We give explicit parametrizations for all the homogeneous Riemannian structures on model spaces of Thurston geometry. As an application, we give all the homogeneous contact metric structures on $3$-dimensional Sasakian space forms.
We study homogeneous geodesics in $4$-dimensional solvable Lie groups $\mathrm{Sol}_0^4$, $\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}$ and $\mathrm{Nil}_4$.
The Ricci tensor field, $\varphi$-Ricci tensor field and the characteristic Jacobi operator on almost Kenmotsu $3$-manifolds are investigated. We give a classification of locally symmetric almost Kenmotsu $3$-manifolds.
We consider magnetic curves corresponding to the Killing magnetic fields in hyperbolic 3-space.