A Finsler metric of constant Gauss curvature K = 1 on 2-sphere
aa r X i v : . [ m a t h . DG ] F e b A Finsler metric of constant Gauss curvature K = 1 on 2-sphere ∗ By I. Masca, S. V. Sabau, H. Shimada
Abstract
We construct a concrete example of constant Gauss curvature K = 1on the 2-sphere having all geodesics closed and of same length. Keywords : Riemannian manifolds Zoll metrics geodesics sectional cur-vature Finsler manifolds flag curvature.
Subclass : 53C60, 53C22
Zoll surfaces are Riemannian metrics all of whose geodesics are simple closedcurves of equal length (see for example [3], [6] and many other authors). Theyare natural generalizations of the metric on the round sphere, having the Gausscurvature of Zoll metrics not constant.More precisely, a Zoll surface of rotation on ( S , g ) is a Riemannian metricon the 2-sphere with canonical spherical coordinates ( r, θ ) ∈ [0 , π ] × (0 , π ] givenby g = [1 + h (cos r )] dr ⊗ dr + sin rdθ ⊗ dθ, (1.1)where h : [ − , → ( − ,
1) is a smooth function such that(i) h is odd function, i.e. h ( − x ) = − h ( x ), for all x ∈ [ − , h ( −
1) = h (1) = 0.Observe that h (0) = 0 from condition (i).It can be checked that this Riemannian has all geodesics closed, of samelength, and with Gauss curvature G ( r ) = 1 + h (cos r ) − cos r · h ′ (cos r )[1 + h (cos r )] (1.2)(see [3] or [7] for detailed computations). ∗ to appear in An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si Mat. (N.S.) M , of a Zoll surface( S , g ) is a smooth manifold diffeomorphic to S .One problem to ask is what kind of natural geometrical structures are carriedby the manifold of geodesics M of a Zoll surface ( S , g ) ? Some of the answers are already known:(i) M ≃ S can be endowed with a symplectic structure, or it can be regardedas a Riemannian manifold with an averaged Riemannian metric (see [3])for details).(ii) M also carries a Finsler structure of constant Gauss curvature K = 1 (see[4], [7] for details).The existance of a such Finsler structure on the manifold of geodesics wasshown for the fist time by Bryant in [4] by using exterior differential systems.He also constructed a Finsler metric of constant Gauss curvature K = 1 on S with the supplementary condition of projectively flatness by using an approachborrowed from algebraic geometry. However, this metrics (called today Bryantmetrics) do not come from Zoll metrics (as pointed out by Bryant himself in [4],p. 254, bottom).The concrete construction of the Finsler surfaces of constant Gauss curvature K = 1 on S using positively curved Zoll metrics, was studied very recently in[7], where one can find a lot of information about these Finsler metrics on S as well as the parametric equation of the indicatrix. However the concrete formof the fundamental function F was not yet explicitely obtained.Without being trivial, observe that the Zoll metrics constructed using thefollowing polynomial function: h ( x ) = ( εx (1 − x ) , < ε < x (1 − x ) n , n > G ( x ) >
0, for x ∈ [ − , K = 1 Finsler metric on the manifold of geodesics M .The aim of the present paper is to find the explicit form of the fundamentalfunction F of such a Finsler metric in the simplest case of h : [ − , → R , h ( x ) = εx (1 − x ) , < ε < . (1.4)We will study the solutions of this equation, find the precise positive realsolution that corresponds to the Finsler metric and write it explicitely in aformula with radicals.It will be seen that this Finsler metric is actually an ( α, β )-metric, albeit aquite complicated one.This Finsler metric, even quite complicated when explicitely written in co-ordinates has some remarkable properties2i) it is of constant Gauss curvature K = 1 on S ,(ii) it has all geodesics closed and of same length,(iii) it is a Finsler surface of revolution, i.e. it is rotationally invariant,(iv) the unit speed geodesics of this Finsler structure, emanating from the somepoint p ∈ M intersect at the same distance from p in the some point, i.e.the cut locus of a point p ∈ M is a point.In the Riemannian case, a metric with these properties is unique and it mustbe the canonical metric of the round sphere, but in the Finsler case, we have afamily of such Finsler metrics, each of them corresponding to an odd function h : [ − , → ( − ,
