Toponogov comparison theorem for open triangles
aa r X i v : . [ m a t h . DG ] M a y Toponogov Comparison Theorem for Open Triangles ∗† Kei KONDO · Minoru TANAKA in memory of the late professor Detlef Gromoll
Abstract
The aim of our article is to generalize the Toponogov comparison theorem to acomplete Riemannian manifold with smooth convex boundary. A geodesic trianglewill be replaced by an open (geodesic) triangle standing on the boundary of themanifold, and a model surface will be replaced by the universal covering surface ofa cylinder of revolution with totally geodesic boundary.
Cohn -Vossen is one of pioneers in global differential geometry. More than seventy yearsago, he investigated the relationship between the total curvature and the Riemannianstructure of complete open surfaces. He has given big influence to many geometers whoresearch in global differential geometry, although he studied only 2-dimensional manifoldsin [CV1] and [CV2]. For example, Cohn -Vossen proved the following theorem known asa splitting theorem :
Theorem 1.1 ([CV2, Satz 5])
If a complete Riemannian -manifold has non-negativeGaussian curvature and admits a straight line, then its universal covering space is iso-metric to Euclidean plane. Toponogov ([T2]) generalized this splitting theorem for any dimensional complete Rie-mannian manifolds with non-negative sectional curvature by making use of the Toponogovcomparison theorem ([T1]). It is well known that the Toponogov comparison theorem hasproduced many great classical results, e.g., the maximal diameter theorem by Toponogov([T1]), the structure theorem with positive sectional curvature by Gromoll and Meyer([GM]), and the soul theorem with non-negative sectional curvature by Cheeger and Gro-moll ([CG]). Besides the Toponogov comparison theorem, some techniques originatingfrom Euclidean geometry also play a key role in the references above. The techniquessuch as drawing a circle or a geodesic polygon, and joining two points by a minimal ∗ † Key words and phrases. Cut locus, focal locus, open triangle, radial curvature, Riemannian manifoldwith boundary, Toponogov’s comparison theorem. , ∞ ). Very familiar surfaces such as paraboloidsor 2-sheeted hyperboloids are typical examples of a von Mangoldt surface of revolution.Hence, it is natural to employ a von Mangoldt surface of revolution as a model surface.The reason why a von Mangoldt surface of revolution is used as a model surface lies inthe following property of the surface: Theorem 1.2 ([Tn, Main Theorem])
The cut locus of a point on a von Mangoldt surfaceof revolution is empty or a subray of the meridian opposite to the point.
It would be impossible to prove [IMS, Theorem 1.3] for general surfaces of revolution,because the cut locus of the surface appears as an obstruction, when we draw a geodesictriangle in the model surface. For example, the proof of [KT2, Lemma 4.10] suggests suchan obstruction. In [KT2], the present authors very recently generalized [IMS, Theorem1.3] for a surface of revolution admitting a sector which has no pair of cut points.Our purpose in this article is to establish the Toponogov comparison theorem for Rie-mannian manifolds with convex boundary from the standpoint of the radial curvature ge-ometry.
Now we will introduce the radial curvature geometry for manifolds with boundary :We first introduce our model, which will be later employed as a reference surface ofcomparison theorems in complete Riemannian manifolds with boundaries. Let f M := ( R , d ˜ x ) × m ( R , d ˜ y )be a warped product of two 1-dimensional Euclidean lines ( R , d ˜ x ) and ( R , d ˜ y ), wherethe warping function m : R → (0 , ∞ ) is a positive smooth function satisfying m (0) = 1and m ′ (0) = 0. Then we call e X := n ˜ p ∈ f M ; ˜ x (˜ p ) ≥ o a model surface . Since m ′ (0) = 0, the boundary ∂ e X := { ˜ p ∈ e X ; ˜ x (˜ p ) = 0 } e X is totally geodesic . The metric ˜ g of e X is expressed as(1.1) ˜ g = d ˜ x + m (˜ x ) d ˜ y on [0 , ∞ ) × R . The function G ◦ ˜ µ : [0 , ∞ ) → R is called the radial curvature function of e X , where we denote by G the Gaussian curvature of e X , and by ˜ µ any ray emanatingperpendicularly from ∂ e X (notice that such ˜ µ will be called a ∂ e X -ray). Remark that m : [0 , ∞ ) → R satisfies the differential equation m ′′ ( t ) + G (˜ µ ( t )) m ( t ) = 0with initial conditions m (0) = 1 and m ′ (0) = 0. We define a sector e X ( θ ) := ˜ y − ((0 , θ ))in e X for each constant number θ >
0. Since a map (˜ p, ˜ q ) → (˜ p, ˜ q + c ), c ∈ R , over e X isan isometry, e X ( θ ) is isometric to ˜ y − ( c, c + θ ) for all c ∈ R .Hereafter, let ( X, ∂X ) denote a complete Riemannian n -dimensional manifold X witha smooth boundary ∂X . We say that ∂X is convex , if all eigenvalues of the shape operator A ξ of ∂X are non-negative in the inward vector ξ normal to ∂X . Notice that our sign of A ξ differs from [S]. That is, for each p ∈ ∂X and v ∈ T p ∂X , A ξ ( v ) = − ( ∇ v N ) ⊤ holds. Here, we denote by N a local extension of ξ , and by ∇ the Riemannian connectionon X .For a positive constant l , a unit speed geodesic segment µ : [0 , l ] → X emanating from ∂X is called a ∂X -segment if d ( ∂X, µ ( t )) = t on [0 , l ]. If µ : [0 , l ] → X is a ∂X -segmentfor all l >
0, we call µ a ∂X -ray . Here, we denote by d ( ∂X, · ) the distance function to ∂X induced from the Riemannian structure of X . Notice that a ∂X -segment is orthogonal to ∂X by the first variation formula, and so a ∂X -ray is too.For any fixed two points p, q ∈ X \ ∂X , an open triangle OT( ∂X, p, q ) := ( ∂X, p, q ; γ, µ , µ )in X is defined by two ∂X -segments µ i : [0 , l i ] → X , i = 1 ,
2, a minimal geodesic segment γ : [0 , d ( p, q )] → X , and ∂X such that µ ( l ) = γ (0) = p, µ ( l ) = γ ( d ( p, q )) = q. In this article, whenever an open triangle OT( ∂X, p, q ) = ( ∂X, p, q ; γ, µ , µ ) in X isgiven, ( ∂X, p, q ; γ, µ , µ ), as a symbol,always means that the minimal geodesic segment γ is the side opposite to ∂X emanatingfrom p to q , and that the ∂X -segments µ , µ are sides emanating from ∂X to p , q ,respectively. 3 X, ∂X ) is said to have the radial curvature ( with respect to ∂X ) bounded from belowby that of ( e X, ∂ e X ) if, for every ∂X -segment µ : [0 , l ) → X , the sectional curvature K X of X satisfies K X ( σ t ) ≥ G (˜ µ ( t ))for all t ∈ [0 , l ) and all 2-dimensional linear spaces σ t spanned by µ ′ ( t ) and a tangent vectorto X at µ ( t ). For example, if the Riemannian metric of e X is d ˜ x + d ˜ y , or d ˜ x +cosh (˜ x ) d ˜ y ,then G (˜ µ ( t )) = 0, or G (˜ µ ( t )) = −
1, respectively. Furthermore, the radial curvaturemay change signs wildly (e.g., [KT1, Example 1.2], [TK]).Our main theorem is now stated as follows :
Toponogov’s comparison theorem for open triangles.
Let ( X, ∂X ) be a completeconnected Riemannian n -dimensional manifold X with smooth convex boundary ∂X whoseradial curvature is bounded from below by that of a model surface ( e X, ∂ e X ) with its metric ( ) . Assume that e X admits a sector e X ( θ ) which has no pair of cut points. Then, forevery open triangle OT( ∂X, p, q ) = ( ∂X, p, q ; γ, µ , µ ) in X with d ( µ (0) , µ (0)) < θ , there exists an open triangle OT( ∂ e X, ˜ p, ˜ q ) = ( ∂ e X, ˜ p, ˜ q ; ˜ γ, ˜ µ , ˜ µ ) in e X ( θ ) such that d ( ∂ e X , ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X, ˜ q ) = d ( ∂X, q ) and that ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q, d ( µ (0) , µ (0)) ≥ d (˜ µ (0) , ˜ µ (0)) . Furthermore, if d ( µ (0) , µ (0)) = d (˜ µ (0) , ˜ µ (0)) holds, then ∠ p = ∠ ˜ p, ∠ q = ∠ ˜ q hold. Here ∠ p denotes the angle between two vectors γ ′ (0) and − µ ′ ( d ( ∂X, p )) in T p X . Notice that we do not assume that ∂X is connected in our main theorem. Moreover,remark that the opposite side γ of OT( ∂X, p, q ) does not meet ∂X (Lemma 6.1 in Section6). A related result for our main theorem is [MS, Theorem 3.4] of Mashiko and Shiohama.In [MS], they treat a pair ( M, N ) of a complete connected Riemannian manifold M and acompact connected totally geodesic hypersurface N of M such that the radial curvaturewith respect to N is bounded from below by that of the model (( a, b ) × m N, N ), where( a, b ) denotes an interval, in their sense. Note that the radial curvature with respect to N is bounded from below by that of our model ([0 , ∞ ) , d ˜ x ) × m ( R , d ˜ y ), if it is boundedfrom below by that of their model (( a, b ) × m N, N ). Thus, our Toponogov comparisontheorem for open triangles is applicable to the pair (
M, N ).We first prove the Toponogov comparison theorem for thin open triangles (see Defi-nition 2.1 for the definition of thin open triangles). The first variation formula and somefundamental properties of the second variation formula will play key roles when we prove4he Toponogov comparison theorem for thin open triangles. This new technique gives anew and sophisticated way of the proof of the original Toponogov comparison theorem.It was clarified in [KT2] that the Toponogov comparison theorem holds for any geodesictriangles, if the Toponogov comparison theorem holds for thin geodesic triangles. This isalso true for open triangles, if we imitate new techniques developed in [KT2].There are many examples of model surfaces satisfying the assumption in our maintheorem. For example, it is clear that a model surface ( e X, ∂ e X ) with the metric d ˜ x + d ˜ y ,or d ˜ x + cosh (˜ x ) d ˜ y has no pair of cut points in a sector e X ( θ ) for each constant θ > Example 1.3
Let f M := ( R , dt ) × m ( S , dθ ) be a warped product of a 1-dimensionalEuclidean line ( R , dt ) and a unit circle ( S , dθ ) satisfying the next three conditions:(C–1) The warping function m : R → (0 , ∞ ) is a smooth even function satisfying m (0) =1 and m ′ (0) = 0.(C–2) The radial curvature function G (˜ µ ( t )) = − m ′′ ( t ) /m ( t ) is non-increasing on [0 , ∞ ).(C–3) m ′ ( t ) = 0 on R \ { } .Tamura ([Tm]) proved that the cut locus of a point ˜ p ∈ f M with θ (˜ p ) = 0 is the unionof the meridian θ = π opposite to θ = 0 and a subarc of the parallel t = − t (˜ p ). Now,we introduce the Riemannian universal covering surface c M := ( R × m R , d ˜ x + m (˜ x ) d ˜ y )of ( f M , dt + m ( t ) dθ ). It follows from Tamura’s theorem above that the half space e X := ([0 , ∞ ) × m R , d ˜ x + m (˜ x ) d ˜ y ) of c M has no pair of cut points in a sector e X ( θ ) foreach constant θ >
0. For example, a model surface with the metric d ˜ x + ( e − ˜ x ) d ˜ y isone of such models.We discuss applications of the Toponogov comparison theorem for open triangles in[KT3], which are splitting theorems of two types. Also, the Toponogov comparison theo-rem for open triangles in a weak form is discussed in the article.In the following sections, all geodesics will be normalized, unless otherwise stated. Acknowledgements.
We are very grateful to Professor Ryosuke Ichida for his helpfulcomments on the very first draft of this article. Finally, we would like to express toProfessor Detlef Gromoll our deepest gratitude for his comment [G] upon our work onradial curvature geometry ([KT1], [KT2]).
