(Locally) shortest arcs of special sub-Riemannian metric on the Lie group SO(3)
aa r X i v : . [ m a t h . DG ] O c t UDK 519.46 + 514.763 + 512.81 + 519.9 + 517.911MSC 22E30, 49J15, 53C17 (LOCALLY) SHORTEST ARCS OF SPECIAL SUB-RIEMANNIANMETRIC ON THE LIE GROUP SO (3) V. N. BERESTOVSKII, I. A. ZUBAREVA,
Abstract.
The authors find geodesics, shortest arcs, diameter, cut locus,and conjugate sets for left-invariant sub-Riemannian metric on the Liegroup SO (3) , under condition that the metric is right-invariant relative tothe Lie subgroup SO (2) ⊂ SO (3) . Keywords and phrases: geodesic, left-invariant sub-Riemannian metric, Liealgebra, Lie group, shortest arc.
Introduction
In paper [1] are found exact shapes of sheres of special left-invariant sub-Rieman-nian metric d on three-dimensional Lie groups: Heisenberg group H, SO (3) and SL ( R ) . In the last two cases one can give the following natural geometric descriptionof the metric d. The Lie groups SO (3) and SL ( R ) / ± E can be interpreted astransitive groups of preserving orientation isometries of unit euclidean sphere S inthree-dimensional Euclidean space and of the Lobachevskii plane L with Gaussiancurvature − and hence as spaces S and L of unit tangent vectors over thesesurfaces. The spaces S and L admit Riemannian metric (scalar product) g bySasaki (see [2] or section K in Besse book [3]). In addition, canonical projections p : ( S , g ) → S and p : ( L , g ) → L (or, which is equivalent, p : SO (3) → SO (3) /SO (2) and p : SL ( R ) / ± E → SL ( R ) /SO (2) are Riemannian submersions [3]. The metric d is defined by (totally nonholonomic) left-invariant distribution D on SO (3) and SL ( R ) / ± E , which is orthogonal to fibers of Riemannian submersion p, and restriction of scalar product g to D. Moreover, canonical projections(1) p : ( SO (3) , d ) → S , p : SL ( R ) / ± E → L are submetries [4], natural generalizations of Riemannian submersion. The dis-tribution D on S and L is nothing other than the restriction to S and L ofhorizontal distribution of Levi-Civita connection [3] for S and L . Therefore undermentioned identifications of SO (3) and SL ( R ) / ± E with S and L , any smoothpath c = c ( t ) , ≤ t ≤ t , in SO (3) and SL ( R ) / ± E , tangent to the distribution D, is realized as parallel translation of the vector c (0) ∈ S and c (0) ∈ L alongprojection p ( c ( t )) , ≤ t ≤ t . The first author was partially supported by the Russian Foundation for Basic Research (Grant14-01-00068-a) and a grant of the Government of the Russian Federation for the State Support ofScientific Research (Agreement №14.B25.31.0029).
It follows from here and the Gauss-Bonnet theorem [5] for S and L that canon-ical projection (to the base of fibration-submersion) of a geodesic in ( SO (3) , d ) or ( SL ( R ) / ± E , d ) must be a solution of Dido’s isoperimetric problem ( isoperimetrix )on the base S or L , while a geodesic is a horizontal lift of an isoperimetrix in S or L . Using this fact, submetries (1) and the suggestion that an isoperimetrix in S or L must have constant geodesic curvature, the authors of paper [1] deducedexact shapes of spheres without searching geodesics and shortest arcs.In this paper, with the help of mentioned interpretation of geodesics, generalmethods of paper [6], and the Gauss-Bonnet theorem for S , we find geodesics,shortest arcs, the diameter, cut locus, and conjugate sets in ( SO (3) , d ) . Formulas,analogous to (10) and (21), are obtained in paper [7], but we apply other methodsand give detailed proofs. 1.
