Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. III
aa r X i v : . [ m a t h . DG ] F e b Total Curvatures of Model Surfaces ControlTopology of Complete Open Manifolds withRadial Curvature Bounded Below. III ∗† Kei KONDO · Minoru TANAKA
Dedicated to Professor K. Shiohama on the occasion of his seventieth birthday.
Abstract
This article is the third in a series of our investigation on a complete non-compact connected Riemannian manifold M . In the first series [KT1], we showedthat all Busemann functions on an M which is not less curved than a von Mangoldtsurface of revolution f M are exhaustions, if the total curvature of f M is greater than π . A von Mangoldt surface of revolution is, by definition, a complete surface ofrevolution homeomorphic to R whose Gaussian curvature is non-increasing alongeach meridian. Our purpose of this series is to generalize the main theorem in [KT1]to an M which is not less curved than a more general surface of revolution. The Gauss–Bonnet theorem says that the total curvature c ( S ) of a compact Riemannian2-dimensional manifold S is a topological invariant, i.e., c ( S ) = 2 πχ ( S ) . Here χ ( S ) denotes the Euler characteristic of S .In 1935, Cohn -Vossen generalized the Gauss–Bonnet theorem for complete non-compact Riemannian 2-dimensional manifolds as follows :
Theorem 1.1 ([CV1, Satz 6])
If a connected, complete non-compact, finitely connectedRiemannian -manifold M admits a total curvature c ( M ) , then, c ( M ) ≤ πχ ( M ) holds. Here χ ( M ) denotes the Euler characteristic of M . ∗ Mathematics Subject Classification (2000) : 53C20, 53C21. † Keywords : Busemann function, radial curvature, total curvature c ( M ) is not a topological invariant anymore. But 2 πχ ( M ) − c ( M ) is a geometric invariant depending only on the ends of M , which is a consequencefrom the isoperimetric inequalities (see [SST, Theorem 5.2.1]).In 1984, Shiohama proved the next result peculiar to geometry of total curvature onsurfaces : Theorem 1.2 ([S, Main Theorem])
Let M be a connected, complete non-compact, finitelyconnected and oriented Riemannian -manifold with one end. If the total curvature c ( M ) satisfies c ( M ) > (2 χ ( M ) − π, then all Busemann functions on M are exhaustions. In particular, if the total curvatureof M is greater than π , then M is homeomorphic to R and also all Busemann functionsare exhaustions. Here the
Busemann function F γ : M −→ R of a ray γ in a complete non-compactRiemannian (any dimensional) manifold M is, by definition, F γ ( x ) := lim t →∞ { t − d ( x, γ ( t )) } , and a function ϕ : M −→ R is called an exhaustion, if ϕ − ( −∞ , a ] is compact for all a ∈ R .Theorem 1.2 was generalized to higher-dimensional manifolds in [KT1]. Roughlyspeaking, it was proved in [KT1] that all Busemann functions on a complete non-compactconnected Riemannian manifold not less curved than a von Mangoldt surface of revolution f M are exhaustions, if the total curvature of f M is greater than π (The theorem will belater stated in full detail as Theorem 1.4 in this article).A von Mangoldt surface of revolution is, by definition, a complete surface of revolutionhomeomorphic to R whose Gaussian curvature is non-increasing along each meridian.The monotonicity of the Gaussian curvature of a von Mangoldt surface of revolution looksrestrictive, but very familiar surfaces such as a paraboloid or a 2-sheeted hyperboloid arevon Mangoldt surfaces of revolution.Although Cohn -Vossen restricted himself to 2-dimensional manifolds, he has devel-oped fundamental techniques, such as drawing a circle or a geodesic polygon, and joiningtwo points by a minimal geodesic segment, to investigate the structures of complete Rie-mannian 2-dimensional manifolds. We, Riemannian geometers, should be awed by thefact that such