Here, we sketch in the organization from Sections 3 to 8, because we need many lemmasfor proving our main theorem. 5hroughout this section, let (
X, ∂X ) be a complete connected Riemannian n -manifold X with smooth convex boundary ∂X whose radial curvature is bounded from below bythat of a model surface ( e X, ∂ e X ).Our main purpose in Sections 3 to 5 is to prove the following lemma, which is one offundamental lemmas to establish the Toponogov comparison theorem for open triangles(Theorem 8.4) : Lemma on thin open triangles.
For every thin open triangle
OT( ∂X, p, q ) in X , thereexists an open triangle OT( ∂ e X , ˜ p, ˜ q ) in e X such that d ( ∂ e X , ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X, ˜ q ) = d ( ∂X, q ) and that ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q. Thin open triangles are defined as follows:
Definition 2.1 (Thin open triangle)
An open triangle OT( ∂X, p, q ) in (
X, ∂X ) iscalled a thin open triangle , if(TOT–1) the opposite side γ of OT( ∂X, p, q ) to ∂X emanating from p to q is containedin a normal convex neighborhood in X \ ∂X , and(TOT–2) L ( γ ) < inj(˜ q s ) for all s ∈ [0 , d ( p, q )],where L ( γ ) denotes the length of γ , and ˜ q s denotes a point in e X with d ( ∂ e X, ˜ q s ) = d ( ∂X, γ ( s ))for each s ∈ [0 , d ( p, q )].Here, the injectivity radius inj(˜ p ) of a point ˜ p ∈ e X is the supremum of r > q ∈ e X with d (˜ p, ˜ q ) < r , there exists a unique minimal geodesic segment joining˜ p to ˜ q . Remark that, for each point ˜ p ∈ e X \ ∂ e X , the inequality inj(˜ p ) > d ( ∂ e X, ˜ p ) holds,if ˜ p is sufficiently close to ∂ e X .Hence, Sections 3 and 4 are set up to prove Lemma on thin open triangles (Lemma5.8) : In Section 3, we investigate the relationship between minimal geodesic segments ina complete connected Riemannian manifold X with smooth boundary ∂X and the focalcut locus of ∂X (Lemma 3.5). Section 4 is the heart of this article, i.e., we have KeyLemma (Lemma 4.5) of this article. Here Lemma 4.5 is a comparison theorem of theRauch type on length of ∂X -segments in variations of a ∂X -segment. We also have a rare application of the Warner comparison theorem in the proofs of Lemmas 4.3 and 4.4.Notice that Lemmas 3.5 and 4.5 are indispensable for us to prove Lemma on thin opentriangles.In Section 5, we prove Lemma on thin open triangles, using Lemmas 3.5 and 4.5.6n Section 6, we see, without curvature assumption, that the opposite side of any opentriangle to ∂X does not meet ∂X , if ∂X is convex.In Section 7, we establish the Alexandrov convexity (Lemma 7.3). In the proof ofLemma 7.3, we may understand that it is a very important property that the oppositeside of an open triangle to the boundary in a model surface is unique (i.e., we can notprove the equation (7.17) in the proof of Lemma 7.3 without this property). In order toprove Lemma 7.3, we have to treat a non-differentiable Lipschitz function. It follows fromDini’s theorem ([D]) that, for any Lipschitz function f on [ a, b ], f is differentiable almosteverywhere, and Z ba f ′ ( t ) dt = f ( b ) − f ( a )holds. Note that the Cantor-Lebesgue function g on [0 ,
1] is differentiable almost every-where, but 0 = Z g ′ ( t ) dt < g (1) − g (0) = 1 . Cohn -Vossen applied in [CV1] and [CV2] the properties above to global differential ge-ometry.In Section 8, we prove our main theorem, the Toponogov comparison theorem for opentriangles, by using new techniques established in [KT2, Section 4] and Lemma 7.3. ∂X Our purpose of this section is to investigate the relationship between minimal geodesicsegments in a complete connected Riemannian manifold with smooth boundary and thefocal cut locus of the boundary (see Lemma 3.5). It will be clarified, by using Lemma 3.5,in Section 5 that the cut locus of the manifold is not an obstruction at all when we drawa corresponding open triangle in a model surface for each open triangle in the manifold.Throughout this section, let (
X, ∂X ) denote a complete connected Riemannian n -manifold X with smooth boundary ∂X .First, we will recall the definitions of ∂X -Jacobi fields, focal loci of ∂X , and cut lociof ∂X , which are used throughout this article. Definition 3.1 ( ∂X -Jacobi field) Let µ : [0 , ∞ ) → X be a unit speed geodesic ema-nating perpendicularly from ∂X . A Jacobi field J ∂X along µ is called a ∂X -Jacobi field ,if J ∂X satisfies J ∂X (0) ∈ T µ (0) ∂X, J ′ ∂X (0) + A µ ′ (0) ( J ∂X (0)) ∈ ( T µ (0) ∂X ) ⊥ . Here J ′ denotes the covariant derivative of J along µ , and A µ ′ (0) denotes the shape operatorof ∂X . Definition 3.2 (Focal locus of ∂X ) A point µ ( t ), t = 0, is called a focal point of ∂X along a unit speed geodesic µ : [0 , ∞ ) → X emanating perpendicularly from ∂X , if there7xists a non-zero ∂X -Jacobi field J ∂X along µ such that J ∂X ( t ) = 0. The focal locus Foc( ∂X ) of ∂X is the union of the focal points of ∂X along all of the unit speed geodesicsemanating perpendicularly from ∂X . Definition 3.3 (Cut locus of ∂X ) Let µ : [0 , l ] → X be a ∂X -segment. The endpoint µ ( l ) of µ ([0 , l ]) is called a cut point of ∂X along µ , if any extended geodesic¯ µ : [0 , l ] −→ X of µ , l > l , is not a ∂X -segment anymore. The cut locus Cut( ∂X ) of ∂X is the union of the cut points of ∂X along all of the ∂X -segments.Set FC( ∂X ) := Foc( ∂X ) ∩ Cut( ∂X ) . We then call FC( ∂X ) the focal cut locus of ∂X .From the similar argument in [IT1], we have the following lemma. Lemma 3.4 (see [IT1, Lemma 2])
The Hausdorff dimension of
FC( ∂X ) is at most n − .In particular, H n − (FC( ∂X )) = 0 . Here H n − denotes the ( n − -dimensional Hausdorffmeasure. An open neighborhood U ( q ) of q ∈ X is called a normal convex neighborhood of q , if,for any points q , q ∈ U ( q ), there exists a unique minimal geodesic segment σ joining q to q such that the segment σ is contained in U ( q ). Then, the following lemma followsfrom Lemma 3.4. Lemma 3.5
Assume that p Foc( ∂X ) , q Cut( p ) , and γ ([0 , d ( p, q )]) ∩ ∂X = ∅ , where γ denotes the minimal geodesic segment joining p to q . Then, for each v ∈ S n − q := { v ∈ T q X ; k v k = 1 } , there exists a sequence { γ i : [0 , l i ] → X } i ∈ N of minimal geodesic segments γ i emanating from p = γ i (0) convergent to γ such that γ i ([0 , l i ]) ∩ FC( ∂X ) = ∅ and lim i →∞ k exp − q ( γ i ( l i )) k exp − q ( γ i ( l i )) = v. Here exp − q denotes the local inverse of the exp q on a normal convex neighborhood U ( q ) of q disjoint from ∂X .Proof. Let { q j } j ∈ N denote a sequence of points q j ∈ U ( q ) convergent to q such that q j Cut( p ) , α j ([0 , d ( p, q j )]) ∩ ∂X = ∅ , j →∞ k exp − q ( q j ) k exp − q ( q j ) = v. Here α j : [0 , d ( p, q j )] → X denotes the minimal geodesic segment emanating from p = α j (0) to q j . We will prove that, for each q j , there exists a sequence { γ ( j ) i : [0 , l ( j ) i ] → X } i ∈ N of minimal geodesic segments γ ( j ) i emanating from p = γ ( j ) i (0) convergent to α j such that(3.1) γ ( j ) i ([0 , l ( j ) i ]) ∩ FC( ∂X ) = ∅ . It is sufficient to prove the existence of the sequence γ ( j ) i for each j ∈ N , because it iseasy to prove the existence of the sequence { γ i : [0 , l i ] → X } i ∈ N in our lemma by takinga subsequence of { γ ( j ) i : [0 , l ( j ) i ] → X } i, j ∈ N .Choose any q j and fix it. Since p is not a focal point of ∂X , there exists a normalconvex neighborhood B ε ( p ) of p with radius 2 ε such that(3.2) B ε ( p ) ∩ Foc( ∂X ) = ∅ . Since q j is not a cut point of p , there exist two numbers l j > d ( p, q j ), θ j >
0, and aneighborhood U j around q j such that U j is diffeomorphic to V α ′ j (0) ( θ j ) × ( ε, l j ). Here weset V α ′ j (0) ( θ j ) := (cid:8) w j ∈ T p X ; k w j k = 1 , ∠ ( w j , α ′ j (0)) < θ j (cid:9) . Here, the diffeomorphism Φ j from V α ′ j (0) ( θ j ) × ( ε, l j ) onto U j is given byΦ j ( w j , s ) := exp p ( s w j ) . Since Φ − j is Lipschitz, the map Π j := P j ◦ Φ − j : U j → V α ′ j (0) ( θ j ) is also Lipschitz, where P j : V α ′ j (0) ( θ j ) × ( ε, l j ) → V α ′ j (0) ( θ j ) denotes the projection to the first factor. Therefore,it follows from Lemma 3.4 that H n − (Π j ( U j ∩ FC( ∂X ))) = 0 . This implies that there exists a sequence { w ( j ) i } i ∈ N of elements w ( j ) i ∈ V α ′ j (0) ( θ j ) convergentto α ′ j (0) such that(3.3) w ( j ) i Π j ( U j ∩ FC( ∂X ))for each i ∈ N . Let { l ( j ) i } i ∈ N be a sequence of numbers l ( j ) i ∈ (0 , l j ) convergent to d ( p, q j ).By setting γ ( j ) i ( s ) := exp p ( s w ( j ) i ) , s ∈ [0 , l ( j ) i ] , for each i ∈ N , it follows from (3.2) and (3.3) that we get a sequence of minimal geodesicsegments γ ( j ) i emanating from p = γ ( j ) i (0) convergent to α j satisfying (3.1). ✷ Length of ∂X -segments in variations Our purpose of this section is to prove a comparison theorem (Lemma 4.5) of the Rauchtype on length of ∂X -segments in variations of a ∂X -segment, by using the second varia-tion formula and the Warner comparison theorem. As a result, readers might be surprisedby, and would realize, as Gromoll once suggested, that we may still understand a globalmatter on a Riemannian manifold by the second variation, because Lemma on thin opentriangles (Lemma 5.8), proved by Lemmas 3.5 and 4.5, plays an important role in theproof of the Toponogov comparison theorem for open triangles (see Section 8).Throughout this section, let ( X, ∂X ) denote a complete connected Riemannian n -manifold X with smooth convex boundary ∂X whose radial curvature is bounded frombelow by the radial curvature function G of a model surface ( e X, ∂ e X ) with its metric (1.1).Take any point r ∈ X \ ( ∂X ∪ Foc( ∂X )), and fix it. Then there exists a positivenumber ε := ε ( r ) such that(4.1) B ε ( r ) ∩ (Foc( ∂X ) ∪ ∂X ) = ∅ , where B ε ( r ) denotes the normal convex neighborhood of r with radius 2 ε . Take anypoint p ∈ B ε ( r ), and fix it. Let µ : [0 , l ] → X denote a ∂X -segment to p = µ ( l ). By(4.1), we may find a number ε ∈ (0 , ε ] independent of the choice of p and an openneighborhood U around lµ ′ (0) such thatexp ⊥ : U → B ε ( p )is a diffeomorphism. Here exp ⊥ denotes the normal exponential map on the normal bundleof ∂X . Let ξ : R → S n − p be a unit speed geodesic on S n − p emanating from µ ′ ( l ) = ξ (0),where S n − p := { v ∈ T p X ; k v k = 1 } . Notice that ∠ ( µ ′ ( l ) , ξ ( θ )) = | θ | for all θ ∈ [ − π, π ].From now on, we assume that the curve ξ and its parameter value θ ∈ [ − π, π ] are alsofixed. Then, we get a minimal geodesic segment c emanating from p = c (0) defined by c ( s ) := exp p ( s ξ ( θ ))for all s ∈ ( − ε , ε ). Thus, we get a geodesic variation ϕ : [0 , l ] × ( − ε , ε ) → X of µ defined by ϕ ( t, s ) := exp ⊥ (cid:18) tl v ( s ) (cid:19) , where we set v ( s ) := (cid:0) exp ⊥ | U (cid:1) − ( c ( s )). For each s ∈ ( − ε , ε ), c ( s ) is joined by a geodesicsegment ϕ s ( · ) := ϕ ( · , s ) emanating perpendicularly from ∂X . By setting J ∂X ( t ) := ∂ϕ∂s ( t, , we get a ∂X -Jacobi field J ∂X along µ . It is clear that(4.2) J ∂X ( l ) = c ′ (0) . Then, we first get the following lemma. 10 emma 4.1
For each t ∈ [0 , l ] , an orthogonal component Y ∂X ( t ) of J ∂X ( t ) with respectto µ ′ ( t ) is given by Y ∂X ( t ) := J ∂X ( t ) − cos θl t µ ′ ( t ) . Proof.
