From Metric Properties to a Rigidity Conjecture on Finsler Manifolds via a Geodesic Detecting Algorithm Alexandru Kristály and Ágoston Róth Abstract. The aim of this paper is to consider Busemann-type inequalities on Finsler manifolds. We actually formulate a rigidity conjecture: any Finsler manifold which is a Busemann NPC space is Berwaldian. This statement is supported by some theoretical results and numerical examples. The presented examples are ob- tained by using evolutionary techniques (genetic algorithms) which can be used for detecting geodesics on a large class of not necessarily reversible Finsler manifolds. 2000 Mathematics Subject Classi…cation: 58B20, 53C60, 53C70, 32Q05. Keywords: Busemann NPC space, Finsler manifold, Berwald space, numerical detection of geodesics via genetic algorithms. 1 Introduction In the forties, Busemann developed a synthetic geometry of G-spaces, see [3]. These spaces possess the essential geometric properties of Finsler manifolds but they have no necessarily a di¤erential structure. Nowadays, Busemann’scurvature is referred to non-positively curved (quasi-)metric spaces, called Busemann NPC spaces; this notion of non-positive curvature requires that in small geodesic triangles the length of a side is at least the twice of the geodesic distance of the mid-points of the other two sides, see [3, p. 237]. Note that on Riemannian manifolds the non-positivity of the sectional curvature characterizes Busemann’snon-positivity curvature. In the …fties, Aleksandrov introduced independently another notion of curvature on metric spaces, a stronger one than that of Busemann, see Jost [5, Section 2.3]. Nevertheless, on Riemannian manifolds the Aleksandrov curvature condition holds if and only if the sectional curvature is non-positive, see Bridson-Hae‡iger[2, Theorem 1A.6]. In the Finsler case the picture is quite rigid: if on a reversible Finsler manifold (M; F ) the Aleksandrov curvature condition holds (on the induced metric space by (M; F )) then (M; F ) should be Riemannian, see [2, Proposition 1.14]. Having these facts in our mind, a natural question arises: What about Finsler manifolds that are Busemann NPC spaces? Clearly, we are thinking …rst to those Finsler manifolds which have non-positive ‡ag curvature. However, within this class, we easily …nd examples which are not Busemann NPC spaces. For instance, the Hilbert metric of the interior of a simple, closed and strictly convex curve c in the Euclidean plane de…nes a Finsler metric with constant ‡ag curvature 1, but Busemann’s curvature condition holds if and only if c is an ellipse, see Kelly-Straus [6]. In the latter case, the Hilbert metric becomes a Riemannian one. The above example suggests we face again a rigidity result as in the Aleksandrov case. However, a recent result of Kozma-Kristály-Varga [8] entitle us to believe that the appropriate class of Finsler manifolds is the Berwald one, a larger class than Riemannian manifolds. Namely, we formulate the following Conjecture. Let (M; F ) be a (not necessarily reversible) Finsler manifold such that (M; dF ) is a Busemann NPC space. Then (M; F ) is a Berwald space. The aim of the present paper is to recall some theoretical developments and certain numerical examples in order to support the above Conjecture. In addition, we propose a new numerical approach to detect minimal geodesics on Finsler manifolds by using evolutionary techniques. Note that the presented numerical approach can be e¢ ciently used for verifying di¤erent metric relations on a large class of Finsler manifolds exploiting the basic property of Finslerian-geodesics that they locally minimize the integral length. The paper is organized as follows. In the next section we recall the notion of Busemann NPC (quasi-)metric spaces and basic elements of Finsler manifolds. Section 3 deals with the Conjecture from both theoretical and numerical point of view. Section 4 is independent from the previous ones and describes the detection of geodesics on Finsler manifolds by using genetic algorithms. 