Results 1 
9 of
9
The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line. arXiv:1209.2836
"... Abstract. In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space Diff1(R) equipped with the homogenous 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 ..."
Abstract

Cited by 10 (9 self)
 Add to MetaCart
(Show Context)
Abstract. In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space Diff1(R) equipped with the homogenous 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 L2metric. Here Diff1(R) denotes the extension of the group of all either compactly supported, rapidly decreasing or H∞diffeomorphisms, that allows for a shift towards infinity. In particular this result provides an analytic solution formula for the corresponding geodesic equation, the nonperiodic HunterSaxton equation. In addition we show that one can obtain a similar result for the twocomponent HunterSaxton equation and discuss the case of the nonhomogenous Sobolev one metric which is related to the CamassaHolm equation. 1.
SOBOLEV METRICS ON SHAPE SPACE, II: WEIGHTED SOBOLEV METRICS AND ALMOST LOCAL METRICS
"... Abstract. In continuation of [5] we discuss metrics of the form G P ∫ p ∑ ( ) ( f (h,k) = Φi Vol(f) g (Pi)fh,k M i=0) vol(f ∗ g) on the space of immersions Imm(M,N) and on shape space Bi(M,N) = Imm(M,N)/Diff(M). Here (N,g) is a complete Riemannian manifold, M is a compact manifold, f: M → N is an ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
Abstract. In continuation of [5] we discuss metrics of the form G P ∫ p ∑ ( ) ( f (h,k) = Φi Vol(f) g (Pi)fh,k M i=0) vol(f ∗ g) on the space of immersions Imm(M,N) and on shape space Bi(M,N) = Imm(M,N)/Diff(M). Here (N,g) is a complete Riemannian manifold, M is a compact manifold, f: M → N is an immersion, h and k are tangent vectors to f in the space of immersions, f∗g is the induced Riemannian metric on M, vol(f∗g) is the induced volume density on M, Vol(f) = ∫ M vol(f∗g), Φi are positive realvalued functions, and (Pi)f are operators like some power of the Laplacian ∆f∗g. We derive the geodesic equations for these metrics and show that they are sometimes wellposed with the geodesic exponential mapping a local diffeomorphism. The new aspect here are the weights Φi(Vol(f)) which we use to construct scale invariant metrics and order 0 metrics with positive geodesic distance. We treat several concrete special cases in detail. 1.
Rtransforms for Sobolev H2metrics on spaces of plane curves
 Geometry, Imaging and Computing
"... Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H2metric witho ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
(Show Context)
Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H2metric without zero order terms. We find isometries (called Rtransforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We also show how to utilise the isometries to compute geodesics numerically.
A ZOO OF DIFFEOMORPHISM GROUPS ON R n
"... Abstract. We consider the groups DiffB(Rn), DiffH∞(Rn), and DiffS(Rn) of smooth diffeomorphisms on Rn which differ from the identity by a function whichisineitherB (boundedinallderivatives), H ∞ = ⋂ k≥0Hk, orS (rapidly decreasing). We show that all these groups are smooth regular Lie groups. 1. ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We consider the groups DiffB(Rn), DiffH∞(Rn), and DiffS(Rn) of smooth diffeomorphisms on Rn which differ from the identity by a function whichisineitherB (boundedinallderivatives), H ∞ = ⋂ k≥0Hk, orS (rapidly decreasing). We show that all these groups are smooth regular Lie groups. 1.
RTRANSFORMS FOR SOBOLEV H 2METRICS ON SPACES OF PLANE CURVES
"... Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H 2 metric with ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H 2 metric without zero order terms. We find isometries (called Rtransforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We demonstrate the value of using Rtransforms by some numerical experiments.
on Rn which differ from the identity by a mapping in B[M] (global Denjoy–
"... Abstract. Let C[M] be a (local) Denjoy–Carleman class of Beurling or Roumieu type, where the weight sequence M = (Mk) is logconvex and has moderate growth. We prove that the groups DiffB[M](Rn), ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Let C[M] be a (local) Denjoy–Carleman class of Beurling or Roumieu type, where the weight sequence M = (Mk) is logconvex and has moderate growth. We prove that the groups DiffB[M](Rn),
(Gelfand–Shilov), or
"... Abstract. Let C [M] be a (local) Denjoy–Carleman class of Beurling or Roumieu type, where the weight sequence M = (Mk) is logconvex and has moderate growth. We prove that the groups DiffB [M](Rn), DiffW [M],p(Rn), DiffS [M] ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Let C [M] be a (local) Denjoy–Carleman class of Beurling or Roumieu type, where the weight sequence M = (Mk) is logconvex and has moderate growth. We prove that the groups DiffB [M](Rn), DiffW [M],p(Rn), DiffS [M]