Jacobi vector fields along geodesics in glued manifolds and application

Jacobi vector fields along geodesics in glued manifolds and application

Nobuhiro Innami

Department of Mathematics Faculty of Science, Niigata University Niigata, 950-2181, JAPAN

1 Introduction

LetMα,α∈Λ, be complete connected Riemannian manifolds which are glued at their boundaries. We call such a manifold M =∪_α∈ΛM_α a glued Riemannian manifold. Geodesics in a glued Riemannian manifold M are by definition locally minimizing curves in M. The variation vector fields through geodesics satisfy the Jacobi equation in each component manifold. We find how Jacobi vector fields along geodesics change when they pass across the boundary of a component manifold into the neighboring component. These are the contents of Section 2 to 6 which are seen in [7], and generalized by Takiguchi ([10], [11], [12]). As an application we show the Toponogov comparison theorem for glued surfaces (Section 7.1), and apply it to a minimal network problem in glued surfaces (Section 7.2).

2 Definitions

2.1 Glued Riemannian manifolds

Busemann and Phadke ([1]) have made glued G-surfaces M such that there exist points around which any geodesic circles are not convex inM. Glued surfaces are often used as intuitive examples in some papers and literature on Riemannian geometry of geodesics. The surfaces of things are often made up of smooth surfaces with boundary. Taking those examples into consideration and thinking ones in Section 5, we give the definition of glued Riemannian manifolds.

A complete connected one-dimensional glued Riemannian manifold M is by definition a piecewise smooth Riemannian manifold which is, therefore, isometric to a closed curve with suitable length or an interval in the real line.

We assume for the inductive method thatcomplete glued Riemannian manifolds M with dimensionn−1 are defined.

LetM be a connected topological manifold with dimensionnand boundaryB (possibly B = ∅). We say that (M, g) is a complete glued Riemannian manifold with boundary B having a decomposition Γ : M =∪α∈ΛMα if the decomposition

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Γ satisfies the following.

(1) Each (Mα, gα),gα=g|Mα, is a smooth complete Riemannian manifold with boundaryBαand dimensionn.

(2) (IntMα)∩Mβ=∅forα6=β∈Λ where IntMα is the interior ofMα.

(3) Each connected component of the boundary Bα of Mα with Riemannian metricgα|Bα is also a glued Riemannian manifold with dimensionn−1 for anyα∈Λ.

(4) IfMαandMβ are glued atp∈Bα∩Bβ, there is a neighborhoodU ofpinM such thatU = (U∩Mα)∪(U∩Mβ) and both Bα andBβ are differentiable aroundpas hypersurfases in Mα andMβ, thenTpBα =TpBβ andgα=gβ

onTpBα.

(5) For any pointp∈M there exist finitely manyα∈Λ with p∈Mα.

2.2 The law of passage and reflection

Let M be a glued Riemannian manifold with boundary B and decomposition M =∪α∈ΛMα. Let B^t=∪α∈ΛBα. Let Nαbe the inward unit normal vector field toBαin Mαfor eachα∈Λ. EachNαis defined on the set of points at whichBα

is differentiable. For any point p∈ B^t where B^t is differentiable we are going to define the lawQof passage and reflection depending on whetherp6∈B or p∈B.

If p ∈ Bα∩Bβ for some α 6=β ∈ Λ, then the map Qαβ : TpMα → TpMβ is defined as

Qαβ(X) =X−gα(X, Nα)Nα−gα(X, Nα)Nβ

for any tangent vectorX ∈TpMα. We call it thelaw of passage atpfrom Mα to Mβ.

If p∈ B, and, hence, p∈Bα for a singleα∈Λ, then the map Qα:TpMα → TpMα is defined as

Qα(X) =X−2gα(X, Nα)Nα

for any tangent vectorX ∈TpMα. We call it thelaw of reflection atptoB. The lawQαof reflection is considered to be a special case of the lawQααof passage.

We may simply write Q instead ofQαβ and Qα. The law Q comes from the condition of straightness and isometry, that is,

(1) gβ(Q(X), Y) = gα(X, Y), and gβ(Q(X), Nβ) = −gα(X, Nα) for any Y ∈ TpBα=TpBβ andX ∈TpMα,

(2) gβ(Q(X), Q(X)) =gα(X, X) for anyX∈TpMα.

