5,167 research outputs found
Classification results on surfaces in the isotropic 3-space
The isotropic 3-space I^3 which is one of the Cayley--Klein spaces is
obtained from the Euclidean space by substituting the usual Euclidean distance
with the isotropic distance. In the present paper, we give several
classifications on the surfaces in I^3 with the constant relative curvature
(analogue of the Gaussian curvature) and the constant isotropic mean curvature.
In particular, we classify the helicoidal surfaces in I^3 with constant
curvature and analyze some special curves on these.Comment: 12 pages, 2 figure
Strong-field tidal distortions of rotating black holes: III. Embeddings in hyperbolic 3-space
In previous work, we developed tools for quantifying the tidal distortion of
a black hole's event horizon due to an orbiting companion. These tools use
techniques which require large mass ratios (companion mass much smaller
than black hole mass ), but can be used for arbitrary bound orbits, and for
any black hole spin. We also showed how to visualize these distorted black
holes by embedding their horizons in a global Euclidean 3-space,
. Such visualizations illustrate interesting and important
information about horizon dynamics. Unfortunately, we could not visualize black
holes with spin parameter : such holes cannot
be globally embedded into . In this paper, we overcome this
difficulty by showing how to embed the horizons of tidally distorted Kerr black
holes in a hyperbolic 3-space, . We use black hole perturbation
theory to compute the Gaussian curvatures of tidally distorted event horizons,
from which we build a two-dimensional metric of their distorted horizons. We
develop a numerical method for embedding the tidally distorted horizons in
. As an application, we give a sequence of embeddings into
of a tidally interacting black hole with spin . A
small amplitude, high frequency oscillation seen in previous work shows up
particularly clearly in these embeddings.Comment: 10 pages, 6 figure
Cone fields and topological sampling in manifolds with bounded curvature
Often noisy point clouds are given as an approximation of a particular
compact set of interest. A finite point cloud is a compact set. This paper
proves a reconstruction theorem which gives a sufficient condition, as a bound
on the Hausdorff distance between two compact sets, for when certain offsets of
these two sets are homotopic in terms of the absence of {\mu}-critical points
in an annular region. Since an offset of a set deformation retracts to the set
itself provided that there are no critical points of the distance function
nearby, we can use this theorem to show when the offset of a point cloud is
homotopy equivalent to the set it is sampled from. The ambient space can be any
Riemannian manifold but we focus on ambient manifolds which have nowhere
negative curvature. In the process, we prove stability theorems for
{\mu}-critical points when the ambient space is a manifold.Comment: 20 pages, 3 figure
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