Cut locus and optimal synthesis in the sub-riemannian problem on the group of motions of a plane

(1)

DOI:10.1051/cocv/2010005 www.esaim-cocv.org

CUT LOCUS AND OPTIMAL SYNTHESIS IN THE SUB-RIEMANNIAN PROBLEM ON THE GROUP OF MOTIONS OF A PLANE

^∗

Yuri L. Sachkov

¹

Abstract. The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI:

10.1051/cocv/2009004; Sachkov,ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed.

Mathematics Subject Classification.49J15, 93B29, 93C10, 53C17, 22E30.

Received May 28, 2009.

Published online March 24, 2010.

1. Introduction

This work completes the study of the left-invariant sub-Riemannian problem on the group of motions of a plane SE(2) = R²SO(2) started in [7,10]. In visual geometric terms, this problem can be stated as follows:

given two unit vectorsv0= (cosθ0,sinθ0),v1= (cosθ1,sinθ1) attached respectively at two given points (x0, y0), (x1, y1) in the plane, one should ﬁnd an optimal motion in the plane that transfers the vectorv0to the vectorv1, see Figure1. The vector can move forward or backward and rotate simultaneously. The required motion should be optimal in the sense of minimal length in the space (x, y, θ), whereθis the slope of the moving vector.

The corresponding optimal control problem reads as follows:

x˙ =u1cosθ, y˙=u1sinθ, θ˙=u2, (1.1) q= (x, y, θ)∈M =R²_x,y×S_θ¹, u= (u1, u2)∈R², (1.2) q(0) =q0= (0,0,0), q(t1) =q1= (x1, y1, θ1), (1.3) l=

_t₁

0

u²1+u²2dt→min, (1.4)

Keywords and phrases.Optimal control, sub-Riemannian geometry, diﬀerential-geometric methods, left-invariant problem, group of motions of a plane, rototranslations, cut locus, optimal synthesis.

∗The author is partially supported by Russian Foundation for Basic Research, Project No. 09-01-00246-a, and by the Program of Presidium of Russian Academy of Sciences “Mathematical control theory”.

1 Program Systems Institute, Pereslavl-Zalessky, Russia.sachkov@sys.botik.ru

Article published by EDP Sciences c EDP Sciences, SMAI 2010

(2)

q

0

= ( x

0

, y

0

, θ

0

)

x y

θ q

1

= ( x

1

, y

1

, θ

1

)

Figure 1. Problem statement.

or, equivalently,

J = 1 2

_t₁

0

(u²1+u²2) dt→min. (1.5)

This problem has important relations to vision [4,8,9], robotics [6], and diﬀusion equation on SE(2) [2].

Notice that before this work a global description of the cut locus and optimal synthesis was known for left-invariant sub-Riemannian problems on the following Lie groups only: the Heisenberg group (Vershik and Gershkovich [11]), and SO(3), SU(2)∼=S³, SL(2) (Boscain and Rossi [3]).

First we recall the main results of the previous works [7,10]. In paper [7] the normal Hamiltonian system of Pontryagin Maximum Principle was written in a triangular form in appropriate coordinates on cotangent bundleT^∗M, so that its vertical subsystem takes the form of mathematical pendulum:

γ˙ =c, c˙=−sinγ, (γ, c)∈C∼= (2S_γ¹)×Rc, (1.6) x˙ = sinγ

2cosθ, y˙= sinγ

2 sinθ, θ˙=−cosγ

2· (1.7)

The phase cylinder of pendulum (1.6) decomposes into invariant subsets according to values of the energy E=c²/2−cosγ:

C= 5 i=1

C_i,

C1={λ∈C|E∈(−1,1)}, (1.8)

C2={λ∈C|E∈(1,+∞)}, (1.9)

C3={λ∈C|E= 1, c= 0}, (1.10)

C4={λ∈C|E=−1}={(γ, c)∈C|γ= 2πn, c= 0}, (1.11) C5={λ∈C|E= 1, c= 0}={(γ, c)∈C|γ=π+ 2πn, c= 0}. (1.12)

(3)

In the subsetsC1,C2,C3 elliptic coordinates (ϕ, k) that rectify the ﬂow of the pendulum were introduced: ϕis the phase, andka reparameterized energy of pendulum (1.6):

k=

(E+ 1)/2 inC1∪C3, k=

2/(E+ 1) inC2.

