90 research outputs found
Sub-Riemannian geodesics on nested principal bundles
We study the interplay between geodesics on two non-holono\-mic systems that
are related by the action of a Lie group on them. After some geometric
preliminaries, we use the Hamiltonian formalism to write the parametric form of
geodesics. We present several geometric examples, including a non-holonomic
structure on the Gromoll-Meyer exotic sphere and twistor space.Comment: 10 page
On the Alexandrov Topology of sub-Lorentzian Manifolds
It is commonly known that in Riemannian and sub-Riemannian Geometry, the
metric tensor on a manifold defines a distance function. In Lorentzian
Geometry, instead of a distance function it provides causal relations and the
Lorentzian time-separation function. Both lead to the definition of the
Alexandrov topology, which is linked to the property of strong causality of a
space-time. We studied three possible ways to define the Alexandrov topology on
sub-Lorentzian manifolds, which usually give different topologies, but agree in
the Lorentzian case. We investigated their relationships to each other and the
manifold's original topology and their link to causality.Comment: 20 page
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
In this paper, we study the geodesic flow of a right-invariant metric induced
by a general Fourier multiplier on the diffeomorphism group of the circle and
on some of its homogeneous spaces. This study covers in particular
right-invariant metrics induced by Sobolev norms of fractional order. We show
that, under a certain condition on the symbol of the inertia operator (which is
satisfied for the fractional Sobolev norm for ), the
corresponding initial value problem is well-posed in the smooth category and
that the Riemannian exponential map is a smooth local diffeomorphism.
Paradigmatic examples of our general setting cover, besides all traditional
Euler equations induced by a local inertia operator, the Constantin-Lax-Majda
equation, and the Euler-Weil-Petersson equation.Comment: 40 pages. Corrected typos and improved redactio
Conformal loop ensembles and the stress-energy tensor
We give a construction of the stress-energy tensor of conformal field theory
(CFT) as a local "object" in conformal loop ensembles CLE_\kappa, for all
values of \kappa in the dilute regime 8/3 < \kappa <= 4 (corresponding to the
central charges 0 < c <= 1, and including all CFT minimal models). We provide a
quick introduction to CLE, a mathematical theory for random loops in simply
connected domains with properties of conformal invariance, developed by
Sheffield and Werner (2006). We consider its extension to more general regions
of definition, and make various hypotheses that are needed for our construction
and expected to hold for CLE in the dilute regime. Using this, we identify the
stress-energy tensor in the context of CLE. This is done by deriving its
associated conformal Ward identities for single insertions in CLE probability
functions, along with the appropriate boundary conditions on simply connected
domains; its properties under conformal maps, involving the Schwarzian
derivative; and its one-point average in terms of the "relative partition
function." Part of the construction is in the same spirit as, but widely
generalizes, that found in the context of SLE_{8/3} by the author, Riva and
Cardy (2006), which only dealt with the case of zero central charge in simply
connected hyperbolic regions. We do not use the explicit construction of the
CLE probability measure, but only its defining and expected general properties.Comment: 49 pages, 3 figures. This is a concatenated, reduced and simplified
version of arXiv:0903.0372 and (especially) arXiv:0908.151
The rolling problem: overview and challenges
In the present paper we give a historical account -ranging from classical to
modern results- of the problem of rolling two Riemannian manifolds one on the
other, with the restrictions that they cannot instantaneously slip or spin one
with respect to the other. On the way we show how this problem has profited
from the development of intrinsic Riemannian geometry, from geometric control
theory and sub-Riemannian geometry. We also mention how other areas -such as
robotics and interpolation theory- have employed the rolling model.Comment: 20 page
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