The global behaviour of Finsler geodesics. Applications Sorin V. Sabau Tokai University, Sapporo, Japan Sorin V. Sabau The global behaviour of Finsler geodesics. Applications 1. Finsler metrics Definition A Finsler norm, or metric, on a real smooth, n-dimensional manifold M is a function F : TM ! [0; 1) that is positive and smooth on TMg = TMnf0g, has the homogeneity property F (x; λv) = λF (x; v), for all λ > 0 and all v 2 Tx M, having also the strong convexity property that the Hessian matrix 1 @2F 2 g = ij 2 @y i @y j is positive definite at any point u = (x i ; y i ) 2 TMg. (M; F ) is called a Finsler manifold or Finsler structure. The Finsler structure is called absolute homogeneous if F (x; −y) = F (x; y) because this leads to the homogeneity condition F (x; λy) = jλjF (x; y), for any λ 2 R. We don't need this assumption in the present talk. By means of the Finsler fundamental function F one defines the indicatrix bundle (or the Finslerian unit sphere bundle) by S SM := x2M Sx M, where Sx M := fy 2 M j F (x; y) = 1g. At each x 2 M we also have the indicatrix at x Sx M := fv 2 Tx M j F (x; v) = 1g = SM \ Tx M which is a smooth, closed, strictly convex hypersurface in Tx M. Remark. The fundamental function F of a Finsler structure (M; F ) determines and it is determined by the (tangent) indicatrix, or the total space of the unit tangent bundle of SM := fu 2 TM : F (u) = 1g which is a smooth hypersurface of TM. Remark. The fundamental function F of a Finsler structure (M; F ) determines and it is determined by the (tangent) indicatrix, or the total space of the unit tangent bundle of SM := fu 2 TM : F (u) = 1g which is a smooth hypersurface of TM. At each x 2 M we also have the indicatrix at x Sx M := fv 2 Tx M j F (x; v) = 1g = SM \ Tx M which is a smooth, closed, strictly convex hypersurface in Tx M. Riemannian vs. Finslerian unit circles. Let γ :[a; b] ! M be a regular piecewise C 1-curve in M, and let a := t0 < t1 < ··· < tk := b be a partition of [a; b] such that γj[ti−1;ti ] is smooth for each interval [ti−1; ti ], i 2 f1; 2;:::; kg. Definition The forward integral length of γ is given by k Z ti + X Lγ := F (γ(t); γ_ (t))dt; i=1 ti−1 dγ whereγ _ = is the tangent vector along the curve γj . dt [ti−1;ti ] Proposition + 0 b (L ) (0) =gγ_ (b)(γ; U)ja k Xh − + i + g − (_γ(t ); U(ti )) − g + (_γ(t ); U(ti )) γ_ (ti ) i γ_ (ti ) i i=1 Z b − gγ_ (Dγ_ γ;_ U)dt; a where Dγ_ is the covariant derivative along γ with respect to the Chern connection and γ is arc length parametrized. Definition A regular piecewise C 1-curve γ on a Finsler manifold is called a forward geodesic if (L+)0(0) = 0 for all piecewise C 1-variations of γ that keep its ends fixed. In terms of Chern connection a constant speed geodesic is characterized by the condition Dγ_ γ_ = 0. Definition Likely, a regular piecewise C 1-curve γ on a Finsler manifold is called a backward geodesic if (L−)0(0) = 0 for all piecewise C 1-variations of γ that keep its ends fixed, where k Z ti − X Lγ := F (γ(t); −γ_ (t))dt i=1 ti−1 is the backward integral length of γ. Obviously in the Riemannian case forward geodesics and backward geodesics coincide so this distinction is superfluous. For any two points p, q on M, let us denote by Ωp;q the set of all piecewise C 1-curves γ :[a; b] ! M such that γ(a) = p and γ(b) = q. Proposition The map + d : M × M ! [0; 1); d(p; q) := inf Lγ γ2Ωp;q gives the Finslerian distance on M. It can be easily seen that d is in general a quasi-distance, i.e., it has the properties 1 d(p; q) ≥ 0, with equality if and only if p = q; 2 d(p; q) ≤ d(p; r) + d(r; q), with equality if and only if r lies on a minimal geodesic segment joining from p to q (triangle inequality). The reverse distance d(q; p) is actually the Finslerian distance induced by the backward integral length. Remark In the case where (M; F ) is absolutely homogeneous, the symmetry condition d(p; q) = d(q; p) holds and therefore (M; d) is a genuine metric space. We do not assume this symmetry condition in the present talk. Definition A sequence of points fxi g ⊂ M, on a