The Hamiltonian system (1.6), (1.7) was integrated in Jacobi’s functions [12]. The equation of pendulum (1.6) has a discrete group of symmetriesG={Id, ε¹, . . . , ε⁷}=Z2×Z2×Z2 generated by reﬂections in the axes of coordinatesγ,c, and translations (γ, c)→(γ+ 2π, c). Reﬂectionsεⁱare symmetries of the exponential mapping

Exp : N=C×R+→M, Exp(λ, t) =qt. The main result of work [7] is an upper bound on cut time

tcut= sup{t1>0|qs is optimal fors∈[0, t1]}

along extremal trajectoriesq_s. It is based on the fact that a sub-Riemannian geodesic cannot be optimal after a Maxwell point,i.e., a point where two distinct geodesics of equal sub-Riemannian length meet one another.

A natural idea is to look for Maxwell points corresponding to discrete symmetries of the exponential mapping.

For each extremal trajectory q_s = Exp(λ, s), we described Maxwell times tⁿ_εi(λ), i = 1, . . . ,7, n = 1,2, . . ., corresponding to discrete symmetriesεⁱ. The following upper bound was proved in work [7]:

tcut(λ)≤t(λ), λ∈C, (1.13)

where t(λ) = min(t¹_εi(λ)) is the ﬁrst Maxwell time corresponding to the group of symmetries G. We recall the explicit deﬁnition of the function t(λ) below in equations (2.1)–(2.5).

In work [10], the local optimality of sub-Riemannian geodesics was completely characterized. Extremal trajectories corresponding to oscillating pendulum (i.e., to λ∈C1) do not have conjugate points, thus they are locally optimal forever. In the case of rotating pendulum (λ∈ C2) the ﬁrst conjugate time is bounded from below and from above by the ﬁrst Maxwell timest¹_ε² andt¹_ε⁵ respectively. For critical values of energy of the pendulum, there are no conjugate points. As a consequence, the following bound was proved in Theorem 2.5 [10]:

t(λ)≤t^conj1 (λ), λ∈C. (1.14)

Also, in work [10] the global optimality of geodesics was studied. We constructed open dense domains in preimage and image of exponential mapping and proved that the exponential mapping transform these strata diﬀeomorphically. As a consequence, we showed that inequality (1.13) is in fact an equality.

In this work we obtain our further results for problem (1.1)–(1.5). We consider in detail the action of the exponential mapping at the boundary of the 3-dimensional diﬀeomorphic domains. This boundary is decomposed into smooth strata of dimension 2, 1, 0 so that restriction of exponential mapping to these strata is a diﬀeomorphism (Sect.2). These results provide a detailed description of the global structure of the exponential mapping (Thm.3.1). The optimal synthesis is constructed in Theorem3.2.

In Theorems3.4and3.5we characterize the global structure of the Maxwell set (the set of pointsq1connected by more than one optimal trajectory with the initial pointq0), and the cut locus (the set of points where extremal trajectories lose optimality). For each point of the Maxwell set there are exactly two optimal trajectories. The cut locus has three connected components Cut⁺_loc, Cut⁻_loc, and Cutglob. The initial point q0 is contained in the closure of the local components Cut^±_loc, and is separated from the global component Cutglob. The global component admits a simple description:

Cutglob={q= (x, y, θ)∈M |θ=π},

(4)

while the local components are subsets of the Moebius strip:

Cut^±_loc⊂ {q= (x, y, θ)∈M |xcos(θ/2) +ysin(θ/2) = 0}

deﬁned by some inequalities, see Theorem 3.5. Embedding of the cut locus in the solid torus is shown at Figure18.

In Section4 we present explicit optimal solutions for special boundary conditions.

2. Structure of exponential mapping at the boundary of open strata

We recall some more deﬁnitions and notation introduced in the previous works [7,10]. The function t : C →(0,+∞] on the phase cylinder of pendulum (2S¹_γ)×Rc=C=∪⁵_i=1Ci that evaluates the cut time along sub-Riemannian geodesics Exp(λ, t),t∈C, is deﬁned as follows:

λ∈C1 ⇒ t(λ) = 2K(k), (2.1)

λ∈C2 ⇒ t(λ) = 2kp¹1(k), (2.2)

λ∈C3 ⇒ t(λ) = +∞, (2.3)

λ∈C4 ⇒ t(λ) =π, (2.4)