techniques are ever now not only useful, but also powerful for investigatingthe topology of any dimensional complete Riemannian manifolds.Furthermore, as pointed out in the preface of [SST], it took more than thirty years toobtain higher-dimensional extensions of Cohn -Vossen’s results for complete non-compactRiemannian 2-dimensional manifolds. They are the splitting theorem by Toponogov [To],the structure theorem with positive sectional curvature by Gromoll and Meyer [GM], andthe soul theorem with non-negative sectional curvature by Cheeger and Gromoll [CG].2ence, it requires many years and is also very difficult to generalize some results peculiarto geometry of surfaces to any dimensional complete Riemannian manifolds. In fact, onemay find such results in [SST], which have not been generalized in higher dimensions yet.Our purpose of this article is to generalize the main theorem in [KT1] to a completenon-compact connected Riemannian manifold not less curved than a more general surfaceof revolution. To state this precisely, we will begin on the definition of a non-compactmodel surface of revolution.Let f M denote a complete 2-dimensional Riemannian manifold homeomorphic to R with a base point ˜ p ∈ f M . Then, we call the pair ( f M , ˜ p ) a non-compact model surface ofrevolution if its Riemannian metric d ˜ s is expressed in terms of geodesic polar coordinatesaround ˜ p as d ˜ s = dt + f ( t ) dθ , ( t, θ ) ∈ (0 , ∞ ) × S p. (1.1)Here f : (0 , ∞ ) −→ R is a positive smooth function which is extensible to a smooth oddfunction around 0, and S p := { v ∈ T ˜ p f M | k v k = 1 } . The function G ◦ ˜ γ : [0 , ∞ ) −→ R is called the radial curvature function of ( f M , ˜ p ), where we denote by G the Gaussiancurvature of f M , and by ˜ γ any meridian emanating from ˜ p = ˜ γ (0). Remark that f satisfies the differential equation f ′′ ( t ) + G (˜ γ ( t )) f ( t ) = 0with initial conditions f (0) = 0 and f ′ (0) = 1. For each constant number δ >
0, a sector e V ( δ ) ⊂ f M is defined by e V ( δ ) := n ˜ x ∈ f M | < θ (˜ x ) < δ o . Notice that the n -dimensional model surfaces of revolution are defined similarly, and theyare completely classified in [KK].The total curvature c ( f M ) of ( f M , ˜ p ) is formally defined as the improper integral, i.e., c ( f M ) := Z f M G + ◦ t d f M + Z f M G − ◦ t d f M if Z f M G + ◦ t d f M < ∞ , or Z f M G − ◦ t d f M > −∞ . Here we set G + ( t ) := max { G (˜ γ ( t )) , } = G + | G | G − ( t ) := min { G (˜ γ ( t )) , } = G − | G | . Notice that G = G + ◦ t + G − ◦ t . If c ( f M ) exists, c ( f M ) = 2 π (1 − lim t →∞ f ′ ( t )) holds, since d f M = f dtdθ and f ′ (0) = 1. By Theorem 1.1, c ( f M ) ≤ π c ( f M ) > −∞ means that f M admits a finite total curvature (if c ( f M ) exists).Let ( M, p ) be a complete non-compact n -dimensional Riemannian manifold with abase point p ∈ M . We say that ( M, p ) has radial curvature at the base point p boundedfrom below by that of a non-compact model surface of revolution ( f M , ˜ p ) if, along every unitspeed minimal geodesic γ : [0 , a ) −→ M emanating from p = γ (0), its sectional curvature K M satisfies K M ( σ t ) ≥ G (˜ γ ( t ))for all t ∈ [0 , a ) and all 2-dimensional linear spaces σ t spanned by γ ′ ( t ) and a tangentvector to M at γ ( t ). Notice that, if the Riemannian metric of f M is dt + t dθ , or dt + sinh t dθ , then G (˜ γ ( t )) = 0, or G (˜ γ ( t )) = −
