J-Trajectories in 4-Dimensional Solvable Lie Group $$\mathrm {Sol}_0^4$$

Parallel and totally umbilical hypersurfaces of the four‐dimensional Thurston geometry Sol04$\text{Sol}^4_0$

“…One computes the sectional curvatures from Equation (2.7), using that 𝐸 3 , 𝐸 4 , and 𝑏𝐸 1 − 𝑎𝐸 2 form a tangent frame for this hypersurface. □ mentioned in [7], and this agrees with Theorem 1 in [15]. Namely, we have…”

“…Remark The totally geodesic hypersurfaces in Corollary 3.4 are factors of the warped product representations of $$\mathrm{Sol}^4_0$$ mentioned in [7], and this agrees with Theorem 1 in [15]. Namely, we have $$$$\begin{align*} \mathrm{Sol}^4_0&\cong \mathbb {H}^3(-1) \times _{e^{2t}} \mathbb {R}\cong {\left\lbrace M(x,y,z,t) \in \mathrm{Sol}^4_0\mid z = c \right\rbrace} \times _{e^{2t}} \mathbb {R}, \\ \mathrm{Sol}^4_0&\cong \mathbb {H}^2(-4) \times _{e^{-t}} \mathbb {R}^2 \cong (\mathbb {H}^2(-4) \times _{e^{-t}} \mathbb {R}) \times _{e^{-t}} \mathbb {R}\\ &\cong {\left\lbrace M(x,y,z,t) \in \mathrm{Sol}^4_0\mid ax + by = c \right\rbrace} \times _{e^{-t}} \mathbb {R}.$$…”

Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups

“…Remark 5.1. In our previous work [21] we chose the complex structure J := −J − which is compatible to the geometric structure (see [84]). The resulting Hermitian surface (Sol 4 0 , g, J) is a globally conformal Kähler surface with Lee form −2dt.…”

Killing Magnetic Curves in $\: \mathbb{H}^{3}$

“…In hyperbolic 3-space, even if magnetic field V is non-trivial, magnetic curves may be geodesics. This phenomena happens when V = 0 along a geodesic γ (This phenomena does not occur for Kähler magnetic curves, or more generally for J-trajectories [14,15]).…”