λ∈C5 ⇒ t(λ) = +∞, (2.5)

where p¹1(k) is the ﬁrst positive root of the equation cnp(E(p)−p)− dnpsnp = 0. Here and below we use Jacobi’s functions cn, sn, dn, E, and the complete elliptic integral of the ﬁrst kindK [12]. Further,

M=M \ {q0},

N ={(λ, t)∈N |t≤t(λ)}, N_i=C_i×R+, i= 1, . . . ,5,

N ={(λ, t)∈ ∪³_i=1N_i|t <t(λ), snτ cnτ = 0},

N={(λ, t)∈ ∪³_i=1Ni|t=t(λ) or snτ cnτ = 0} ∪N4∪N5, N4=N∩N4,

M ={q∈M |R1(q)R2(q) sinθ= 0}, M={q∈M |R1(q)R2(q) sinθ= 0}, where

R1=ycosθ

2−xsinθ

2, R2=xcosθ

2 +ysinθ 2·

Along with the coordinates (k, ϕ, t), we use in the domainsN1,N2,N3 also the coordinates (k, p, τ):

(λ, t)∈N1∪N3 ⇒ τ = (2ϕ+t)/2, p=t/2, (λ, t)∈N2 ⇒ τ= (2ϕ+t)/(2k), p=t/(2k).

In work [10] we proved that Exp : N →M is a diﬀeomorphism, and that Exp(N)⊂M. In this section we describe the action of the exponential mapping

Exp : N=N \N →M=M\M.

(5)

2.1. Decomposition of the set N

Consider the following subsets of the setN:

Ncut={(λ, t)∈N |t=t(λ)}, (2.6)

Nconj={(λ, t)∈N2|t=t(λ), snτ = 0}, (2.7) NMax=Ncut\Nconj, Nrest=N\Ncut.

The meaning of the subscripts in Ncut, Nconj, NMax, and Nrest is the following: we will show that Exp(Ncut) is the cut locus, Exp(NMax) is the ﬁrst Maxwell set, Exp(Nconj) is the intersection of the cut locus with the conjugate locus (caustic), and Exp(Nrest) has no special meaning in this problem (so it contains all the rest strata), see Theorem3.4. We have the following decompositions:

N =NN, N=NcutNrest, Ncut=NMaxNconj, (2.8) here and below we denote bythe union of disjoint sets.

In order to study the structure of the exponential mapping at the setN, we need a further decomposition into subsetsN_i,i= 1, . . . ,58, deﬁned by Table1.

Images of the projections

N_i∩ {t <t(λ), snτ cnτ = 0} → {p= 0}, (k, τ, p)→(k, τ,0), N_i∩ {t=t(λ)} → {p= 0}, (k, τ, p)→(k, τ,0),

are shown respectively in Figures2 and3.

Table1 provides a deﬁnition of the setsN_i, e.g., the second column of this table means that N1 ={(λ, t)∈N |λ∈C1⁰, τ ∈(0, K), p=K, k∈(0,1)}.

Here we use the following decomposition of the sets C_i into connected components:

C1=∪¹_i=0C1ⁱ, C1ⁱ ={(γ, c)∈C1|sgn(cos(γ/2)) = (−1)ⁱ}, i= 0,1, C2=C2⁺∪C2⁻, C2^± ={(γ, c)∈C2|sgnc=±1},

C3=∪¹_i=0(C3ⁱ⁺∪C3ⁱ⁻),

C3^i± ={(γ, c)∈C3|sgn(cos(γ/2)) = (−1)ⁱ, sgnc=±1}, i= 0,1, C4=∪¹_i=0C4ⁱ, C4ⁱ ={(γ, c)∈C|γ= 2πi, c= 0}, i= 0,1,

C5=∪¹_i=0C5ⁱ, C5ⁱ ={(γ, c)∈C|γ=π+ 2πi, c= 0}, i= 0,1.

Introduce the following index sets for numeration of the subsetsN_i:

I={1, . . . ,58}, C={1, . . . ,34}, J ={26,28,30,32}, (2.9)

R={35, . . . ,58}, X=C\J. (2.10)

Notice thatI=CR,J ⊂C.

Lemma 2.1.

(1) We have N_i∩N_j=∅ for any distincti, j∈I.

(6)

Table 1. Deﬁnition of setsN_i.

