GROVE SHIOHAMA TYPE SPHERE THEOREM IN FINSLER GEOMETRY

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1 Kondo, K. Osaka J. Math. 52 (2015), GROVE SHIOHAMA TYPE SPHERE THEOREM IN FINSLER GEOMETRY KEI KONDO (Received September 30, 2014) Abstract From radial curvature geometry s standpoint, we prove a few sphere theorems of the Grove Shiohama type for certain classes of compact Finsler manifolds. 1. Introduction Beyond a doubt, one of the most beautiful theorems in global Riemannian geometry is the diameter sphere theorem of Grove and Shiohama [3]. In their proof, Toponogov s comparison theorem (TCT) was very first applied seriously together with the critical point theory, introduced by themselves, of distance functions. That is, if a complete Riemannian manifold X has a critical point, say q ¾ X Ò {p}, of the distance function d p to a point p ¾ X, then q is the cut point of p. And hence d p is not differentiable at q. However, they overcame the analytical obstruction by applying the original TCT to the triangle (pxy) with the interior angle (pxy) 2 at x. That is the point, i.e., they took the manifold into their hands by directly drawing segments on it. Our purpose of this article is to prove a sphere theorem of the Grove Shiohama type for a certain class of forward complete Finsler manifolds whose radial flag curvatures are bounded below by 1. Of course, our major tools to prove it are a TCT for such a class and the critical point theory, more precisely, Gromov s isotopy lemma ([2]). Such a TCT is easily proved by modifying the TCT established in [6] (see Section 2 in this article), and the isotopy lemma holds from a similar argument to the Riemannian case. The fact that, compared with the Riemannian case, there are few theorems on the relationship between the topology and the curvature of a Finsler manifold is the worthy of note. E.g., Shen s finiteness theorem ([10]), Rademacher s quarter pinched sphere theorem ([9]), and the finiteness of topological type and a diffeomorphism theorem to Euclidean spaces of the author with Ohta and Tanaka in [7]. To state our sphere theorems of the Grove Shiohama type in Finsler case, we will introduce several notions in the geometry and radial curvature geometry: Let (M, F, p) denote a pair of a forward complete, connected, n-dimensional C ½ -Finsler manifold (M, F) with a base point p ¾ M, and dï M M [0,½) denote the distance function 2010 Mathematics Subject Classification. Primary 53C60; Secondary 53C21, 53C22.

2 1144 K. KONDO induced from F. Remark that the reversibility F( Ú) F(Ú) is not assumed in general, and hence d(x, y) d(y, x) is allowed. For a local coordinate (x i ) i1 n of an open subset O M, let (xi, Ú j ) i, n j1 be the coordinate of the tangent bundle TO over O such that Ú Ï n j1 Ú j x j x, x ¾ O. For each Ú ¾ T x M Ò {0}, the positive-definite n n matrix (g i j (Ú)) n i, j1 Ï 1 2 (F 2 ) n 2 (Ú) Ú i Ú j i, j1 provides us the Riemannian structure g Ú of T x M by ¼ ½ g Ú n a i n x, b j i x x j x i1 j1 Ï n i, j1 g i j (Ú)a i b j. This is a Riemannian approximation of F in the direction Ú. For two linearly independent vectors Ú, Û ¾ T x M Ò {0}, the flag curvature is defined by where R Ú K M (Ú, Û) Ï g Ú (R Ú (Û, Ú)Ú, Û) g Ú (Ú, Ú)g Ú (Û, Û) g Ú (Ú, Û) 2, denotes the curvature tensor induced from the Chern connection. Remark that K M (Ú, Û) depends on the flag {sú tû s, t ¾ Ê}, and also on the flag pole {sú s 0}. Given Ú, Û ¾ T x M Ò {0}, define the tangent curvature by T M (Ú, Û) Ï g X (D Y Y Y (x) D X Y Y (x), X(x)), where the vector fields X, Y are extensions of Ú, Û, and D Û Ú X(x) denotes the covariant derivative of X by Ú with reference vector Û. Independence of T M (Ú, Û) from the choices of X, Y is easily checked. Note that T M 0 if and only if M is of Berwald type (see [11, Propositions 7.2.2, ]). In Berwald spaces, for any x, y ¾ M, the tangent spaces (T x M, F Tx M) and (T y M, F Ty M) are mutually linearly isometric (cf. [1, Chapter 10]). In this sense, T M measures the variety of tangent Minkowski normed spaces. Let ÉM be a complete 2-dimensional Riemannian manifold, which is homeomorphic to Ê 2 if ÉM is non-compact, or to Ë 2 if ÉM is compact. Fix a base point Ép ¾ ÉM. Then, we call the pair ( ÉM, Ép) a model surface of revolution if its Riemannian metric dés 2 is expressed in terms of the geodesic polar coordinate around Ép as dés 2 dt 2 f (t) 2 d 2, (t, ) ¾ (0, a) Ë 1 Ép,