1, respectively.For this definition, the radial curvature geometry looks artificial, but this is not thecase , i.e., we can construct a model surface of revolution for any complete Riemannianmanifold with an arbitrary given point as a base point (see [KT2, Lemma 5.1]). Theexistence of a ( f M , ˜ p ) is therefore very natural on the above definition.Now, we are in a point where we will state our main theorem : Let R M denote the setof all rays on M and R p the set of all rays emanating from p . Moreover, for each γ ∈ R M ,let Π( γ ) denote the set of all α ∈ R p which is a limit ray of the sequence of minimalgeodesic segments joining p to γ ( t i ) for some divergent sequence { t i } . Hence, α ∈ Π( γ ) isan asymptotic ray to γ emanating from p . Notice that Π( γ ) = { γ } , if γ ∈ R p .We set A p := { γ ′ (0) ∈ S n − p | γ ∈ R p } , where S n − p := { v ∈ T p M | k v k = 1 } , and denote by diam( A p ) the diameter of A p . Asubset S of A p is said to be a δ -covering of A p , if A p ⊂ [ v ∈ S B δ ( v ) , where B δ ( v ) := (cid:8) w ∈ S n − p | ∠ ( v, w ) ≤ δ (cid:9) . Main Theorem
Let ( M, p ) be a complete non-compact connected Riemannian n -manifold M whose radial curvature at the base point p is bounded from below by that of a non-compact model surface of revolution ( f M , ˜ p ) . Assume that (MT–1) c ( f M ) > π , and (MT–2) f M has no pair of cut points in a sector e V ( δ ) for some δ ∈ (0 , π ] .Then, for any γ , γ , . . . , γ k ∈ R M such that { α ′ (0) ∈ S n − p | α ∈ S ki =1 Π( γ i ) } is a δ -covering of A p , max { F γ i | i = 1 , , . . . , k } is an exhaustion. Moreover, if diam( A p ) ≤ δ , then F γ is an exhaustion for all γ ∈ R M . f M is non-negative everywhere. In fact, the model surface in [KT1, Example 1.2] satisfies bothproperties (MT–1) and (MT–2), but lim t →∞ G ◦ ˜ γ ( t ) = −∞ for each meridian ˜ γ .If a non-compact model surface of revolution f M admits a finite total curvature, then,for each ε >
0, there exists a compact subset e K ε of f M such that Z f M \ e K ε | G | d f M < ε . Hence, we might conjecture that the Gaussian curvature of f M should be almost flatoutside of a compact subset of f M . The following theorem shows that this conjecture is false and that the radial curvature function G ( t ) may change signs wildly. Theorem 1.3 ([TK])
Let ( f M , ˜ p ) be a non-compact model surface of revolution with itsmetric (1.1). If f M admits −∞ < c ( f M ) < π, then, for any ε > , there exists a model surface of revolution ( c M , b p ) with its metric b g = dt + m ( t ) dθ , ( t, θ ) ∈ (0 , ∞ ) × S b p, satisfying the differential equation m ′′ ( t ) + b G ( t ) m ( t ) = 0 with initial conditions m (0) = 0 and m ′ (0) = 1 , and admitting a finite total curvature c ( c M ) such that (1) (cid:13)(cid:13)(cid:13) G (˜ γ ( t )) − b G ( t ) (cid:13)(cid:13)(cid:13) L ≤ ε , (2) c ( f M ) ≥ c ( c M ) ≥ c ( f M ) − ε (respectively c ( f M ) + ε ≥ c ( c M ) ≥ c ( f M ) ), (3) G (˜ γ ( t )) ≥ b G ( t ) (respectively b G ( t ) ≥ G (˜ γ ( t )) ) on [0 , ∞ ) , and (4) lim inf t →∞ b G ( t ) = −∞ (respectively lim sup t →∞ b G ( t ) = ∞ ). The property (MT–2) is satisfied by a von Mangoldt surface of revolution, i.e., e V ( π )has no pair of cut points. In fact, it was proved in [T] that the cut locus of a point on a vonMangoldt surface of revolution is empty or a subray of the meridian opposite to the point.The assumption (MT–2) is not strong . For example, consider a non-compact modelsurface of revolution whose radial curvature function is non-increasing (or non-positive)along a subray of a meridian. If the surface admits a finite total curvature, then thesurface admits a sector which has no pair of cut points (see [KT2, Sector Theorem]). Wedo not know if (MT–2) can be removed from Main Theorem or not.Since it is clear that diam( A p ) ≤ π , as a corollary to Main Theorem, we get Theorem 1.4 ([KT1, Main Theorem])
Let ( M, p ) be a complete non-compact Rieman-nian n -manifold M whose radial curvature at the base point p is bounded from below bythat of a non-compact von Mangoldt surface of revolution ( M ∗ , p ∗ ) . If c ( M ∗ ) > π , thenall Busemann functions on M are exhaustions. π/ Acknowledgements.