1) with some supplementary properties such as G ( r ) >
0. Wesingle out one such Finsler metric in the present paper.We point out that in the case of the function (1.3) with n = 2, the implicitindicatrix equation is an 8-th order algebraic equation in F . The famous Abel-Ruffini theorem says that there is no solution in radicals to a general equationof degree 5 or higther, with arbitrary coefficients, hence a similar approch withthe present paper is impossible. New algebraic and analytical notices are neededand we will leave this case for the future research. Let us start with the odd function h in (1.4), and observe that the Gausscurvature of the corresponding Zoll metric g on S is G ( x ) = − εx + 1( εx − ε − , (2.1)that is obviously positive on [ − , < ε < .Moreover, let us recall the following essential result from [7]. Theorem 2.1
The parametric equations of the indicatrix of the correspondingFinsler metric of constant flag curvature K = 1 are v ( r ) = ± R · p sin r − c v ( r ) = 1 + h (cos r )cos r − p sin r − c Z rr c sin s cos s h − cos s · h ′ (cos s )1 + h (cos s ) ih h (cos s ) p sin s − c i ds. (2.2)Observe that the sign of v is different from [7], but this makes no differencefor our research. We will compute now the algebraic equation for the Finslermetric F . This result follows from a more general result in [7], but we computeit here directly for the sake of completeness.3et us start by remarking that, by using the function h in (1.4), and someelementary straightforward computations, the second equation in (2.2) reads v = 1 + ε sin r cos r cos r (cid:20) sin r − c cos R cos r + 2 ε (sin r − c ) (cid:21) = cos r cos R − ε (sin r − c ) + εc , (2.3)where c = sin R . This leads tocos r cos R = v + ε (sin r − c ) − εc , (2.4)and from here, it follows that for this choice of h , the implicit equation of theindicatrix curve of F is1 − v cos R = ( v + εv cos R − εc ) , (2.5)where ( R, θ ) ∈ (cid:2) − π , π (cid:3) × [0 , π ) are the coordinates on the manifold of geodesics M ≃ S , ( v , v ) in the coordinates of the fiber T ( R,θ ) M , and c = sin R .In order to obtain the algebraic equation for the fundamental function of thisFinsler space F , we simply substitute v i by v i F , i ∈ { , } , since the indicatrixequation is given by F = 1. It follows F − v cos R = ( εc F − εv cos R − F v ) F , (2.6)and after some computations, and the use of c = sin R , we put this algebraicequation in the canonical form AF + BF + CF + DF + E = 0 , (2.7)where A := 1 − ε (1 − c ) c B := 2 εc (1 − c ) v C := (2 c ε − c ε + 2 c ε − v − (1 − c ) v D := − ε (1 − c ) v v E := − ε v (1 − c ) . (2.8)We need to find a real positive solution of this 4-th order algebraic equation.Obviously, by using the substitution c = sin R , there formulas can be written interms of the coordinate R , but we do not need the explicit form for these.A glance at the coefficients of the equation (2.7) gives the following. Lemma 2.2
For any ε ∈ (0 , ) , the coefficients A , C , E of (2.7) satisfy A > , C < , E < . (2.9)4f we think the equation (2.7) as defininig a Minkowski type metric in eachtangent plane T ( R, Θ) M , then it is clear that its coefficients are constants andhence we can consider the function f : R −→ R , f ( x ) = Ax + Bx + Cx + Dx + E . Observe that1. f (0) = E <
0, and2. lim x →±∞ f ( x ) = + ∞ .Since f is a smooth (hence continuous) function, we obtain Proposition 2.3
The algebraic equation (2.7) has at least two real solutions,one non-positive and one non-negative.
