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The Global Behaviour of Finsler Geodesics. Applications

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The Global Behaviour of Finsler Geodesics. Applications 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.
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