Since J ∂X is a Jacobi field along µ , there exist constant numbers a and b satisfying (cid:10) J ∂X ( t ) , µ ′ ( t ) (cid:11) = at + b for all t ∈ [0 , l ]. Since J ∂X (0) is orthogonal to µ ′ (0), we see b = 0. Furthermore, by (4.2),we see a = cos θl . Thus, we get (cid:10) J ∂X ( t ) , µ ′ ( t ) (cid:11) = cos θl t for all t ∈ [0 , l ]. Hence, the Jacobi field Y ∂X along µ defined by Y ∂X ( t ) := J ∂X ( t ) − cos θl t µ ′ ( t )is orthogonal to µ ′ ( t ) on [0 , l ]. ✷ In this article, we denote by I l∂X ( V, W ) := I l ( V, W ) − (cid:10) A µ ′ (0) ( V (0)) , W (0) (cid:11) the index form with respect to µ | [0 , l ] for piecewise C ∞ vector fields V, W along µ | [0 , l ] ,where we set I l ( V, W ) := Z l (cid:8)(cid:10) V ′ , W ′ (cid:11) − (cid:10) R ( µ ′ , V ) µ ′ , W (cid:11)(cid:9) dt, which is a symmetric bilinear form. The following lemma is clear from the first and secondvariation formulas and Lemma 4.1. Lemma 4.2
The equalities L ′ (0) = cos θ and L ′′ (0) = I l∂X ( Y ∂X , Y ∂X ) hold. Here L ( s ) denotes the length of the geodesic segment ϕ s ( · ) emanating perpendicularly from ∂X . Now, choose any sufficiently small number λ > e X λ , ∂ e X λ ) denote amodel surface with its metric ˜ g λ = d ˜ x + m λ (˜ x ) d ˜ y on [0 , ∞ ) × R . Here the positive smooth function m λ satisfies the differential equation m ′′ λ + ( G − λ ) m λ = 0 , m λ (0) = 1 , m ′ λ (0) = 0 , where G denotes the radial curvature function of ( e X, ∂ e X ). Thus, the radial curvature of( X, ∂X ) is greater than G λ := G − λ . Take any point ˜ p in e X λ \ ∂ e X λ satisfying d ( ∂ e X λ , ˜ p ) = d ( ∂X, p ) = d ( ∂X, µ ( l )) = l. p .Let ˜ µ λ : [0 , l ] → e X λ denote a ∂ e X λ -segment to ˜ p , and let e E λ denote a unit parallelvector field along ˜ µ λ orthogonal to ˜ µ λ . Then, we define a ∂ e X λ -Jacobi field e Z λ along ˜ µ λ by e Z λ ( t ) := 1 m λ ( l ) m λ ( t ) e E λ ( t ) . Furthermore, by the same definition, we denote also by I l ( · , · ) the symmetric bilinearform for piecewise C ∞ vector fields along ˜ µ λ | [0 , l ] . Then, we have the following lemma. Lemma 4.3 I l ( e Z λ , e Z λ ) ≥ I l∂X ( Z ∂X , Z ∂X ) + λm λ ( l ) Z l m λ ( t ) dt holds for all ∂X -Jacobi fields Z ∂X along µ orthogonal to µ with k Z ∂X ( l ) k = 1 .Proof. We can prove this lemma by the argument in the proof of the Warner comparisontheorem [W]. For completeness, we will give a proof here. Let E be a unit parallel vectorfield along µ orthogonal to µ such that E ( l ) = Z ∂X ( l ) , where Z ∂X denotes a ∂X -Jacobi field along µ orthogonal to µ . Set W ( t ) := 1 m λ ( l ) m λ ( t ) E ( t ) . Since K X ( σ t ) ≥ G (˜ µ ( t )) > G λ (˜ µ λ ( t )) = G (˜ µ ( t )) − λ , we have I l ( e Z λ , e Z λ ) = Z l (cid:26)D e Z ′ λ , e Z ′ λ E − G λ (˜ µ λ ( t )) (cid:13)(cid:13)(cid:13) e Z λ (cid:13)(cid:13)(cid:13) (cid:27) dt (4.3) = Z l (cid:8) h W ′ , W ′ i − ( G (˜ µ ( t )) − λ ) k W k (cid:9) dt ≥ Z l (cid:8) h W ′ , W ′ i − K X ( σ t ) k W k (cid:9) dt + λ Z l k W k dt = I l ( W, W ) + λm λ ( l ) Z l m λ ( t ) dt. Since Z ∂X is the ∂X -Jacobi field with Z ∂X ( l ) = E ( l ) = W ( l ), it follows from [S, Lemma2.10 in Chapter III] that(4.4) I l ( W, W ) − (cid:10) A µ ′ (0) ( W (0)) , W (0) (cid:11) = I l∂X ( W, W ) ≥ I l∂X ( Z ∂X , Z ∂X ) . Since (cid:10) A µ ′ (0) ( W (0)) , W (0) (cid:11) ≥
0, we get, by (4.3) and (4.4), I l ( e Z λ , e Z λ ) ≥ I l∂X ( Z ∂X , Z ∂X ) + (cid:10) A µ ′ (0) ( W (0)) , W (0) (cid:11) + λm λ ( l ) Z l m λ ( t ) dt ≥ I l∂X ( Z ∂X , Z ∂X ) + λm λ ( l ) Z l m λ ( t ) dt. ✷ c λ : ( − ε , ε ) → e X λ denote the minimal geodesic segment emanating from ˜ p =˜ c λ (0) corresponding to the minimal geodesic segment c ( s ) = exp p ( s ξ ( θ )) , s ∈ ( − ε , ε )in B ε ( p ) ⊂ X . Without loss of generality, we may assume that B ε (˜ p ) ∩ ∂ e X λ = ∅ . Weconsider a geodesic variation ˜ ϕ ( λ ) : [0 , l ] × ( − ε , ε ) → e X λ of ˜ µ λ defined by˜ ϕ ( λ ) ( t, s ) := exp ⊥ (cid:18) tl ˜ v λ ( s ) (cid:19) , where we set ˜ v λ ( s ) := (cid:0) exp ⊥ (cid:1) − (˜ c λ ( s )). By setting e J λ ( t ) := ∂ ˜ ϕ ( λ ) ∂s ( t, , we get a ∂ e X λ -Jacobi field e J λ along ˜ µ λ . As well as above, e J λ ( l ) = ˜ c ′ λ (0) holds, and theJacobi field e Y λ along along ˜ µ λ defined by e Y λ ( t ) := e J λ ( t ) − cos θl t ˜ µ ′ λ ( t )is orthogonal to ˜ µ ′ λ ( t ) on [0 , l ]. Lemma 4.4
There exists a number λ := λ ( l , ε ) > depending on l and ε such that,for any λ ∈ (0 , λ ) , any unit speed geodesic ξ on S n − p emanating from µ ′ ( l ) , and any θ ∈ (0 , π ) , the inequality I l ( e Y λ , e Y λ ) − I l∂X ( Y ∂X , Y ∂X ) ≥ λ C sin θ holds. Here C is a constant number given by C := 12 m ( l ) Z l m ( t ) dt, where l := d ( ∂X, r ) .Proof. Since e Y λ ( l ) = ˜ c ′ λ (0) − cos θ ˜ µ ′ λ ( l ) = ± sin θ · e E λ ( l ) = ± sin θ · e Z λ ( l ) , and since e Y λ is a ∂ e X λ -Jacobi field orthogonal to ˜ µ λ , we see e Y λ ( t ) = ± sin θ · e Z λ ( t )on [0 , l ]. Notice that any ∂ e X λ -Jacobi field orthogonal to ˜ µ λ is equal to a e Z λ ( t ), a ∈ R .Hence, we have(4.5) I l ( e Y λ , e Y λ ) = sin θ · I l ( e Z λ , e Z λ ) . Similarly, we have, by Lemma 4.1, Y ∂X ( t ) = sin θ · Z ∂X ( t )13or some ∂X -Jacobi field Z ∂X along µ orthogonal to µ with k Z ∂X ( l ) k = 1. Hence, we have(4.6) I l∂X ( Y ∂X , Y ∂X ) = sin θ · I l∂X ( Z ∂X , Z ∂X ) . By combining (4.5) and (4.6), we get, by Lemma 4.3, I l ( e Y λ , e Y λ ) − I l∂X ( Y ∂X , Y ∂X ) = sin θ n I l ( e Z λ , e Z λ ) − I l∂X ( Z ∂X , Z ∂X ) o (4.7) ≥ λ sin θm λ ( l ) Z l m λ ( t ) dt. On the other hand, since lim λ ↓ m λ ( t ) = m ( t ) and | l − l | < ε , we may find a number λ > m λ ( l ) Z l m λ ( t ) dt > m ( l ) Z l m ( t ) dt (4.8)for all λ ∈ (0 , λ ). From (4.7) and (4.8), we have proved this lemma. ✷ Lemma 4.5 (Key lemma)
For each λ ∈ (0 , λ ) , there exists a number δ := δ ( λ ) ∈ (0 , ε ) such that, for any p ∈ B ε ( r ) , any unit speed geodesic ξ on S n − p emanating from µ ′ ( l ) , any θ ∈ [0 , π ] , and any λ ∈ (0 , λ ) , the inequality L ( s ) ≤ e L λ ( s ) holds for all s ∈ [0 , δ ] , and the equality occurs if and only if s = 0 , θ = 0 , or θ = π .Here e L λ ( s ) denotes the length of the geodesic segment ˜ ϕ ( λ ) s ( · ) = ˜ ϕ ( λ ) ( · , s ) emanatingperpendicularly from ∂ e X λ to ˜ c λ ( s ) .Proof. Although the angle θ has been fixed in the arguments above of this section, weconsider here that θ is a variable. Hence, we denote L ( s ) by L ( s, θ ), which is a smoothfunction of two variables s and θ and depends smoothly on p , ξ (0), and ξ ′ (0). Furthermore,we define the reminder term R ( s, θ ) of the Taylor expansion of L ( s, θ ) about s = 0 by(4.9) R ( s, θ ) := L ( s, θ ) − (cid:26) L (0 , θ ) + L ′ (0 , θ ) s + 12! L ′′ (0 , θ ) s (cid:27) , where we set L ′ (0 , θ ) := ∂L∂s (0 , θ ) and L ′′ (0 , θ ) := ∂ L∂s (0 , θ ) . From (4.9), Lemma 4.2, and the equation (4.6) in the proof of Lemma 4.4, we have L ( s, θ ) = l + s cos θ + s I l∂X ( Y ∂X , Y ∂X ) + R ( s, θ )(4.10) = l + s cos θ + s sin θ I l∂X ( Z ∂X , Z ∂X ) + R ( s, θ ) . (4.11) 14t is clear that R (0 , θ ) = ∂ R ∂s (0 , θ ) = ∂ R ∂s (0 , θ ) = 0 . Hence, there exists a smooth function R ( s, θ ) depending smoothly on p , ξ (0), and ξ ′ (0)such that(4.12) R ( s, θ ) = R ( s, θ ) s . Since B ε ( r ) ∩ Foc( ∂X ) = ∅ , the geodesic ϕ s ( · ) is locally minimal for each s ∈ ( − ε , ε ).Hence, we may assume that the triangle inequalities(4.13) L ( s, θ ) ≤ l + s = L ( s, L ( s, π − θ ) ≥ l − s = L ( s, π )hold for all sufficiently small | θ | and all s ∈ [0 , ε ). The equations (4.13) and (4.14) meanthat, for each s ∈ (0 , ε ), the function L ( s, · ) attains a local maximum (resp. minimum)at θ = 0 (resp. θ = π ). Hence, by (4.11) and (4.12),(4.15) ∂ R ∂θ ( s,
0) = ∂ R ∂θ ( s, π ) = 0for each s ∈ [0 , ε ). Since R ( s,
0) = R ( s, π ) = 0 holds on [0 , ε ), we see, by (4.15), thatthere exists a smooth function R ( s, θ ) such that(4.16) R ( s, θ ) = R ( s, θ ) θ ( π − θ ) . By (4.12) and (4.16), we have(4.17) R ( s, θ ) = R ( s, θ ) θ ( π − θ ) s for all θ ∈ [0 , π ] and all s ∈ [0 , ε ). On the other hand, since R depends continuously on p , ξ (0), and ξ ′ (0), there exists a constant C > |R ( s, θ ) | ≤ C holds for all p ∈ B ε ( r ), all ξ on S n − p , all θ ∈ [0 , π ], and all s ∈ [0 , ε / |R ( s, θ ) | ≤ C θ ( π − θ ) s for all p ∈ B ε ( r ), all ξ on S n − p , all θ ∈ [0 , π ] and all s ∈ [0 , ε / L ( s, θ ) ≤ l + s cos θ + s I l∂X ( Y ∂X , Y ∂X ) + C θ ( π − θ ) s .