Taking into account now the Vieta’s formulae for (2.7), that is F F F F = EA < , (2.10)it results that there are only two situations possible:1. there are two real solutions of opposite signs (+ , − ), and two complexconjugate solutions, or2. all four solutions are real, having the signs (+ − −− ) or (+ + + − ). Let us recall that there are several equivalent methods available to solve 4-th order algebraic equation, like Ferrari’s method, Galois’ method and maybeothers.We will work in the following with the Galois’ method (see [5] for a verysimple exposition). It is well-known that over C , the algebraic equation (2.7)has four solutions F k , k ∈ { , , , } .We start by making the substitution F = X − B A , (3.1)such that the equation (2.7) reduces to X + αX + βX + γ = 0 , (3.2)where α , β , γ are some constants that can be written in terms of the initialcoefficients of (2.7), but we don’t need their explicit form.This is called the reduced quartic polynomial equation . It is clear that over C , this equation must have four solutions.5 emark 3.1 We mention that some more computation shows that the coef-ficients of the reduced equation (3.2) are polynomials in the fiber coordinates ( v , v ) with the coefficients depending only on R , that is constants in a tangentplane T ( R, Θ) M . More precisely, we get α = α v + α v β = β v v + β v γ = γ v + γ v + γ v v . (3.3)Let us define the following expressions z := ( X + X )( X + X ) z := ( X + X )( X + X ) z := ( X + X )( X + X ) , (3.4)where X k , k ∈ { , , , } , are the solutions (some of them might be complex) ofthe quartic equation (3.2), and let us consider the cubic equation having theseexpressions as solutions, that is( z − z )( z − z )( z − z ) = 0 . (3.5)Some computation shows this can be written as z − αz + ( α − γ ) z + β = 0 , (3.6)where α , β , γ are the coefficients of (3.2).This equation is called the resolvent equation of the equation (3.2).Applying Cardano’s formulas (see [5]) for the resolvent equation (3.6), weget the solutions z = P + Q + 2 α z = ωP + ω Q + 2 α − (cid:16) P + Q − α (cid:17) + i √
32 ( P − Q ) z = ω P + ωQ + 2 α − (cid:16) P + Q − α (cid:17) − i √
32 ( P − Q ) . Here, ω is the cubic root of the unity, and P = r A (6) + q B (12) , (3.8) Q = r A (6) − q B (12) , where A (6) and B (12) are some homogeneous polynomials of orders 6 and 12 inthe fiber coordinates ( v , v ), respectively.6he resolvent is used in the following way to obtain the solutions of theoriginal quartic equation.If we consider0 = ( X + X ) + ( X + X ) and z = ( X + X )( X + X ) (3.9)it follows ( X + X = √− z X + X = −√− z . (3.10)Similarly we get ( X + X = √− z X + X = −√− z , (3.11)and ( X + X = √− z X + X = −√− z . (3.12)By putting all these equations together, we obtain a linear system of sixequations with four unknown that can be studied by elementary means. Indeed,the coefficients matrix is (3.13)and its rank is clearly 4. It is elementary to see that the solutions can beobtained by solving the linear system X X X X = √− z √− z √− z −√− z . (3.14)The solution of this system reads X = 12 (cid:2) √− z + √− z + √− z (cid:3) X = 12 (cid:2) √− z − √− z − √− z (cid:3) X = 12 (cid:2) −√− z + √− z − √− z (cid:3) X = 12 (cid:2) −√− z − √− z + √− z (cid:3) . (3.15)We obtain the main result. 