1 2 Preliminaries 2.1 Busemann NPC spaces Let (M; d) be a quasi-metric space (i.e., d is not necessarily symmetric). A continuous curve c : [0; r] M with c (0) = p; c (r) = q is a shortest geodesic, if l (c) = d (p; q), where l (c) is the generalized length of !c and it is de…ned by n l (c) = sup d (c (ti 1) ; c (ti)) : 0 = t0 < t1 < ::: < tn = r; n N : 2 (i=1 ) X We say that (M; d) is a locally geodesic (length) space if for every point p M there is a neighborhood 2 Np M such that for every two points x; y Np there exists a shortest geodesic joining them. In the sequel, we always assume that the shortest geodesics2 are parametrized proportionally to the arc length, i.e., l c = t l (c) ; t [0; r] : j[0;t] 8 2 De…nition 1 (Busemann NPC space) A locally geodesic space (M; d) is said to be a Busemann non-positive curvature space (shortly, Busemann NPC space), if for every p M there exists a neighborhood Np M 2 such that for any two shortest geodesics c1; c2 : [0; 1] M with c1 (0) = c2 (0) Np and with endpoints ! 2 c1 (1) ; c2 (1) Np we have 2 1 1 2d c1 ; c1 d(c1 (1) ; c2 (1)): (1) 2 2 2.2 Finsler manifolds Let M be a connected m-dimensional C1 manifold and let TM = p M TpM be its tangent bundle. If the 2 continuous function F : TM [0; ) satis…es the conditions that it is C1 on TM 0 ; F (p;ty) = tF (p; y) for ! 1 S nf g 1 2 all t 0 and y TpM; i.e., F is positively homogeneous of degree one; and the matrix gij(y) := F (y) 2 2 yiyj is positive de…nite for all y TM 0 ; then we say that (M; F ) is a Finsler manifold. If F is absolutely homogeneous, then (M; F ) is2 said ton be f greversible. Let TM be the pull-back of the tangent bundle TM by : TM 0 M: Unlike the Levi-Civita connection in Riemann geometry, there is no unique natural connectionn in f g the ! Finsler case. Among these k connections on T M; we choose the Chern connection whose coe¢ cients are denoted by ij; see Bao-Chern- Shen [1, p. 38]. This connection induces the curvature tensor, denoted by R; see [1, Chapter 3]. The Chern connection de…nes the covariant derivative DVU of a vector …eld U in the direction V TpM: Since, in i 2 general, the Chern connection coe¢ cients jk in natural coordinates have a directional dependence, we must say explicitly that DVU is de…ned with a …xed reference vector. In particular, let c : [0; r] M be a smooth k i ! curve with velocity …eld T = T(t) = c_(t) = T k=1;m : Suppose that U = U i=1;m and W are vector …elds de…ned along c: We de…ne D U with reference vector W as T i dU j k i @ DTU = + U T (jk)(c;W) i ; dt @p c(t) j @ where i is a basis of Tc(t)M: A C1 curve c : [0; r] M; with velocity T = c_ is a (Finslerian) @p c(t) j i=1;m ! geodesicn if o T D = 0 with reference vector T. (2) T F (T) If the Finslerian speed of the geodesic c is constant, then (2) becomes d2ci dcj dck + i = 0; i = 1; :::; m = dim M: (3) dt2 dt dt jk (c;T) In the sequel, we assume that the geodesics have constant Finslerian speed. 2 A Finsler manifold (M; F ) is said to be forward (resp. backward) geodesically complete if every geodesic c : [0; 1] M can be extended to a geodesic de…ned on [0; ) (resp. ( ; 1]). ! 1 1 Let c : [0; r] M be a piecewise C1 curve. Its integral length is de…ned as ! r LF (c) = F (c (t) ; c_ (t)) dt: Z0 For p; q M, denote by (p; q) the set of all piecewise C1 curves c : [0; r] M such that c (0) = p and 2 [0;r] ! c (r) = q. De…ne the map dF : M M [0; ) by ! 1 dF (p; q) = inf LF (c) : (4) c (p;q) 2 [0;r] Of course, we have dF (p; q) 0; where equality holds if and only if p = q; and the triangle inequality holds, i.e., dF (p0; p2) dF (p0; p1) + dF (p1; p2) for every p0; p1; p2 M: In general, since F is only a 2 positive homogeneous function, dF (p; q) = dF (q; p); thus, (M; dF ) is only a quasi-metric space. If (M; g) is a 6 Riemannian manifold, we will use the notation dg instead of dF which becomes a usual metric function. Busemann and Mayer [4, Theorem 2, p. 186] proved that the generalized length l(c) and the integral length LF (c) of any (piecewise) C1 curve coincide for Finsler manifolds. Let (p; y) TM 0 and let V be a section of the pulled-back bundle TM. Then, 2 n f g g(p;y) (R (V; y) y; V) K (y; V) = 2 (5) g (y; y) g (V; V) g (y; V) (p;y) (p;y) (p;y) is the ‡ag curvature with ‡ag y and transverse edge V. Here, i j 1 2 i j g(p;y) := gij(p;y)dp dp := F dp dp ; p M; y TpM; (6) 2 i j 2 2 y y is the Riemannian metric on the pulled-back bundle TM.