The first named author would like to express to Professor S. Ohtahis deepest gratitude for his helpful comments on the first version of our main theoremin the differential topology seminar at Kyoto university, 14th July, 2009.
This section is set up as a preliminary to the proof of Main Theorem (Theorem 3.6) in thenext section. Throughout this section, let ( f M , ˜ p ) denote a non-compact model surface ofrevolution which admits a total curvature c ( f M ) > π . Lemma 2.1
There exists a positive number r such that Z V G d f M > π + 2Λ holds for all open set V ⊂ f M containing B r (˜ p ) as a subset. Here we set Λ := c ( f M ) − π . Proof.
Since c ( f M ) is finite, for each positive number ε , there exists a positive number r ε such that Z f M \ B rε (˜ p ) | G | d f M < ε holds. In particular, for ε := Λ , there exists a positive number r such that Z f M \ B r (˜ p ) | G | d f M < Λ (2.1)Let V ⊂ f M be an open set containing B r (˜ p ) as subset. It is clear that Z V G d f M ≥ c ( f M ) − Z f M \ V | G | d f M ≥ c ( f M ) − Z f M \ B r (˜ p ) | G | d f M (2.2)By (2.1) and (2.2), we get Z V G d f M > π + 2Λ . ✷ c ( f M ) > π , it follows from Cohn -Vossen’s theorem [CV2, Satz 5] that f M has nostraight line. Thus, by [SST, Lemma 6.1.1], the next lemma is clear : Lemma 2.2
There exists a number r > r such that no ray emanating from a point in f M \ B r (˜ p ) passes through B r (˜ p ) . Lemma 2.3
For each ˜ q ∈ f M \ B r (˜ p ) , there exists a number r > r such that, for any ˜ x ∈ f M \ B r (˜ p ) , ∠ (˜ p ˜ q ˜ x ) ≥ π . Here ∠ (˜ p ˜ q ˜ x ) denotes the angle at the vertex ˜ q of the geodesic triangle △ (˜ p ˜ q ˜ x ) .Proof. Take any point ˜ q ∈ f M \ B r (˜ p ) and fix it. Let V ˜ q denote the connected componentof f M \ [ e γ ∈R ˜ q e γ ([0 , ∞ ))containing B r (˜ p ), where R ˜ q denotes the set of all rays emanating from ˜ q . Notice thatthe existence of V ˜ q is guaranteed by Lemma 2.2, and that the boundary ∂V ˜ q consists oftwo rays e α + , e α − ∈ R ˜ q , which might be the same. From Lemma 2.1, c ( V ˜ q ) := Z V ˜ q G d f M > π + 2Λ holds. On the other hand, since V ˜ q does not admit a ray in R ˜ q , it follows from [SST,Lemma 6.1.3] that c ( V ˜ q ) equals the interior angle at ˜ q of V ˜ q . Hence, the interior angle at˜ q of V ˜ q is greater than π . Therefore, we get ∠ ( e α ′ + (0) , e α ′− (0)) = 2 π − c ( V ˜ q ) < π − . Since V ˜ q does not admit a ray in R ˜ q and e α + , e α − are symmetric under the reflection withrespect to the meridian µ ˜ q passing through ˜ q ,max { ∠ ( e γ ′ (0) , µ ′ ˜ q ( d (˜ p, ˜ q ))) | e γ ∈ R ˜ q } = ∠ ( e α ′ + (0) , µ ′ ˜ q ( d (˜ p, ˜ q )))= ∠ ( e α ′− (0) , µ ′ ˜ q ( d (˜ p, ˜ q ))) < π − Λ . (2.3)In particular, by (2.3), ∠ ( e γ ′ (0) , µ ′ ˜ q ( d (˜ p, ˜ q ))) < π − Λ holds for all e γ ∈ R ˜ q .Let e α : [0 , d (˜ q, ˜ x )] −→ f M denote a minimal geodesic segment joining ˜ q to a point ˜ x ∈ f M . If d (˜ q, ˜ x ) is sufficient large, then e α ′ (0) is close to some e γ ′ (0), e γ ∈ R ˜ q . Therefore, thereexists a number r > r such that, for any minimal geodesic