We notice that Q(Nα) =−Nβ andQ(Y) =Y for any Y ∈TpBα.

2.3 Geodesics

Letc: [a, b]→M be a curve ( any interval is possible as a domain ) andJc = {s∈[a, b]|c(s)∈B^t}. We say that a curvec isregular ifJc has no accumulation point in [a, b]. For a regular curveclet a mapT :Jc→Jc be given by

T(s) = min{u∈Jc|s < u}

for anys∈Jc. The mapT is like aceiling function in the billiard ball problems.

We say that a regular curve γ : [a, b] → M parametrized by arc-length is a geodesic curve inM if it satisfies the following conditions for anys∈Jγ.

(1) γ|[s, T(s)] is a geodesic curve in Mα in the usual sense ifγ([s, T(s)])⊂Mα. (2) Ifγ(s)∈Bαfor someα∈Λ, thenBαis differentiable atγ(s) and ˙γ(s−0)6∈

T_γ(s)Bαwhere ˙γ(t) is the tangent vector ofγ att.

(3) ˙γ(s+ 0) =Q( ˙γ(s−0))

3 Variation vector fields

LetM be a glued Riemannian manifold with boundaryB and letq∈Bα∩Bβ

(possibly α=β) be a point satisfying the condition (4) in Section 2.1. Let Xβ ∈ TqMβ. We define a map Pβ:Xβ⊥→TqBβ as

Pβ(v) =v− gβ(v, Nβ) gβ(Xβ, Nβ)Xβ,

whereXβ⊥={w∈TqMβ|gβ(w, Xβ) = 0}. Let^ξSbe the second fundamental form with respect to the unit normal vector field Nξ to Bξ which satisfies by definition that

ξ∇ZNξ =−^ξSq(Z)

for any tangent vector Z ∈TqBα where ξ =α, β, and ^ξ∇ is the Levi-Civita con- nection with respect togξ. Notice that^ξS(Z)∈TqBξ and^ξS is a symmetric linear transformation ofTqBα=TqBβ. We define a linear mapA(Xβ) :Xβ⊥→Xβ⊥ as

A(Xβ)(v) =gβ(Xβ, Nβ)(^αS+^βS)◦Pβ(v)−gβ(Xβ,(^αS+^βS)◦Pβ(v))Nβ

for any tangent vector v∈Xβ⊥.

Lemma 1. The map A(Xβ)is symmetric.

We will see the reason why A is defined as above. We first observe that the difference between^α∇ and^β∇aroundq∈Bα∩Bβ.

Lemma 2. LetY ∈T_qB_α andX a tangent vector field toM_α defined around qin Bα. Then, we get the equation

β∇YQ(X)−Q(^α∇YX)

= gβ(Q(X),(^αS+^βS)(Y))Nβ−gβ(Q(X), Nβ)(^αS+^βS)(Y).

Let γ : [a, b] → M be a geodesic curve with γ(t0) = q, γ([a, t0−0]) ⊂ Mα, γ([t0+ 0, b])⊂Mβ. We write Xα(t) = ˙γ(t) for a≤t ≤t0−0 and Xβ(t) = ˙γ(t) fort0+ 0≤t≤b. Consider a variation ϕ: [a, b]×(−ε, ε)→Mα∪Mβ such that ϕ(t,0) =γ(t) andϕ_s=ϕ(·, s) is a geodesic curve for eachsand the functiont₀(s) of the parameters at which the geodesic curves pass across or reflect is smooth fors.

LetYα(t) be the variation vector field fora≤t≤t0−0 andYβ(t) fort0+ 0≤t≤b.

Then, we have the following.

Lemma 3. The variation vector fieldYξ satisfies the following.