3 SPHERE THEOREM IN FINSLER GEOMETRY 1145 where 0 a ½, fï (0,a)Ê denotes a positive smooth function which is extensible to a smooth odd function around 0, and Ë 1 Ép Ï {Ú ¾ T Ép ÉM Ú 1}. Define the radial curvature function GÏ [0, a) Ê such that G(t) is the Gaussian curvature at É (t), where É Ï [0, a) ÉM is any (unit speed) meridian emanating from Ép. Note that f satisfies the differential equation f ¼¼ G f 0 with initial conditions f (0) 0 and f ¼ (0) 1. It is clear that, if f (t) t, sin t, and sinh t, then ÉM Ê 2, Ë 2, and À 2 ( 1), respectively. We call ( ÉM, Ép) a von Mangoldt surface if G is non-increasing on [0, a). A round sphere is the only compact, smooth von Mangoldt surface, i.e., f satisfies lim ta f ¼ (t) 1. If a von Mangoldt surface has the property a ½ and if it is not a round sphere, then lim ta f (t) 0 and lim ta f ¼ (t) 1. Therefore, such a surface ( ÉM, Ép) has a singular point, say Éq ¾ ÉM, at the maximal distance from Ép ¾ ÉM such that d(ép, Éq) a, and hence ÉM is an Alexandrov space. Its shape can be understood as a balloon (See [4, Example 1.2]). On the other hand, paraboloids and 2-sheeted hyperboloids are typical examples of non-compact von Mangoldt surfaces. An atypical example of such a surface is found in [8, Example 1.2]. We say that a Finsler manifold (M, F, p) has the radial flag curvature bounded below by that of a model surface of revolution ( ÉM, Ép) if, along every unit speed minimal geodesic Ï [0, l) M emanating from p, we have K M (È (t), Û) G(t) for all t ¾ [0,l) and Û ¾ T (t) M linearly independent to È (t). Also, we say that (M, F, p) has the radial tangent curvature bounded below by a constant Æ ¾ ( ½,0] if, along every unit speed minimal geodesic Ï [0, l) M emanating from p, T M (È (t), Û) Æ for all Û ¾ T (t) M. We set Br (p) Ï {x ¾ M d(p, x) r}, (1.1) G p (x) Ï {È (l) ¾ T x M is a minimal geodesic segment from p to x}, where l Ï d(p, x), L m (c) Ï Ê a 0 max{f(èc), F( Èc)} ds, and rad p Ï sup x¾m d(p, x). Now, our main result is stated as follows: Theorem 1.1. Let (M, F, p) be a compact connected n-dimensional C ½ -Finsler manifold whose radial flag curvature is bounded below by 1 and radial tangent curvature is equal to 0. Assume that (1) F(Û) 2 g Ú (Û, Û) for all x ¾ B 2 (p), Ú ¾ G p(x), and Û ¾ T x M; (2) g Ú (Û, Û) F(Û) 2 for all x ¾ M Ò B 2 (p), Ú ¾ G p(x) and Û ¾ T x M; (3) the reverse curve Æc(s)Ï c(a s) of c is geodesic and L m (c)rad p for all minimal geodesic segments cï [0, a] M Ò {p}. If rad p 2, then M is homeomorphic to the sphere Ë n.

4 1146 K. KONDO REMARK 1.2. We give here related remarks for Theorem 1.1: Except for L m (c) rad p, all conditions in the theorem are sufficient ones that make our TCTs hold (see Corollary 2.8 and Lemma 2.9 in Section 2). The biggest obstruction when we establish TCTs in Finsler geometry is the covariant derivative even though F is reversible. Thanks to the conditions (1) and (2), we can overcome the obstruction, i.e., by the (1) and f ¼ (t)cost 0 on [0, 2), we can transplant the strictly convexness of B 2 (Ép) ÉM Ë 2 to B 2 (p) (See Section 4), where the convexity on B 2(Ép) arises from the positive second fundamental form for f ¼ 0 on [0,2). As well as, the strictly concaveness of ÉMÒB 2 (Ép) is transplanted to MÒB t 0 (p) by the (2) and f ¼ (t)cost 0 on (2,]. Note that one may construct non-riemannian spaces satisfying (1) and (2) (cf. [7, Example 1.3]). The geodesic property on Æc in the condition (3) and T M (È (t), Û) 0 just only imply g È (D È Èc Èc, È )0. Note that D È Èc Èc 0 in general. We can replace L m (c)rad p in (3) with the following weaker assumption: L m (c) for c satisfying c([0, a])(m Ò B 2 (p)) ; rad p for c emanating from q ¾ B rad p (p) to any point in B 2 (p). Here, note that B rad p (p) {q} (see Lemma 3.4). Remark that diam(m) from the Bonnet Myers theorem ([1, Theorem 7.7.1]). We can remove the (3) in Theorem 1.1 as follows: Corollary 1.3. Let (M, F, p) be a compact connected n-dimensional C ½ -Finsler manifold whose radial flag curvature is bounded below by 1 and radial tangent curvature is equal to 0. Assume that (1) F(Û) 2 g Ú (Û, Û) for all x ¾ B 2 (p), Ú ¾ G p(x), and Û ¾ T x M; (2) g Ú (Û, Û) F(Û) 2 for all x ¾ M Ò B 2 (p), Ú ¾ G p(x) and Û ¾ T x M. If F is reversible and diam(m) rad p 2, then M is homeomorphic to Ë n. If F is of Berwald type, the geodesic property on Æc (of the (3)) and T M (È (t), Û) 0 in Theorem 1.1 are automatically satisfied. Hence, we have one more corollary: Corollary 1.4. Let (M, F, p) be a compact connected n-dimensional C ½ -Berwald space whose radial flag curvature is bounded below by 1. Assume that (1) F(Û) 2 g Ú (Û, Û) for all x ¾ B 2 (p), Ú ¾ G p(x), and Û ¾ T x M; (2) g Ú (Û, Û) F(Û) 2 for all x ¾ M Ò B 2 (p), Ú ¾ G p(x) and Û ¾ T x M; (3) L m (c) rad p for all minimal geodesic segments cï [0, a] M Ò {p}. If rad p 2, then M is homeomorphic to Ë n.