15y applying the same argument above for e L λ ( s ) = e L λ ( s, θ ), there exists a constant C > e L λ ( s, θ ) ≥ l + s cos θ + s I l ( e Y λ , e Y λ ) − C θ ( π − θ ) s holds for all θ ∈ [0 , π ] and all s ∈ [0 , ε / e L λ ( s, θ ) − L ( s, θ ) ≥ s n I l ( e Y λ , e Y λ ) − I l∂X ( Y ∂X , Y ∂X ) o − ( C + C ) θ ( π − θ ) s (4.22) ≥ λC sin θ s − ( C + C ) θ ( π − θ ) s ≥ λC sin θ s − C θ ( π − θ ) s holds for all θ ∈ [0 , π ] and all s ∈ [0 , ε / C := max { C , C } .Since x sin x < π x ∈ (0 , π/ θ sin θ · ( π − θ ) < π · ( π − θ ) < π , π/ π − θ sin θ · θ = π − θ sin( π − θ ) · θ < π · θ < π π/ , π ). Hence, by (4.23) and (4.24), we see(4.25) θ ( π − θ )sin θ < π , π ). If we define δ := min (cid:26) ε , λC π C (cid:27) (cid:16) ≤ ε < ε (cid:17) , then, by (4.25), λC sin θ s − C θ ( π − θ ) s = ( s · sin θ ) " λC − C (cid:26) θ ( π − θ )sin θ (cid:27) s (4.26) > ( s · sin θ ) (cid:0) λC − π C s (cid:1) ≥ s ∈ [0 , δ ] and all θ ∈ (0 , π ). Therefore, by (4.22) and (4.26), the proof iscompleted. ✷ Thin open triangles
Throughout this section, let ( e X, ∂ e X ) denote a model surface with its metric (1.1). Lemma 5.1
Let ˜ µ : [0 , l ] → e X be a ∂ e X -segment. Then, for each < s < min { inj(˜ µ ( l )) , l } , the function d ( ∂ e X , exp ˜ µ ( l ) ( s ˜ ξ ( θ )) is strictly increasing on [0 , π ] . Here ˜ ξ : R → S µ ( l ) := { ˜ v ∈ T ˜ µ ( l ) e X ; k ˜ v k = 1 } denotes a unit speed geodesic segment on S µ ( l ) emanating from − ˜ µ ′ ( l ) = ˜ ξ (0) .Proof. This lemma is clear from the first variation formula. ✷ The next lemma is a direct consequence of the Clairaut relation ([SST, Theorem 7.1.2])and the first variational formula :
Lemma 5.2
For each constant c ≥ , and each point ˜ p ∈ e X , d (˜ p, ˜ τ c ( s )) is strictly in-creasing on [˜ y (˜ p ) , ∞ ) . Here ˜ τ c ( s ) := ( c, s ) ∈ e X denote the arc of ˜ x = c . By Lemma 5.2, we have
Lemma 5.3
Let
OT( ∂ e X, ˜ p , ˜ q ) and OT( ∂ e X , ˜ p , ˜ q ) be open triangles in e X such that (5.1) d ( ∂ e X, ˜ q ) = d ( ∂ e X, ˜ p ) , and that (5.2) ∠ ˜ q + ∠ ˜ p ≤ π. If d (˜ p , ˜ q ) + d (˜ p , ˜ q ) < inj(˜ p ) , then there exists an open triangle OT( ∂ e X , ˜ p, ˜ q ) such that (5.3) d ( ∂ e X , ˜ p ) = d ( ∂ e X, ˜ p ) , d (˜ p, ˜ q ) = d (˜ p , ˜ q ) + d (˜ p , ˜ q ) , d ( ∂ e X, ˜ q ) = d ( ∂ e X, ˜ q ) , and that (5.4) ∠ ˜ p ≥ ∠ ˜ p. Proof.
Let OT( ∂ e X, ˜ p , ˜ q ) := ( ∂ e X , ˜ p , ˜ q ; ˜ γ , ˜ µ (1)1 , ˜ µ (1)2 )and OT( ∂ e X, ˜ p , ˜ q ) := ( ∂ e X , ˜ p , ˜ q ; ˜ γ , ˜ µ (2)1 , ˜ µ (2)2 )17e open triangles in e X satisfying (5.1) and (5.2), and we fix them. By (5.1), we mayassume that OT( ∂ e X , ˜ p , ˜ q ) is adjacent to OT( ∂ e X , ˜ p , ˜ q ) as a common side ˜ µ (1)2 = ˜ µ (2)1 ,and that ˜ y (˜ p ) < ˜ y (˜ q ) = ˜ y (˜ p ) < ˜ y (˜ q ). Choose any number a ∈ ( d (˜ p , ˜ q ) + d (˜ p , ˜ q ) , inj(˜ p )) , and fix it. We will introduce geodesic polar coordinates ( r, θ ) around ˜ p on B a (˜ p ) suchthat θ = 0 on ˜ µ (1)1 ∩ B a (˜ p ), and that 0 < θ (˜ q ) ≤ θ (˜ q ) ≤ π . Notice that˜ q ∈ B a (˜ p ) . In fact, from the triangle inequality, we have d (˜ p , ˜ q ) ≤ d (˜ p , ˜ q ) + d (˜ q , ˜ q ) = d (˜ p , ˜ q ) + d (˜ p , ˜ q ) < a. Since there is nothing to prove if ∠ ˜ q + ∠ ˜ p = π , we may assume, by (5.2), that(5.5) ∠ ˜ q + ∠ ˜ p < π. Hence, ˜ q is in A (˜ p ), where A (˜ p ) is a domain defined by A (˜ p ) := B a (˜ p ) ∩ θ − (0 , θ (˜ q )) . Let ˜ τ : [˜ y (˜ q ) , ∞ ) → e X be an arc of ˜ x = ˜ x (˜ q ) emanating from ˜ q = ˜ τ (˜ y (˜ q )) ∈ A (˜ p )given by ˜ τ ( s ) := (˜ x (˜ q ) , s ). By (5.5), we get d (˜ p , ˜ τ (˜ y (˜ q ))) < d (˜ p , ˜ q ) + d (˜ q , ˜ q ) = d (˜ p , ˜ q ) + d (˜ p , ˜ q ) < a. Since lim s →∞ d (˜ p , ˜ τ ( s )) = ∞ , it follows from the intermediate value theorem that thereexists a number s ∈ (˜ y (˜ q ) , ∞ ) satisfying d (˜ p , ˜ τ ( s )) = a , and furthermore that thereexists a number s ∈ (˜ y (˜ q ) , s ) satisfying(5.6) d (˜ p , ˜ τ ( s )) = d (˜ p , ˜ q ) + d (˜ q , ˜ q ) = d (˜ p , ˜ q ) + d (˜ p , ˜ q ) . We will prove that the subarc ˜ τ | [˜ y (˜ q ) , s ] is contained in A (˜ p ). Suppose that there existsa number s ∈ (˜ y (˜ q ) , s ] such that ˜ τ ( s ) is not in A (˜ p ). Since the subarc ˜ τ | [˜ y (˜ q ) , s ] liesin B a (˜ p ), there exists s ∈ (˜ y (˜ q ) , s ] such that(5.7) θ (˜ τ ( s )) = θ (˜ q ) . Since ˜ y (˜ q ) < s , we have, by Lemma 5.2,(5.8) d (˜ q , ˜ q ) < d (˜ q , ˜ τ ( s )) . By (5.7), we see that the geodesic extension ˜ σ : [0 , d (˜ p , ˜ τ ( s ))] → e X of ˜ γ meets ˜ τ at˜ τ ( s ) = ˜ σ ( d (˜ p , ˜ τ ( s ))). Notice that the geodesic segment ˜ σ is minimal, since˜ τ ( s ) ∈ B a (˜ p ) ⊂ B inj(˜ p ) (˜ p ) . d (˜ p , ˜ τ ( s )) = d (˜ p , ˜ q ) + d (˜ q , ˜ τ ( s )) > d (˜ p , ˜ q ) + d (˜ q , ˜ q ) = d (˜ p , ˜ τ ( s )) . On the other hand, since s < s , it follows from Lemma 5.2 that(5.10) d (˜ p , ˜ τ ( s )) < d (˜ p , ˜ τ ( s )) . The equation (5.10) contradicts the equation (5.9). Therefore, we have proved that thesubarc ˜ τ | [˜ y (˜ q ) , s ] is contained in A (˜ p ).Since the minimal geodesic segment ˜ γ : [0 , d (˜ p , ˜ τ ( s ))] → e X joining ˜ p to ˜ τ ( s ) lies inthe closure of A (˜ p ), the inequality ∠ ˜ p ≥ ∠ ˜ γ ′ (0) , − d ˜ µ (1)1 dt ( d ( ∂ e X, ˜ p )) ! holds. Hence, it is clear that the open triangle OT( ∂ e X , ˜ p, ˜ q ) := ( ∂ e X, ˜ p, ˜ q ; ∂ e X , ˜ p , ˜ τ ( s ))satisfies (5.3) and (5.4) in our lemma. ✷ Hereafter, let (
X, ∂X ) be a complete connected Riemannian n -manifold X with smooth convex boundary ∂X whose radial curvature is bounded from below by that of ( e X, ∂ e X ).Let λ denote the positive number guaranteed in Lemma 4.4. Choose any number λ ∈ (0 , λ ) and fix it. In the following, for the λ , we also denote by ( e X λ , ∂ e X λ ) a modelsurface with its metric ˜ g λ = d ˜ x + m λ (˜ x ) d ˜ y on [0 , ∞ ) × R . Here the positive smooth function m λ satisfies the differential equation m ′′ λ + ( G − λ ) m λ = 0with initial conditions m λ (0) = 1 and m ′ λ (0) = 0, where G denotes the radial curvaturefunction of ( e X, ∂ e X ). Then, the next lemma is clear from Lemmas 4.5 and 5.1: Lemma 5.4
Let p be a point in X \ ( ∂X ∪ Foc( ∂X )) , and δ ( p ) the number δ guaranteedin Lemma 4.5 to the point r := p . Then, for any q ∈ X with d ( p, q ) < δ ( p ) , there existsan open triangle OT( ∂ e X λ , ˜ p, ˜ q ) in e X λ corresponding to the triangle OT( ∂X, p, q ) in X such that (5.11) d ( ∂ e X λ , ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X λ , ˜ q ) = d ( ∂X, q ) and that (5.12) ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q. By Lemmas 5.3 and 5.4, we have the following lemma.19 emma 5.5
For every thin open triangle
OT( ∂X, p, q ) in X with γ ∩ Foc( ∂X ) = ∅ , thereexists an open triangle OT( ∂ e X λ , ˜ p, ˜ q ) in e X λ such that (5.13) d ( ∂ e X λ , ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X λ , ˜ q ) = d ( ∂X, q ) and that (5.14) ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q. Here γ denotes the opposite side of OT( ∂X, p, q ) to ∂X emanating from p to q .Proof. It is sufficient to prove that(5.15) max S = d ( p, q ) , where S denotes the set of all s ∈ [0 , d ( p, q )] such that there exists an open triangleOT( ∂ e X λ , ˜ p, ˜ γ ( s )) ⊂ e X λ corresponding to the triangle OT( ∂X, p, γ ( s )) ⊂ X satisfying(5.13) and (5.14) for q = γ ( s ). From Lemma 5.4, it is clear that S is non-empty. Supposingthat s := max S < d ( p, q ), we will get a contradiction. Since s ∈ S , there exists an opentriangle OT( ∂ e X λ , ˜ p , ˜ q ) ⊂ e X λ corresponding to OT( ∂X, p, γ ( s )) ⊂ X such that (5.13)and (5.14) hold for q = γ ( s ), ˜ p = ˜ p , and ˜ q = ˜ q . In particular,(5.16) ∠ p ≥ ∠ ˜ p , ∠ ( ∂X, γ ( s ) , p ) ≥ ∠ ˜ q , where ∠ ( ∂X, γ ( s ) , p ) denotes the angle between two sides joining γ ( s ) to ∂X and p forming the triangle OT( ∂X, p, γ ( s )). Let δ ( γ ( s )) denote the number δ guaranteed inLemma 4.5 to the point r := γ ( s ). Choose a sufficiently small number ε with0 < ε < min { δ ( γ ( s )) , d ( p, q ) − s } , and fix it. By Lemma 5.4, we have an open triangle OT( ∂ e X λ , ˜ p , ˜ q ) ⊂ e X λ correspondingto OT( ∂X, γ ( s ) , γ ( s + ε )) ⊂ X such that (5.13) and (5.14) hold for p = γ ( s ), q = γ ( s + ε ), ˜ p = ˜ p , and ˜ q = ˜ q . In particular,(5.17) ∠ ( ∂X, γ ( s ) , γ ( s + ε )) ≥ ∠ ˜ p , ∠ γ ( s + ε ) ≥ ∠ ˜ q . Since ∠ ( ∂X, γ ( s ) , p ) + ∠ ( ∂X, γ ( s ) , γ ( s + ε )) = π, we get, by (5.16) and (5.17), ∠ ˜ q + ∠ ˜ p ≤ π. Since OT( ∂X, p, q ) is a thin open triangle,min { inj(˜ p ) , inj(˜ q ) } ≥ L ( γ ) ≥ d (˜ p , ˜ q ) + d (˜ p , ˜ q )holds. Thus, if we apply Lemma 5.3 twice for the pair OT( ∂ e X λ , ˜ p , ˜ q ) and OT( ∂ e X λ , ˜ p , ˜ q ),we get two open triangles OT( ∂ e X λ , b p, b γ ( s + ε )) and OT( ∂ e X λ , ˜ p, ˜ γ ( s + ε )) in e X λ suchthat(5.18) ∠ ˜ p ≥ ∠ b p, ∠ ˜ q ≥ ∠ ˜ γ ( s + ε ) . ∂ e X λ , b p, b γ ( s + ε )) and OT( ∂ e X λ , ˜ p, ˜ γ ( s + ε )) are isometric, weobtain(5.19) ∠ b p = ∠ ˜ p. Hence, by (5.16), (5.17), (5.18), and (5.19), both open triangles OT( ∂X, p, γ ( s + ε )) andOT( ∂ e X λ , ˜ p, ˜ γ ( s + ε )) satisfy (5.13) for q = γ ( s + ε ) and ˜ q = ˜ γ ( s + ε ) and ∠ p ≥ ∠ ˜ p, ∠ γ ( s + ε ) ≥ ∠ ˜ γ ( s + ε ) . This implies that s + ε is in S . This contradicts the fact that s is the maximum of S .Hence (5.15) holds. ✷ Lemma 5.6