N_i N1 N2 N3 N4 N5 N6 N7 N8

λ C1⁰ C1⁰ C1⁰ C1⁰ C1¹ C1¹ C1¹ C1¹

τ (0, K) (K,2K) (2K,3K) (3K,4K) (0, K) (K,2K) (2K,3K) (3K,4K)

p K K K K K K K K

N_i N9 N10 N11 N12 N13 N14 N15 N16

λ C2⁺ C2⁺ C2⁺ C2⁺ C2⁻ C2⁻ C2⁻ C2⁻

τ (3K,4K) (0, K) (K,2K) (2K,3K) (−3K,−2K) (−2K,−K) (−K,0) (0, K) p p¹1 p¹1 p¹1 p¹1 p¹1 p¹1 p¹1 p¹1

N_i N17 N18 N19 N20 N21 N22 N23 N24 N25 N26 N27 N28 N29 N30

λ C1⁰ C1⁰ C1⁰ C1⁰ C1¹ C1¹ C1¹ C1¹ C2⁺ C2⁺ C2⁺ C2⁺ C2⁻ C2⁻

τ 0 K 2K 3K 0 K 2K 3K 3K 0 K 2K K −2K

p K K K K K K K K p¹1 p¹1 p¹1 p¹1 p¹1 p¹1

N_i N31 N32 N35 N36 N37 N38 N39 N40 N41 N42

λ C2⁻ C2⁻ C1⁰ C1⁰ C1⁰ C1⁰ C1¹ C1¹ C1¹ C1¹

τ −K 0 0 K 2K 3K 0 K 2K 3K

p p¹1 p¹1 (0, K) (0, K) (0, K) (0, K) (0, K) (0, K) (0, K) (0, K) N_i N47 N48 N49 N50 N51 N52 N53 N54

λ C3⁰⁺ C3⁰⁻ C3¹⁺ C3¹⁻ C2⁺ C2⁺ C2⁺ C2⁺

τ 0 0 0 0 3K 0 K 2K

p (0,+∞) (0,+∞) (0,+∞) (0,+∞) (0, p¹1) (0, p¹1) (0, p¹1) (0, p¹1) N_i N55 N56 N57 N58

λ C2⁻ C2⁻ C2⁻ C2⁻

τ K −2K −K 0

p (0, p¹1) (0, p¹1) (0, p¹1) (0, p¹1) N_i N33 N34 N43 N44 N45 N46

λ C4⁰ C4¹ C4⁰ C4¹ C5⁰ C5¹

t π π (0, π) (0, π) (0,+∞) (0,+∞)

(2) There are the following decompositions of subsets of the setN:

Ncut=∪i∈CN_i, Nconj=∪i∈JN_i, Nrest=∪i∈RN_i, thus

NMax=∪i∈XN_i, (2.11)

N=i∈IN_i. (2.12)

Proof. Both statements (1) and (2) follow directly from Table1, deﬁnitions of the setsN, NMax, Ncut, Nconj,

Nrest, and decompositions (2.8).

(7)

N35

N38 N36

N37

N52

N51

N55 N58

N44

N39

N43 N42 N40

N41

N54

N53

N57 N56

N47

N48 N50

N49

N45

N46 γ

c

Figure 2. N_i∩ {t <t(λ),snτ cnτ = 0}.

N1

N17

N4

N33

N20 N18

N2

N3 N19

N26 N10

N9

N25

N29 N16 N32 N15

N34

N1

N17 N8 N21 N5

N33 N24 N22

N6

N7 N23

N28 N12

N11

N27

N31 N14 N30 N13

γ c

Figure 3. N_i∩ {t=t(λ)}.

2.2. Exponential mapping of the sets N

₃₅

, N

₄₇

, N

₂₆

, N

₅₂ 2.2.1. Exponential mapping of the set N35

In order to describe the image Exp(N35 ), we will need the following function:

R²1(θ) = 2(artanh(sin(θ/2))−sin(θ/2)), θ∈[0, π). (2.13)

(8)

0 π

θ

R1

Figure 4. Plot ofR1=R²1(θ).

0 π 2 p

k n2

n1

n3

n4

1 N35

Figure 5. DomainN35 .

π 2πθ

R1

m2

m1

m4

m3

M35

Figure 6. Domain M35 .

It is obvious thatR²1∈C^∞[0, π),R²1(0) = 0,R²1(θ)>0 forθ∈(0, π), lim_θ→π−0R²1(θ) = +∞, and dR²1

dθ (0) = 0. (2.14)

A plot of the functionR²1(θ) is given at Figure4.