(1) ^ξ∇Xξ

ξ∇XξYξ+Rξ(Yξ, Xξ)Xξ = 0, (2) Q(Yα(t0)) =Yβ(t0),

(3) Q(^α∇X_αYα(t0))−^β∇X_βYβ(t0) =A(Xβ(t0))(Yβo(t0)),

whereξ=α, β and Rξ is the Riemannian curvature tensor and Yβo is the perpen- dicular component ofYβ toXβ(t0). Further, if gα(Yα(a),Xα(a)) = 0, then

Xξ⊥Yξ for ξ=α, β.

We can show many properties which perpendicular Jacobi vector fields along geodesic curves have in a smooth Riemannian manifold.

4 The passage and mirror equation

In this section we show the passage and mirror equation and make the relation between S = ^αS+^βS and A clear. Let λH denote the maximal eigenvalue of a symmetric linear transformationH.

Letγ: [a, b]→M be a geodesic curve andY a vector field alongγ. We callY aJacobi vector field alongγif it satisfies (1) – (3) in Lemma 3. Lett1∈[a, b]. We say thatγ(t2) is a conjugate point to γ(t1) along γ if there is a nontrivial Jacobi vector field alongγwithY(t1) = 0 andY(t2) = 0. We say thatγ(t2) is afocal point to γ(t₁) along γ if there is a nontrivial perpendicular Jacobi vector field alongγ with∇γ˙Y(t1) = 0,Y(t1)6= 0 and Y(t2) = 0.

Lemma 4. (The passage and mirror equation) LetM_αandM_β be flat Riemannian manifolds with boundary Bα and Bβ, respectively, such that Mα is glued to Mβ

around q ∈ Bα∩Bβ in Bα∩Bβ. Let γ : [0, t0] → Mα∪Mβ be a geodesic curve passing acrossBα∩Bβ at only one pointq=γ(a). Suppose γmeets at the angle θ to the tangent space TqBα. Ifγ(t0) is the first conjugate point toγ(0)along γ and b=t0−a, then we have the inequality

λSq

sinθ ≥λ_{A( ˙γ(a+0))} = 1 a+1

b,

where Sq =^αSq+^βSq. The equality sign is true in the first inequality if and only if there are the eigenvectors ofA( ˙γ(a+ 0))andSq with eigenvalues λ_{A( ˙γ(a+0))} and λSq in the subspace spanned by {Nβ,γ(a˙ + 0)}. In particular, the equality sign is always true if the dimension ofM is two.

We can also show the following lemmas which are straightforward modifications of Lemma 4.

Lemma 5. If the flat Riemannian manifolds in Lemma 4 are replaced by the man- ifolds of constant curvature k² (k > 0), then the a and b in Lemma 4 change to

1

ktankaand 1

ktankb, respectively.

Lemma 6. If the flat Riemannian manifolds in Lemma 4 are replaced by the man- ifolds of constant curvature −k² (k >0), then thea and b in Lemma 4 change to

1

ktanhka and 1

ktanhkb, respectively.

We show the relation betweenSq =^αSq+^βSq andA(X).

Lemma 7. Let Mα and Mβ be (n+1)-dimensional Riemannian manifolds with boundaryBαandBβ, respectively, such thatMαis glued toMβaroundq∈Bα∩Bβ. LetX ∈TqMβ and letX meet at the angleθtoTqBβ. Then, the following are true.

(1) If the dimension ofM is two, thenA(X) =κα+κβ

sinθ , whereκξ is the geodesic curvature ofBξ atq forξ=α, β.

(2) Sq = 0 if and only ifA(X) = 0.

(3) If Sq ≤0, thenA(X)≤0 andtrA(X)≥ 1 sinθtrSq. (4) If Sq ≥0, thenA(X)≥0 andtrA(X)≤ 1

sinθtrSq. (5) If Sq =λI, thentrA(X) = λ

sinθ(1 + (n−1) sin²θ).

Here trSq is by definition the trace ofSq.

Let both Mα and Mβ be submanifolds in a Riemannian manifold ˜M of class C^∞. It is natural to ask what happens to S =^αS+^βS ifTqMα =TqMβ as the tangent spaces of submanifolds in ˜M for any point q ∈ Bα∩Bβ. The following lemma answers this question.