Preliminaries
Let us recall that the Lie group Gl ( n ) = Gl ( R n ) consists of all real ( n × n ) − matrices g = ( g ij ) , i, j = 1 , . . . n , such that det g = 0 , and the Lie subgroup Gl ( n ) (the con-nected component of the unit e in Gl ( n ) ) is defined by condition det g > . It isnaturally to consider both groups as open submanifolds in R n with coordinates g ij ,i, j = 1 , . . . n. Their Lie algebra gl ( n ) = Gl ( n ) e := Gl ( n ) e = R n is the set of all real ( n × n ) -matrices with usual structure of vector space and Lie bracket(2) [ a, b ] = ab − ba ; a, b ∈ gl ( n ) . Let e ij ∈ gl ( n ) , i, j = 1 , . . . n , be a matrix which has 1 in i -th row and j -th columnand 0 in all other places. Lin( a, b ) denotes linear span of vectors a, b . As an auxiliarytool we shall use standard scalar product ( · , · ) on the Lie algebra gl ( n ) = R n for n = 3 . By definition, the Euclidean space E n is R n with standard scalar product ( x, y ) = x T y, where x, y ∈ R n are regarded as vector-columns and T denotes hereand later the transposition of matrices. The Lie group SO ( n ) = O ( n ) ∩ Gl ( n ) of all orthogonal matrices with the de-terminant 1 is a connected Lie subgroup in в Gl ( n ) . Its Lie algebra ( so ( n ) , [ · , · ]) is a Lie subalgebra of the Lie algebra ( gl ( n ) , [ · , · ]) , consisting of all skew-symmetricmatrices.Let G and H be Lie groups with Lie algebras g and h ; φ : G → H is a Lie groupshomomorphism. Then(3) φ ◦ exp g = exp h ◦ dφ e , moreover,(4) dφ e : ( g , [ · , · ]) → ( h , [ · , · ]) is a Lie algebra homomorphism (see lemma 1.12 in [9]). If g ∈ G then I( g ) : G → G, where I( g )( g ) = g gg − is inner automorphism of the Lie group G. Consequently,
Ad( g ) := d I( g ) e ∈ Gl ( g ) is automorphism of the Lie algebra g and d Ad e ( v ) :=ad( v ) := [ v, · ] for v ∈ g [9]. Therefore, on the ground of formula (3),(5) I( g ) ◦ exp = exp ◦ Ad( g ) , ROUP SO (3) (6) Ad(exp g ( v )) = exp gl ( g ) (ad( v )) , v ∈ g . In case of left-invariant sub-Riemannian metrics on Lie groups, every geodesic is aleft shift of some geodesic which starts at the unit. Thus later we shall consider onlygeodesics with unit origin. Theorem 5 in paper [6] implies the following theorem.
Theorem 1.
Let G be a connected Lie subgroup of the Lie group SO ( n ) ⊂ Gl ( n ) with the Lie algebra g , D is totally nonholonomic left-invariant distribution on G, ascalar product h· , ·i on D ( e ) is proportional to restriction of the scalar product ( · , · ) (to D ( e ) ). Then parametrized by arclength normal geodesic (i.e. locally shortest arc) γ = γ ( t ) , t ∈ ( − a, a ) ⊂ R , γ (0) = e, on ( G, d ) with left-invariant sub-Riemannianmetric d , defined by distribution D and scalar product h· , ·i on D ( e ) , satisfies thesystem of ordinary differential equations (7) · γ ( t ) = γ ( t ) u ( t ) , u ( t ) ∈ D ( e ) ⊂ g , h u ( t ) , u ( t ) i ≡ , (8) · u ( t )+ · v ( t ) = − [ u ( t ) , v ( t )] , where u = u ( t ) , v = v ( t ) ∈ g , ( v ( t ) , D ( e )) ≡ , t ∈ ( − a, a ) ⊂ R , are some real-analytic vector functions. Geodesics of special left-invariant sub-Riemannian metric on theLie group SO (3) Theorem 2.
