Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds

1994 ◽  
Vol 117 (1) ◽  
pp. 207-225 ◽  
Author(s):  
Ara Basmajian
Keyword(s):  
Hyperbolic Manifolds ◽  
Totally Geodesic ◽  
Tubular Neighborhoods ◽  
Totally Geodesic Hypersurfaces

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.

Chains that realize the Gromov invariant of hyperbolic manifolds

1997 ◽  
Vol 17 (3) ◽  
pp. 643-648 ◽  
Author(s):  
DOUGLAS JUNGREIS
Keyword(s):  
Hyperbolic Manifold ◽  
Homology Class ◽  
Hyperbolic Manifolds ◽  
Uniform Measure ◽  
Geodesic Boundary ◽  
Maximal Volume ◽  
Totally Geodesic ◽  
Gromov Norm

For any closed hyperbolic manifold of dimension $n \geq 3$, suppose a sequence of $n$-cycles representing the fundamental homology class have norms converging to the Gromov invariant. We show that this sequence must converge to the uniform measure on the space of maximal-volume ideal simplices. As a corollary, we show that for a hyperbolic $n$-manifold $L$ ($n \geq 3$) with totally-geodesic boundary, the Gromov norm of ($L,\partial L$) is strictly greater than the volume of $L$ divided by the maximal volume of an ideal $n$-simplex.

scholarly journals Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

2021 ◽  
Vol 17 (0) ◽  
pp. 401
Author(s):  
Dubi Kelmer ◽  
Hee Oh
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifold ◽  
General Theorem ◽  
Hyperbolic Manifolds ◽  
Quantitative Estimates ◽  
Geometrically Finite ◽  
Logarithm Law ◽  
Tubular Neighborhoods ◽  
Logarithm Laws ◽  
Metric Balls

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{M} $\end{document}</tex-math></inline-formula> be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.</p>

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