Schwarzian derivative and Numata Finsler structures

The flag curvature of the Numata Finsler structures is shown to admit a nontrivial prolongation to the one-dimensional case, revealing an unexpected link with the Schwarzian derivative of the diffeomorphisms associated with these Finsler structures. Mathematics Subject Classification 2000: 58B20, 53A55 1 Finsler structures in a nutshell 1.1 Finsler metrics A Finsler structure is a pair (M,F ) where M is a smooth, n-dimensional, manifold and F : TM → R a given function whose restriction to the slit tangent bundle TM \M = {(x, y) ∈ TM | y ∈ TxM \{0}} is strictly positive, smooth, and positively homogeneous of degree one, i.e., F (x, λy) = λF (x, y) for all λ > 0; one furthermore demands that the n× n vertical Hessian matrix with entries gij(x, y) = ( 1 2 F 2 ) yiyj be positive definite, (gij) > 0. See [1]. These quantities are (positively) homogeneous of degree zero, and the fundamental tensor g = gij(x, y)dx i ⊗ dx (1.1) defines a sphere’s worth of Riemannian metrics on each TxM parametrized by the direction of y. See [2]. The distinguished “vector field” l = l ∂ ∂xi , where l(x, y) = y F (x, y) , (1.2) actually a section of π(TM) where π : TM \M → M is the natural projection, is such that g(l, l) = 1. mailto: [email protected] UMR 6207 du CNRS associée aux Universités d’Aix-Marseille I et II et Université du Sud Toulon-Var; Laboratoire affilié à la FRUMAM-FR2291 1 ha l-0 02 56 29 4, v er si on 2 3 Ju l 2 00 8 There is a wealth of Finsler structures, apart from the special case of Riemannian structures (M, g) for which F (x, y) = √ gij(x)yy. For instance, the so-called Randers metrics F (x, y) = √ aij(x)yy + bi(x)y i (1.3) satisfy all previous requirements if a = aij(x)dx i ⊗ dx is a Riemann metric and if the 1-form b = bi(x)dx i is such that a(x)bi(x)bj(x) < 1 for all x ∈ M . 1.2 Flag curvature Unlike the Riemannian case, there is no canonical linear Finsler connection on π(TM). An example, though, is provided by the Chern connection ω j = Γ i jk(x, y)dx k which is uniquely defined by the following requirements [1]: (i) it is symmetric, Γjk = Γ i kj , and (ii) it almost transports the metric tensor, i.e., dgij −ω k i gjk −ω k j gik = 2Cijkδy , with δy = dy +N i jdx j , where the N i j(x, y) = Γ i jky k are the components of the non linear connection associated with the Chern connection, and the Cijk(x, y) = 1 2 (gij)yk those of the Cartan tensor, specific to Finsler geometry. Using the “horizontal covariant derivatives” δ/δx = ∂/∂x−N j i ∂/∂y , one expresses the (horizontal-horizontal part of the) Chern curvature by R i j kl = δ δxk Γjl + Γ i mkΓ m jl − (k ↔ l), (1.4) and the flag curvature (associated with the flag l ∧ v defined by v ∈ TxM) by K(x, y, v) = Rikv v g(v, v)− g(l, v) , where Rik = l Rjikl l . (1.5) One says that a Finsler structure is of scalar curvature ifK(x, y, v) does not depend on the vector v, i.e., if Rik = K(x, y)hik, (1.6) with hik = gik− lilk the components of the “angular metric”, where li = gijl (= Fyi). See [1, 2]. 2 Numata Finsler structures 2.1 The Numata metric Numata [4] has proved that metrics of the form F (x, y) = √ qij(y)yy + bi(x)y , on TM where M ⊂ R, 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) = √ δijyy + fxiy , (2.1) where M = {