Let be given the basis (9) a = e − e , b = e − e , c = e − e of the Lie algebra so (3) , D ( e ) = Lin( a, b ) , and scalar product h· , ·i on D ( e ) with or-thonormal basis a, b. Then left-invariant distribution D on the Lie group SO (3) withgiven D ( e ) is totally nonholonomic and the pair ( D ( e ) , h· , ·i ) defines left-invariantsub-Riemannian metric d on SO (3) . Moreover, any parametrized by arclength ge-odesic γ = γ ( t ) , t ∈ R , in ( SO (3) , d ) with condition γ (0) = e is a product of two1-parameter subgroups: (10) γ ( t ) = exp( t (cos φ a + sin φ b + βc )) exp( − tβc ) , where φ , β are some arbitrary constants.Proof. It follows from formulae (2) and (9) that(11) [ a, b ] = c, [ b, c ] = a, [ c, a ] = b. This implies the first statement of theorem.It is clear that on D ( e ) (12) h· , ·i = 12 ( · , · ) . In consequence of theorem 3 in [6] every geodesic on 3-dimensional Lie group withleft-invariant sub-Riemannian metric is normal. Then it follows from theorem 1 thatone can apply ODE (7),(8) to find geodesics γ = γ ( t ) , t ∈ R , in ( SO (3) , d ) . V. N. BERESTOVSKII, I. A. ZUBAREVA,
It is clear that(13) u ( t ) = cos φ ( t ) a + sin φ ( t ) b, v ( t ) = β ( t ) c, and the identity (8) is written in the form − [cos φ ( t ) a + sin φ ( t ) b, β ( t ) c ] = · φ ( t )( − sin φ ( t ) a + cos φ ( t ) b )+ · β ( t ) c. In consequence of (11), expression in the left part of equality is equal to β ( t )(cos φ ( t ) b − sin φ ( t ) a ) . We get identities · β ( t ) = 0 , · φ ( t ) = β ( t ) . Hence(14) β = β ( t ) = const , φ ( t ) = βt + φ . In view of (7), (13), and (14), it must be(15) · γ ( t ) = γ ( t )(cos( βt + φ ) a + sin( βt + φ ) b ) . Let us prove that (10) is a solution of ODE (15). One can easily deduce fromformulae (11) equalities(16) (ad( c )) = a, (ad( b )) = − b, (ad( a )) = c, where ( f ) denotes the matrix of linear map f : so (3) → so (3) in the base a, b, c ; later ( f ) is identified with f . On the ground of formulae (6), (16), (14), (13), · γ ( t ) = exp( t (cos φ a + sin φ b + βc ))(cos φ a + sin φ b + βc ) exp( − tβc )+ γ ( t )( − βc ) = γ ( t ) exp( tβc )(cos φ a + sin φ b + βc ) exp( − tβc ) + γ ( t )( − βc ) = γ ( t ) exp( tβc )(cos φ a + sin φ b ) exp( − tβc ) + γ ( t )( βc ) + γ ( t )( − βc ) = γ ( t ) · [Ad(exp( tβc ))(cos φ a + sin φ b )] = γ ( t ) · [exp(ad( tβc ))(cos φ a + sin φ b )] = γ ( t ) · [exp( tβ (ad( c )))(cos φ a + sin φ b )] = γ ( t ) · [(exp( tβa ))(cos φ a + sin φ b )] = γ ( t ) · (cos( βt + φ ) a + sin( βt + φ ) b ) = γ ( t ) u ( t ) . (cid:3) Remark 1.
Both 1-parameter subgroups from formula (10) are nowhere tangent todistribution D for β = 0 so that any their interval has infinite length in metric d. Remark 2.
On p. 258 in book [8] , A.A.Agrachev and Yu.L.Sachkov proved that,analogously to formula (10), every normal trajectory (geodesic) of left-invariant sub-Riemannian metric, defined by a distribution with corank 1, on a compact Lie group,starting at the unit, is a product of no more than two 1-parameter subgroups. Let usremind that any geodesic of left-invariant sub-Riemannian metric on 3-dimensionalLie group is normal.
Proposition 1.
Let γ ( t ) , t ∈ R , be geodesic in ( SO (2 , , d ) defined by formula(10). Then for any t ∈ R , (17) γ ( t ) − γ ( t ) = exp(( t − t )(cos( βt + φ ) a +sin( βt + φ ) b + βc )) exp( − ( t − t ) βc ) . ROUP SO (3) Proof.
On the basis of formulae (5), (6), (16), γ ( t ) − γ ( t ) = exp( t βc ) exp( − t (cos φ a + sin φ b + βc )) · exp( t (cos φ a + sin φ b + βc )) exp( − tβc ) =exp( t βc ) exp(( t − t )(cos φ a + sin φ b + βc )) exp( − t βc ) exp( − ( t − t ) βc ) =[I(exp( t βc ))(exp(( t − t )(cos φ a + sin φ b + βc )))] · exp( − ( t − t ) βc ) =exp[Ad(exp( t βc )(( t − t )(cos φ a + sin φ b + βc ))] · exp( − ( t − t ) βc ) =exp[exp(ad( t βc ))(( t − t )(cos φ a + sin φ b + βc ))] · exp( − ( t − t ) βc ) =exp[exp( t βa )(( t − t )(cos φ a + sin φ b + βc ))] · exp( − ( t − t ) βc ) =exp(( t − t )(cos( βt + φ ) a + sin( βt + φ ) b + βc )) · exp( − ( t − t ) βc ) . (cid:3) Remark 3.
