame of rank \(r\) along \(\gamma\) such that \(\nabla_TT=\kappa_1\nu_1\), \(\nabla_T\nu_1=-\kappa_1T+\kappa_2\nu_2\), \(\nabla_T\nu_j=-\kappa_{j}\nu_{j-1}+\kappa_{j+1}\nu_{j+1}\) for \(j=2,\dots,r-2\), \(\nabla_T\nu_{r-1}=-\kappa_{r-1}\nu_{r-2}\), where \(\kappa_1\), \(\kappa_2\),\(\dots\),\(\kappa_{r-1}\) are positive \(C^\infty\) functions on the manifold, then \(\gamma\) is a Frenet curve of osculating order \(r\). Moreover, \(\kappa_j\) is called the \(j\)th curvature of \(\gamma\). A geodesic is a Frenet curve of osculating order 1, a Riemannian circle is a Frenet curve of osculating order 2 with constant curvature \(\kappa_1\), and a helix of order \(r\) is defined as a Frenet curve of osculating order \(r\) such that all the curvatures \(\kappa_j\) are constant. If a \((2n+1)\)-dimensional almost contact metric manifold \((\overline M,\varphi,\xi,\eta,\overline g)\) is normal, that is its Nijenhuis tensor \(N_\varphi\) vanishes, then \(M\) is called a Sasakian manifold. If \(\gamma(s)\) is a curve on \(M\) parametrized by an arc-length, then the contact angle \(\theta\) of \(\gamma\) is a function defined as the angle between the tangent vector field and the characteristic vector field \(\xi\), that is, \(\cos\theta=\overline g(\dot\gamma,\xi_{|\gamma(s)})\). In particular, if the contact angle is equal to \(\pi/2\), then the curve \(\gamma\) is called a Legendre curve. The Kähler structure on \(\mathbb{C}\) induces the canonical Sasakian structure \((\varphi,\xi,\eta,\overline g)\) on \(\mathbb S^3=\{p=(z,w);\;|z|^2+|w|^2=1\}\subset\mathbb C^2\), where \(\overline g\) is the usual metric induced from the Euclidean metric of \(\mathbb R^4\equiv\mathbb C^2\), the characteristic vector field \(\xi\) is defined by \(\xi_p=J_0p=i\,p\in T_p\mathbb S^3\) for any \(p\in\mathbb S^3\), and a \((1,1)\)-tensor field \(\varphi\) and 1-form \(\eta\) are defined by splitting \(J_0X\) into the tangential and the normal components, respectively, \(J_0X=\varphi X-\eta(X)p\) for any \(X\in T_p\mathbb S^3\). If \(M=\mathbb R\times\mathbb S^3\) is the product manifold, \(t\) is the global coordinate on \(\mathbb R\), then a vector field on \(M\) is defined as \(\left(f\frac{d}{dt},X\right)\), where \(X\) is tangent to \(\mathbb S^3\) and \(f\) is a \(C^\infty\) function on \(M\), the metric \(g\) is the product metric \(g=dt^2+\overline g\), and an almost complex structure \(J\) is defined by \(J\left(f\frac{d}{dt},X\right)=\left(\eta(X)\frac{d}{dt},\varphi X-f\xi\right)\). It is known that \(J\) is integrable but \((M,g,J)\) is not Kähler. On a Kähler manifold \((M,g,J)\) the fundamental 2-form \(\Omega_J\) is closed, hence it defines a Kähler magnetic field. The Lorentz force corresponding to \(\Omega_J\) is precisely \(J\). So, a Kähler magnetic trajectory on \(M\) is a solution of the Lorentz equation \((*)\). Since \(M=\mathbb R\times\mathbb S^3\) is not Kähler, its fundamental 2-form \(\Omega_J\) does not define a magnetic field on \(M\). However, a parametrized smooth curve \(\gamma:I\subset\mathbb R\to\mathbb R\times\mathbb S^3\), \(\gamma(s)=(t(s),c(s))\) on \(M\), where \(c\) is a curve on \(\mathbb S^3\), is called a \(J\)-trajectory with the strength \(q\neq 0\) on \(M\) if it satisfies the second order Lorentz differential equation \((*)\). In this paper, the authors study the geometry of \(J\)-trajectories on the product manifold \(M=\mathbb R\times\mathbb S^3\). They prove that if \(\gamma:I\subset\mathbb R\to\mathbb R\times\mathbb S^3\) is a smooth curve, parametrized by arc-length, in the product space \(M\), then \(\gamma\) is a \(J\)-trajectory with strength \(q\neq 0\) if and only if it satisfies the differential equations \(\ddot t=q\,\eta(\dot c)\), \(\overline\nabla_{\dot c}\dot c=q(-\dot c\,\xi+\varphi\dot c)\) of motion for a \(J\)-trajectory with strength \(q\), where \(c\) is a curve on \(\mathbb S^3\) and \(t\) is the global coordinate on \(\mathbb R\), and \(\gamma\) has the first two curvatures constant, \(\kappa_1=|q|\) and \(\kappa_1=|\varphi\,\dot c|\). Moreover, the third curvature is \(\kappa_3=|\eta(\dot c)|\). Also, the authors show that if \(\gamma:I\subset\mathbb R\to\mathbb R\times\mathbb S^3\) is a \(J\)-trajectory with strength \(q\neq 0\) in the product space \(M\), parametrized by arc-length, then \(\gamma\) is a helix of order 3 if and only if \(\gamma\) is a Legendre contact magnetic curve on \(\mathbb S^3\) with strength \(q\), and \(\gamma\) is a Riemannian circle if and only if \(\gamma\) is parametrized by \(t(s)=a(1-\cos(qs))+b\sin(qs)\) and \(c(s)=\cos\varpi(s)p_0+\sin\varpi(s)\xi_0\). Finally, they show that if \(\gamma(s)=(t(s),c(s))\) is a \(J\)-trajectory with strength \(q\neq 0\) in the product space \(M\) such that \(\dot c\) is parallel to \(\xi\), then the curve \(\gamma\) is always periodic and its fundamental period is \(\frac{2\pi}{q}\), and if the curve \(c\) is a Legendre curve on \(\mathbb S^3\), then the curve \(\gamma\) is periodic if and only if \(\frac{q}{\sqrt{q^2+4}}\) is a rational number.
scientific article; zbMATH DE number 6948935
scientific article