segment e α : [0 , d (˜ q, ˜ x )] −→ f M joining ˜ q to ˜ x with d (˜ q, ˜ x ) > r , ∠ ( e α ′ (0) , µ ′ ˜ q ( d (˜ p, ˜ q ))) < π − Λ . (2.4)7he equation (2.4) implies that ∠ (˜ p ˜ q ˜ x ) ≥ π for all ˜ x ∈ f M \ B r (˜ p ). ✷ Our purpose of this section is to prove Main Theorem (Theorem 3.6). In the proof of thetheorem, we will apply a new type of the Toponogov comparison theorem. The compar-ison theorem was established by the present authors as generalization of the comparisontheorem in conventional comparison geometry, which is stated as follows :
A New Type of Toponogov Comparison Theorem ([KT2, Theorem 4.12])
Let ( M, p ) be a complete non-compact Riemannian manifold M whose radial curvatureat the base point p is bounded from below by that of a non-compact model surface ofrevolution ( f M , ˜ p ) . If ( f M , ˜ p ) admits a sector e V ( δ ) , δ ∈ (0 , π ] , having no pair of cutpoints, then, for every geodesic triangle △ ( pxy ) in ( M, p ) with ∠ ( xpy ) < δ , there existsa geodesic triangle e △ ( pxy ) := △ (˜ p ˜ x ˜ y ) in e V ( δ ) such that d (˜ p, ˜ x ) = d ( p, x ) , d (˜ p, ˜ y ) = d ( p, y ) , d (˜ x, ˜ y ) = d ( x, y ) (3.1) and that ∠ ( xpy ) ≥ ∠ (˜ x ˜ p ˜ y ) , ∠ ( pxy ) ≥ ∠ (˜ p ˜ x ˜ y ) , ∠ ( pyx ) ≥ ∠ (˜ p ˜ y ˜ x ) . (3.2) Here ∠ ( pxy ) denotes the angle between the minimal geodesic segments from x to p and y forming the triangle △ ( pxy ) . Remark 3.1
In [KT3], the present authors very recently generalized, from the radialcurvature geometry’s standpoint, the Toponogov comparison theorem to a complete Rie-mannian manifold with smooth convex boundary.Hereafter, let (
M, p ) denote a complete non-compact Riemannian n -manifold M whoseradial curvature at the base point p is bounded from below by that of a non-compact modelsurface of revolution ( f M , ˜ p ) with its metric (1.1), R M the set of all rays on M , and R p the set of all rays emanating from p . Moreover, for each γ ∈ R M , let Π( γ ) denote the setof all α ∈ R p which is a limit ray of the sequence of minimal geodesic segments joining p to γ ( t i ) for some divergent sequence { t i } . Furthermore, we assume that(MTI–1) c ( f M ) > π , and(MTI–2) f M has no pair of cut points in a sector e V ( δ ) for some δ ∈ (0 , π ].8 emma 3.2 Let γ ∈ R M and α : [0 , d ( p, q )] −→ M a minimal geodesic segment joining p to a point q ∈ M \ B r ( p ) such that ∠ ( α ′ (0) , β ′ γ (0)) < δ for some β γ ∈ Π( γ ) . Then, ∠ ( σ ′ (0) , α ′ ( d ( p, q ))) ≤ π − Λ holds for a ray σ emanating from q asymptotic to γ . Here Λ and r denote the positivenumbers guaranteed in Lemmas 2.1 and 2.2, respectively.Proof. Since β γ ∈ Π( γ ), there exists a divergent sequence { t i } such that the sequence ofminimal geodesic segments β i : [0 , d ( p, γ ( t i ))] −→ M joining p to γ ( t i ) convergent to β γ .Since lim t → ∠ ( β ′ i (0) , β ′ γ (0)) = 0, there is a number i ∈ N such that ∠ ( β ′ i (0) , α ′ (0)) < δ for all i ≥ i . Thus, by the new type of the Toponogov comparison theorem, there existsa geodesic triangle e △ ( pγ ( t i ) q ) ⊂ e V ( δ ) corresponding to the