Lemma 8. Let M˜ be a Riemannian manifold of class C^∞. Suppose a glued Rie- mannian manifoldM =∪α∈ΛMα is immersed in M˜ and its component manifolds are of class C^∞ in M˜ as submanifolds. If TqMα = TqMβ at any point q near a pointp∈Bα∩Bβat whichBα andBβ are differentiable, then we have the equation

αSq+^βSq = 0.

Therefore,A(X) = 0for any tangent vectorX ∈TqMβ with X 6∈TqBβ. 5 Examples

In this section we give some examples which help us in understanding the notion of glued Riemannian manifolds and the definitions ofQandA(X).

5.1 Surfaces of cylinders

Let M = M1 ∪M2∪M3 be a union of the following three surfaces in the Euclidean spaceE³:

M1 = {(x, y,0)|x²+y²≤1},

M2 = {(x, y, z)|x²+y²= 1,0≤z≤1}, M3 = {(x, y,1)|x²+y²≤1},

and gα, α = 1,2,3, are induced Riemannian metrics from the natural Euclidean metric ofE³. Then, we see that

B1 = {(x, y,0)|x²+y²= 1}, B3 = {(x, y,1)|x²+y²= 1}, B2 = B1∪B3,

and, hence, B =∅, B^t =B₁∪B₃ =B⁰. For any pointp∈ B₁∩B₂ we see that TpM1 is thexy-plane,TpM2is the hyperplane through pand perpendicular to the vector from 0 to p, N1(p) = −p, N2(p) = (0,0,1) which may be considered as a vector atp, andTp(B1∩B2) =TpB1 is the tangent line toB1throughp.

Let a unit vectorX1 ∈ TpM1 with g1(X1,−N1) = sinθ > 0, Then, Q(X1) is the unit vector X2 in TpM2 with g2(X2, N2) = sinθ > 0. Since B1 and B3 are unit circles, we see that ¹Sp =I and ³Sq = I for any point p ∈B1 and q ∈ B3, respectively. ConcerningB2we also see that²S= 0. Further,A(X2) = 1

sinθI, and A(X1) = 1

sinθI whereI is the identity map.

5.2 Surfaces of cones

Letcbe a positive and

M1 = {(x, y,0)|x²+y²≤1},

M2 = {(x, y, z)|x=tcosθ, y=tsinθ,

z= (1−t)c,0≤t≤1,0≤θ≤π}, M3 = {(x, y, z)|x=tcosθ, y=tsinθ,

z= (1−t)c,0≤t≤1, π≤θ≤2π}

with induced Riemannian metrics from the natural Euclidean metric of E³. Then, B1 is a unit circle, each of B2 and B3 consists of a half circle and two segments.

M₂ is glued to M₃ at two segments of their boundary, and to M₁ at a half circle.

M1 is glued toM2 andM3 at their half circles.

Letp= (0,0, c)∈B2∩B3. BothB2andB3 are not differentiable at the vertex p, so thatpcan not be in the interior point of any geodesic curve in our sense.

If p ∈ B2∩B3 and B2∩B3 is differentiable around p, then ²Sp = ³Sp = 0.

Hence,A(X) = 0 for any vector X∈TpM2(andTpM3).

If p ∈ B1∩B2 and B1∩B2 is differentiable atp, then ¹Sp = I, and ²Sp =

√ 1

1 +c²I. Hence,A(X) = µ 1

√1 +c² + 1

¶ 1

sinθIwhereX meets at the angleθto TpB1.

6 Comparison theorems

Let M be a glued Riemannian manifold with dimension n + 1. Let γ : (−∞,∞)−→M be a geodesic withγ(ai)∈B^t for· · ·< a−1< a0< a1<· · ·. Let e1, . . . , en+1 = ˙γ(0) be an orthonormal basis at γ(0) ande1(t), . . . , en+1(t) = ˙γ(t) the parallel vector fields alongγsuch thatej(0) =ej,Q(ej(ai−0)) =ej(ai+ 0) for j= 1, . . . , n+1. LetR(t) be the matrix representation ofR(·,γ(t)) ˙˙ γ(t) with respect to the basis{ej(t)} andAi the one of A( ˙γ(ai+ 0)). IfY(t) =P_n

j=1Yj(t)ej(t) is a

perpendicular variation vector field through geodesics, then we see from Lemma 3 that

(1) Y⁰⁰(t) +R(t)Y(t) = 0 fort6=ai, (JR).

