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Parallel and totally geodesic hypersurfaces of 5-dimensional 2-step homogeneous nilmanifolds

Mehri Nasehi (2016)

Czechoslovak Mathematical Journal

In this paper we study parallel and totally geodesic hypersurfaces of two-step homogeneous nilmanifolds of dimension five. We give the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces. Moreover, we prove that two-step homogeneous nilmanifolds of dimension five which have one-dimensional centre never admit parallel hypersurfaces. Also we prove that the only two-step homogeneous nilmanifolds of dimension five which admit totally geodesic hypersurfaces...

Parallel and totally geodesic hypersurfaces of solvable Lie groups

Mehri Nasehi (2016)

Archivum Mathematicum

In this paper we consider special examples of homogeneous spaces of arbitrary odd dimension which are given in [5] and [16]. We obtain the complete classification and explicitly describe parallel and totally geodesic hypersurfaces of these spaces in both Riemannian and Lorentzian cases.

Preferred parameterisations on homogeneous curves

Michael Eastwood, Jan Slovák (2004)

Commentationes Mathematicae Universitatis Carolinae

We show how to specify preferred parameterisations on a homogeneous curve in an arbitrary homogeneous space. We apply these results to limit the natural parameters on distinguished curves in parabolic geometries.

Projective invariant metrics and open convex regular cones. I

Fabio Podestà (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this work we give a characterization of the projective invariant pseudometric P , introduced by H. Wu, for a particular class of real 𝐂 ∞ -manifolds; in view of this result, we study the group of projective transformations for the same class of manifolds and we determine the integrated pseudodistance p of P in open convex regular cones of ℝ n , endowed with the characteristic metric.

Projective invariant metrics and open convex regular cones. II

Fabio Podestà (1987)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The aim of this work, which continues Part I with the same title, is to study a class of projective transformations of open, convex, regular cones in ℝ n and to prove a structure theorem for affine transformations of a restricted class of cones; we conclude with a version of the Schwarz Lemma holding for affine transformations.

Projective reparametrization of homogeneous curves

Boris Doubrov (2005)

Archivum Mathematicum

We study the conditions when locally homogeneous curves in homogeneous spaces admit a natural projective parameter. In particular, we prove that this is always the case for trajectories of homogeneous nilpotent elements in parabolic spaces. On algebraic level this corresponds to the generalization of Morozov–Jacobson theorem to graded semisimple Lie algebras.

Projective spaces of second order.

Andrzej Miernowski, Witold Mozgawa (1997)

Collectanea Mathematica

Grassmannians of higher order appeared for the first time in a paper of A. Szybiak in the context of the Cartan method of moving frame. In the present paper we consider a special case of higher order Grassmannian, the projective space of second order. We introduce the projective group of second order acting on this space, derive its Maurer-Cartan equations and show that our generalized projective space is a homogeneous space of this group.

Projective structure, SL ˜ ( 3 , ℝ ) and the symplectic Dirac operator

Marie Holíková, Libor Křižka, Petr Somberg (2016)

Archivum Mathematicum

Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is ˜ ( 3 , ) .

Pseudo-Riemannian weakly symmetric manifolds of low dimension

Bo Zhang, Zhiqi Chen, Shaoqiang Deng (2019)

Czechoslovak Mathematical Journal

We give a classification of pseudo-Riemannian weakly symmetric manifolds in dimensions 2 and 3 , based on the algebraic approach of such spaces through the notion of a pseudo-Riemannian weakly symmetric Lie algebra. We also study the general symmetry of reductive 3 -dimensional pseudo-Riemannian weakly symmetric spaces and particularly prove that a 3 -dimensional reductive 2 -fold symmetric pseudo-Riemannian manifold must be globally symmetric.

Pseudo-symmetric contact 3-manifolds III

Jong Taek Cho, Jun-ichi Inoguchi, Ji-Eun Lee (2009)

Colloquium Mathematicae

A trans-Sasakian 3-manifold is pseudo-symmetric if and only if it is η-Einstein. In particular, a quasi-Sasakian 3-manifold is pseudo-symmetric if and only if it is a coKähler manifold or a homothetic Sasakian manifold. Some examples of non-Sasakian pseudo-symmetric contact 3-manifolds are exhibited.

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