triangle △ ( pγ ( t i ) q ), i ≥ i ,such that (3.1) holds for x = γ ( t i ) and y = q , and that ∠ ( − α ′ ( d ( p, q )) , σ ′ i (0)) ≥ ∠ (˜ p ˜ q ˜ γ ( t i )) . Here σ i : [0 , d ( q, γ ( t i ))] −→ M denotes a minimal geodesic segment joining q to γ ( t i ). ByLemma 2.3, we get ∠ ( − α ′ ( d ( p, q )) , σ ′ i (0)) ≥ π for sufficiently large i . Hence, ∠ ( − α ′ ( d ( p, q )) , σ ′ (0)) ≥ π where σ denotes a limit ray of the sequence { σ i } , which is asymptotic to γ . ✷ Hereafter, let F γ denote a Busemann function of a γ ∈ R M . Notice that, by thedefinition of F γ , | F γ ( x ) − F γ ( y ) | ≤ d ( x, y ) holds for all x, y ∈ M , i.e., F γ is Lipschitzcontinuous with Lipschitz constant 1. Hence, F γ is differentiable except for a measurezero set. Moreover, we have Proposition 3.3 ([KT1, Theorem 3.1])
Let γ be a ray on a complete non-compact Rie-mannian manifold M . Then, F γ is differentiable at a point q ∈ M if and only if thereexists a unique ray emanating from q asymptotic to γ . Moreover, the gradient vector of F γ at a differentiable point q equals the velocity vector of the unique ray asymptotic to γ . Lemma 3.4
Let γ ∈ R M and α : [0 , d ( p, q )] −→ M a minimal geodesic segment joining p to a point q ∈ M \ B r ( p ) such that ∠ ( α ′ (0) , β ′ γ (0)) < δ for some β γ ∈ Π( γ ) . If F γ is differentiable at α ( t ) for almost all t ∈ ( a, b ) ⊂ ( r , d ( p, q )] ,then F γ ( α ( b )) − F γ ( α ( a )) ≥ ( b − a ) sin Λ . roof. Assume that F γ is differentiable at α ( t ), t ∈ ( a, b ). By Lemma 3.2 and Propo-sition 3.3, we get ∠ (( ∇ F γ ) α ( t ) , α ′ ( t )) ≤ π − Λ Hence, for almost all t ∈ ( a, b ), ddt F γ ( α ( t )) = h ( ∇ F γ ) α ( t ) , α ′ ( t ) i = cos (cid:0) ∠ (( ∇ F γ ) α ( t ) , α ′ ( t )) (cid:1) ≥ sin Λ It follows from Dini’s theorem [D] (cf. [H, Section 2.3], [WZ, Theorem 7.29]) that F γ ( α ( b )) − F γ ( α ( a )) = Z ba ddt F γ ( α ( t )) dt ≥ ( b − a ) sin Λ . ✷ Lemma 3.5
Let γ ∈ R M and α : [0 , d ( p, q )] −→ M a minimal geodesic segment joining p to a point q ∈ M \ B r ( p ) such that ∠ ( α ′ (0) , β ′ γ (0)) ≤ δ for some β γ ∈ Π( γ ) . Then, F γ ( q ) − F γ ( α ( r )) ≥ ( d ( p, q ) − r ) sin Λ (3.3) holds.Proof. First, we will prove (3.3) under the assumption that ∠ ( α ′ (0) , β ′ γ (0)) < δ . The general case will be completed by the limit argument. If we prove that, for each t ∈ ( r , d ( p, q )), there exists a number ε > F γ ( α ( t )) − F γ ( α ( s )) ≥ ( t − s ) sin Λ (3.4)holds for all s, t ∈ ( t − ε , t + ε ) with s < t , then the equation (3.3) is clear.Take any t ∈ ( r , d ( p, q )), and fix it. Since α is minimal on [0 , d ( p, q )], α ( t ) is nota cut point of p = α (0). Hence, there exist an open neighborhood U ⊂ S n − p around α ′ (0), an open neighborhood U around α ( t ), and an open interval ( t − ε , t + ε ) suchthat U × ( t − ε , t + ε ) is diffeomorphic to U by a map ϕ , where ϕ − ( v, t ) := exp p ( tv ).Since F γ ◦ ϕ − is Lipschitz, it follows from Rademacher’s theorem (cf. [Mo]) that thereexists a set E ⊂ T p M of Lebesgue measure zero such that F γ ◦ ϕ − is differentiable