For every thin open triangle
OT( ∂X, p, q ) in X with (5.20) p Foc( ∂X ) , there exists an open triangle OT( ∂ e X λ , ˜ p, ˜ q ) in e X λ such that (5.21) d ( ∂ e X λ , ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X λ , ˜ q ) = d ( ∂X, q ) and that (5.22) ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q. Proof.
Let OT( ∂X, p, q ) := ( ∂X, p, q ; γ, µ , µ ) be a thin open triangle in X satisfying(5.20), and we fix it. Since p is not a focal point of ∂X , and q is not a cut point of p ,it follows from Lemma 3.5 that there exists a sequence { γ i : [0 , l i ] → X } i ∈ N of minimalgeodesic segments γ i emanating from p = γ i (0) convergent to the opposite side γ ofOT( ∂X, p, q ) to ∂X such that γ i ([0 , l i ]) ∩ FC( ∂X ) = ∅ , and that lim i →∞ k exp − q ( γ i ( l i )) k exp − q ( γ i ( l i )) = − γ ′ ( l ) , where l := d ( p, q ). Then, we may find a sufficiently large i ∈ N such that an opentriangle OT( ∂X, p, γ i ( l i )) = ( ∂X, p, γ i ( l i ) ; γ i , µ , η i ) is thin in X for each i ≥ i . Hereeach η i is a ∂X -segment to γ i ( l i ). Choose any i ≥ i and fix it. By Lemma 5.5, thereexists an open triangle OT( ∂ e X λ , ˜ p, ˜ γ i ( l i )) = ( ∂ e X λ , ˜ p, ˜ γ i ( l i ) ; ˜ γ i , ˜ µ , ˜ η i ) ⊂ e X λ correspondingto OT( ∂X, p, γ i ( l i )) such that (5.13) hold for q = γ i ( l i ), and(5.23) ∠ ( − µ ′ ( d ( ∂X, p )) , γ ′ i (0)) ≥ ∠ ( − ˜ µ ′ ( d ( ∂X, p )) , ˜ γ ′ i (0)) , (5.24) ∠ ( η ′ i ( d ( ∂X, γ i ( l i ))) , γ ′ i ( l i )) ≥ ∠ (˜ η ′ i ( d ( ∂X, γ i ( l i ))) , ˜ γ ′ i ( l i )) . Since lim i →∞ γ ′ i (0) = γ ′ (0),(5.25) ∠ p = lim i →∞ ∠ ( − µ ′ ( d ( ∂X, p )) , γ ′ i (0)) .
21n the other hand,(5.26) ∠ q ≥ lim sup i →∞ ∠ ( η ′ i ( d ( ∂X, γ i ( l i ))) , γ ′ i ( l i ))holds by [IT2, Lemma 2.1]. Then, from (5.23), (5.24), (5.25), (5.26), it follows that ∠ p ≥ lim i →∞ ∠ ( − ˜ µ ′ ( d ( ∂X, p )) , ˜ γ ′ i (0)) , and that ∠ q ≥ lim i →∞ ∠ (˜ η ′ i ( d ( ∂X, γ i ( l i ))) , ˜ γ ′ i ( l i )) . By taking the limit of the sequence OT( ∂ e X λ , ˜ p, ˜ γ i ( l i )), we therefore get an open triangleOT( ∂ e X λ , ˜ p, ˜ q ) corresponding to OT( ∂X, p, q ) ⊂ X such that (5.21) and (5.22) hold. ✷ Lemma 5.7
For every thin open triangle
OT( ∂X, p, q ) in X , there exists an open triangle OT( ∂ e X λ , ˜ p, ˜ q ) in e X λ such that (5.27) d ( ∂ e X λ , ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X λ , ˜ q ) = d ( ∂X, q ) and that (5.28) ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q. Proof.
Let OT( ∂X, p, q ) := ( ∂X, p, q ; γ, µ , µ ) be a thin open triangle in X , and we fixit. Take any sufficiently small ε > q is not in Cut( p ε ), where we set p ε := µ ( d ( ∂X, p ) − ε ). Let µ ε denote the restriction of µ , i.e., µ ε ( t ) := µ ( t ) on [0 , d ( ∂X, p ) − ε ]. Without loss of generality, we may assume that the open triangle OT( ∂X, p ε , q ) =( ∂X, p ε , q ; γ ε , µ ε , µ ) is thin. Here γ ε : [0 , l ε ] → X denotes the minimal geodesic segmentemanating from p ε = γ ε (0) to q = γ ε ( l ε ). Since p ε Foc( ∂X ), it follows from Lemma5.6 that there exists an open triangle OT( ∂ e X λ , ˜ p ε , ˜ q ) = ( ∂ e X λ , ˜ p ε , ˜ q ; ˜ γ ε , ˜ µ ε , ˜ µ ) ⊂ e X λ corresponding to OT( ∂X, p ε , q ) such that (5.21) holds for p = p ε , and that ∠ ( − µ ′ ε ( d ( ∂X, p ε )) , γ ′ ε (0)) ≥ ∠ ( − ˜ µ ′ ε ( d ( ∂X, p ε )) , ˜ γ ′ ε (0)) , ∠ ( µ ′ ( d ( ∂X, q )) , γ ′ ε ( l ε )) ≥ ∠ (˜ µ ′ ( d ( ∂X, q )) , ˜ γ ′ ε ( l ε )) . Since lim ε ↓ γ ε = γ , we have ∠ p = lim ε ↓ ∠ ( − µ ′ ε ( d ( ∂X, p ε )) , γ ′ ε (0))and ∠ q = lim ε ↓ ∠ ( µ ′ ( d ( ∂X, q )) , γ ′ ε ( l ε )) . If ε goes to 0, we therefore get an open triangle OT( ∂ e X λ , ˜ p, ˜ q ) ⊂ e X λ corresponding toOT( ∂X, p, q ) ⊂ X such that (5.27) and (5.28) hold. ✷ By taking the limit of λ , it follows from Lemma 5.7 that we have the lemma on thinopen triangles: 22 emma 5.8 (Lemma on thin open triangles) For every thin open triangle
OT( ∂X, p, q ) in X , there exists an open triangle OT( ∂ e X, ˜ p, ˜ q ) in e X such that (5.29) d ( ∂ e X , ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X, ˜ q ) = d ( ∂X, q ) and that (5.30) ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q. Remark 5.9
From Section 4 and this section, it has been clarified that we can proveLemma on thin open triangles by the second variation formula and the Warner comparisontheorem. But, we can not do by the first variation formula and the Berger comparisontheorem. ∂X of an open triangle In Definition 2.1, the opposite side to ∂X of a thin open triangle is defined not to meetthe boundary. In this section, we will show that the opposite side to ∂X of any opentriangle on any complete connected Riemannian manifold X with smooth boundary ∂X does not meet ∂X , if ∂X is convex. Lemma 6.1
Let ( X, ∂X ) be a complete connected Riemannian n -dimensional manifold X with smooth boundary ∂X whose radial curvature is bounded from below by that of amodel surface ( e X, ∂ e X ) . If ∂X is convex, then, for any open triangle OT( ∂X, p, q ) in X ,the opposite side γ of OT( ∂X, p, q ) to ∂X emanating from p to q does not meet ∂X .Proof. Suppose that γ intersects ∂X at γ ( s ) for some s ∈ (0 , d ( p, q )). Without loss ofgenerality, we may assume that γ ((0 , s )) ∩ ∂X = ∅ . Since γ intersects ∂X at γ ( s ), γ is tangent to ∂X at γ ( s ).It is well-known that each point of e X admits a normal convex neighborhood. Hence,there exists a constant C > q s ) > C for all s ∈ [0 , s ], where ˜ q s denotesa point in e X satisfying d ( ∂ e X , ˜ q s ) = d ( ∂X, γ ( s )) . By this property, we may choose a number s ∈ (0 , s ) in such a way that L ( γ | [ s , s ] ) = s − s < inj(˜ q s )for all s ∈ [ s , s ]. Therefore, for each s ∈ [ s , s ), any open triangle OT( ∂X, γ ( s ) , γ ( s ))in X is thin, if s ∈ [ s , s ) \ { s } is sufficiently close to s .Let S denote the set of all s ∈ ( s , s ) such that there exists an open triangleOT( ∂ e X , ˜ γ ( s ) , ˜ γ ( s )) ⊂ e X corresponding to the triangle OT( ∂X, γ ( s ) , γ ( s )) ⊂ X sat-isfying(6.1) d ( ∂ e X, ˜ γ ( s )) = d ( ∂X, γ ( s )) , d (˜ γ ( s ) , ˜ γ ( s )) = s − s , d ( ∂ e X, ˜ γ ( s )) = d ( ∂X, γ ( s )) , ∠ γ ( s ) ≥ ∠ ˜ γ ( s ) , ∠ γ ( s ) ≥ ∠ ˜ γ ( s ) . By Lemma 5.8, the set S is non-empty. By the similar argument in the proof of Lemma5.5, we see that sup S = s . Hence, there exists a decreasing sequence { ε i } i ∈ N convergentto 0 such that s − ε i is in S for all i ∈ N . For each i ∈ N , there exists an open triangleOT( ∂ e X , ˜ γ ( s ) , ˜ γ ( s − ε i )) ⊂ e X corresponding to the triangle OT( ∂X, γ ( s ) , γ ( s − ε i )) ⊂ X such that (6.1) and (6.2) hold for γ ( s ) = γ ( s − ε i ). Since γ is tangent to ∂X at γ ( s ),we have lim s ↑ s ∠ γ ( s ) = π . Thus, by (6.2) for γ ( s ) = γ ( s − ε i ), we getlim sup i →∞ ∠ ˜ γ ( s − ε i ) ≤ π . If lim inf i →∞ ∠ ˜ γ ( s − ε i ) < π ∂ e X of OT( ∂ e X , ˜ γ ( s ) , ˜ γ ( s − ε i )) meets ∂ e X for some s ∈ ( s , s )sufficiently close to s , which contradicts the fact that ∂ e X is totally geodesic. Hence,lim i →∞ ∠ ˜ γ ( s − ε i ) = π ∂ e X of the limit open triangle OT( ∂ e X , ˜ γ ( s ) , ˜ γ ( s − ε i ))as i → ∞ is tangent to ∂ e X . This is also a contradiction, since ∂ e X is totally geodesic.Therefore, γ does not meet ∂X . ✷ By the same argument as in the proof of [KT2, Lemma 5.1], we have the followinglemma.