Deﬁne the following subset of the setM, see Figures 6and14:

M35 ={q∈M |θ∈(π,2π), R1∈(0, R²1(2π−θ)), R2= 0}.

Lemma 2.2. The mapping Exp : N35 →M35 is a diﬀeomorphism of 2-dimensional manifolds.

To be more precise, we state that Exp(N35 ) = M35 and Exp|_N

35 is a diﬀeomorphism of the manifoldN35

onto the manifoldM35 . Below we will write such statements brieﬂy as in Lemma2.2.

Proof. Formulas (5.2)–(5.6) [7] imply that in the domainN35 we have the following:

sin(θ/2) = snp , cos(θ/2) =−cnp , (2.15)

R1= 2(p−E(p))/k, R2= 0. (2.16)

(9)

By Theorem 2.5 [10] (see (1.14)), the restriction Exp|_N

35 is nondegenerate. Thus the set Exp(N35 ) is an open connected domain in the 2-dimensional manifold

S={q∈M |θ∈(π,2π), R1>0, R2= 0}.

On the other hand, the set N35 is an open connected simply connected domain in the 2-dimensional manifold T ={ν= (λ, t)∈N1|τ= 0, p∈(0, K), k∈(0,1)}.

In the topology ofT, we have

∂N35 =∪⁴_i=1ni,

n1={ν∈N1|τ = 0, p= 0, k∈[0,1]}, n2={ν∈N1|τ = 0, p∈[0, π/2], k= 0}, n3={ν∈N1|τ = 0, p=K(k), k∈[0,1)}, n4={ν∈N1|τ = 0, p∈[0,+∞), k= 1]}, see Figure5.

It follows from formulas (2.15) and (2.16) that

Exp(n1) =m1={q∈M |θ= 2π, R1= 0, R2= 0}, Exp(n2) =m2={q∈M |θ∈[π,2π], R1= 0, R2= 0}, Exp(n3) =m3={q∈M |θ=π, R1>0, R2= 0},

Exp(n4) =m4={q∈M |θ∈[π,2π], R1=R²1(2π−θ), R2= 0}, moreover,∂M35 =∪⁴_i=1mi, see Figure6.

Now we show that Exp(N35 )⊂M35 and Exp : N35 →M35 is a diﬀeomorphism.

(a) We show that Exp(N35 )∩M35 =∅. Formulas (2.15) and (2.16) give the following asymptotics ask→0:

θ= 2π−2p+o(1), R1=k(p/2−(sin 2p)/4) +o(k).

There exists (p, k) close to (π/2,0) such that the corresponding point (θ, R1) is arbitrarily close to (0,0), with θ >0,R1>0. Thus there existsν∈N35 such that Exp(ν)∈M35 .

(b) We show that Exp(N35 )=S. Formulas (2.15) and (2.16) yield the following chain:

θ→2π−0 ⇒ snp →0 ⇒ p→0 ⇒ R1→0.

Thus there exists q∈S\Exp(N35 ).

(c) We prove that Exp(N35 )⊂M35 . By contradiction, suppose that there exists a pointq1∈Exp(N35 )\M35 . Since the mapping Exp|_N

35 is nondegenerate, we can choose this point such thatq1∈Exp(N35 )\cl(M35 ).

Choose any point q2 ∈ S \cl(M35 ). Connect the points q1, q2 by a continuous curve in S, and find at this curve a point q3 ∈ S\Exp(N35 ), q3 ∈/ cl(M35 ) such that there exists a converging sequence qⁿ → q3, qⁿ = Exp(νⁿ) ∈ Exp(N35 ). Further, there exist a subsequence νⁿⁱ ∈ N35 converging to a finite or infinite limit. If νⁿⁱ →ν¯ ∈ N35 , then q3 = Exp(¯ν)∈ int Exp(N35 ) by nondegeneracy of Exp|_N

35, a contradiction. If νⁿⁱ →ν¯∈∂N35 , then

q3= Exp(¯ν)∈Exp(∂N35 ) =∂M35 ⊂cl(M35 ),

(10)

a contradiction. Finally, if νⁿⁱ → ∞, then at this sequence kⁿⁱ → 1−0, pⁿⁱ → ∞, thus R1(qⁿⁱ) → ∞, a contradiction.