5 SPHERE THEOREM IN FINSLER GEOMETRY TCTs To prove Theorem 1.1, we need Toponogov s comparison theorems (TCT) in Finsler geometry. In [6], we recently established a TCT for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface. In this section, we modify the TCT in the case where a model surface is the unit sphere Angles, triangles, and a counterexample. Let (M, F, p) be a forward complete, connected C ½ -Finsler manifold with a base point p ¾ M, and denote by d its distance function. It follows from the Hopf Rinow theorem that the forward completeness guarantees that any two points in M can be joined by a minimal geodesic segment. Owing to d(x, y) d(y, x) generally, we need a distance with the symmetric property to define the angles : Define d m (x, y) Ï max{d(x, y), d(y, x)}. Since d(p, x) d(p, y) d m (x, y), we may define the angles with respect to d m as follows. DEFINITION 2.1 (Angles). Let cï [0, a] M be a unit speed minimal geodesic segment (i.e., F(Èc) 1) with p c([0, a]). The forward and the backward angles (pc(s)c(a)), (pc(s)c(0)) ¾ [0, ] at c(s) are defined via d(p, c(s cos (pc(s)c(a)) Ï h)) d(p, c(s)) lim h0 d m (c(s), c(s h)) d(p, c(s)) d(p, c(s h)) cos (pc(s)c(0)) Ï lim h0 d m (c(s h), c(s)) REMARK 2.2. The limits in Definition 2.1 are as follows: d(p, c(s h)) d(p, c(s)) lim h0 d m (c(s), c(s h)) d(p, c(s)) d(p, c(s h)) lim h0 d m (c(s h), c(s)) for s ¾ [0, a), for s ¾ (0, a]. 1 min{g Ú(Ú, Èc(s)) Ú ¾ G p (c(s))}, 1 max{g Ú(Ú, Èc(s)) Ú ¾ G p (c(s))} where Ï max{1, F( Èc(s))}. These are, of course, in [ 1, 1] (see [6, Lemma 2.2]). DEFINITION 2.3 (Forward triangles). For three distinct points p, x, y ¾ M, ( px, py) Ï (p, x, yá,, c) will denote the forward triangle consisting of unit speed minimal geodesic segments emanating from p to x, from p to y, and c from x to y. Then the corresponding

6 1148 K. KONDO Fig. 1. The forward angle. Fig. 2. The forward triangle. interior angles x, y at the vertices x, y are defined by x Ï (pc(0)c(a)), y Ï (pc(a)c(0)), respectively, where a Ï d(x, y). DEFINITION 2.4 (Comparison triangles). Fix a model surface of revolution ( ÉM, Ép). Given a forward triangle ( px, py)(p, x, yá,, c) M, a geodesic triangle (ÉpÉxÉy) ÉM is called its comparison triangle if Éd(Ép, Éx) d(p, x), É d(ép, Éy) d(p, y), É d(éx, Éy) L m (c) hold, where L m (c) Ê d(x,y) 0 max{f(èc), F( Èc)} ds.

7 SPHERE THEOREM IN FINSLER GEOMETRY 1149 There are many forward triangles admitting their comparison triangles, but TCT does not always hold for all of them: EXAMPLE 2.5 ([5]). For an even number q, let M be Ê 2 with the l q -norm. Then, M is Minkowskian. Take a forward triangle ( px, py) M, where p Ï (0, 0), x Ï (1, 0), y Ï (0, 1) ¾ M, and let c(t) Ï (1 t, t) denote the side of ( px, py) joining x to y. Assume that q is sufficiently large. Then, we observe that both angles x and y are nearly 0, respectively. We are able to think of (Ê 2, Ép) as a reference surface for M, because flag curvature K M 0. It is clear that ( px, py) admits its comparison triangle (ÉpÉxÉy) Ê 2. Since ( px, py) is nearly equilateral, (ÉpÉxÉy) is too. Hence, x Éx and y Éy hold. Therefore, TCT does not hold for the ( px, py) Modified TCTs. From Example 2.5, we understand that some strong conditions are demanded to establish a TCT in Finsler geometry. Taking this into account, we have the following: Theorem 2.6 ([6, Theorem 1.2]). Assume that (M, F, p) is a forward complete, connected C ½ -Finsler manifold whose radial flag curvature is bounded below by that of a von Mangoldt surface ( ÉM, Ép) satisfying f ¼ () 0 and G() 0 for unique ¾ (0, ½). Let ( px, py) (p, x, yá,, c) M be a forward triangle satisfying that, for some open neighborhood N (c) of c, (1) c([0, d(x, y)]) M Ò B (p); (2) g Ú (Û, Û) F(Û) 2 for all z ¾ N (c), Ú ¾ G p (z) and Û ¾ T z M; (3) T M (Ú, Û) 0 for all z ¾ N (c), Ú ¾ G p (z) and Û ¾ T z M, and the reverse curve Æc(s) Ï c(d(x, y) s) of c is also geodesic. If such ( px, py) admits a comparison triangle (ÉpÉxÉy) ÉM, then we have x Éx and y Éy. REMARK 2.7. The application of Theorem 2.6 will be referred to [7]. Here, we proved the finite topological type and a diffeomorphism theorem to Ê n. Corollary 2.8. Assume that (M, F, p) is a compact connected C ½ -Finsler manifold whose radial flag curvature is bounded below by 1. Let ( px, py) (p, x, yá,, c) M be a forward triangle satisfying that, for some open neighborhood N (c) of c, (1) c([0, d(x, y)]) M Ò B 2 (p); (2) g Ú (Û, Û) F(Û) 2 for all z ¾ N (c), Ú ¾ G p (z) and Û ¾ T z M; (3) T M (Ú, Û) 0 for all z ¾ N (c), Ú ¾ G p (z) and Û ¾ T z M, and the reverse curve Æc(s) Ï c(d(x, y) s) of c is also geodesic.

