Some Hyperbolic 3-Manifolds with Totally Geodesic Boundary

2010 ◽  
Vol 17 (03) ◽  
pp. 457-468 ◽  
Author(s):  
Agnese Ilaria Telloni
Keyword(s):  
Fundamental Groups ◽  
Isometry Groups ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Cyclic Coverings

We construct a family of compact hyperbolic 3-manifolds with totally geodesic boundary, depending on three integer parameters. Then we determine geometric presentations of the fundamental groups of these manifolds and prove that they are cyclic coverings of the 3-ball branched along a specified tangle with two components. Finally, we give a classification of these manifolds up to homeomorphism (resp., isometry), and determine their isometry groups.

scholarly journals Examples and classification of Riemannian submersions satisfying a basic equality

2005 ◽  
Vol 72 (3) ◽  
pp. 391-402 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Unit Sphere ◽  
Isometric Immersion ◽  
Riemannian Submersion ◽  
Sharp Inequality ◽  
Riemannian Submersions ◽  
Equality Case ◽  
Totally Geodesic ◽  
Basic Equality

In an earlier article we obtain a sharp inequality for an arbitrary isometric immersion from a Riemannian manifold admitting a Riemannian submersion with totally geodesic fibres into a unit sphere. In this article we investigate the immersions which satisfy the equality case of the inequality. As a by-product, we discover a new characterisation of Cartan hypersurface in S4.

scholarly journals Almost Complex Surfaces in the Nearly Kähler SL(2,ℝ) × SL(2,ℝ)

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1160
Author(s):  
Elsa Ghandour ◽  
Luc Vrancken
Keyword(s):  
Complete Classification ◽  
Second Fundamental Form ◽  
Complex Surfaces ◽  
Kähler Structure ◽  
Totally Geodesic ◽  
Parallel Second Fundamental Form ◽  
Almost Complex ◽  
Nearly Kähler ◽  
Nearly Kähler Structure

The space S L ( 2 , R ) × S L ( 2 , R ) admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form.

scholarly journals On the cyclic coverings of the knot 52

1999 ◽  
Vol 42 (3) ◽  
pp. 575-587 ◽  
Author(s):  
P. Bandieri ◽  
A. C. Kim ◽  
M. Mulazzani
Keyword(s):  
Fundamental Groups ◽  
Branched Coverings ◽  
Homology Groups ◽  
Cyclic Coverings

We construct a family of hyperbolic 3-manifolds whose fundamental groups admit a cyclic presentation. We prove that all these manifolds are cyclic branched coverings of S3 over the knot 52 and we compute their homology groups. Moreover, we show that thecyclic presentations correspond to spines of the manifolds.

scholarly journals Octonions, triality, the exceptional Lie algebra F4 and polar actions on the Cayley hyperbolic plane

2020 ◽  
Vol 31 (07) ◽  
pp. 2050051
Author(s):  
Andreas Kollross
Keyword(s):  
Projective Plane ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Hyperbolic Plane ◽  
Isometry Groups ◽  
Totally Geodesic ◽  
Polar Actions ◽  
Exceptional Lie Algebra ◽  
Explicit Formulae ◽  
Lie Brackets

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras [Formula: see text] and [Formula: see text], i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we determine all polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

scholarly journals On the classification of rank-two representations of quasiprojective fundamental groups

2008 ◽  
Vol 144 (5) ◽  
pp. 1271-1331 ◽  
Author(s):  
Kevin Corlette ◽  
Carlos Simpson
Keyword(s):  
Fundamental Groups ◽  
Monodromy At Infinity ◽  
Dense Representation

AbstractSuppose that X is a smooth quasiprojective variety over ℂ and ρ:π1(X,x)→SL(2,ℂ) is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X→Y with Y either a Deligne–Mumford (DM) curve or a Shimura modular stack.

Classification of compact lorentzian 2-orbifolds with noncompact full isometry groups

2012 ◽  
Vol 53 (6) ◽  
pp. 1037-1050 ◽  
Author(s):  
N. I. Zhukova ◽  
E. A. Rogozhina
Keyword(s):  
Isometry Groups

scholarly journals GROUPS OF LOCALLY-FLAT DISK KNOTS AND NON-LOCALLY-FLAT SPHERE KNOTS

2005 ◽  
Vol 14 (02) ◽  
pp. 189-215 ◽  
Author(s):  
GREG FRIEDMAN
Keyword(s):  
Fundamental Group ◽  
Piecewise Linear ◽  
Complete Classification ◽  
Fundamental Groups ◽  
Singular Set ◽  
Flat Disk ◽  
Set Of Points

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the "boundary knot" that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classification of all such group pairs in dimension ≥6. For more general knots, we also obtain complete classifications of these group pairs under certain restrictions on the singularities. Finally, we use spinning constructions to realize further examples of boundary knot groups.

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