Consequently, Exp(N35 )⊂M35 .

(d) The mapping Exp : N35 →M35 is a diﬀeomorphism since Exp|_N

35 is nondegenerate and proper, and

N35 ,M35 are connected and simply connected.

2.2.2. Exponential mapping of the set N47

Deﬁne the following subset ofM, see Figure14:

M47 ={q∈M |θ∈(π,2π), R1=R²1(2π−θ), R2= 0}.

Proof. We pass to the limitk→1−0 in formulas (2.15) and (2.16) and obtain forν∈N47 : sin(θ/2) = tanhp, cos(θ/2) =−1/coshp, R1= 2(p−tanhp), R2= 0.

This coordinate representation shows that Exp : N47 →M47 is a diﬀeomorphism.

Before the study of Exp|_N

52, postponed till the next subsection, we need to consider the setN26 contained in the boundary of N52 . In order to parameterize regularly the image Exp(N26 ), we introduce the necessary functions.

Recall that the functionp=p¹1(k),k∈[0,1), is the ﬁrst positive root of the functionf1(p) = cnp(E(p)−p)−

dnpsnp, see equation (5.12) and Corollary 5.1 [7]. Deﬁne the function

v1¹(k) = am(p¹1(k), k), k∈[0,1). (2.17) Lemma 2.4.

(1) The number v=v¹1(k)is the ﬁrst positive root of the function h1(v, k) =E(v, k)−F(v, k)−

1−k²sin²v tanv, k∈[0,1).

(2) v1¹∈C^∞[0,1).

(3) v1¹(k)∈(π/2, π)for k∈(0,1); moreover, v1¹(0) =π.

(4) The functionv¹1(k)is strictly decreasing at the segmentk∈[0,1).

(5) lim_v→1−0v¹1(k) =π/2, thus settingv1¹(1) =π/2, we obtainv1¹∈C[0,1].

(6) v1¹(k) =π−(π/2)k²+o(k²),k→+0.

Proof. (1) follows from (2.17) sincep=p¹1 is the ﬁrst positive root of the functionf1(p).

(2) follows sincep¹1∈C^∞[0,1) by Lemma 5.3 [7].

(3) follows sincep¹1∈(K,2K) andp¹1(0) =π, see Corollary 5.1 [7].

(4) We have forv∈(π/2, π]:

∂ h1

∂ v =−

1−k²sin²v/cos²v <0,

∂ h1

∂ k =− k

1−k²(E(v, k)−

1−k²sin²vtanv)<0.

Thus dv¹1

dk =−∂h1/∂k

∂h1/∂v <0 for k∈[0,1).

(11)

0 π 2

v

1 k π

Figure 7. Plot ofv =v1¹(k).

0 π θ

R1

Γ1

Figure 8. The curve Γ1.

(5) Monotonicity and boundedness ofv¹1(k) imply that there exists a limit limk→1−0v1¹(k) = ¯v∈[π/2, π). If v¯∈(π/2, π), then ask→1−0

h1(v¹1(k), k)→ ¯v

0

(|cost| −1/|cost|) dt−

1−sin²¯vtan ¯v=∞, which contradicts the identityh1(v1¹(k), k)≡0,k∈[0,1). Thus ¯v=π/2.

(6) As (k, v)→(0, π), we haveh1(v, k) =v−π+ (π/2)k²+o(k²+ (v−π)²), thusv¹1(k) =π−(π/2)k²+o(k²),

k→+0.

A plot of the functionv¹1(k) is given in Figure7.

Deﬁne the curve Γ1⊂S={q∈M |θ∈(0, π), R1>0, R2= 0} given parametrically as follows:

θ= 2 arcsin(ksinv1¹(k)), (2.18)

R1= 2(F(v¹1(k), k)−E(v¹1(k), k)), k∈[0,1), (2.19) see Figure8.

Lemma 2.5.

(1) The function ksinv1¹(k) is strictly increasing as k ∈ [0,1], thus the function θ = θ(k), k ∈ [0,1], determined by(2.18)has an inverse functionk=k1¹(θ),θ∈[0, π].

(2) k¹1∈C[0, π]∩C^∞[0, π).

(3) The functionk1¹(θ)is strictly increasing as θ∈[0, π].