8 1150 K. KONDO Fig. 3. The forward triangle of TCT. If such ( px, py) admits a comparison triangle (ÉpÉxÉy) in (Ë 2, Ép), then we have x Éx and y Éy. Here, (Ë 2, Ép) denotes the unit sphere, i.e., its Riemannian metric dés 2 is expressed as dés 2 dt 2 f (t) 2 d 2, (t, ) ¾ (0, ) Ë 1 Ép, such that f (t) sin t. Proof. In Theorem 2.6, f ¼ (t) 0 on (, ½), because f ¼ () 0 and G() 0 for unique ¾ (0, ½). Hence, the corollary is immediate from Theorem 2.6, since f ¼ (t) cos t 0 on (2, ) and f ¼ (2) 0 for unique 2 ¾ (0, ). Lemma 2.9. Assume that (M, F, p) is a compact connected C ½ -Finsler manifold whose radial flag curvature is bounded below by 1. Let ( px, py)(p, x, yá,, c) M be a forward triangle satisfying that, for some open neighborhood N (c) of c, (1) c([0, d(x, y)]) B 2 (p)ò{p}; (2) F(Û) 2 g Ú (Û, Û) for all z ¾ N (c), Ú ¾ G p (z) and Û ¾ T z M; (3) T M (Ú, Û) 0 for all z ¾ N (c), Ú ¾ G p (z) and Û ¾ T z M, and the reverse curve Æc(s) Ï c(d(x, y) s) of c is also geodesic. If such ( px, py) admits a comparison triangle (ÉpÉxÉy) in (Ë 2, Ép), then we have x Éx and y Éy. Proof. Hence, one can prove this lemma by the almost similar argument as that in [6]. See Section 4 in this article for a detailed explanation of that. Set Ï max{f(û), F( Û)}. The assumption (2) yields 2 g Ú (Û, Û).

9 SPHERE THEOREM IN FINSLER GEOMETRY Proof of Theorem 1.1 Let (M, F, p) be the same as that in Theorem 1.1. Hence, our model surface as a reference surface is the unit sphere (Ë 2, Ép), i.e., its Riemannian metric dés 2 is expressed as dés 2 dt 2 f (t) 2 d 2, (t, ) ¾ (0, ) Ë 1 Ép, such that f (t) sin t. Lemma 3.1. The set Ë Ò 2 B t (Ép) is strictly convex for all t ¾ (2, ), i.e., for any distinct two points Éx, Éy ¾ B t (Ép) and minimal geodesic segment ÉcÏ [0, a] Ë 2 between them, we have Éc((0, a)) Ë Ò 2 B t (Ép), where a Ï d(éx, É Éy). Proof. Use the second variation formula. Hereafter, by the Bonnet Myers theorem ([1, Theorem 7.7.1]), we may assume, without loss of generality, (3.1) rad p. Lemma 3.2 (Key lemma). For any distinct two points x, y ¾ M Ò B 2 (p), then c([0, d(x, y)]) B 2 (p) holds for all minimal geodesic segments c emanating from x to y. In particular, the set M Ò B 2 (p) is convex. Suppose that c([0, d(x, y)]) B 2 (p) for some minimal geodesic Proof. segment c emanating from x to y. Then, we consider five cases: CASE 1: Assume that there exist s 0, s 1, s 2 ¾ [0, d(x, y)) with 0 s 0 s 1 s 2 such that c([s 0, s 1 )) M Ò B 2 (p), c([s 1, s 2 ]) B 2 (p). For sufficiently small 0 with s 1 s 0, take the forward triangle ( pc(s 0 ), pc(s 1 )) M. Note that c([s 0, s 1 ]) M Ò B 2 (p). Since d(p, c(s 0)) 2 and d(p, c(s 1 )) 2, we have, by the assumption and (3.1), that d(p, c(s 0 )) d(p, c(s 1 )) d m (c(s 0 ), c(s 1 )) L m (c) d(p, c(s 0 ))d(p, c(s 1 )), and hence ( pc(s 0 ), pc(s 1 )) admits a comparison triangle (Ép c(s0 ) c(s1 )) Ë 2.

10 1152 K. KONDO Fig. 4. The limit argument. By Corollary 2.8, we have (pc(s 1 )c(s 0 )) c(s1 ). It follows from [13, Proposition 2.1] (see Fig. 4 1 ) that 2 (pc(s 1 )c(s 0 )) lim 0 Set (Ép c(s0 ) c(s1 )) Ï lim 0 (Ép (pc(s1 )c(s 0 )) c(s1 ). c(s0 ) c(s1 )), and let ÉÏ [0, d( É c(s0 ), c(s1 ))] Ë 2 denote the side of (Ép c(s0 ) c(s1 )) joining c(s0 ) to c(s1 ). If c(s1 ) 2, then É([0, d( É c(s0 ), c(s1 ))]) B 2 (Ép) because B 2 (Ép) is geodesic. This contradicts d(ép, É c(s0 )) 2. If c(s1 ) 2, then there exists a ¾ (0, d( É c(s0 ), c(s1 ))) such that ¾ B 2 (Ép). This contradicts the structure of the cut locus of Ë 2 because (É(a)Ép c(s1 )) and B 2 (Ép) is geodesic. CASE 2: Assume that there exist s 3, s 4, s 5 ¾ (0, d(x, y)] with 0 s 3 s 4 s 5 such that c([s 3, s 4 ]) B 2 (p), c((s 4, s 5 ]) M Ò B 2 (p). 1 In the right picture of Fig. 4, the circle marks on the segments emanating from p to c(s 1 ) mean that the segments have the same length.