Finsler manifold (M; F ), is called a forward Cauchy sequence if for any " > 0, there exists N = N(") > 0 such that for all N ≤ i < j we have d(xi ; xj ) < ". A sequence of points fxi g ⊂ M is called a backward Cauchy sequence if for any " > 0, there exists N = N(") > 0 such that for all N ≤ i < j we have d(xj ; xi ) < ". The Finsler space (M; F ) is called forward (backward) complete with respect to the Finsler distance d if and only if every forward (backward) Cauchy sequence converges, respectively. Definition A Finsler manifold (M; F ) is called forward (backward) geodesically complete if and only if any short geodesic γ :[a; b) ! M can be extended to a long geodesic γ :[a; 1) ! M (γ :(−∞; b]) ! M). The equivalence between forward completeness as metric space and geodesically completeness is given by the Finslerian version of Hopf-Rinow Theorem. In the Finsler case, unlikely the Riemannian counterpart, forward completeness is not equivalent to backward one, except the case when M is compact. A Finsler metric that is forward and backward complete is called bi-complete. Remark Even though the exponential map is quite similar with the correspondent notion in Riemannian geometry, we point out two distinguished properties 1 1 expx is only C at the zero section of TM, i.e. for each 1 fixed x, the map expx y is C with respect to y 2 Tx M, and C 1 away from it. Its derivative at the zero section is the identity map (Whitehead); 2 2 expx is C at the zero section of TM if and only if the Finsler structure is of Berwald type. In this case exp is actually C 1 on entire TM (Akbar-Zadeh). Definitions. Let γy (t) be the unit speed geodesic from p 2 M with initial velocity y. 1 the conjugate value cy of y : cy := supfrjno point γy (t); t 2 [0; r] is conjugate to pg, 2 the first conjugate point of p along γy : γy (cy ); 3 the conjugate radius at p : cp := infy2SpM cy ; 4 the conjugate locus of p : Conp := fγy (cy )jy 2 SpM; cy < 1g; Definitions. 1 the cut value iy of y : iy := supfrj the geodesic segment γyj[0;r] is globally minimizing g; 2 the cut point of p along γy : γy (iy ), for iy < 1; 3 the injectivity radius at p : ip := infy2SpM iy ; 4 the cut locus of p : C(p) := fγy (iy )jy 2 SpM; iy < 1g. Remark. Namely, if q 2 C(p), then at least one of the following must hold 1 q is the first conjugate point of p along γy ; 2 there exists (at least) two distinct geodesics af the same length from p to q. Properties of the geodesics 1 The cut point of p along γ must occur either before, or exactly at, the first conjugate point. 2 The geodesic γyj[0;r] is the unique minimizer of arc length among all piecewise C 1 curves with fixed end points, for any r < iy . 3 The 'unique minimizer property' will fail at the cut point if it happens before the first conjugate point. Properties of the geodesics 1 The cut point of p along γ must occur either before, or exactly at, the first conjugate point. 2 The geodesic γyj[0;r] is the unique minimizer of arc length among all piecewise C 1 curves with fixed end points, for any r < iy . 3 The 'unique minimizer property' will fail at the cut point if it happens before the first conjugate point. Remark. Namely, if q 2 C(p), then at least one of the following must hold 1 q is the first conjugate point of p along γy ; 2 there exists (at least) two distinct geodesics af the same length from p to q. 1 The notion of cut locus was introduced and studied for the first time by H. Poincare in 1905 for the Riemannian case. 2 In the case of a two dimensional analytical sphere, S. B. Myers has proved in 1935 that the cut locus of a point is a finite tree in both Riemannian and Finslerian cases. 3 In the case of an analytic Riemannian manifold, M. Buchner has shown the triangulability of the cut locus of a point p, and has determined its local structure for the low dimensional case in 1977 and 1978, respectively. 4 The cut locus of a point can have a very complicated structure. For example, H. Gluck and D. Singer have constructed a C 1 Riemannian manifold that has a point whose cut locus is not triangulable.