7 heorem 3.2 The fundamental function F of the K = 1 Finsler metric on S induced by h in (1.4) is F = 12 "s − (cid:18) P + Q + 2 α (cid:19) + vuut − (cid:18) P + Q − α (cid:19) + s(cid:18) P + Q − α (cid:19) + 3( P − Q ) − B A , (3.16) where P , Q are given in the formulas (3.8) , α in (3.3) , and A , B in (2.8) ,respectively. Indeed, observe that the Vieta’s formulas for the resolvent equation (3.6) give z z z = − β < , that is, if all the solutions of the resolvent (3.6) are real, the following cases arepossible1. if there is only a real solution, this solution is negative, or2. if all three solutions are real, then either all of them are negative, or twosolutions are positive and one negative.However, the case when two solutions are positive and one negative cannothappen because of the following argument. Let us assume, for instance, that z > , z > , z < . Then,1. if all solutions are different, i.e. z = z , then from (3.15) it follows thatall solutions X i , i ∈ { , , , } are complex, i.e. the equation (3.2) cannothave real solutions, and hence the same for the equation (2.7). Of coursethis is not possible since we know from geometrical reasons (see [7]) thatthere exists a real Finsler metric of constant flag curvature K = 1 whoseindicatrix is given by (2.2) for the function h chosen by us, or2. there is a double solution z = z . Then (3.15) implies that X , X arecomplex conjugate solutions and X = X is a double real solution thatmust be positive since it is given by radicals. Using (3.1) it follows thatsame thing can be said about F i = X i − B A , i ∈ { , , , } , as well,namely, F , F are complex conjugate solutions and F = F is a doublereal solution.However, this implies F F F F > X = 12 "s − (cid:18) P + Q + 2 α (cid:19) + vuut − (cid:18) P + Q − α (cid:19) + s(cid:18) P + Q − α (cid:19) + 3( P − Q ) (3.17)is the solution of (3.2) that we are looking for.By using the transformation (3.1), the conclusion follows. Remark 3.3
It worth pointing out that the Finsler metric F in (3.16) belongsto the family of so-called ( α, β ) -metrics. Indeed, if we consider ( α, β ) such that v := α − β cos R and v := β , aftersome computations we get that A (6) and B (12) are homogeneous polynomialsof degree six and twelve in α and β , hence the Finsler metric in (3.16) canbe written as an ( α, β ) metric. This fact can be regarded as a consequence ofa more general result in [8]. Please do not confuse the α and β used in thisRemark with the α and β in (3.3). One legitimate question to ask if the Finsler metric we have computed in thepresent paper is indeed of constant flag curvature K = 1.Let us recall that a Finsler metric on a real smooth, n -dimensional manifold M is a function F : T M → [0 , ∞ ) that is positive and smooth on g T M = T M \{ } , has the homogeneity property F ( x, λv ) = λF ( x, v ), for all λ > v ∈ T x M , having also the strong convexity property that the Hessian matrix g ij = 12 ∂ F ∂y i ∂y j (4.1)is positive definite at any point u = ( x i , y i ) ∈ g T M .The fundamental function F of a Finsler structure ( M, F ) determines and itis determined by the (tangent) indicatrix , or the total space of the unit tangentbundle of F , namely Σ F := { u ∈ T M : F ( u ) = 1 } which is a smooth hypersurface of T M . At each x ∈ M we also have the indicatrix at x Σ x := { v ∈ T x M | F ( x, v ) = 1 } = Σ F ∩ T x M T x M .To give a Finsler structure ( M, F ) is therefore equivalent to giving a smoothhypersurface Σ ⊂ T M for which the canonical projection π : Σ → M is asurjective submersion and having the property that for each x ∈ M , the π -fiberΣ x = π − ( x ) is strictly convex including the origin O x ∈ T x M .In the present paper we have started with a Zoll metric ( S , g ) given in(1.1) determined by the odd function h in (1.4). It is easy to see that theRiemannian metric g given in (1.1) with h in (1.4) is indeed a Zoll metric. Thisfollows immediately from [3]. Indeed, it is elementary to see that the Darbouxtheorem 4.11 in [3] is satisfied by the function f = 1 + h .The fundamental geometrical property of Zoll metrics is that they have allgeodesics closed and of same length. This implies that the space of geodesics,that is the topological space where a point represents an oriented geodesic, isa smooth differentiable manifold. Observe that this property is not true ingeneral for an arbitrary Riemannian metric. Therefore it is natural to constructa Finsler metric on the manifold of geodesics M of a Zoll metric ( S , g ) suchthat the indicatrices of this Finsler metric coincide with the closed geodesics of g . If we denote by U g S = Σ the unit sphere bundle of the Zoll metric ( S , g ),then on Σ there exists a canonical g -orthogonal coframe { θ , θ , θ } uniquellydetermined by the Riemannian metric g . We recall here this construction for thesake of completeness. Let us consider a g -orthonormal basis f , f of the tangentspace T x S and denote by α , α the dual co-frame in T ∗ x S . Then the coframe { θ , θ , θ } is easily obtained, the one-forms { θ , θ } are the tautological formsof α , α on T ∗ M , while θ is the Levi-Civita connection form.Then this moving frame satisfies the structure equations dθ = θ ∧ θ dθ = θ ∧ θ dθ = Gθ ∧ θ , (4.2)where G is the Gauss curvature of g regarded as function of one variable on Σ.Since we are using the odd function h in (1.4), the Gauss curvature is given by(2.1). A quick glance at the formula (2.1) convinces that g has positive Gausscurvature G > S .Let us consider the following coframe changing ω ω ω = √ G − √ G θ θ θ (4.3)and by computing the structure equations of the coframe { ω , ω , ω } , we obtainthe structure equations dω = − Iω ∧ ω + ω ∧ ω dω = ω ∧ ω dω = ω ∧ ω − Jω ∧ ω , (4.4)10here we denote I := 12 G √ G G θ , J := − G √ G G θ . We point out that here, the function G is the one in (2.1). The subscripts hererepresent the directional derivatives with respect to the given co-frame, that is df = f θ θ + f θ θ + f θ θ , for any smooth function f : Σ → R .The 3-manifold Σ endowed with the coframe { ω , ω , ω } is called a gener-alized Finsler manifold with the invariants I , J , K = 1 (see [2], [7], etc.).Let us observe the following relations between the structures induced by { θ , θ , θ } and { ω , ω , ω } : • the geodesic foliation { θ = 0 , θ = 0 } of g coincides with the foliation { ω = 0 , ω = 0 } (the indicatrix foliation); • the indicatrix foliation { θ = 0 , θ = 0 } of g coincides with the foliation { ω = 0 , ω = 0 } (the geodesic foliation).Let us consider now the manifold of geodesics M of the Zoll metric ( S , g ).This is a 2-dimensional smooth manifold diffeomorphic to S . The followingdiagram is commutative Σ λ ~ ~ ⑦⑦⑦⑦⑦⑦⑦ π ! ! ❈❈❈❈❈❈❈❈ ι / / T MS M Here λ : Σ → S is the projection that identifies a leaf of the foliation { θ =0 , θ = 0 } with a point on S , and π : Σ → M is the projection that identifies aleaf of the foliation { θ = 0 , θ = 0 } with a point on the manifold of geodesics M . It is clear that any Finsler structure on a manifold M induces a generalizedFinsler structure { ω , ω , ω } on the indicatrix bundle, where { ω , ω , ω } is thecanonical orthonormal moving co-frame on Σ determined by F (see [2], or anyother textbook of Finsler geometry).However, clearly not every generalized Finsler structure leads to a Finslermetric. We will show however that the generalized Finsler structure constructedhere actually gives a classical Finsler structure on M .We recall here a fundamental result by Bryant (see [4]). Proposition 4.1