11 SPHERE THEOREM IN FINSLER GEOMETRY 1153 pc(s 4 ), Consider the forward triangle ( pc(s 5 )) M, where 0 is sufficiently small with s 5 s 4. Applying the similar limit argument in Case 1 to ( pc(s 4 ), pc(s 5 )), we have the triangle (Ép c(s4 ) c(s5 )) Ï lim 0 (Ép c(s4 ) c(s5 )) satisfying c(s4 ) 2. The angle condition yields the same contradiction as that in Case 1. CASE 3: Assume that there exist s 0,s 1,s 2 ¾ [0,d(x, y)] with s 0 s 1 s 2 such that c([0, d(x, y)]) B 2 (p) {c(s 1)}, c((s 0, s 1 )), c((s 1, s 2 )) M Ò B 2 (p). Then, we get a contradiction from the same argument as Case 1, or Case 2. CASE 4: Assume that there exist s 0, s 1 ¾ (0, d(x, y)) with s 0 s 1 such that and that c((s 0, s 1 )) B 2 (p)ò{p}, c(s 0), c(s 1 ) ¾ B 2 (p), (3.2) (pc(s0 )c(s 1 )) 2, (pc(s1 )c(s 0 )) 2. Take a subdivision r 0 Ï s 0 r 1 r k 1 r k s 1 of [s 0,s 1 ] such that ( pc(r i 1 ), pc(r i )) admits a comparison triangle É Ï(Ép i c(ri 1 ) c(ri ))Ë 2 for each i 1,2,,k. Applying Lemma 2.9 to ( pc(r i 1 ), pc(r i )), but for each i 2, 3,, k 1, we have (3.3) c(ri 1 ) (Ép c(ri 1 ) pc(r 0 ), For sufficiently small, Æ 0 with r 1 r 0 c(ri )), c(ri ) (Ép c(ri ) c(ri 1 )). and Æ r k r k 1, take two forward triangles ( pc(r 1 )), ( pc(r k 1 ), pc(r k Æ)) M. Note that these two triangles admit their comparison triangles É Ï (Ép c(r0 ) c(r1 )), É Æ Ï (Ép c(rk 1 ) c(rk Æ)) Ë 2, respectively. Without loss of generality, we may assume É 1 lim 0 É and É k lim Æ0 É Æ because lim 0 É and lim Æ0 É Æ are isometric to É 1 and É k, respectively. By Lemma 2.9, c(r0 ) (Ép c(r0 ) c(r1 )), c(r1 ) (Ép c(r1 ) c(r0 )), and that c(r k 1 ) (Ép c(rk 1 ) c(rk Æ)), c(rk Æ) (Ép c(rk Æ) c(rk 1 )). Hence, it follows from (3.2) and [13, Proposition 2.1] that (3.4) c(r0 ) (Ép c(r0 ) c(r1 )), c(r1 ) (Ép c(r1 ) c(r0 )), and that (3.5) 2 c(r 0 ) lim 0 c(rk 1 ) (Ép c(rk 1 ) c(rk )), 2 c(r k ) lim Æ0 c(rk Æ) (Ép c(rk ) c(rk 1 )).

12 1154 K. KONDO Fig. 5. The limit argument in (3.4). Starting from É 1, we inductively draw a geodesic triangle É i1 Ë 2 which is adjacent to É i c(r1 )) ( c(rk )). Since c(r i ) c(r i ) for each i 1, 2,, k 1, we obtain, by so as to have a common side Ép c(ri ), where 0 Ï ( c(r0 )) ( (3.3), (3.4), (3.5), (3.6) (Ép c(ri ) c(ri 1 ))(Ép c(ri ) c(ri1 )). Let Ï Ç [0, L m (c [s0,s 1 ])] Ë 2 denote the broken geodesic segment consisting of minimal c(ri 1 ) to c(ri ), i 1, 2,, k. Set (s) Ç Ï (t(ç (s)), ( (s))). Ç By (3.6), we have the unit speed, but not necessarily minimal at this moment, geodesic ÉÏ [0, a] Ë 2 emanating from c(r0 ) to c(rk ) and passing under ([0, Ç Lm (c [s0,s 1 ])]), i.e., (É) ¾ [0, ( c(rk ))] on [0, a] and t(ç (s)) t(é(u)) for all (s, u) ¾ (0, L m (c [s0,s 1 ])) (0, a) with (Ç (s)) (É(u)) (see Fig. 6). Since a L m (c [s0,s 1 ]) by the assumption and (3.1), É is minimal with (É(0) Ép É(a)). This contradicts the structure of the cut locus of Ë 2 because B 2 (Ép) is geodesic. CASE 5: Assume that c is passing through p. Take a sequence {c i Ï [0, l i ] geodesic segments from M Ò {p}} i¾æ of minimal geodesic segments c i emanating from x c i (0) convergent to c. Applying the same argument as that in Case 4 to each c i for sufficiently large i, we get a contradiction. Note that lim i½ L m (c i ) L m (c) rad p, but x, y ¾ M Ò B 2 (p).

13 SPHERE THEOREM IN FINSLER GEOMETRY 1155 Fig. 6. The segment É. Therefore, c([0, d(x, y)]) B 2 (p) holds for all minimal geodesic segments c emanating from x to y. The second assertion is clear from the first assertion. Corollary 3.3. Let ( px, py) (p, x, yá,, c) M be a forward triangle for x ¾ M Ò B 2 (p) and y ¾ B 2 (p) satisfying {z} Ï c((0, a)) B 2 (p), where a Ï d(x, y). Then, the forward triangle ( px, pz) M admits its comparison triangle (ÉpÉxÉz) Ë 2 such that x Éx and z Éz. Additionally, if the forward triangle ( pz, py) M admits its comparison triangle (ÉpÉzÉy) Ë 2, then z Éz and y Éy. Proof. triangles. Apply the same limit argument in the proof of Lemma 3.2 to forward Lemma 3.4. The function d(p, ) attains its maximum at a unique point q ¾ M. In particular, M Ò B 2 (p) is a topological disk.