To change a sign of β in (10) is the same as to change a sign of t andto change the angle φ by angle φ ± π. Remark 4.
For any matrix B ∈ SO (2) = exp( R c ) , the map l B ◦ r B − , where l B ismultiplication from the left by B , r B − is multiplication from the right by B − , issimultaneously automorphism Ad B of the Lie algebra ( so (3) , [ · , · ]) , preserving h· , ·i , and automorphism of the Lie group SO (3) , preserving distribution D and metric d. In particular in view of (6), (16) Ad B ( a + βc ) = exp( φ a )( a + βc ) = cos φ a + sin φ b + βc, if (18) B = exp( φ c ) = φ − sin φ φ cos φ . Lemma 1. (19) exp( t ( a + βc )) = I(exp( − ξb ))(exp( t p β a )) , where (20) cos ξ = 1 p β , sin ξ = β p β . Proof.
Taking into account (20), (16), (6), we get t ( a + βc ) = ( t p β (cos ξ · a + sin ξ · c )) = (exp( ξb ))( t p β a ) =(exp(ad( − ξb )))( t p β a ) = Ad(exp( − ξb ))( t p β a ) . Now in consequence of obtained equalities and (5), exp( t ( a + βc )) = exp(Ad( − ξb )( t p β a )) = I(exp( − ξb ))(exp( t p β a )) . (cid:3) V. N. BERESTOVSKII, I. A. ZUBAREVA,
Theorem 3.
The geodesic γ = γ ( t ) of left-invariant sub-Riemannian metric d onthe Lie group SO (3) , defined by formula (10), is equal to (21) − n − m cos ( βt + φ ) − βn sin ( βt + φ ) − m sin ( βt + φ ) + βn cos ( βt + φ ) m cos φ − βn sin φ (1 − β n ) cos βt + βm sin βt − n cos ( βt + φ ) cos φ (1 − β n ) sin βt − βm cos βt − n sin ( βt + φ ) cos φ m sin φ + βn cos φ βm cos βt − (1 − β n ) sin βt − n cos ( βt + φ ) sin φ (1 − β n ) cos βt + βm sin βt − n sin ( βt + φ ) sin φ , where (22) m = sin ( t p β ) p β , n = 1 − cos ( t p β )1 + β . Proof.
Let φ = 0 . Then (10) takes the form γ ( t ) | φ =0 = exp ( t ( a + βc )) exp ( − tβc ) . Using lemma 1, (22) and carrying out routine calculations, we get exp ( t ( a + βc )) =11 + β β p β − β cos t p β − sin t p β t p β cos t p β
00 0 1 × − β p β β = − n − m nβm − n (1 + β ) − mβnβ mβ − nβ . Now, using (10) and (18) for φ = − βt , we get γ ( t ) | φ =0 = − n − m nβm − n (1 + β ) − mβnβ mβ − nβ · βt sin βt − sin βt cos βt = − n − m cos βt − nβ sin βt βn cos βt − m sin βtm (1 − n (1 + β )) cos βt + mβ sin βt (1 − n (1 + β )) sin βt − mβ cos βtβn mβ cos βt − (1 − β n ) sin βt (1 − β n ) cos βt + mβ sin βt . By (18), matrices B = exp( φ ) and exp ( − tβc ) commute. It follows from here andfrom remark 4 that(23) γ ( t ) = B · γ ( t ) | φ =0 · B − . Substitution of formula (18) into the last equality finishes the proof. (cid:3) Shortest arcs on the Lie group ( SO (3) , d ) The group SO (3) is realized as the group of all preserving orientation isometries v → gv ; g ∈ SO (3) , v ∈ S of unit sphere S ⊂ R , whose elements v are regarded as vector-columns. It is notdifficult to check that Lie subgroup SO (2) := { exp sc, s ∈ R } ⊂ SO (3) ROUP SO (3) is the stabilizer of vector v = (1 , , T = e ∈ S with respect to this action.Moreover the group SO (2) acts (simply) transitively by rotations on unit circle S := S ∩ e ⊥ ⊂ S . Therefore S is naturally identified with quotient homogeneous space SO (3) /SO (2) and the group SO (3) itself is diffeomorphic to the space S of all unit tangent vectorsto S . Namely, every element g ∈ SO (3) corresponds to ge ′ , where e ′ is usual paralleltranslation of vector e to point e . Moreover, in consequence of introduction,1) Any segment of a smooth path c = c ( t ) in ( SO (3) , d ) , tangent to distribution D, has the same length as its image relative to canonical projection(24) p : g ∈ SO (3) → ge ∈ S ;
