Pseudo-Riemannian Symmetries on Heisenberg groups

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 $\Gamma$-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.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1201.044

Explicit Formulas for Non-Geodesic Biharmonic Curves of the Heisenberg Group

We consider the biharmonicity condition for maps between Riemannian manifolds (see [BK]), and study the non-geodesic biharmonic curves in the Heisenberg group H_3. First we prove that all of them are helices, and then we obtain explicitly their parametric equations.Comment: 16 pages, 2 figure

Constant angle surfaces in the Lorentzian Heisenberg group

In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics. In particular, we give an explicit local parametrization of these surfaces and we produce some examples.Comment: 13 pages, 8 figure

Geodesicity and Isoclinity Properties for the Tangent Bundle of the Heisenberg Manifold with Sasaki Metric

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 $(H_3,g)$.Comment: 12 page

案例與例外 : 十三妹作為香港專欄作家 = Hong Kong columnist Shi San Mei : an (exceptional) case study

香港專欄作家十三妹在一九五○年代末突然冒起，以隨筆、漫談吸引了大量讀者。她從來不與報紙編輯或其他專欄作家見面，因而予人神秘之感；但她又在專欄里坦然自述身世和生活，恣意批評政治、社會、文化各方面的現象，樹敵無數。十三妹以寫作為生，卻自言與同行格格不入，本文嘗試利用十三妹專欄裏的豐富材料，勾勒五、六○年代香港職業寫作人的形態，並通過十三妹涉及的紛爭，探討當時文學圏裹的權力關係

Pseudo-Riemannian Symmetries on Heisenberg groups

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 $\Gamma$-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

Gamma-oryzanol, a main component of rice bran oil: <i>in vitro</i> studies of its antioxidant properties

Since investigations so far carried out provided an unclear picture of the mechanism of the antioxidant action of gamma-oryzanol, the aim of the present work has been the contribution to the elucidation of its the molecular mechanisms by using in vitro, previously well-characterized, experimental models (such as scavenging of DPPH° ROS scavenging, Fe and azocompound triggered lipoperoxidation) that allow us to study the reactions involved in the complex process of lipoperoxidation