(2) Y(t) is continuous.

(3) Y⁰(ai−0)−Y⁰(ai+ 0) =Ai(Y(ai)).

These properties motivate us to study the solutions of the differential equation (JR) of Jacobi type satisfying the conditions (2) and (3). To study the conjugacy property we use the following index form

I(Y, W) =−

n−1X

i=1

hAi(Y), Wi(ai) + Z _b

a

(hY⁰, W⁰i − hRY, Wi)dt

for piecewise smooth vector valued functionsY,W on [a, b], where we assume that Y,W are not differentiable possibly ata < a1<· · ·< an−1< b.

Lemma 9. Let R1(t) and R2(t) be symmetric (n, n)-matrix valued functions on [a, b] which are possibly discontinuous at a < a₁ <· · · < a_k−1 < b and let A_i and Bi be symmetric(n, n)-matrices for i = 1, . . . , k−1. Suppose Ri(t) ≤R2(t) and Ai≤Bi for anyt∈[a, b] andi= 1, . . . , k−1. If(JR2)is without conjugate points on[a, b], then so is(JR1)in[a, b].

Lemma 10. Let R1(t) and R2(t) be symmetric (n, n)-matrix valued functions on [a, b]which are possibly discontinuous ata < a1<· · ·< ak−1< band letAiandBi

be symmetric(n, n)-matrices fori= 1, . . . , k−1. SupposehR1(t)x, xi ≤ hR2(t)y, yi and hAix, xi ≤ hBiy, yi for any t∈[a, b], i= 1, . . . , k−1 and any x, y ∈Rⁿ with kxk=kyk.

(A) Suppose (JR2) is without conjugate points in [a, b]. If Y1(t) and Y2(t) are solutions of(J_R1)and(J_R2)satisfying the above conditions (2) and (3) with Y1(a) =Y2(a) = 0, kY10(a)k=kY20(a)k, respectively, then kY1(t)k ≥ kY2(t)k for anyt∈[a, b].

(B) Suppose(JR2)is without focal points in[a, b]. IfY1(t)andY2(t)are solutions of(JR1)and(JR2)satisfying the above conditions (2) and (3) withkY1(a)k= kY2(a)k, Y10(a) = Y20(a) = 0, respectively, then kY1(t)k ≥ kY2(t)k for any t∈[a, b].

7 Application

LetM be a glued Riemannian manifold. In this section we assume that (1) M is without boundary.

(2) The dimension ofM is two.

(3) The Gaussian curvature ofM is greater than or euqal to a constant c.

(4) If p ∈ Bα∩Bβ and both Bα and Bβ are differentiable aroundp, then the sumκα(p) +κβ(p) of geodesic curvatures is nonnegative.

(5) Ifp∈ ∪^m_i=1B_αiand eachB_αiare broken atpwith inner angleθ_i, thenP_m

i=1θ_i is less than 2π.

We say that a point satisfying (5) issingular. Any minimal geodesic curve does not pass through any singular point.

7.1 Toponogov comparison theorems

Let T(p, q) be a minimal geodesic segment connecting points p and q, and let M(c) be the 2–dimensional space form with constant curvature c. We call 4p1p2p3 = T(p1, p2) ∪T(p2, p3) ∪T(p3, p1) a geodesic triangle with vertices p1, p2, p3∈M. Let ˜pi, i= 1,2,3, be points inM(c) which satisfy that the lengths of T(˜pi,p˜i+1) are, respectively, equal to the ones of T(pi, pi+1) fori = 1,2,3 and p4=p1. We call4˜p1p˜2p˜3=T(˜p1,p˜2)∪T(˜p2,p˜3)∪T(˜p3,p˜1) thecomparison trian- gle of 4p1p2p3 in M(c). Let T(p0, p1)∪T(p0, p2) be a brokun geodesic segment and αthe angle of T(p0, p1) withT(p0, p2) atp0. We call (T(p0, p1), T(p0, p2), α) a hinge. Let ˜pi, i = 0,1,2, be points in M(c) which satisfy that the lengths of T(˜p₀,p˜_i) are, respectively, equal to the ones of T(p₀, p_i) for i = 1,2. We call (T(˜p0,p˜1), T(˜p0,p˜2), α) thecomparison hingeinM(c).