14 1156 K. KONDO Proof. Suppose that there exist two distinct points x, y ¾ B rad p (p). Then, the forward triangle ( px, py) M admits its comparison triangle (ÉpÉxÉy) Ë 2. Let cï [0, d(x, y)] M and ÉcÏ [0, L m (c)] Ë 2 be sides of ( px, py) and (ÉpÉxÉy) emanating from x to y and from Éx to Éy, respectively. By Corollary 2.8 and Lemma 3.1, d(p, c(s)) rad p holds for all s ¾ (0, d(x, y)). This contradicts the definition of rad p. The second assertion follows from the uniqueness and Lemma 3.2. DEFINITION 3.5. We say that a point x ¾ M is a ( forward) critical point for p ¾ M if, for every Û ¾ T x MÒ{0}, there exists Ú ¾G p (x) such that g Ú (Ú, Û) 0. Here see (1.1) for the definition of G p (x). By similar arguments to the Riemannian case, we have Gromov s isotopy lemma [2]: Lemma 3.6. Given 0 r 1 r 2 ½, if B r 2 (p)ò B r 1 (p) has no critical point for p ¾ M, then B r 2 (p)ò B r 1 (p) is homeomorphic to B r 1 (p) [r 1, r 2 ]. Lemma 3.7. B 2 (p) is a topological disk. Proof. There are no critical point for p in B 2 (p) Ò {p}. In particular, Since M Ò B 2 (p) is convex (Lemma 3.2), B 2 (p) has no critical point for p. Suppose that there exists a critical point x ¾ B 2 (p) Ò {p} for p. Let q ¾ M be the same as that in Lemma 3.4 such that d(p, q) rad p, and cï [0, a] M a unit speed minimal geodesic segment emanating from q to x, where a Ï d(q, x). Then, c([0, a]) B 2 (p). From the cases in the proof of Lemma 3.2, it is sufficient to consider the case where c((0, a)) B 2 (p) is one point, say {q 1 } Ï c((0, a)) B 2 (p). Since both q c(0) and x c(a) are critical points for p, we have (3.7) (pc(0)c(a)) 2, (pc(a)c(0)) 2. Note that c does not pass through p, because, by the definition of critical points, there exist at least two minimal segments emanating from p to x and c is minimal. Now, take a subdivision s 0 Ï 0 s 1 s k 1 s k Ï a of [0, a] such that c(s 1 ) q 1 ¾ B 2 (p) and that ( pc(s i 1 ), pc(s i )) admits a comparison triangle É Ï (Ép i c(si 1 ) c(si )) Ë 2 for each i 2, 3,, k. By Corollary 3.3, ( pc(s 0 ), pc(s 1 )) admits its a comparison triangle É Ï (Ép 1 c(s0 ) c(s1 )) Ë 2 satisfying (3.8) c(s0 ) (Ép c(s0 ) c(s1 )), c(s1 ) (Ép c(s1 ) c(s0 )).

15 Moreover, by Corollary 3.3 again, we have (3.9) c(s1 ) (Ép c(s1 ) c(s2 )), SPHERE THEOREM IN FINSLER GEOMETRY 1157 c(s2 ) (Ép c(s2 ) c(s1 )). In particular, by (3.7) and (3.8), we have (3.10) (Ép c(s0 ) c(s1 )) 2. Applying Lemma 2.9 to ( pc(s i 1 ), pc(s i )) for each i 3, 4,, k, (3.11) c(si 1 ) (Ép c(si 1 ) c(si )), c(si ) (Ép c(si ) In particular, by (3.7) and (3.11), we have (3.12) (Ép c(sk ) c(sk 1 )) 2. c(si 1 )). Starting from É 1, we inductively draw a geodesic triangle É i1 Ë 2 which is adjacent c(s1 )) c(sk )). Since c(s i ) c(s i ) for each i 1, 2,, k 1, we obtain, by (3.11), to É i so as to have a common side Ép c(si ), where 0 Ï ( c(s0 )) ( ( (3.8), (3.9), (3.13) (Ép c(si ) c(si 1 ))(Ép c(si ) c(si1 )). desic segments from c(s k ), respectively. Since É lives under ([0, Ç Lm (c)]), we have, by (3.12) and (3.10), Let Ï Ç [0, L m (c)] Ë 2 denote the broken geodesic segment consisting of minimal geo- c(si 1 ) to c(si ), i 1, 2,, k. Set (r) Ç Ï (t(ç (r)), ( (r))). Ç By (3.13), we have the unit speed geodesic ÉÏ [0, b] Ë 2 emanating from É(0) c(s0 ) to É(b) c(sk ) and passing under ([0, Ç Lm (c)]), i.e., (É) ¾ [0, ( c(sk ))] on [0, b] and t(ç (r)) t(é(u)) for all (r, u) ¾ (0, L m (c)) (0, b) with (Ç (r)) (É(u)). Since b L m (c) by (3.1), É is minimal with ÈÉ (0), ÈÉ (0), where É and É denote minimal geodesic segments (i.e., sub-meridians) emanating from Ép to c(s0 ) and from p to (3.14) ÈÉ(0), ÈÉ (É d(ép, c(s0 ))) 2, ÈÉ(b), ÈÉ (É d(ép, c(sk ))) 2. Since d(ép, É c(s0 )) 2 d(ép, É c(sk )), there exists b 0 ¾ (0,b) such that É(b 0 )¾ B 2 (Ép). Let ÆÏ [0, ] Ë 2 be an extension of É to the antipodal point c(s0 ) to c(s0 ) É(0), and set Æ(u) Ï (t(u), (u)). Since Ë 2 Ò B u (Æ(0)) is strictly convex for all u ¾ (2, ) (by the same proof of Lemma 3.1), (ÈÆ(rad p ), (t) Æ(radp )) 2 holds. This implies (3.15) t ¼ (rad p ) 0,