2) under indicated identification of SO (3) with S , any path c = c ( t ) , ≤ t ≤ t , tangent to distribution D, is realized as parallel vector field in S along p ( c ( t )) , ≤ t ≤ t , with initial unit tangent vector c (0) ∈ S ;
3) By the Gauss-Bonnet theorem [5], under parallel translation in S of non-zerotangent vector along a contour, bounding a region in S with area S < π, thevector turns in the direction of bypass by the angle S. Let us use statements 1) — 3) to find shortest arcs in ( SO (3) , d ) . In consequenceof proposition 1, remark 4, and left invariance of the metric d , it is sufficient toinvestigate segments of geodesics of the form(25) γ ( t ) = exp( t ( a + βc )) exp( − tβc ) , ≤ t ≤ t , and their projections(26) x ( t ) := p ( γ ( t )) = γ ( t ) · e = γ ( t ) · (1 , , T = (1 − n, m, βn ) T , ≤ t ≤ t , to the sphere S , where m , n are defined by formulae (22) (we used formula (21) for φ = 0 ).Since the second factor in (25) lies in SO (2) , then orbits (26) coincide with seg-ments of orbits of 1-parameter subgroup y ( t ) = exp( t ( a + βc )) , t ∈ R . It is not difficult to calculate that ± (1 / p β )( β, , T ∈ S are unit eigen-vectors of matrix a + βc with respect to zero eigenvalue. Consequenly, 1-parametersubgroup y ( t ) , t ∈ R , preserves these vectors. Scalar products of these vectors with e are equal to ± ( β/ p β ) . Then spherical distance from the point e to the axisof these vectors is equal to(27) r = arccos ( | β | / p β ) ≤ π/ . Therefore the orbit { γ ( t ) e = y ( t ) e } is spherical circle of radius r < π/ withunique center (1 / p β )( β, , T , if β = 0 . It is not difficult to see that if β > , then in consequence of theorem 3, curve (26) for t = 2 π/ p β goes around thiscircle, bounding lesser region Ψ of S with this center inside it, one times, leavingthe region Ψ from the left .Let us formulate the Gauss-Bonnet theorem [5]. Let M be two-dimensional ori-ented manifold with Riemannian metric ds , Φ is a region in M, homeomorphic todisc and bounded by closed piece-wise regular curve γ with regular links γ , . . . , γ n , forming angles α , . . . , α n from the side of region Φ . Direction on the curve γ is V. N. BERESTOVSKII, I. A. ZUBAREVA, given so that the region Φ is situated from the right under bypass of the curve inthis direction. Then Theorem 4. (28) n X k =1 Z γ k κds + n X k =1 ( π − α k ) = 2 π − Z Z Φ Kdσ, where κ is geodesic curvature at points of links of the curve, K is Gaussian (sec-tional) curvature of the surface ( M, ds ) , and integration in the right part of equalityis taken by area element of the region Φ . In particular, if γ is a regular curve, then (29) Z γ κds = 2 π − Z Z Φ Kdσ.
Proposition 2.
Geodesic curvature of curve (26) for β > is equal to −| β | . Proof.
In consequence of what has been said, applying equality (29) to circle (26)for β > and t = 2 π/ p β , one needs to take region Φ = S r Ψ in S and K = 1 . Then the left part of (29) is equal to κt . For the right part, we need area σ (Φ) . It is known that in S (30) l ( r, α ) = α sin r, (31) S ( r, α ) = Z r α sin sds = α ch s | r = α (1 − cos r ) , where l ( r, α ) is the length of arc of circle with radius r and central angle α ≤ π, and S ( r, α ) is area of corresponding sector. Then in consequense of (27), σ (Ψ) = 2 π − | β | p β ! ,σ (Φ) = 4 π − σ (Ψ) = 2 π | β | p β ! , πκ p β = 2 π − σ (Φ) = − π | β | p β , κ = −| β | . (cid:3) Proposition 3.
