15,970 research outputs found
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms
The -metric or Fubini-Study metric on the non-linear Grassmannian of all
submanifolds of type in a Riemannian manifold induces geodesic
distance 0. We discuss another metric which involves the mean curvature and
shows that its geodesic distance is a good topological metric. The vanishing
phenomenon for the geodesic distance holds also for all diffeomorphism groups
for the -metric.Comment: 26 pages, LATEX, final versio
Geodesic Distance in Planar Graphs
We derive the exact generating function for planar maps (genus zero
fatgraphs) with vertices of arbitrary even valence and with two marked points
at a fixed geodesic distance. This is done in a purely combinatorial way based
on a bijection with decorated trees, leading to a recursion relation on the
geodesic distance. The latter is solved exactly in terms of discrete
soliton-like expressions, suggesting an underlying integrable structure. We
extract from this solution the fractal dimensions at the various
(multi)-critical points, as well as the precise scaling forms of the continuum
two-point functions and the probability distributions for the geodesic distance
in (multi)-critical random surfaces. The two-point functions are shown to obey
differential equations involving the residues of the KdV hierarchy.Comment: 38 pages, 8 figures, tex, harvmac, eps
Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group
We study Sobolev-type metrics of fractional order on the group
\Diff_c(M) of compactly supported diffeomorphisms of a manifold . We show
that for the important special case the geodesic distance on
\Diff_c(S^1) vanishes if and only if . For other manifolds we
obtain a partial characterization: the geodesic distance on \Diff_c(M)
vanishes for and for ,
with being a compact Riemannian manifold. On the other hand the geodesic
distance on \Diff_c(M) is positive for and
.
For we discuss the geodesic equations for these metrics. For
we obtain some well known PDEs of hydrodynamics: Burgers' equation for ,
the modified Constantin-Lax-Majda equation for and the
Camassa-Holm equation for .Comment: 16 pages. Final versio
Geodesic Distance Histogram Feature for Video Segmentation
This paper proposes a geodesic-distance-based feature that encodes global
information for improved video segmentation algorithms. The feature is a joint
histogram of intensity and geodesic distances, where the geodesic distances are
computed as the shortest paths between superpixels via their boundaries. We
also incorporate adaptive voting weights and spatial pyramid configurations to
include spatial information into the geodesic histogram feature and show that
this further improves results. The feature is generic and can be used as part
of various algorithms. In experiments, we test the geodesic histogram feature
by incorporating it into two existing video segmentation frameworks. This leads
to significantly better performance in 3D video segmentation benchmarks on two
datasets
Geodesic distances in Liouville quantum gravity
In order to study the quantum geometry of random surfaces in Liouville
gravity, we propose a definition of geodesic distance associated to a Gaussian
free field on a regular lattice. This geodesic distance is used to numerically
determine the Hausdorff dimension associated to shortest cycles of 2d quantum
gravity on the torus coupled to conformal matter fields, showing agreement with
a conjectured formula by Y. Watabiki. Finally, the numerical tools are put to
test by quantitatively comparing the distribution of lengths of shortest cycles
to the corresponding distribution in large random triangulations.Comment: 21 pages, 8 figure
- …