Lemma 6.2
For any complete connected Riemannian manifold X with smooth boundary ∂X and any compact subset K of ∂X , there exists a locally Lipschitz function G ( t ) on [0 , ∞ ) such that the radial curvature of X with respect to any ∂X -segment emanatingfrom K is bounded from below by that of the model surface with radial curvature function G ( t ) . Proposition 6.3
Let X be a complete connected Riemannian manifold X with smoothboundary ∂X . If ∂X is convex, then the opposite side to ∂X of any open triangle on X does not meet ∂X .Proof. By the definition of an open triangle, both feet p and q of an open triangleOT( ∂X, p, q ) ⊂ X are contained in a bounded set of ∂X . Hence it is clear from Lemmas6.1 and 6.2 that γ does not meet ∂X . ✷ Alexandrov’s convexity
Our purpose of this section is to establish the Alexandrov convexity (Lemma 7.3).Throughout this section, let ( e X, ∂ e X ) denote a model surface with its metric (1.1). Itfollows from Lemma 5.2 that(7.1) lim s ↓ s d (˜ p, ˜ τ c ( s )) − d (˜ p, ˜ τ c ( s )) s − s ≥ s > ˜ y (˜ p ). The following two lemmas are useful for proving that the function D defined in Lemma 7.3 is locally Lipschitz. In the first lemma, we will prove that theleft-hand term of the equation (7.1) is strictly positive : Lemma 7.1
For each ∂ e X -ray ˜ µ : [0 , ∞ ) → e X and each number a , c > , s > ˜ y (˜ µ (0)) ,there exist numbers ε > and δ > such that (7.2) | d (˜ µ ( a ) , ˜ τ c ( s )) − d (˜ µ ( a ) , ˜ τ c ( s )) | ≥ | s − s | · m ( c ) · sin ε holds for all a ∈ ( a − δ, a + δ ) , c ∈ ( c − δ, c + δ ) , and s ∈ ( s − δ, s + δ ) .Proof. We choose a positive number δ less than min { a , c , s − ˜ y (˜ µ (0)) } , and fix it. Let a, c, s be any numbers in ( a − δ, a + δ ), ( c − δ, c + δ ), and ( s − δ, s + δ ), respectively.Since no minimal geodesic segment joining ˜ µ ( a ) to ˜ τ c ( s ) is perpendicular to ˜ τ c , there existsa positive number ε ∈ (0 , π/
2) such that(7.3) Φ(˜ γ, s ) := ∠ (˜ τ ′ c ( s ) , ˜ γ ′ ( d (˜ µ ( a ) , ˜ τ c ( s )))) ≤ π − ε holds for all a ∈ ( a − δ, a + δ ) c ∈ ( c − δ, c + δ ), s ∈ ( s − δ, s + δ ), and minimalgeodesic segments ˜ γ joining ˜ µ ( a ) to ˜ τ c ( s ). By [IT2, Lemma 2.1] and (7.3), we havelim inf s ↓ s d (˜ µ ( a ) , ˜ τ c ( s )) − d (˜ µ ( a ) , ˜ τ c ( s )) d (˜ τ c ( s ) , ˜ τ c ( s )) ≥ sin ε . This equation implies (7.2) (see the proof of [KT2, Lemma 4.2] for the detail of thisproof). ✷ Notice that, for given a triple ( a, b, c ) of positive numbers a, b, c , there exists an opentriangle OT( ∂ e X , ˜ p, ˜ q ) in e X with ∠ ˜ p, ∠ ˜ q ∈ (0 , π ) satisfying a = d ( ∂ e X, ˜ p ), b = d (˜ p, ˜ q ), and c = d ( ∂ e X , ˜ q ) if and only if( a, b, c ) ∈ T := { ( a, b, c ) ∈ R ; a, c > , | a − c | < b } . By Lemma 5.2, the existence of such a triangle OT( ∂ e X , ˜ p, ˜ q ) = ( ∂ e X , ˜ p, ˜ q ; ˜ γ, ˜ µ , ˜ µ ) isunique up to an isometry except for the opposite side ˜ γ to ∂ e X . Hence,Θ( a, b, c ) := | ˜ y (˜ µ (0)) − ˜ y (˜ µ (0)) | is a well-defined function on the set T . 25 emma 7.2 The function Θ( a, b, c ) is locally Lipschitz.Proof. Choose any point ( a , b , c ) ∈ T , and fix it. Let ˜ µ : [0 , ∞ ) → e X be the ∂ e X -raywith ˜ y (˜ µ (0)) = 0. Moreover, we choose the ∂ e X -ray ˜ µ : [0 , ∞ ) → e X in such a way that d (˜ µ ( a ) , ˜ µ ( c )) = b and ˜ y (˜ µ (0)) >
0. By setting ˜ p := ˜ µ ( a ) and ˜ q := ˜ µ ( c ), wehence get an open triangle OT( ∂ e X, ˜ p , ˜ q ) ⊂ e X with ∠ ˜ p , ∠ ˜ q ∈ (0 , π ) satisfying a = d ( ∂ e X , ˜ p ) , b = d (˜ p , ˜ q ) , c = d ( ∂ e X , ˜ q ) . First we will prove that(7.4) | Θ( a + ∆ a, b , c ) − Θ( a , b , c ) | ≤ m ( c ) sin ε | ∆ a | for all ∆ a ∈ R with | ∆ a | < δ . Here the numbers ε and δ are the constants guaranteed to a , c , and s := ˜ y (˜ µ (0)) > τ c : R → e X be the arc ˜ x = c . Then, wemay find a point ˜ q ∆ a on ˜ τ c satisfying b = d (˜ p ∆ a , ˜ q ∆ a ), where ˜ p ∆ a := ˜ µ ( a + ∆ a ). Thus,we also get an open triangle OT( ∂ e X , ˜ p ∆ a , ˜ q ∆ a ) ⊂ e X with ∠ ˜ p ∆ a , ∠ ˜ q ∆ a ∈ (0 , π ) satisfying a + ∆ a = d ( ∂ e X, ˜ p ∆ a ) , b = d (˜ p ∆ a , ˜ q ∆ a ) , c = d ( ∂ e X, ˜ q ∆ a ) . Let ˜ µ ∆ a denote the side of OT( ∂ e X, ˜ p ∆ a , ˜ q ∆ a ) joining ∂ e X to ˜ q ∆ a . By definition,(7.5) Θ( a + ∆ a, b , c ) = ˜ y (˜ µ ∆ a (0)) , and(7.6) Θ( a , b , c ) = ˜ y (˜ µ (0)) = s . Here we may assume that ˜ y (˜ µ ∆ a (0)) >
0. It is clear from (7.5) and (7.6) that(7.7) | ∆ a Θ | = | ˜ y (˜ µ ∆ a (0)) − s | , where we set ∆ a Θ := Θ( a + ∆ a, b , c ) − Θ( a , b , c ) . Thus, by (7.7), the length | ∆ s | of the subarc of ˜ τ c with end points ˜ q ∆ a and ˜ q is equal to(7.8) | ∆ s | = m ( c ) · | ˜ y (˜ µ ∆ a (0)) − s | = m ( c ) · | ∆ a Θ | . It follows from Lemma 7.1 and (7.8) that(7.9) | d (˜ p , ˜ q ) − d (˜ p , ˜ q ∆ a ) | ≥ | s − ˜ y (˜ µ ∆ a (0)) | · m ( c ) · sin ε = | ∆ s | sin ε . Since b = d (˜ p , ˜ q ) = d (˜ p ∆ a , ˜ q ∆ a ) , we get, by (7.9),(7.10) | d (˜ p ∆ a , ˜ q ∆ a ) − d (˜ p , ˜ q ∆ a ) | ≥ | ∆ s | sin ε .
