LaTeX, 4 pages. Reference added. To appear in Advances in Pure and Applied MathematicsInternational audienceThe flag curvature of the Numata Finsler structures is shown to admit a non\-trivial prolongation to the one-dimensional case, revealing an unexpected link with the Schwarzian derivative of the diffeomorphisms associated with these Finsler structures

Duval, Christian

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Schwarzian derivative and Numata Finsler structuresChristian DuvalTo cite this version:Christian Duval. Schwarzian derivative and Numata Finsler structures. Advances in Pure andApplied Mathematics, De Gruyter, 2010, 1, pp.1. <hal-00256294v2>HAL Id: hal-00256294https://hal.archives-ouvertes.fr/hal-00256294v2Submitted on 3 Jul 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Schwarzian derivativeandNumata Finsler structuresC. DUVAL‡Centre de Physique The´orique, CNRS, Luminy, Case 907F-13288 Marseille Cedex 9 (France)§AbstractThe flag curvature of the Numata Finsler structures is shown to admit a non-trivial prolongation to the one-dimensional case, revealing an unexpected link with theSchwarzian derivative of the diffeomorphisms associated with these Finsler structures.Mathematics Subject Classification 2000: 58B20, 53A551 Finsler structures in a nutshell1.1 Finsler metricsA Finsler structure is a pair (M,F ) where M is a smooth, n-dimensional, manifoldand F : TM → R+ a given function whose restriction to the slit tangent bundleTM \M = {(x, y) ∈ TM | y ∈ TxM \{0}} is strictly positive, smooth, and positivelyhomogeneous of degree one, i.e., F (x, λy) = λF (x, y) for all λ > 0; one furthermoredemands that the n× n vertical Hessian matrix with entries gij(x, y) =(12F 2)yiyjbepositive definite, (gij) > 0. See [1]. These quantities are (positively) homogeneous ofdegree zero, and the fundamental tensorg = gij(x, y)dxi ⊗ dxj (1.1)defines a sphere’s worth of Riemannian metrics on each TxM parametrized by thedirection of y. See [2].The distinguished “vector field”ℓ = ℓi∂∂xi, where ℓi(x, y) =yiF (x, y), (1.2)actually a section of π∗(TM) where π : TM \M → M is the natural projection, issuch that g(ℓ, ℓ) = 1.‡mailto: duval@cpt.univ-mrs.fr§UMR 6207 du CNRS associe´e aux Universite´s d’Aix-Marseille I et II et Universite´ du Sud Toulon-Var;Laboratoire affilie´ a` la FRUMAM-FR22911There is a wealth of Finsler structures, apart from the special case of Riemannianstructures (M, g) for which F (x, y) =√gij(x)yiyj. For instance, the so-called RandersmetricsF (x, y) =√aij(x)yiyj + bi(x)yi (1.3)satisfy all previous requirements if a = aij(x)dxi ⊗ dxj is a Riemann metric and if the1-form b = bi(x)dxi is such that aij(x)bi(x)bj(x) < 1 for all x ∈M .1.2 Flag curvatureUnlike the Riemannian case, there is no canonical linear Finsler connection on π∗(TM).An example, though, is provided by the Chern connection ωij = Γijk(x, y)dxk which isuniquely defined by the following requirements [1]: (i) it is symmetric, Γijk = Γikj , and(ii) it almost transports the metric tensor, i.e., dgij −ωki gjk −ωkj gik = 2Cijkδyk, withδyi = dyi +N ijdxj , where the N ij(x, y) = Γijkyk are the components of the non linearconnection associated with the Chern connection, and the Cijk(x, y) =12(gij)yk thoseof the Cartan tensor, specific to Finsler geometry.Using the “horizontal covariant derivatives” δ/δxi = ∂/∂xi−N ji ∂/∂yj, one expres-ses the (horizontal-horizontal part of the) Chern curvature byR ij kl =δδxkΓijl + ΓimkΓmjl − (k ↔ l), (1.4)and the flag curvature (associated with the flag ℓ ∧ v defined by v ∈ TxM) byK(x, y, v) =Rikvivkg(v, v)− g(ℓ, v)2, where Rik = ℓjRjikℓ ℓℓ. (1.5)One says that a Finsler structure is of scalar curvature ifK(x, y, v) does not dependon the vector v, i.e., ifRik = K(x, y)hik, (1.6)with hik = gik− ℓiℓk the components of the “angular metric”, where ℓi = gijℓj(= Fyi).See [1, 2].2 Numata Finsler structures2.1 The Numata metricNumata [4] has proved that metrics of the form F (x, y) =√qij(y)yiyj + bi(x)yi, onTM whereM ⊂ Rn, with (qij) > 0 and db = 0 are, indeed, of scalar curvature. See [2].Of some interest is the special case qij = δij and b = df with f ∈ C∞(M), viz.,F (x, y) =√δijyiyj + fxiyi, (2.1)whereM ={x ∈ Rn∣∣∣n∑i=1f2xi < 1}. (2.2)2The computation of the flag curvature of this particular Randers metric (1.3) canbe found in [1] and yieldsK(x, y) =341F 4(fxixjyiyj)2−121F 3fxixjxkyiyjyk. (2.3)2.2 Flag curvature & Schwarzian derivativeThe expression (2.3) of the flag curvature of the Numata metric (2.1) holds for n ≥ 2.If n = 1, the left-hand side of (1.6) vanishes along with the curvature (1.4), while itsright-hand vanishes as well since the angular metric has rank zero. For this particulardimension, Equation (1.6) trivially holds true, but tells, however, nothing about theflag curvature K(x, y).At this stage, it is worth noting that (2.3) indeed admits a prolongation to theone-dimensional case; it is therefore tempting to specialize its expression for n = 1.Suppose, thus, that M ⊂ S1 is a nonempty open subset (2.2), so that we haveTM \M = T+M⊔T−M , where T±M = M × R±∗ . The metric (2.1) then readsF (x, y) = |y|+ f ′(x)y, (2.4)using an affine coordinate, x, on S1, with −1 < f ′(x) < +1 (see (2.2)); its restrictionsto T±M are given by F±(x, y) = ϕ′±(x)y > 0, whereϕ′±(x) = f′(x)± 1, (2.5)implying ϕ± ∈ Diff±(S1), with |ϕ′±(x)| < 2 (all x ∈M).The Numata metric (2.4) on T+M , say, is thus associated, via (2.5), to orientation-preserving diffeomorphisms ϕ of S1 such that 0 < ϕ′(x) < 2 (all x ∈M). Given sucha ϕ ∈ Diff+(S1), the fundamental tensor (1.1) retains the form g = ϕ′(x)2dx2 and is,naturally, a Riemannian metric on M .Rewriting Equation (2.3) for T+M , and bearing in mind that y = F (x, y)/ϕ′(x),we readily find that K(x, y) is actually independent of y, namelyK(x) = −121ϕ′(x)2S(ϕ)(x), (2.6)whereS(ϕ)(x) =ϕ′′′(x)ϕ′(x)−32(ϕ′′(x)ϕ′(x))2(2.7)denotes the Schwarzian derivative [5] of the diffeomorphism ϕ of S1. The argumentclearly still holds, mutatis mutandis, for orientation-reversing diffeomorphisms of S1.3We have thus proved theTheorem 2.1. The Numata Finsler structure (M,F ), with metric F given by (2.4)where M ⊂ S1 is defined by (2.2), induces a Riemannian metric, g(ϕ) = ϕ∗(dx2),where ϕ ∈ Diff(S1) is as in (2.5). The flag curvature (2.3) admits a prolongation tothis one-dimensional case and retains the formK = −12S(ϕ)g(ϕ), (2.8)where S(ϕ) = S(ϕ)(x)dx2 is the Schwarzian quadratic differential of ϕ ∈ Diff(S1).As an illustration, the one-dimensional Numata Finsler structures of constant flagcurvature are associated, through (2.5), to the solutions ϕ of (2.8) for K ∈ R, viz.,ϕ±(x) = K−12 arctan(K12 (ax+ b)/(cx+ d)) where a, b, c, d ∈ R with ad− bc = ±1.Let us mention another instance where the Schwarzian derivative is associatedwith curvature, namely the geometry of curves in Lorentzian surfaces of constantcurvature [3].Discussions with P. Foulon are warmly acknowledged.References[1] D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry,GTM 200, Springer, New York, 2004.[2] D. Bao, and C. Robles, “Ricci and Flag Curvatures in Finsler Geometry”, inA Sampler of Riemann-Finsler Geometry, D. Bao, R. L. Bryant, S.-S. Chern, andZ. Shen (Editors), MSRI Publications 50, Cambridge University Press, 2004.[3] C. Duval, and V. Ovsienko, “Lorentzian Worldlines and the Schwarzian Deriva-tive”, Funct. Anal. Applic. 34:2 (2000), 135–137.[4] S. Numata, “On the torsion tensors Rjhk and Phjk of Finsler spaces with a metricds = (gij(dx)dxidxj)1/2 + bi(x)dxi”, Tensor (N.S.) 32 (1978), 27–32.[5] V. Ovsienko, and S. Tabachnikov, Projective Differential Geometry Old And New:From The Schwarzian Derivative To The Cohomology Of Diffeomorphism Group,Cambridge University Press, 2005, and References therein.4

Schwarzian derivative and Numata Finsler structures

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Schwarzian derivative and Numata Finsler structures

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