By using comparison theorems of Lemma 10 and the ideas in [2] and [3] we can have the Toponogov comparison theorems.

Theorem 11. (A) If 4p1p2p3 is a geodesic triangle in M, then the angles of 4p˜1p˜2p˜3 atp˜i is less than or equal to the corresponding ones in 4p1p2p3. (B) If (T(p0, p1), T(p0, p2), α) is a hinge in M then the length of the other

side of 4p0p1p2 is less than or equal to the one of the comparison hinge (T(˜p0,p˜1), T(˜p0,p˜2), α)inM(c), namely,d(p1, p2)≤d(˜p1,p˜2).

We apply this comparison theorem to have a Steiner ratio for a glued Rieman- nian surface. We will see it in the next section.

7.2 Steiner ratio

Let M be a complete Riemannian manifold without boundary. Let P be a finite set of points inM. A shortest network interconnecting P is called a Steiner minimum treewhich is denoted as SMT(P). An SMT(P) may have vertices which are not inP. Such vertices are calledSteiner points. A spanning tree onP is a tree with vertex set P. A shortest spanning tree on P is called a minimum spanning tree onP which is denoted as MST(P). LetL(T) be the total length of edges in a treeT. The Steiner ratio is given by

ρ=ρ(M) = inf

P

L(SM T(P)) L(M ST(P)) Du and Hwang ([4]) have proved that ρ = √

3/2 if M is the Euclidean plane.

This was the affirmative answer of a famous conjecture of Gilbert and Pollak ([5]).

Rubinstein and Weng ([9]) have proved that ρ = √

3/2 if M is a 2-dimensional sphere of constant curvature. Ivanov, Tuzhilin and Cieslik ([8]) have estimated some Steiner ratios for manifolds. In particular, they proved thatρ≤3/4 ifM is a simply connected complete surface of constant curvature -1 without boundary, and thatρ <√

3/2 if M is a surface of constant curvature -1.

LetM be a glued surface satisfying the condition of this section andP a set of npoints inM. Suppose thatP has no singular points. Then, SMT(P) satisfies the following properties.

(1) All terminal points of SMT(P) are points in P.

(2) Any two edges meet at an angle of at least 120^◦ (3) Every Steiner point has degree exactly three.

(4) There are at mostn−2 Steiner points in SMT(P).

We say that a tree T is aSteiner tree if T satisfies (1) to (3). A Steiner tree T is by definitionfull ifT has exactlyn−2 Steiner points. Any Steiner tree can be decomposed into an edge-disjoint union of full Steiner trees.

Any tree T will become a polygonal region with boundary if its edges are re- placed with²-belts as its widen edges. Two terminal points areadjacentinT if they are concecutive on the boundary of the polygonal region. We say that a Steiner tree consists ofconvex cells ifS(p, q)∪T(p, q) is convex for any adjacent terminal points p, q ∈ P where S(p.q) is the subtree from pto q and T(p, q) is a minimal geodesic segment connecting pand q which is homotopic toS(p, q). Our result is the following theorem.

Theorem 12. LetM be a glued Riemannian surface satisfying the condition of this section andP a finite set of points inM. In addition, we assume that the Gaussian

curvature of M is nonnegative. If a minimum Steiner tree of P consists of convex cells, then

L(SM T(P)) L(M ST(P)) ≥

√3 2

To prove this theorem we use the following lemma which is a kind of Toponogov comparison theorem.

Lemma 13. Let M be a simply connected glued surface as in the theorem. Let pi

be a sequence of points inM fori= 1, . . . , n, Let T =∪ⁿ⁻¹_i=1T(pi, pi+1)be a broken geodesic segment andT¯ its copy in the Euclidean planeE²where a copy means that the corresponding edges have the same lengths and the corresponding angles are eqal.

If T∪T(p1, pn)is convex, thend(p1, pn)≤d(¯p1,p¯n).

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