26y the triangle inequality, we have(7.11) | ∆ a | = d (˜ p , ˜ p ∆ a ) ≥ | d (˜ p ∆ a , ˜ q ∆ a ) − d (˜ p , ˜ q ∆ a ) | . By combining the equations (7.8), (7.10), and (7.11), we obtain (7.4). Since Θ( a, b, c ) =Θ( c, b, a ) for all ( a, b, c ) ∈ T , it is clear that(7.12) | Θ( a , b , c + ∆ c ) − Θ( a , b , c ) | ≤ m ( c ) sin ε | ∆ c | for all ∆ c ∈ R with | ∆ c | < δ . We omit the proof of the following inequality, since theproof is similar to that of (7.4) :(7.13) | Θ( a , b + ∆ b, c ) − Θ( a , b , c ) | ≤ m ( c ) sin ε | ∆ b | for all ∆ b ∈ R with | ∆ b | < δ . Therefore, the function Θ( a, b, c ) is locally Lipschitz at( a , b , c ) ∈ T by (7.4), (7.12), and (7.13). ✷ Lemma 7.3 (Alexandrov’s convexity for open triangles)
Let ( X, ∂X ) be a com-plete connected Riemannian n -dimensional manifold X with smooth convex boundary ∂X whose radial curvature is bounded from below by that of ( e X, ∂ e X ) , and let OT( ∂X, p, q ) =( ∂X, p, q ; γ, µ , µ ) be a non-degenerate open triangle in X , i.e., ∠ p, ∠ q ∈ (0 , π ) . Assumethat, for each open triangle OT( ∂X, µ ( at ) , µ ( ct )) = ( ∂X, µ ( at ) , µ ( ct ) ; γ t , µ | [0 , at ] , µ | [0 , ct ] ) , t ∈ (0 , , where a = d ( ∂X, p ) and c = d ( ∂X, q ) , there exists a unique open triangle OT( ∂ e X, ˜ µ ( t )1 ( at ) , ˜ µ ( t )2 ( ct )) = ( ∂ e X , ˜ µ ( t )1 ( at ) , ˜ µ ( t )2 ( ct ) ; ˜ γ t , ˜ µ ( t )1 | [0 , at ] , ˜ µ ( t )2 | [0 , ct ] ) up to an isometry in e X such that (7.14) d ( ∂ e X, ˜ µ ( t )1 ( at )) = d ( ∂X, µ ( at )) , d ( ∂ e X, ˜ µ ( t )2 ( ct )) = d ( ∂X, µ ( ct )) , (7.15) d (˜ µ ( t )1 ( at ) , ˜ µ ( t )2 ( ct )) = d ( µ ( at ) , µ ( ct )) , and that (7.16) ∠ µ ( at ) ≥ ∠ ˜ µ ( t )1 ( at ) , ∠ µ ( ct ) ≥ ∠ ˜ µ ( t )2 ( ct ) . Then, the function D ( t ) := | ˜ y (˜ µ ( t )1 (0)) − ˜ y (˜ µ ( t )2 (0)) | is locally Lipschitz on (0 , , and non-increasing on (0 , . roof. We will state the outline of the proof, since the proof is very similar to [KT2,Lemma 4.4]. If we define a Lipschitz function ϕ on [0 ,
1] by ϕ ( t ) := d ( µ ( at ) , µ ( ct )) , then the function D ( t ) is equal to Θ( at, ϕ ( t ) , ct ). Hence D ( t ) is locally Lipschitz on (0 , D ( t ) is differentiable for almost all t ∈ (0 , t ∈ (0 ,
1) be any numberwhere D ( t ) is differentiable. Then, by the assumption, we have an open triangleOT( ∂ e X , ˜ µ ( t )1 ( at ) , ˜ µ ( t )2 ( ct )) = ( ∂ e X , ˜ µ ( t )1 ( at ) , ˜ µ ( t )2 ( ct ) ; ˜ γ t , ˜ µ ( t )1 | [0 , at ] , ˜ µ ( t )2 | [0 , ct ] )in e X corresponding to the triangleOT( ∂X, µ ( at ) , µ ( ct )) = ( ∂X, µ ( at ) , µ ( ct ) ; γ t , µ | [0 , at ] , µ | [0 , ct ] )in X such that (7.14), (7.15), and (7.16) hold. Since OT( ∂X, µ ( at ) , µ ( ct )) is non-degenerate, we may assume, without loss of generality, that0 = ˜ y (˜ µ ( t )1 ( at )) < ˜ y (˜ µ ( t )2 ( ct )) . Let ˜ µ, ˜ η : [0 , ∞ ) → e X be ∂ e X -rays passing through ˜ µ ( t )1 ( at ) = ˜ µ ( at ), ˜ µ ( t )2 ( ct ) = ˜ η ( ct ),respectively. We define a function e ψ ( t ) := d (˜ µ ( at ) , ˜ η ( ct )) . Since ˜ γ t is unique, we may prove that the function e ψ ( t ) is differentiable at t = t , andthat(7.17) e ψ ′ ( t ) = cos( ∠ ˜ µ ( t )1 ( at )) + cos( ∠ ˜ µ ( t )2 ( ct )) . Indeed, let ˜ z and ˜ z t denote the midpoints of ˜ γ t and ˜ µ ( at )˜ η ( ct ), respectively. Here,˜ µ ( at )˜ η ( ct ) denotes a minimal geodesic segment joining ˜ µ ( at ) to ˜ η ( ct ). Since there existsa unique minimal geodesic segment joining ˜ µ ( t )1 ( at ) = ˜ µ ( at ) to ˜ µ ( t )2 ( ct ) = ˜ η ( ct ),(7.18) lim t → t ˜ z t = ˜ z holds. By the triangle inequality, we have e ψ ( t ) − e ψ ( t ) ≤ d (˜ µ ( at ) , ˜ z ) + d (˜ η ( ct ) , ˜ z ) − d (˜ µ ( at ) , ˜ z ) − d (˜ η ( ct ) , ˜ z )and e ψ ( t ) − e ψ ( t ) ≥ d (˜ µ ( at ) , ˜ z t ) + d (˜ η ( ct ) , ˜ z t ) − d (˜ µ ( at ) , ˜ z t ) − d (˜ η ( ct ) , ˜ z t ) . Hence, lim sup t ↓ t e ψ ( t ) − e ψ ( t ) t − t (7.19) ≤ lim sup t ↓ t d (˜ µ ( at ) , ˜ z ) − d (˜ µ ( at ) , ˜ z ) t − t + lim sup t ↓ t d (˜ η ( ct ) , ˜ z ) − d (˜ η ( ct ) , ˜ z ) t − t t ↓ t e ψ ( t ) − e ψ ( t ) t − t (7.20) ≥ lim inf t ↓ t d (˜ µ ( at ) , ˜ z t ) − d (˜ µ ( at ) , ˜ z t ) t − t + lim inf t ↓ t d (˜ η ( ct ) , ˜ z t ) − d (˜ η ( ct ) , ˜ z t ) t − t hold. From the first variation formula, we have(7.21) lim sup t ↓ t d (˜ µ ( at ) , ˜ z ) − d (˜ µ ( at ) , ˜ z ) t − t = cos( ∠ ˜ µ ( t )1 ( at ))and(7.22) lim sup t ↓ t d (˜ η ( ct ) , ˜ z ) − d (˜ η ( ct ) , ˜ z ) t − t = cos( ∠ ˜ µ ( t )2 ( ct )) . By imitating the proof of [IT2, Lemma 2.1], we obtain(7.23) lim inf t ↓ t d (˜ µ ( at ) , ˜ z t ) − d (˜ µ ( at ) , ˜ z t ) t − t = cos( ∠ ˜ µ ( t )1 ( at ))and(7.24) lim inf t ↓ t d (˜ η ( ct ) , ˜ z t ) − d (˜ η ( ct ) , ˜ z t ) t − t = cos( ∠ ˜ µ ( t )2 ( ct )) . In the above equations, notice (7.18). Combining (7.19), (7.20), (7.21), (7.22), (7.23), and(7.24), we have(7.25) lim t ↓ t e ψ ( t ) − e ψ ( t ) t − t = cos( ∠ ˜ µ ( t )1 ( at )) + cos( ∠ ˜ µ ( t )2 ( ct )) . By the same way, we also see(7.26) lim t ↑ t e ψ ( t ) − e ψ ( t ) t − t = cos( ∠ ˜ µ ( t )1 ( at )) + cos( ∠ ˜ µ ( t )2 ( ct )) . From (7.25) and (7.26), we hence get (7.17). As well as above, since ϕ is differentiable at t = t , we also get(7.27) ϕ ′ ( t ) = cos( ∠ µ ( at )) + cos( ∠ µ ( ct )) . By (7.16), (7.17), and (7.27), we get ϕ ′ ( t ) ≤ e ψ ′ ( t ). Hence, we conclude that D ′ ( t ) ≤ D ′ ( t ) ≤ t ∈ (0 , D ( t ) is non-increasing, since D ( t ) is locally Lipschitz. ✷ Remark 7.4
As pointed out in [KT2, Remark 4.5], it is a very important property that D ( t ) is locally Lipschitz. Without this property, we can not conclude that D ( t ) is non-increasing. 29 Toponogov’s comparison theorem
Our purpose of this section is to prove our main theorem, i.e., the Toponogov comparisontheorem for open triangles (Theorem 8.4), by using new techniques established in [KT2,Section 4] and Lemmas 5.2, 5.8, and 7.3.Throughout this section, let ( e X, ∂ e X ) denote a model surface with its metric (1.1). Lemma 8.1
Let
OT( ∂ e X, ˜ p , ˜ q ) = ( ∂ e X, ˜ p , ˜ q ) ; ˜ γ , ˜ µ (1)1 , ˜ µ (1)2 ) and OT( ∂ e X , ˜ p , ˜ q ) = ( ∂ e X , ˜ p , ˜ q ; ˜ γ , ˜ µ (2)1 , ˜ µ (2)2 ) be open triangles in e X such that (8.1) d ( ∂ e X, ˜ q ) = d ( ∂ e X, ˜ p ) , and that (8.2) ∠ ˜ q + ∠ ˜ p ≤ π. If there exists an open triangle
OT( ∂ e X , ˜ p, ˜ q ) = ( ∂ e X , ˜ p, ˜ q ; ˜ γ, ˜ µ , ˜ µ ) in a sector e X ( θ ) ,which has no pair of cut points, satisfying (8.3) d ( ∂ e X , ˜ p ) = d ( ∂ e X, ˜ p ) , d (˜ p, ˜ q ) = d (˜ p , ˜ q ) + d (˜ p , ˜ q ) , d ( ∂ e X, ˜ q ) = d ( ∂ e X, ˜ q ) , then ∠ ˜ p ≥ ∠ ˜ p, ∠ ˜ q ≥ ∠ ˜ q. Proof.
