1,439 research outputs found
Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line
In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space Diff 1 ( R ) equipped with the homogeneous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat L 2 -metric. Here Diff 1 ( R ) denotes the extension of the group of all compactly supported, rapidly decreasing, or W ∞ , 1 -diffeomorphisms, which allows for a shift toward infinity. Surprisingly, on the non-extended group the Levi-Civita connection does not exist. In particular, this result provides an analytic solution formula for the corresponding geodesic equation, the non-periodic Hunter-Saxton (HS) equation. In addition, we show that one can obtain a similar result for the two-component HS equation and discuss the case of the non-homogeneous Sobolev one metric, which is related to the Camassa-Holm equation
Geometric investigations of a vorticity model equation
This article consists of a detailed geometric study of the one-dimensional
vorticity model equation which is a
particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed
that this equation is the Euler-Arnold equation on
when the latter is endowed with the right-invariant homogeneous
-metric. In this article we prove that the exponential map of
this Riemannian metric is not Fredholm and that the sectional curvature is
locally unbounded. Furthermore, we prove a Beale-Kato-Majda-type blow-up
criterion, which we then use to demonstrate a link to our non-Fredholmness
result. Finally, we extend a blow-up result of Castro-C\'ordoba to the periodic
case and to a much wider class of initial conditions, using a new
generalization of an inequality for Hilbert transforms due to
C\'ordoba-C\'ordoba.Comment: 30 pages; added references; corrected typo
Local and Global Well-posedness of the fractional order EPDiff equation on
Of concern is the study of fractional order Sobolev--type metrics on the
group of -diffeomorphism of and on its Sobolev
completions . It is shown that the
-Sobolev metric induces a strong and smooth Riemannian metric on the
Banach manifolds for . As
a consequence a global well-posedness result of the corresponding geodesic
equations, both on the Banach manifold and on
the smooth regular Fr\'echet-Lie group of all -diffeomorphisms is
obtained. In addition a local existence result for the geodesic equation for
metrics of order is derived.Comment: 37 page
Shape analysis on homogeneous spaces: a generalised SRVT framework
Shape analysis is ubiquitous in problems of pattern and object recognition
and has developed considerably in the last decade. The use of shapes is natural
in applications where one wants to compare curves independently of their
parametrisation. One computationally efficient approach to shape analysis is
based on the Square Root Velocity Transform (SRVT). In this paper we propose a
generalised SRVT framework for shapes on homogeneous manifolds. The method
opens up for a variety of possibilities based on different choices of Lie group
action and giving rise to different Riemannian metrics.Comment: 28 pages; 4 figures, 30 subfigures; notes for proceedings of the Abel
Symposium 2016: "Computation and Combinatorics in Dynamics, Stochastics and
Control". v3: amended the text to improve readability and clarify some
points; updated and added some references; added pseudocode for the dynamic
programming algorithm used. The main results remain unchange
Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds
Given a finite dimensional manifold , the group
of diffeomorphism of which fall
suitably rapidly to the identity, acts on the manifold of submanifolds
on of diffeomorphism type where is a compact manifold with . For a right invariant weak Riemannian metric on
induced by a quite general operator
, we
consider the induced weak Riemannian metric on and we compute its
geodesics and sectional curvature. For that we derive a covariant formula for
curvature in finite and infinite dimensions, we show how it makes O'Neill's
formula very transparent, and we use it finally to compute sectional curvature
on .Comment: 28 pages. In this version some misprints correcte
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