on( U × ( t − ε , t + ε )) \ E . Moreover, for each v ∈ U , we set E v := { t ∈ ( t − ε , t + ε ) | ( v, t ) ∈ E } . Remark that the set E v has also Lebesgue measure zero for all most all v ∈ U (cf. [WZ,Lemma 6.5]). Thus, we may find a sequence { α j } of minimal geodesic segments emanating10rom p converging to α such that each F γ is differentiable at α j ( t ) for almost all t ∈ ( t − ε , t + ε ). By Lemmas 3.2 and 3.4, for each j ∈ N , F γ ( α j ( t )) − F γ ( α j ( s )) ≥ ( t − s ) sin Λ holds for all s, t ∈ ( t − ε , t + ε ) with s < t . Then, by taking the limit, we get (3.4).Assume that ∠ ( α ′ (0) , β ′ γ (0)) = δ . It is clear that there exists a sequence { α i : [0 , ℓ i ] −→ M } of minimal geodesic segments α i emanating from p = α i (0) convergent to α such that ∠ ( α ′ i (0) , β ′ γ (0)) < δ for each i ∈ N . From the argument above, F γ ( α i ( ℓ i )) − F γ ( α i ( r )) ≥ ( ℓ i − r ) sin Λ By taking the limit, we get (3.3). ✷ Set A p := { γ ′ (0) ∈ S n − p | γ ∈ R p } , and denote by diam( A p ) the diameter of A p . Then, we have our main theorem in thisarticle : Theorem 3.6
For any γ , γ , . . . , γ k ∈ R M such that { α ′ (0) ∈ S n − p | α ∈ S ki =1 Π( γ i ) } isa δ -covering of A p , max { F γ i | i = 1 , , . . . , k } is an exhaustion. Moreover, if diam( A p ) ≤ δ , or δ = π , then F γ is an exhaustion for all γ ∈ R M .Proof. Suppose that max { F γ i | i = 1 , , . . . , k } is not an exhaustion, i.e., for some a ∈ R , X := k \ i =1 F − γ i ( −∞ , a ]is non-compact. Hence, there exists a sequence { q j } of points q j ∈ X such thatlim j →∞ d ( p, q j ) = ∞ . Let α j : [0 , d ( p, q j )] −→ M denote a minimal geodesic segment joining p to q j . Sincelim j →∞ d ( p, q j ) = ∞ , there exists a number j ∈ N such that r < d ( p, q j )for all j ≥ j . Furthermore, by choosing an infinite subsequence of { α j } , we may assumethat there exist i ∈ { , , . . . , k } such that, for each j ≥ j , ∠ ( α ′ j (0) , β ′ γ j (0)) ≤ δ β γ j ∈ Π( γ i ). It follows from Lemma 3.5 that F γ i ( q j ) − F γ i ( α j ( r )) ≥ ( d ( p, q j ) − r ) sin Λ for all j ≥ j . Since q j ∈ F − γ i ( −∞ , a ] for all j ≥ j , a − F γ i ( α j ( r )) ≥ ( d ( p, q j ) − r ) sin Λ . Since lim j →∞ d ( p, q j ) = ∞ , we have lim j →∞ F γ i ( α j ( r )) = −∞ . This is impossible, since | F γ i ( p ) − F γ i ( α j ( r )) | ≤ d ( p, α j ( r )) = r for all j ≥ j . Therefore, max { F γ i | i =1 , , . . . , k } is an exhaustion.Next, we will prove the second claim. Assume that diam( A p ) ≤ δ . Since ∠ ( v, w ) ≤ δ for all v, w ∈ A p , it is clear that { v } is a δ -covering of A p for each v ∈ A p . Hence, foreach γ ∈ R M , { α ′ (0) ∈ S n − p | α ∈ Π( γ ) } is a δ -covering of A p . From the argumentabove, this implies that F γ is an exhaustion for all γ ∈ R M . If δ = π , then the claim isclear, since diam( A p ) ≤ π . ✷ From the same argument in [KT1, Section 4.2], we get
Corollary 3.7
The isometry group I ( M ) of M is compact, if diam( A p ) ≤ δ , or δ = π . References [CG] J. Cheeger and D. Gromoll,
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