16 1158 K. KONDO where note that Æ emanates from c(s0 ) to c(s0 ). Since Æ((b 0, rad p )) B 2 (Ép) and f ¼ (t) cos t 0 on (0, 2), it follows from [12, (7.1.15)] that (3.16) t ¼¼ (u) f (t(u)) f ¼ (t(u)) ¼ (t(u)) 2 0 holds on (b 0, rad p ). Hence, by (3.15) and (3.16), t(u) is decreasing on [b 0, rad p ]. Since b ¾ (b 0, rad p ), ÈÉ(b), ÈÉ (t(b)) 2. This contradicts the right inequality in (3.14). Therefore, B 2 (p)ò{p} has no critical point for p. By Lemma 3.6, B 2 (p) is a topological disk. By Lemmas 3.4 and 3.7, M is homeomorphic to Ë n. 4. Appendix: Proof of Lemma 2.9 Let (M, F, p) be a forward complete, connected C ½ -Finsler manifold with a base point p ¾ M, and let d denote its distance function and Cut(p) the cut locus of p. Set B r (x) Ï {y ¾ M d(y, x) r}. Take a point q ¾ M Ò (Cut(p){p}) and small r 0 such that B (q)(cut(p){p}) 2r and that (q)ï (q) B r B r B r (q) is geodesically convex (i.e., any minimal geodesic joining two points in B r (q) is contained in B r (q)). Given a unit speed minimal geodesic segment cï (, ) B r (q), we consider the C ½ -variation ³(t, s) Ï exp p t l exp 1 p (c(s)), (t, s) ¾ [0, l] (, ), where l Ï d(p, c(0)). Since x Ï c(0) Cut(p), there is a unique minimal geodesic segment Ï [0, l] M emanating from p to x. By setting J(t) Ï (³s)(t, 0), we get the Jacobi field J along with J(0) 0 and J(l) Èc(0). Note that J(t) 0 on (0, l] from the minimality of, and that (4.1) J (t) Ï J(t) g È (l)(è (l), Èc(0)) t È (t), l t ¾ [0, l], is the g È -orthogonal component J (t) to È (t) (see [6, Lemma 3.2]). Moreover, since is unique, it follows from the proof of [6, Lemma 2.2] that (4.2) cos (pxc()) cos (pxc( )) 1 g È (l) (È (l), Èc(0)), where Ï max{1, F( Èc(0))}. Hence, (pxc()) (pxc( )). In the following discussion, we set (4.3) Ï (pxc()) (pxc( )).

17 SPHERE THEOREM IN FINSLER GEOMETRY 1159 Hereafter, we assume that the radial flag curvature of (M, F, p) is bounded below by 1. Hence, its model surface is the unit sphere (Ë 2, Ép) with its metric dés 2 dt 2 f (t) 2 d 2, (t, ) ¾ (0, ) Ë 1 Ép, such that f (t) sin t. For small Æ 0 with Æ 1, we set f Æ (t) Ï Ô Ô 1 sin( 1 Æt) 1 Æ on [0,Ô 1 Æ]. Then, f Æ satisfies f ¼¼ Æ (1 Æ) f Æ 0 with f Æ (0)0, f ¼ Æ (0)1. Thus, we have a new sphere (Ë 2 Æ,Éo) with the metric dés 2 Æ dt 2 f Æ (t) 2 d 2 on (0,Ô 1 Æ ) Ë 1 Éo. Since the curvature 1 Æ of (Ë2, Éo) is less than 1, we may also employ Æ (Ë2, Éo) as Æ a reference surface for M. Let c, x c(0), and l d(p, x) be the same in the above. Fix a point Éx ¾ Ë É 2 with Æ d Æ (Éo, Éx) l, where d É Æ denotes the distance function induced from dés 2. Let Æ be the minimal geodesic segment from Éo to Éx, and take a unit parallel É Ï [0, l] Ë 2 Æ vector field E É along É orthogonal to ÈÉ. Define the Jacobi field X É along É by (4.4) É X(t) Ï 1 f Æ (l) f Æ(t) É E(t). Lemma 4.1 ([6, Lemma 3.4]). For any Jacobi field X along which is g È - orthogonal to È and satisfies X(0) 0 and g È (l) (X(l), X(l)) 1, we have ÉI l ( É X, É X) I l (X, X) Æ f Æ (l) 2 l 0 f Æ (t) 2 dt. Here, I l and É Il denote the index forms with respect to [0,l] and É [0,l], respectively. Fix a geodesic ÉcÏ (, ) Ë 2 Æ with Éc(0) Éx such that ÈÉ (l), ÈÉc(0), ÈÉc Ï max{1, F( Èc(0))}, where is as that in (4.3). Consider the geodesic variation ɳ(t, s) Ï exp Éo t l exp 1 Éo (Éc(s)), (t, s) ¾ [0, l] (, ). By setting J(t) É Ï (ɳs)(t, 0), we get the Jacobi field J É along É with J(0) É 0 and ÉJ(l) ÈÉc(0). And the Jacobi field ÉJ (t) Ï J(t) É Å «ÈÉ (l), ÈÉc(0) along É is orthogonal to ÈÉ (t) on [0, l]. l t ÈÉ (t)

18 1160 K. KONDO Lemma 4.2. Assume that (1) B 2r (q) B 2 (p); (2) F(Ú) 2 g È (l) (Ú, Ú) for all Ú ¾ T x M. If ¾ (0, ), then there exists Æ 1 Ï Æ 1 ( f, r) 0 such that, for any Æ ¾ (0, Æ 1 ), ÉI l ( É J, É J ) I l (J, J ) ÆC 1 g È (l) (J (l), J (l)) 0 holds, where C 1 Ï 1(2 f (l 0 ) 2 ) Ê l 0 0 f (t) 2 dt and l 0 Ï d(p, q). Proof. By the assumption (2) in this lemma, (4.5) 2 g È (l) (Èc(0), Èc(0)). Indeed, (4.5) is immediate in the case where 1. If F( Èc(0)), then Èc(0) 1 g È (l) F( Èc(0)), Èc(0) F( Èc(0)) 1 F( Èc(0)) g 2 È (l)(èc(0), Èc(0)). By (4.2) and (4.3), g È (l) (È (l), Èc(0)) cos. Then, J É (l) sin X(l) É holds, where X É is the same as that in (4.4). Since both J É and X É are Jacobi fields on Ë 2, Æ ÉJ (t) sin X(t) É on [0, l]. Hence (4.6) É Il ( É J, É J ) ( sin ) 2 É Il ( É X, É X). On the other hand, it follows from (4.1) and (4.5) that g È (l) (J (l), J (l)) g È (l) (Èc(0), Èc(0)) ( cos ) 2 ( sin ) 2. Then, we get a constant a Ï ( sin ) 2 g È (l) (J (l), J (l)) 0. Since g È (l) (J (l), J (l)) 0 for ¾ (0, ), we have, by Lemma 4.1, hence (4.7) ÉI l ( É X, É X) I l (J, J ) g È (l) (J (l), J (l)) Æ f Æ (l) 2 l I l (J, J ) g È (l) (J (l), J (l)) ÉIl ( X, É X) É Æ l f Æ (l) 2 0 By (4.6) and (4.7), 0 f Æ (t) 2 dt, {a ( sin ) 2 }É Il ( É X, É X) Æ g È (l)(j (l), J (l)) f Æ (l) 2 ÉI l ( É J, É J ) I l (J, J ) a É Il ( É X, É X) Æ g È (l)(j (l), J (l)) Æ g È (l)(j (l), J (l)) f Æ (l) 2 f Æ (l) 2 l 0 f Æ (t) 2, f Æ (t) 2 dt l 0 l 0 f Æ (t) 2 dt. f Æ (t) 2