The necessary and sufficient condition for a generalized Finslersurface (Σ , ω ) to be realizable as a classical Finsler structure on a surface are1. the leaves of the foliation { ω = 0 , ω = 0 } are compact;2. it is amenable, i.e. the space of leaves of the foliation { ω = 0 , ω = 0 } is a differentiable manifold M ; . the canonical immersion ι : Σ → T M , given by ι ( u ) = π ∗ ,u (ˆ e ) , is one-to-one on each π -fiber Σ x ,where we denote by (ˆ e , ˆ e , ˆ e ) the dual frame of the coframing ( ω , ω , ω ) . In the same source it is pointed out that if for example the { ω = 0 , ω = 0 } leaves are not compact, or even in the case they are, if they are ramified, or ifthe curves Σ x winds around origin in T x M , in any of these cases, the generalizedFinsler surface structure is not realizable as a classical Finsler surface.Obviously all conditions in Proposition 4.1 are satisfied by Σ endowed withthe coframe (4.3) due to the properties of the Zoll metric g (see [4] and [7]).It follows that the generalized Finsler metric defined by the coframe (4.3) isactually a classical Finsler metric on the manifold M of geodesics of the Zollmetric constructed by using the function h in (1.4).Let us summarize our construction here. • We start with a specific Zoll metric ( S , g ) given by (1.1) with h in (1.4),with the Gauss curvature G given by (2.1). Observe that G > • We consider the g -orthonormal coframe { θ , θ , θ } on the unit spherebundle Σ := U g S . • We construct the coframe { ω , ω , ω } on Σ by (4.3). Taking the exteriorderivatives of the forms in this coframe, it is easy to see that this is ageneralized Finsler structure on Σ with the invariants I , J , K = 1. • The generalized Finsler structure (Σ; ω , ω , ω ) gives a classical Finslerstructure on the manifold of geodesics M of g , with flag curvature K = 1. • The Finsler structure obtained in this way is determined by the indicatrixcurve in each tangent space T x M , which is the curve (2.2) obtained as theembedding of a g -geodesic in T M .This explaines why the Finsler metric constructed above is of constant flagcurvature K = 1. Observe that in our case, v in (2.2) is given by v = 1 + ε sin r cos r cos r − p sin r − c Z rR sin s cos s p sin s − c (1 + 2 ε cos s ) ds. The following formulas are useful for computing the above integral. Z sin s cos s p sin s − c ds = 1cos R p sin s − c cos s + constant.
12y integration by parts, we get Z rR sin s cos s p sin s − c (1 + 2 ε cos s ) ds = 1cos R " p sin r − c cos r (1 + 2 ε cos r ) + 6 ε Z rR sin s cos s p sin s − c ds . The following formulas are useful when dealing with complex numbers.If z = r (cos θ + i sin θ ), then z = ±√ r (cid:18) cos θ i sin θ (cid:19) . If ¯ z is the conjugate of z , then¯ z = ±√ r (cid:18) cos θ − i sin θ (cid:19) . It follows z + ¯ z ∈ {± √ r cos θ , ± i √ r sin θ } . Let us choose z + ¯ z = 2 √ r cos θ in the previous formula. If z = a + bi = re iθ , r := √ a + b , tan θ := ba , then z + ¯ z = √ · q a + p a + b . Finally, if we let a := − (cid:18) P + Q − α (cid:19) , b := √
32 ( P − Q ) , then z + ¯ z = √ · vuut − (cid:18) P + Q − α (cid:19) + 12 s(cid:18) P + Q − α (cid:19) + 3( P − Q ) . Acknowledgements: The authors are greatful to K. Shibuya and U. Sombonfor many useful discussions. 13 eferences [1] D. Bao, S. S. Chern, Z. Shen,
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Ioana Monica Mas¸caColegiul Nat¸ional ”Andrei S¸aguna”, Bras¸ov
E-mail: [email protected]
Sorin V. Sabau (Corresponding author)
Departament of Biology, Tokai University, Sapporo, Japan
E-mail: [email protected]
Hideo ShimadaTokai University, Sapporo, Japan