Let us assume that projection (26) of geodesic segment (25), where β = 0 , has no self-intersection, i.e. ≤ t < π/ p β , S ( t ) = S ( t , β ) is areaof lesser curvilinear digon P in S , bounded by segment (26) and shortest segment [ x (0) x ( t )] of a length r = r ( t ) in S , ψ = ψ ( t , β ) is interior angle of the digon P .Then (32) r = arccos((1 − n )( t )) , r ′ ( t ) = cos ψ = m p n (2 − n ) , S ( t ) = 2 ψ − | β | t . ROUP SO (3) Moreover S ′ ( t ) > , if t >
0; 0 < ψ ≤ π/ , if < t ≤ π/ p β , and π/ <ψ < π, if π/ p β < t ≤ π/ p β . Proof.
The first equality in (32) is a corollary of (26) and known formula for distancein spherical geometry, the second one is a well-known statement of Riemanniangeometry (on existence of strong angle), the third equality is result of differentiationof first equality in (32). Inequalities for the angle are evident. In consequence ofremark 3 one can assume that β > . Segment [ x (0) x ( t )] has geodesic curvature . Then, with taking into account
Φ = S r P and proposition 2, equation (28) iswritten in the form −| β | t + (2 π − (4 π − ψ )) = 2 π − (4 π − S ( t )) . Consequently, S ( t ) = 2 ψ − | β | t . From here and (31) follow relations S ′ ( t ) = 2 ψ ′ ( t ) − | β | = (1 − cos r ) ψ ′ ( t ) , (33) ψ ′ ( t ) = | β | r = | β | − n , (34) S ′ ( t ) = | β | (cid:18) − n − (cid:19) ( t ) > , < t < π p β . (cid:3) Lemma 2. If β = 0 and t = π, then (25) is noncontinuable shortest arc.Proof. In this case γ ( t ) = exp( ta ) . Then γ (2 π ) = e and, consequently, γ ( π ) = γ ( − π ) . Therefore geodesic segment γ ( t ) , ≤ t ≤ t , is not shortest arc for t > t = π . On the other hand, canonical projection p : ( SO (3) , d ) → S (see (1) и (24)) isa submetry, moreover γ ( π ) = − ( e + e ) + e , p ( γ ( π )) = γ ( π ) e = − e , i.e. path p ( γ ( t )) , ≤ t ≤ π, is shortest connection in S of diametrally oppositepoints e and − e . Then (25) is noncontinuable shortest arc. (cid:3) Proposition 4. β = 0 then geodesic segment (25) is noncontinuable shortestarc when its projection (26) is a) one time passing circle C bounding disc with area S ( t ) ≤ π or b) curve without self-intersections bounding together with the shortestarc [ x (0) x ( t )] in S digon P in S with area S ( t ) = π .2) For every β = 0 there is unique t > such that one of conditions a) or b) issatisfied; a) is satisfied only if | β | ≥ / √ . Proof.
1) a) It is clear that γ ( t ) ∈ SO (2) . Then in consequence of remark 4, segmentof geodesic (10) for the same β and any φ under t ∈ [0 , t ] joins the same points as(25). Consequently every continuation of the segment (25) is not a shortest arc.Let us suppose that there exists a shortest arc γ ( t ) , ≤ t ≤ t < t , in ( SO (3) , d ) which joins points γ (0) = e and γ ( t ) . Then projection x ( t ) = p ( γ ( t )) , ≤ t ≤ t , is one time passing circle C in S with length t < t and therefore bounds a discwith area S ( t ) < S ( t ) ≤ π. Consequently on the ground of the Gauss-Bonnet theorem, results of parallel translations of nonzero vectors along C and C in S aredifferent. Then γ ( t ) = γ ( t ) in view of geometric interpretation of geodesics in ( SO (3) , d ) , given in introduction, a contradiction.b) Let P ′ be a digon, symmetric to the digon P relative to segment [ x (0) x ( t )] . Since S ( t ) = π then by the Gauss-Bonnet theorem, results of parallel translations in S of tangent vectors along closed paths, bounding P and P ′ , are equal. Therefore onthe ground of remarks 3, 4 and geometric interpretation of geodesics in ( SO (3) , d ) , given in introduction, a curve in S , symmetric to the projection (26) of segment (25)relative to segment [ x (0) x ( t )] , is presented in the form p ( γ ( t )) , ≤ t ≤ t , where γ is a geodesic in ( SO (3) , d ) such that γ (0) = γ (0) , γ ( t ) = γ ( t ) . Consequentlyevery continuation of the segment (25) is not a shortest arc.Let us suppose that there is a shortest arc γ ( t ) , ≤ t ≤ t < t , in ( SO (3) , d ) , joining points γ (0) = e and γ ( t ) . Then in consequence of remarks 3 and 4 we canassume that curves (26) and x ( t ) = p ( γ ( t )) , ≤ t ≤ t , lie on the one side ofthe shortest arc [ x (0) x ( t )] and join ends of this shortest arc. Consequently thedigon P and digon P , bounded by the shortest arc [ x (0) x ( t )] and the curve x ( t ) , ≤ t ≤ t , are convex, moreover intersection of their boundaries is the shortestarc [ x (0) x ( t )] , because t < t . Therefore in view of last inequality the curve x ( t ) , < t < t , lies inside P and S ( t ) < S ( t ) = π, where S ( t ) is area of the digon P . Consequently on the ground of the Gauss-Bonnet theorem, results of paralleltranslations of nonzero tangent vectors along boundaries of P and P in S aredifferent. Then γ ( t ) = γ ( t ) in view of geometric interpretation of geodesics in ( SO (3) , d ) , given in introduction, a contradiction.2) On the ground of last equality in (32), the condition a) is fulfilled only if t = 2 π/ p β , ψ ( t ) = π and S π p β ! = 2 π − | β | π p β ≤ π ⇔ | β | ≥ √ . If < | β | < √ , then in consequence of proposition 3 there exists unique t > forwhich the condition b) is satisfied. (cid:3) Later for every number β = 0 we shall find a number t = t ( β ) , satisfyingconditions of proposition 4.I) If | β | ≥ √ , then t = 2 π/ p β . II) If < | β | < √ , then S (2 π/ p β ) < π ⇒ S ( π/ p β ) < π, (35) S ( t ) = π ⇒ π/ p β < t < π/ p β . Therefore in consequence of proposition 3, π/ < ψ ( t ) < π and(36) S ( t ) = π ⇔ ψ − | β | t = π ⇔ | β | t ψ − π/ . ROUP SO (3) In consequence of (32) and (22), cos ψ = m p n (2 − n ) = p β sin( t p β ) q (1 − cos( t p β ))(1 + cos( t p β ) + 2 β ) = p β cos( t p β / q cos ( t p β /
2) + β . We get from here, (36), inequalities for t and ψ that sin ψ = p − cos ψ = | β | sin( t p β / q cos ( t p β /
2) + β , (37) sin (cid:18) | β | t (cid:19) = sin (cid:16) ψ − π (cid:17) = − cos ψ = − p β cos( t p β / q cos ( t p β /
2) + β , (38) cos (cid:18) | β | t (cid:19) = cos (cid:16) ψ − π (cid:17) = sin ψ = | β | sin( t p β / q cos ( t p β /
2) + β , (39) < | β | t < π. Theorem 5.
Conditions a),b) of proposition 4 define a continuous function t = t ( | β | ) , increasing under ≤ | β | ≤ / √ and decreasing under / √ ≤ | β | < + ∞ .Proof. The second statement is evident. The first statement is true, because dt /d | β | > under < | β | < / √ in consequence of (36), (33), (22), (35): t + | β | dt d | β | = 2 ψ ′ ( t ) · dt d | β | = 2 | β | − n · dt d | β | ,t = | β | n − n · dt d | β | = | β | sin ( t p β / β + cos ( t p β / · dt d | β | . (cid:3) Theorem 6. diam( SO (3) , d ) = π √ . Proof.
It follows from theorem 5 that maximal length of shortest arc is attainedunder β = 1 / and it is equal to π √ . This implies needed statement. (cid:3)
Remark 5.
Statement of theorem 6 is a particular case of the first statement oftheorem 2 from paper [1] . Cut locus and conjugate sets in ( SO (3) , d ) Unlike Riemannian manifolds, exponential map
Exp and its restriction
Exp x forsub-Riemannian manifold ( M, d ) without abnormal geodesics (as in the case of ( SO (3) , d ) ) are defined not on T M and T x M but only on D and D ( x ) , where D is distribution on M, taking part in definition of d. Otherwise cut locus and con-jugate sets for such sub-Riemannian manifold are defined in the same way as forRiemannian one [10].
