I am working on the paper "magnetic curves corresponding to Killing magnetic field $E^3$" and I need some help. As I understand, vector field $V$ on manifold $M$ is a Killing vector field if it satisfies $$\langle \nabla_Y V, Z\rangle + \langle \nabla_ZV, Y\rangle =0$$ for every vector fields $Y, Z$ on $M$. Some of the Killing vector field examples are also given such as $\partial _x$ which I can verify directly.
It is also said that for any Killing vector field $V$ corresponding Killing magnetic field can be computed by the formula $$F_V=\iota_Vdv_g,$$ where $\iota$ is inner product and $$\langle X\times Y, Z\rangle =dv_g(X,Y,Z).$$ I do not really understand what is the meaning of that formula and how to interpret and apply it to Killing vector field? For example given Killing vector field $V=-y \partial_ x +x\partial_y$ the Killing magnetic field corresponding to it found as $$F_V=-(xdx+ydy) \wedge dz.$$ Can somebody please explain me clearly how is it found since I have just start working on this topic and I am not very familiar some concepts?
Thanks for your help.
$F_V=\iota_Vdv_g$, this equation is valid in 3-dimensional spaces. It basically says that 2-forms are in bijective correspondence with vector fields. By the way, $\iota$ is used for interior product not for "inner product". See this question to see the relation between them.
Consider a vector field $V \in \mathfrak{X}(M^3)$ for a 3-dimensional manifold $(M^3,g)$. Take its metric equivalent 1-form denoted by $V^\flat$ and then apply Hodge star to this 1-form $\star V^\flat$ which is obviously a 2-form. Now, this 2-form can also be expressed by using the interior product $\iota_V$, as $\star V^\flat=\iota_V dv_g$, where $dv_g$ is the volume form of $(M^3,g)$.
Let's label volume form $dv_g$ as $\Omega$ and take a general vector field $ V=P \partial_x + Q \partial_y + N\partial_z $ $ \in \mathfrak{X}(\mathbb{R}^3)$, where $P$, $Q$ and $N$ are smooth functions in $\mathbb{R}^3$. Since we have the Euclidean metric, corresponding 1-form becomes $V^\flat=P dx+ Q dy + N dz $. Applying the Hodge star and then taking the exterior derivative we get,
$d\star V^\flat=(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial N}{\partial z}) dx \wedge dy \wedge dz=div(V)~\Omega $
Note that the partial derivative terms correspond to divergence of our vector field ($ div(V)$). Recall Cartan's formula, $\mathcal{L}_V\Omega=\iota_V d\Omega +d (\iota_V \Omega)$. The first term including $d\Omega$ is vanishing obviously ($\Omega$ being volume element or top form) and we already obtained the second term above. Hence we get,
$\mathcal{L}_V\Omega= d (\iota_V \Omega)=div(V)~\Omega$.
which implies that the 2-form $\star V^\flat=\iota_V \Omega$ is closed if and only if $div(V)=0$, i.e., the volume element is invariant under the local flows of V. This result allows us to define magnetic fields in 3-dimensions as divergence-free vector fields. Particularly, if $V$ is Killing then $div(V)=0$ condition is automatically satisfied since Killing vector fields are divergence-free by definition.
The second part of your question: $ \langle X\times Y, Z\rangle =dv_g(X,Y,Z).$ This is nothing but scalar triple product in 3-dimensions $ \mathbf {Z} \cdot (\mathbf {X} \times \mathbf {Y} ) $ which is equal to the signed volume of parallelepiped. You can check the geometric interpretation here . It is used to show certain relations in contact manifolds but considering your questions here those relations do not immediately concern you at the moment I think. I would suggest this nice paper$^3$ for more details.
The last part of your question: Given the Killing vector field as $ V=-y \partial_ x +x\partial_y$ , how do we find the corresponding Killing magnetic field?
Actually, it is obtained just by applying the interior product with respect to the given vector field.
$F_V=\iota_V dv_g = \iota_{(-y \partial_ x +x\partial_y)} dx \wedge dy \wedge dz =(-y \partial_ x +x\partial_y) \lrcorner ~ dx \wedge dy \wedge dz = -y dy \wedge dz -x dx \wedge dz $
$\implies F_V=-(xdx+ydy) \wedge dz. $
[3] Cabrerizo, J. L.; Fernández, M.; Gómez, J. S., On the existence of almost contact structure and the contact magnetic field, Acta Math. Hung. 125, No. 1-2, 191-199 (2009). ZBL1212.53112.