19 SPHERE THEOREM IN FINSLER GEOMETRY 1161 where note that a 0, and that É Il ( É X, É X) Ô 1 Æ tan( Ô 1 Æl) 0, because l 2 2 Ô 1 Æ by the assumption (1) in this lemma. Since l l 0 max{d(q, x), d(x, q)} r, and since l, l 0 2 (from the (1)), taking smaller Æ 1 ( f, r) 0 if necessary, we get the desired assertion in this lemma for all Æ ¾ (0, Æ 1 ). Lemma 4.3. Assume that (1) B 2r (q) B 2 (p); (2) F(Ú) 2 g È (l) (Ú, Ú) for all Ú ¾ T x M; (3) T M (È (l), Èc(0)) 0. For each Æ ¾ (0, Æ 1 ), ¾ (0, 2), if ¾ [, ], then there exists ¼ Ï ¼ (M, l, f,, Æ, ) ¾ (0, ) such that L(s) É L(s) holds for all s ¾ [ ¼, ¼ ]. Here, L(s) Ï d(p, c(s)) and É L(s) Ï É d Æ (Éo, Éc(s)). Proof. We will state the outline of the proof, since the proof is very similar to [6, Lemma 3.6] thanks to Lemma 4.2. Set R(s) Ï L(s) {L(0) L ¼ (0)sL ¼¼ (0)s 2 2}. Then, there exists C 2 Ï C 2 (M, l) 0 such that L(s) L(0) L ¼ (0)s 1 2 L¼¼ (0)s 2 R(s) l s cos s2 2 I l(j, J )C 2 s 3. Note that L ¼ (0) cos and L ¼¼ (0) I l (J, J ) hold by [6, Lemma 3.3], (4.2), (4.3), and the assumption (3) in this lemma. Similarly, ÉL(s) l s cos s2 2 É I l ( É J, É J ) C 3 s 3 for some C 3 Ï C 3 ( f, l) 0 and all s ¾ (, ). Since g È (l) (J (l), J (l)) 0 for all ¾ [, ], there exists C 4 Ï C 4 (M, ) 0 such that g È (l) (J (l), J (l)) C 4 0. From Lemma 4.2, É L(s) L(s) s 2 {ÆC 1 C 4 2(C 2 C 3 )s}2 holds. Therefore, we get L(s) É L(s) for all s ¾ [ ¼, ¼ ], if ¼ Ï min{, ÆC 1 C 4 2(C 2 C 3 )}. Thanks to Lemma 4.3 and the structure of Ë 2 Æ, we may prove Lemma 2.9 by the same arguments in Sections 4, 5, and 6 in [6]. REMARK 4.4. Although we do not consider cases of 0, or in Lemma 4.3, Lemma 2.9 holds in cases of x, y 0, or x 0, y because the reverse curve Æc of the geodesic segment c is geodesic. ACKNOWLEDGEMENTS. discussions. I would like to thank Professor M. Tanaka for helpful

20 1162 K. KONDO References [1] D. Bao, S.-S. Chern and Z. Shen: An Introduction to Riemann Finsler Geometry, Springer, New York, [2] M. Gromov: Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), [3] K. Grove and K. Shiohama: A generalized sphere theorem, Ann. of Math. (2) 106 (1977), [4] K. Kondo: Radius sphere theorems for compact manifolds with radial curvature bounded below, Tokyo J. Math. 30 (2007), [5] K. Kondo and S. Ohta: Private communications, (2012). [6] K. Kondo, S. Ohta and M. Tanaka: A Toponogov type triangle comparison theorem in Finsler geometry, preprint (2012), arxiv: [7] K. Kondo, S. Ohta and M. Tanaka: Topology of complete Finsler manifolds with radial flag curvature bounded below, Kyushu J. Math. 68 (2014), [8] K. Kondo and M. Tanaka: Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below, I, Math. Ann. 351 (2011), [9] H.-B. Rademacher: A sphere theorem for non-reversible Finsler metrics, Math. Ann. 328 (2004), [10] Z. Shen: Volume comparison and its applications in Riemann Finsler geometry, Adv. Math. 128 (1997), [11] Z. Shen: Lectures on Finsler Geometry, World Sci. Publishing, Singapore, [12] K. Shiohama, T. Shioya and M. Tanaka: The Geometry of Total Curvature on Complete Open Surfaces, Cambridge Tracts in Mathematics 159, Cambridge Univ. Press, Cambridge, [13] M. Tanaka and S.V. Sabau: The cut locus and distance function from a closed subset of a Finsler manifold, preprint (2012), arxiv: Department of Mathematical Science Yamaguchi University Yamaguchi City, Yamaguchi Pref Japan keikondo@yamaguchi-u.ac.jp