scholarly journals Parallel and totally geodesic hypersurfaces of non‐reductive homogeneous four‐manifolds

2020 ◽  
Vol 293 (9) ◽  
pp. 1707-1729
Author(s):  
Giovanni Calvaruso ◽  
Reinier Storm ◽  
Joeri Van der Veken
Keyword(s):  
Totally Geodesic ◽  
Totally Geodesic Hypersurfaces ◽  
Four Manifolds

On six-dimensional Vaisman — Gray submanifolds of the octave algebra

2019 ◽  
pp. 29-35
Author(s):  
M. Banaru
Keyword(s):  
Structural Equations ◽  
Totally Geodesic ◽  
Hermitian Manifolds ◽  
Cayley Algebra ◽  
Almost Hermitian Manifolds ◽  
Almost Contact Metric ◽  
Totally Geodesic Hypersurfaces ◽  
Metric Structures ◽  
Nearly Cosymplectic ◽  
Kählerian Manifolds

The W1 W4 class of almost Hermitian manifolds (in accordance with the Gray — Hervella classification) is usually named as the class of Vaisman — Gray manifolds. This class contains all Kählerian, nearly Kählerian and locally conformal Kählerian manifolds. As it is known, Vaisman — Gray manifolds are invariant under the conformal transformations of the metric. A criterion in the terms of the configuration tensor for an arbitrary six-dimensional submanifold of Cayley algebra to belong to the Vaisman — Gray class of almost Hermitian manifolds is established. The Cartan structural equations of the almost contact metric structures induced on oriented hypersurfaces of six-dimensional Vaisman — Gray submanifolds of the octave algebra are obtained. It is proved that totally geodesic hypersurfaces of six-dimensional Vaisman — Gray submanifolds of Cayley algebra admit nearly cosymplectic structures (or Endo structures). This result is a generalization of the previously proved fact that totally geodesic hypersurfaces of nearly Kählerian manifolds also admit nearly cosymplectic structures.

scholarly journals Bending in the space of quasi-Fuchsian structures

1991 ◽  
Vol 33 (1) ◽  
pp. 41-49 ◽  
Author(s):  
Christos Kourouniotis
Keyword(s):  
Hyperbolic Manifold ◽  
Geodesic Lamination ◽  
Closed Curves ◽  
Totally Geodesic ◽  
Geodesic Laminations ◽  
Totally Geodesic Hypersurfaces ◽  
Definition Of

In [2] I described the deformation of bending a hyperbolic manifold along an embedded totally geodesic hypersurface. As I remarked there, the deformation is particularly interesting in the case of a surface, because a surface contains many embedded totally geodesic hypersurfaces, namely simple closed curves, along which it is possible to bend. Furthermore, for a surface it is possible to extend the definition of bending to the case of a geodesic lamination, by using the fact that the set of simple closed geodesies is dense in the space of geodesic laminations. This direction has been developed by Epstein and Marden in [1].

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