Force free conformal motions of the sphere

Salvai, Marcos

doi:10.1016/s0926-2245(02)00061-x

Cited by 6 publications

(18 citation statements)

References 7 publications

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“…is a quadratic form on T g G and gives a Riemannian metric on G. The verification of the analogous assertions in the conformal case can be found in [7].…”

Section: A Riemannian Metric On the Configuration Spacementioning

confidence: 72%

“…In particular, if n is even, no geodesic of K through the identity (except the constant one) is a geodesic of G. Remark. Part (b) of Theorem 3 contrasts strongly with the conformal situation in [7], where it is proved that K is totally geodesic in the group of directly conformal transformations of S n endowed with the kinetic energy metric.…”

Section: Force Free Projective Motionsmentioning

confidence: 97%

“…In this subsection and the next one we give some definitions and statements that are analogous to those given for the conformal (instead of the projective) situation in [7]. We present them here for the sake of completeness.…”

Section: The Energy Of Projective Motionsmentioning

confidence: 99%

“…The force free conformal motions of the sphere S n have been studied by the second author in [7], with the aid of a suitable Riemannian metric on S O o (n + 1, 1), an analogue of the classical description of the force free motions of a rigid body in Euclidean space using an invariant metric on S O (3) [1, Appendix 2]. Other applications of the same technique to the study of the dynamics of a rigid body in the hyperbolic spaces of dimensions 2 and 3 can be found in [2,5,6].…”

Section: Introductionmentioning

confidence: 99%

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Force free projective motions of the sphere

Lazarte¹,

Salvai²,

Will³

2007

Journal of Geometry and Physics

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Suppose that the sphere S n has initially a homogeneous distribution of mass and let G be the Lie group of orientation preserving projective diffeomorphisms of S n . A projective motion of the sphere, that is, a smooth curve in G, is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of S n and, more generally, examples of subgroups H of G such that a force free motion initially tangent to H remains in H for all time (in contrast with the previously studied case for conformal motions, this property does not hold for H = S O n+1 ). The main tool is a Riemannian metric on G, which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy.

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“…is a quadratic form on T g G and gives a Riemannian metric on G. The verification of the analogous assertions in the conformal case can be found in [7].…”

Section: A Riemannian Metric On the Configuration Spacementioning

confidence: 72%

Section: Force Free Projective Motionsmentioning

confidence: 97%

Section: The Energy Of Projective Motionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Force free projective motions of the sphere

Lazarte¹,

Salvai²,

Will³

2007

Journal of Geometry and Physics

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“…Hence, v s is the vector field on S 1 obtained by orthogonal projection of the constant vector field ζ → z (s)/r(s) along S 1 . It is well-known (see for instance [5]) that v s is the vector field on the circle (thought of as the imaginary boundary of the hyperbolic disc D) associated to a one parameter group of transvections of D through zero. That is, v s = X s for a unique X s ∈ p. Hence, the curve φ t in M given by Lemma 10 (setting G = M) will be tangent to the distribution D. Clearly γ t • φ t projects to γ t for all t.…”

Section: The Holonomy Of a Closed Path In π(C)mentioning

confidence: 99%

Geodesic paths of circles in the plane

Salvai¹

2010

Rev Mat Complut

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Let E be the Fréchet space of all positively oriented embeddings of the circle in R 2 and let E/ ∼ be the quotient of E modulo orientation preserving diffeomorphisms of the circle. Let π : E → E/ ∼ be the canonical projection and let C denote the space of all constant speed circles. We study geodesics in C and π(C) endowed with the Riemannian metrics induced from the canonical weak Riemannian metrics on E and E/ ∼ , respectively. We also study the holonomy of closed paths in π(C).

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Polar factorization of conformal and projective maps of the sphere in the sense of optimal mass transport

Godoy

Salvai

2018

Isr. J. Math.

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Let M be a compact Riemannian manifold and let µ, d be the associated measure and distance on M . Robert McCann obtained, generalizing results for the Euclidean case by Yann Brenier, the polar factorization of Borel maps S : M → M pushing forward µ to a measure ν: each S factors uniquely a.e. into the composition S = T • U , where U : M → M is volume preserving and T : M → M is the optimal map transporting µ to ν with respect to the cost function d 2 /2.In this article we study the polar factorization of conformal and projective maps of the sphere S n . For conformal maps, which may be identified with elements of O o (1, n + 1), we prove that the polar factorization in the sense of optimal mass transport coincides with the algebraic polar factorization (Cartan decomposition) of this Lie group. For the projective case, where the group GL + (n + 1) is involved, we find necessary and sufficient conditions for these two factorizations to agree. This work was partially supported by Conicet (PIP 112-2011-01-00670), Foncyt (PICT 2010 cat 1 proyecto 1716) Secyt Univ. Nac. Córdoba between µ and ν. In the following, when it is clear from the context, we call it simply optimal.An important particular case is the following: Let M be a compact oriented Riemannian manifold and let µ = vol be the Riemannian measure on M and d the associated distance and consider the cost c (p, q) = 1 2 d (p, q) 2 . Let S : M → M be a Borel map pushing forward µ to a measure ν. Robert McCann proved in [12], generalizing results for the Euclidean case by Brenier [3], that S factors uniquely a.e. into the composition S = T • U , where U : M → M is volume preserving and T : M → M is the optimal map transporting µ to ν. This is called the polar factorization of S in the sense of optimal mass transport. For the sake of brevity we call it the Brenier-McCann polar factorization of S.

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Force free conformal motions of the sphere

Cited by 6 publications

References 7 publications

Force free projective motions of the sphere

Force free projective motions of the sphere

Geodesic paths of circles in the plane

Polar factorization of conformal and projective maps of the sphere in the sense of optimal mass transport

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