(4) The curve Γ1 is a graph of the function

R1=R¹1(θ), θ∈[0, π],

R¹1(θ) = 2(F(v1¹(k), k)−E(v1¹(k), k)), k=k¹1(θ). (2.20) (5) R¹1∈C[0, π]∩C^∞(0, π).

(6) R¹1(θ) =√³

π/2 θ²^/³+o(θ²^/³),θ→+0.

Proof. (1) Ask∈[0,1], we have:

v1¹(k) ↓, v¹1(k)∈[π/2, π],

sinv1¹(k) ↑, ksinv1¹(k) ↑, 2 arcsin(kv1¹(k)) ↑.

(12)

(2) follows from items (2) and (5) of Lemma2.4.

(3) follows from item (1) of this lemma.

(4) follows from (2.18) and (2.19).

(5) follows from item (2) of this lemma.

(6) Ask→+0, we have

v1¹(k) =π−(π/2)k²+o(k²), sinv1¹(k) = (π/2)k²+o(k²), and for the functions (2.18), (2.19)

θ=πk³+o(k³), R1= (π/2)k²+o(k²). Thus asθ→+0, we have

k¹1(θ) =³

θ/π+o(√³

θ), R1¹(θ) =√³

π/2 θ²^/³+o(θ²^/³).

Deﬁne the following subset ofM, see Figure14:

M26 ={q∈M |θ∈(π,2π), R1=R¹1(2π−θ), R2= 0}.

Proof. Forν∈N26 we obtain from formulas (5.7)–(5.12) [7]:

sin(θ/2) =ksnp¹1(k) =ksinv¹1(k), cos(θ/2) =−dnp¹1(k) =−

1−k²sin²v¹1(k),

R1= 2(p¹1(k)−E(p¹1(k))) = 2(F(v¹1(k), k)−E(v¹1(k), k)), R2= 0.

Thus Exp(N26 ) =M26 . Moreover, the mapping Exp : N26 →M26 decomposes into the chain N26

(∗)

→ Γ1 (∗∗)

→ M26 ,

(∗) : k→(θ= 2 arcsin(ksinv1¹(k)), R1= 2(F(v1¹(k), k)−E(v¹1(k), k)), R2= 0), (∗∗) : (θ, R1, R2)→(2π−θ, R1, R2).

The mapping (∗) is a diﬀeomorphism by Lemma2.5. Thus Exp : N26 →M26 is a diﬀeomorphism.

Lemma 2.7.

(1) The functions R¹1(θ),R²1(θ)deﬁned in (2.20)and(2.13)satisfy the inequality R²1(θ)< R¹1(θ), θ∈(0, π).

(2) The mapping Exp : N52 →M52 is a diﬀeomorphism of 2-dimensional manifolds, where M52 ={q∈M |θ∈(π,2π), R1∈(R²1(2π−θ), R¹1(2π−θ)), R2= 0}, see Figure 14.

(13)

0 π 2 π v

k n2

n1

n3

n4

1 N52

Figure 9. DomainN52 .

π 2π

θ

R1

Γ2=M26

γ2=M47

M35

M52

P0

Figure 10. Curvesγ2 and Γ2. Proof. We have

N52 ={ν ∈N2⁺|τ= 0, v∈(0, v1¹(k)), k∈(0,1)}, wherev= am(p, k). Thus

N52 ⊂T ={ν∈N2⁺|τ= 0, v∈[0, π], k∈[0,1]}, and in the 2-dimensional topology ofT

∂N52 =∪⁴_i=1ni,

n1={ν ∈N2⁺|τ= 0, v= 0, k∈[0,1]}, n2={ν ∈N2⁺|τ= 0, v∈[0, π], k= 0}, n3={ν ∈N2⁺|τ= 0, v=v1¹(k), k∈[0,1]}, n4={ν ∈N2⁺|τ= 0, v∈[0, π/2], k= 1}, see Figure9.

By formulas (5.7)–(5.12) [7], the exponential mapping in the domainN52 reads as follows:

sin(θ/2) =ksinv, cos(θ/2) =−

1−k²sin²v, R1= 2(F(v, k)−E(v, k)), R2= 0.

Thus

Exp(N52 )⊂S={q∈M |θ∈(π,2π), R1>0, R2= 0}, Exp(n1) = Exp(n2) =P0={q∈M |θ= 2π, R1= 0, R2= 0}, Exp(n3) =M26 = Γ2, Γ2:=M26 ,

Exp(n4) =M47 =γ2, γ2:=M47 . By Theorem 2.5 [10] (see (1.14)), the mapping Exp|_N

52 is nondegenerate, thus Exp(N52 ) is an open connected domain inS, with∂Exp(N52 )⊂Exp(∂N52 ) = Γ2∪γ2.