Definition 1.
Cut locus C ( x ) (respectively conjugate set S ( x ) ) for a point x insub-Riemannian manifolds M (without abnormal geodesics) is the set of ends of allnoncontinuable beyond its ends shortest arcs starting at the point x (respectively,image of the set of critical points of the map Exp x with respect to Exp x ). Theorem 7.
For every element g ∈ ( SO (3) , d ) , C ( g ) = gC ( e ) and S ( g ) = gS ( e ) . Moreover S ( g ) ⊂ C ( g ) , (40) C ( e ) = { γ β ( t ( β )) : β ∈ R } , (41) S ( e ) = { γ β ( t ( β )) : β ≥ / } = SO (2) r { e } ; S ( e ) is diffeomorphic to R ; (42) S ( e ) = S ( e ) ∪ { e } = SO (2) ,S ( e ) is diffeomorphic to circle S ; (43) C ( e ) r S ( e ) = ( C ( e ) r S ( e )) ∪ (cid:26) γ β ( t ( β )) = γ − β ( t ( − β )) : β = 1 √ (cid:27) ,C ( e ) r S ( e ) is diffeomorphic to real projective plane RP ; C ( e ) is homeomorphic to RP ∪ R , where RP ∩ R is one-point set; C ( e ) is homeomorphic to RP ∪ S , where RP ∩ S is one-point set.Proof. First statement is a corollary of left invariance of the metric d on SO (3) . Inclusion S ( g ) ⊂ C ( g ) , formulae (40), (41), equality in brace from (43), and dif-feomorphism S ( e ) ∼ = R are corollaries from the proof of proposition 4 and remark4. Formula (42) and diffeomorphism S ( e ) ∼ = S follow from formula (41). Equality(43) follows from formulae (40), (41); C ( e ) r S ( e ) ∼ = RP follows from equalities γ ( β,φ ) ( t ( β )) = γ ( − β, − βt + φ + π ) ( t ( − β )) при β ≤ / . Now it is not difficult toprove remaining statements. (cid:3)
Remark 6.
It follows from (42) and equalities C ( g ) = gC ( e ) , S ( g ) = gS ( e ) that g ∈ gSO (2) = S ( g ) ⊂ C ( g ) for all g ∈ SO (3) , while x / ∈ C ( x ) and x / ∈ S ( x ) for anypoint x of arbitrary smooth Riemannian manifold. This constitutes radical differenceof Riemannian and sub-Riemannian manifolds. ROUP SO (3) References [1]
Berestovskii V.N., Zubareva I.A. , "Shapes of spheres of special nonholonomic left-invariantintrinsic metrics on some Lie groups", Siber. Math. J., 42, No. 4, 613-628 (2001).[2]
Sasaki S. , "On the differential geometry of tangent bundles of Riemannian manifolds", TohokuMath. J., 10, No. 3, 338-354 (1958); II: 14, No 2, 146-155 (1962).[3]
Besse A.L. , Manifolds all of whose geodesics are closed, Springer-Verlag, Berlin, Heidelberg,New York (1978).[4]
Berestovskii V.N., Guijarro L. , "A metric characterization of Riemannin submersions", Ann.Global Anal. Geom., 18, No 6, 577-588 (2000).[5]
Pogorelov A.V. , Differential geometry, Nordhof (1959).[6]
Berestovskii V.N. , "Universal methods of the search of normal geodesics on Lie groups withleft-invariant sub-Riemannian metric", Siber. Math. J., 55, No 5, 783-791 (2014).[7]
Boscain U., Rossi F. , "Invariant Carnot-Carath´eodory metrics on S , SO (3) , SL (2) , and lensspaces", SIAM J. Control Optim., 47, No 4, 1851-1878 (2008).[8] Agrachev A.A., Sachkov Yu.L. , Geometric control theory [in Russian], Fizmatlit, Moscow(2005).[9]
Helgason S. , Differential geometry and symmetric spaces, Academic Press, New York (1962).[10]
Gromoll D., Klingenberg W., Meyer W. , Riemannsche geometrie im grossen, Springer-Verlag,Heidelberg (1968).
V.N.BerestovskiiThe Sobolev Institute of Mathematics SD RAS, Novosibirsk, Russia
E-mail address : [email protected] I.A.ZubarevaOmsk Department of the Sobolev Institute of Mathematics, Omsk, Russia
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