In this work, we study helix spacelike and timelike surfaces in the Lorentzian Berger sphere $s_{varepsilon}^3$, that is the three-dimensional sphere endowed with a $1$-parameter family of Lorentzian metrics, obtained by deforming the round metric on
$s^3$ along the fibers of the Hopf fibration $s^3to s^2({1}/{2})$ by $-varepsilon^2$. Our main result provides a characterization of the helix surfaces in $s_{varepsilon}^3$ using the symmetries of the ambient space and a general helix in $s_{varepsilon}^3$, with axis the infinitesimal generator of the Hopf fibers. Also, we construct some explicit examples of helix surfaces in $s_{varepsilon}^3$.
We show that $phi$-invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least $2$ are
all minimal. We prove that an odd-dimensional $phi$-invariant submanifold of a metric contact pair with orthogonal characteristic foliations inherits a contact form with an almost contact metric structure, and this induced structure is contact metric if and only if the submanifold is tangent to one Reeb vector field and orthogonal to the other one. Furthermore we show that the leaves of the two characteristic foliations of the differentials of the contact pair are minimal. We also prove that when one Reeb vector field is Killing and spans one characteristic foliation, the metric contact pair is a product of a contact metric manifold with $mathbb{R}$.
The notion of $Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on $Z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $g$ of $G$ admits a $Gamm
a$-grading where $Gamma$ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group $mathbb{H}_3$ adapted to the symmetries of a $Gamma$-symmetric structure on $mathbb{H}_3$. We prove that the classification of $z$-symmetric Riemannian and Lorentzian metrics on $mathbb{H}_3$ corresponds to the classification of left-invariant Riemannian and Lorentzian metrics, up to isometry. We study also the $Z_2^k$-symmetric structures on $G/H$ when $G$ is the $(2p+1)$-dimensional Heisenberg group for $k geq 1$. This gives examples of non riemannian symmetric spaces. When $k geq 1$, we show that there exists a family of flat and torsion free affine connections adapted to the $Z_2^k$-symmetric structures.
Flag manifolds are in general not symmetric spaces. But they are provided with a structure of $mathbb{Z}_2^k$-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. We detail for the flag ma
nifold $SO(5)/SO(2)times SO(2) times SO(1)$ what are the conditions for a metric adapted to the $mathbb{Z}_2^2$-symmetric structure to be naturally reductive.
The notion of $Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on $z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $g$ of $G$ admits a $Gamm
a$-grading where $Gamma$ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group $mathbb{H}_3$ adapted to the symmetries of a $Gamma$-symmetric structure on $mathbb{H}_3$. We prove that the classification of $z_2^2$-symmetric Riemannian and Lorentzian metrics on $mathbb{H}_3$ corresponds to the classification of left invariant Riemannian and Lorentzian metrics, up to isometries. This gives examples of non-symmetric Lorentzian homogeneous spaces.
We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form $omega$ from the Heisenberg manifold $(
H_3,g)$ to $(TH_3,g^S)$ are not totally geodesic, and the distributions $F^H=L(E_1^H,E_2^H)$ and $F^V=L(E_1^V,E_2^V)$ are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold $(H_3,g)$, are geodesics in the tangent bundle endowed with the Sasaki metric $(TH_3,g^s)$, if and only if the curves considered on the base manifold are geodesics. Then, we get two particular examples of geodesics from $(TH_3,g^s)$, which are not horizontal or natural lifts of geodesics from the base manifold $(H_3,g)$.