The curves Γ2 andγ2 intersect one another at the pointP0. We show that they have no other intersection points. By contradiction, assume that the curves Γ2 andγ2have intersection points distinct fromP0, then the domain Exp(N52 ) is bounded by ﬁnite arcs of the curves Γ2 and γ2, i.e., there exists a point P1 ∈ γ2∩Γ2, P1 = P0, such that ∂Exp(N52 ) = P0γ2P1 ∪ P0Γ2P1. Then Exp(N52 ) does not contain the curves γ2, Γ2.

(14)

0 M46

M55

M51 M53

M45

M57

M29=M31

M25=M27

R2=x R1=y

Figure 11. Decomposition of surface{θ= 0}.

0 M19=M21

M3=M8

M2=M5 M1=M6

M17=M23

M4=M7

M20=M24

M18=M22

M33=M34

R1

R2

Figure 12. Decomposition of surface {θ=π}.

This is a contradiction to the diﬀeomorphic property of the mapping Exp : n3 = N26 → Γ2 = M26 and Exp : n4=N47 →γ2=M47 , see Lemmas2.6and2.3respectively.

Consequently,γ2∩Γ2=P0, and the domain Exp(N52 ) is bounded by the curvesγ2, Γ2. The equalities dR¹1

dθ (0) = +∞, dR²1

dθ (0) = 0 (see Lem. 2.5 and Eq. (2.14)) imply that R²1(θ) < R¹1(θ) for suﬃciently smallθ >0. Further, the representations

Γ2=M26 ={q∈M |θ∈(π,2π), R1=R¹1(2π−θ), R2= 0}, γ2=M47 ={q∈M |θ∈(π,2π), R1=R²1(2π−θ), R2= 0}

imply the required inequality

R²1(θ)< R¹1(θ), θ∈(0, π), and the equality Exp(N52 ) =M52 .

Since the mapping Exp : N52 → M52 is nondegenerate and proper, and the domains N52 , M52 are open, connected, and simply connected, it follows that this mapping is a diﬀeomorphism.

The mutual disposition of the curvesγ2=M47 and Γ2=M26 is shown in Figures10and14.

2.3. Decomposition of the set M

Now we have the functions R²1(θ)< R¹1(θ) required for deﬁnition of the following decomposition:

M =∪⁵⁸_i=1M_i, (2.21)

where the subsets M_i are deﬁned by Table2. Notice that some of the sets M_i coincide between themselves, unlike the setsN_i, see (2.12). A precise deﬁnition of coincidingM_i is given below in Theorem3.1, item (1).

The structure of decomposition (2.21) in the surfaces {θ = 0}, {θ = π}, {R1 = 0}, {R2 = 0} is shown respectively in Figures11,12,13and14.

2.4. Exponential mapping of the sets N

₃₆

, N

₅₃

, N

₁₈

, N

₃₃

, N

₃₄

, N

₁₇

, N

₂₇

, N

₁

, N

₁₀ 2.4.1. Exponential mapping of the set N36

(15)

0 θ 2π

π

R2

M46

M40

M18=M22

M40

M44

M43

M33=M34

M42

M20=M24

M38

M45

Figure 13. Decomposition of surface{R1= 0}.

0 θ2π

π

R1

Γ1

γ1

γ2

Γ2

Γ4

γ4

γ3

Γ3

M11=M12

M25=M27

M28

M54

M49 M39

M19=M21

M37

M48

M58

M32

M17=M23

M35 M47

M52 M26

M9=M10

M29=M31

M13=M14

M30

M56

M50

M41

M15=M16

M33=M34

M43

M44

Figure 14. Decomposition of surface{R2= 0}.

Proof. By formulas (5.2)–(5.6) [7], exponential mapping in the domainN36 reads as follows:

sin(θ/2) =

1−k²snp /dnp , cos(θ/2) =−cnp /dnp , R1= 0, R2=−2f2(p, k)/(kdnp), wheref2(p, k) =k²cnpsnp+ dnp(p−E(p))>0 by Lemma 5.2 [7], thus Exp(N36 )⊂M36 .