By (8.1), we may assume that OT( ∂ e X, ˜ p , ˜ q ) is adjacent to OT( ∂ e X , ˜ p , ˜ q ) so asto have a common side ˜ µ (1)2 = ˜ µ (2)1 , i.e., ˜ q = ˜ p . We may also assume that0 = ˜ y (˜ p ) < ˜ y (˜ q ) = ˜ y (˜ p ) < ˜ y (˜ q ) . Furthermore, we may assume that ˜ p = ˜ p and ˜ y (˜ q ) >
0. Remark that ˜ µ = ˜ µ (1)1 . If ∠ ˜ q + ∠ ˜ p = π holds, then there is nothing to prove. Thus, by (8.2), we may assumethat ∠ ˜ q + ∠ ˜ p < π . Hence, from the triangle inequality and (8.3), we see(8.4) d (˜ p , ˜ q ) < d (˜ p, ˜ q ) = d (˜ p , ˜ q ) . Since ˜ q ∈ e X ( θ ), it follows from Lemma 5.2 and (8.4) that˜ y (˜ q ) < ˜ y (˜ q ) < θ . Thus, ˜ γ lies in e X ( θ ). Since e X ( θ ) has no pair of cut points, the geodesic extension˜ σ of ˜ γ does not intersect the side ˜ γ except for ˜ q . We will prove that ˜ σ does notintersect ˜ τ ([˜ y (˜ q ) , ˜ y (˜ q )]), where ˜ τ denotes ˜ τ ( t ) := (˜ x (˜ q ) , t ) ∈ e X . Suppose that ˜ σ intersect˜ τ ([˜ y (˜ q ) , ˜ y (˜ q )]) at a point ˜ σ ( s ). From Lemma 5.2, we have(8.5) d (˜ q , ˜ q ) < d (˜ q , ˜ σ ( s )) . σ ( s ) = ˜ q , since ˜ γ does not meet ˜ σ except for ˜ q . Thus, by (8.3) and (8.5), d (˜ p , ˜ q ) < d (˜ p , ˜ q ) + d (˜ q , ˜ σ ( s )) = d (˜ p , ˜ σ ( s )) . Hence, by applying Lemma 5.2 again, we get ˜ y (˜ q ) < ˜ y (˜ σ ( s )). This is impossible,since ˜ σ ( s ) ∈ ˜ τ ((˜ y (˜ q ) , ˜ y (˜ q )]). Therefore, we have proved that ˜ σ does not intersect˜ τ ([˜ y (˜ q ) , ˜ y (˜ q )]).If the extension ˜ σ intersects ∂ e X at a point ˜ σ ( s ) in e X ( θ ), then we denote by e A ( θ )the domain bounded by ˜ µ and ˜ σ ([0 , s ]). If ˜ σ does not intersect ∂ e X in e X ( θ ), then ˜ σ intersects the ∂ e X -ray ˜ y = θ at a point ˜ σ ( s ). In this case, e A ( θ ) denotes the domainbounded by ˜ µ , ˜ σ ([0 , s ]), and the ∂ e X -segment to ˜ σ ( s ). By the argument above, thepoint ˜ q lies in the domain e A ( θ ). Hence, the opposite side ˜ γ of OT( ∂ e X, ˜ p, ˜ q ) to ∂ e X must lie in the closure of e A ( θ ), since e X ( θ ) has no pair of cut points. In particular, ∠ ˜ p ≥ ∠ ˜ p is now clear. By repeating the same argument above for the pair of opentriangles OT( ∂ e X , ˜ q , ˜ p ) and OT( ∂ e X , ˜ q , ˜ p ), we also get ∠ ˜ q ≥ ∠ ˜ q . ✷ From now on, we denote by (
X, ∂X ) a complete connected Riemannian n -dimensionalmanifold X with smooth convex boundary ∂X whose radial curvature is bounded frombelow by that of ( e X, ∂ e X ). Lemma 8.2
If an open triangle
OT( ∂X, p, q ) = ( ∂X, p, q ; γ, µ , µ ) in X admits an opentriangle OT( ∂ e X, ˜ p, ˜ q ) = ( ∂ e X, ˜ p, ˜ q ; ˜ γ, ˜ µ , ˜ µ ) in a sector e X ( θ ) satisfying (8.6) d ( ∂ e X, ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X , ˜ q ) = d ( ∂X, q ) , then, for any s ∈ (0 , d ( p, q )) , there exists an open triangle OT( ∂ e X , ˜ p, ˜ γ ( s )) in e X ( θ ) satisfying ( ) for q = γ ( s ) and ˜ q = ˜ γ ( s ) .Proof. It is clear from Lemmas 5.8 and 8.1. See also the proof of [KT2, Lemma 4.9]. ✷ Proposition 8.3
Let
OT( ∂X, p, q ) = ( ∂X, p, q ; γ, µ , µ ) be an open triangle in X . Then,there exists an open triangle OT( ∂ e X, ˜ p, ˜ q ) = ( ∂ e X, ˜ p, ˜ q ; e γ, e µ , e µ ) in e X satisfying (8.7) d ( ∂ e X, ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X , ˜ q ) = d ( ∂X, q ) . Furthermore, if the
OT( ∂ e X , ˜ p, ˜ q ) lies in a sector e X ( θ ) , which has no pair of cut points,then (8.8) ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q. Proof.
Since µ (resp. µ ) is the ∂X -segment to p (resp. to q ), we obtain c ≤ a + b, a ≤ b + c. Here we set a := d ( ∂X, p ), b := d ( p, q ), and c := d ( ∂X, q ). Hence, we have | a − c | ≤ b .Choose any point ˜ p ∈ e X satisfying d ( ∂ e X, ˜ p ) = a , and fix the point. Since we have d (˜ p, ˜ τ c ( s )) = | a − c | at s = ˜ y (˜ p ) and lim s →∞ d (˜ p, ˜ τ c ( s )) = ∞ , we may find a number31 ≥ ˜ y (˜ p ) such that d (˜ p, ˜ τ c ( s )) = b . Here ˜ τ c denotes the arc ˜ x = c , i.e., ˜ τ c ( s ) = ( c, s ) ∈ e X .Putting ˜ q := ˜ τ c ( s ), we therefore find a triangle OT( ∂ e X , ˜ p, ˜ q ) satisfying (8.7).Hereafter, we assume that the OT( ∂ e X , ˜ p, ˜ q ) lies in the sector e X ( θ ). Let S be theset of all s ∈ (0 , d ( p, q )) such that there exists an open triangle OT( ∂ e X, ˜ p, ˜ γ ( s )) ⊂ e X ( θ )corresponding to the triangle OT( ∂X, p, γ ( s )) ⊂ X satisfying (8.7) and (8.8) for q = γ ( s )and ˜ q = ˜ γ ( s ). Since OT( ∂X, p, γ ( ε )) ⊂ X is a thin open triangle in X for any sufficientlysmall ε >
0, it follows from Lemma 5.8 that S is non-empty. Since there is nothing toprove in the case where sup S = d ( p, q ), we then suppose that s := sup S < d ( p, q ) . Since s ∈ S , there exists an open triangle OT( ∂ e X , ˜ p , ˜ q ) ⊂ e X ( θ ) corresponding tothe triangle OT( ∂X, p, γ ( s )) ⊂ X such that (8.7) and (8.8) hold for q = γ ( s ), ˜ p = ˜ p ,and ˜ q = ˜ q . Choose any ε ∈ (0 , d ( p, q ) − s ) in such a way that the open triangleOT( ∂X, γ ( s ) , γ ( s + ε )) ⊂ X is thin. From Lemma 5.8, there exists an open triangleOT( ∂ e X , ˜ p , ˜ q ) ⊂ e X corresponding to the OT( ∂X, γ ( s ) , γ ( s + ε )) ⊂ X such that (8.7)and (8.8) hold for p = γ ( s ), q = γ ( s + ε ), ˜ p = ˜ p , and ˜ q = ˜ q . It is clear thatthe pair of open triangles OT( ∂ e X , ˜ p , ˜ q ) and OT( ∂ e X, ˜ p , ˜ q ) satisfy (8.1) and (8.2) inLemma 8.1. For this pair, it is clear from Lemma 8.2 that there exists an open triangleOT( ∂ e X , b p, b q ) ⊂ e X ( θ ) such that (8.3) holds for ˜ p = b p and ˜ q = b q . This implies that s + ε ∈ S . This therefore contradicts the fact that s = sup S . ✷ Theorem 8.4 (Toponogov’s comparison theorem for open triangles)
Let ( X, ∂X ) be a complete connected Riemannian n -dimensional manifold X with smooth convex bound-ary ∂X whose radial curvature is bounded from below by that of a model surface ( e X, ∂ e X ) with its metric ( ) . Assume that e X admits a sector e X ( θ ) which has no pair of cutpoints. Then, for every open triangle OT( ∂X, p, q ) = ( ∂X, p, q ; γ, µ , µ ) in X with d ( µ (0) , µ (0)) < θ , there exists an open triangle OT( ∂ e X, ˜ p, ˜ q ) = ( ∂ e X, ˜ p, ˜ q ; ˜ γ, ˜ µ , ˜ µ ) in e X ( θ ) such that (8.9) d ( ∂ e X , ˜ p ) = d ( ∂X, p ) , d (˜ p, ˜ q ) = d ( p, q ) , d ( ∂ e X, ˜ q ) = d ( ∂X, q ) and that (8.10) ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q, d ( µ (0) , µ (0)) ≥ d (˜ µ (0) , ˜ µ (0)) . Furthermore, if d ( µ (0) , µ (0)) = d (˜ µ (0) , ˜ µ (0)) holds, then ∠ p = ∠ ˜ p, ∠ q = ∠ ˜ q hold. roof. Since the claim of our theorem is trivial for degenerate open triangles, we assumethat the open triangle OT( ∂X, p, q ) is not degenerate. Here, we make use of the samenotations used in Lemma 7.3 and its proof.Applying the triangle inequality to the open triangle OT( ∂X, µ ( at ) , µ ( ct )) ⊂ X , wesee(8.11) ϕ ( t ) − ( a + c ) t ≤ d ( µ (0) , µ (0)) ≤ ϕ ( t ) + ( a + c ) t for all t ∈ (0 , a := d ( ∂X, p ), c := d ( ∂X, q ), and ϕ ( t ) := d ( µ ( at ) , µ ( ct )). By thefirst assertion of Proposition 8.3, for each t ∈ (0 , ∂ e X, ˜ µ ( t )1 ( at ) , ˜ µ ( t )2 ( ct )) = ( ∂ e X , ˜ µ ( t )1 ( at ) , ˜ µ ( t )2 ( ct ) ; ˜ γ t , ˜ µ ( t )1 | [0 , at ] , ˜ µ ( t )2 | [0 , ct ] )in e X which has the same side lengths as the OT( ∂X, µ ( at ) , µ ( ct )). Thus, as well as(8.11), we see(8.12) ϕ ( t ) − ( a + c ) t ≤ d (˜ µ ( t )1 (0) , ˜ µ ( t )2 (0)) ≤ ϕ ( t ) + ( a + c ) t for all t ∈ (0 , d ( µ (0) , µ (0)) − a + c ) t ≤ d (˜ µ ( t )1 (0) , ˜ µ ( t )2 (0)) ≤ d ( µ (0) , µ (0)) + 2( a + c ) t for all t ∈ (0 , d ( µ (0) , µ (0)) < θ , it follows from (8.13) that there exists anumber ε > d (˜ µ ( t )1 (0) , ˜ µ ( t )2 (0)) < θ holds on (0 , ε ). Hence,(8.15) OT( ∂ e X , ˜ µ ( t )1 ( at ) , ˜ µ ( t )2 ( ct )) ⊂ e X ( θ )for each t ∈ (0 , ε ). By the second assertion of Proposition 8.3, we get(8.16) ∠ µ ( at ) ≥ ∠ ˜ µ ( t )1 ( at ) , ∠ µ ( ct ) ≥ ∠ ˜ µ ( t )2 ( ct )for each t ∈ (0 , ε ). Since e X ( θ ) has no pair of cut points, it follows from (8.15) that the op-posite side ˜ γ t of OT( ∂ e X , ˜ µ ( t )1 ( at ) , ˜ µ ( t )2 ( ct )) to ∂ e X is unique for all t ∈ (0 , ε ). From Lemma7.3 and (8.14), it follows that the function D ( t ) = d (˜ µ ( t )1 (0) , ˜ µ ( t )2 (0)) is non-increasing on(0 , ε ) and D ( t ) < θ holds on (0 , ε ). Thus, we finally see that D ( t ) is non-increasing on(0 , D ( t ) < θ holds on (0 , , ∂ e X , ˜ p, ˜ q ) = ( ∂ e X , ˜ p, ˜ q ; ˜ γ, ˜ µ , ˜ µ ) := OT( ∂ e X , ˜ µ (1)1 ( a ) , ˜ µ (1)2 ( c ))in e X ( θ ), we get(8.17) ∠ p ≥ ∠ ˜ p, ∠ q ≥ ∠ ˜ q. D ( t ) = d (˜ µ ( t )1 (0) , ˜ µ ( t )2 (0)) ≤ d ( µ (0) , µ (0)) + 2( a + c ) t holds on (0 , D ( t ) is non-increasing on (0 , D (1) = d (˜ µ (0) , ˜ µ (0)) ≤ d ( µ (0) , µ (0)) + 2( a + c ) t on (0 , d ( µ (0) , µ (0)) ≥ d (˜ µ (0) , ˜ µ (0)) . By (8.17) and (8.19), the open triangle OT( ∂ e X , ˜ p, ˜ q ) is therefore an open triangle satisfyingconditions (8.9) and (8.10).Assume that d ( µ (0) , µ (0)) = d (˜ µ (0) , ˜ µ (0)) = D (1) holds. By (8.18), D ( t ) ≤ a + c ) t + D (1)holds on (0 , t ↓ D ( t ) ≤ D (1) . Hence, D ( t ) must be constant on (0 , D ( t ) is non-increasing on (0 , ∠ µ ( at ) = ∠ ˜ µ ( t )1 ( at ) and ∠ µ ( ct ) = ∠ ˜ µ ( t )2 ( ct ) holdon (0 , ∠ p = ∠ ˜ p and ∠ q = ∠ ˜ q . ✷ References [CG] J. Cheeger and D. Gromoll,
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