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Revista de la Unión Matemática Argentina

versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.47 n.2 Bahía Blanca jul./dic. 2006

 

Geodesics of the space of oriented lines of euclidean space

Marcos Salvai

Abstract: For n = 3 or n = 7 let 𝕋n be the space of oriented lines in ℝn . In a previous article we characterized up to equivalence the metrics on  n 𝕋 which are invariant by the induced transitive action of a connected closed subgroup of the group of Euclidean motions (they exist only in such dimensions and are pseudo-Riemannian of split type) and described explicitly their geodesics. In this short note we present the geometric meaning of the latter being null, time- or space-like.

On the other hand, it is well-known that 𝕋n is diffeomorphic to G (Hn ) , the space of all oriented geodesics of the n -dimensional hyperbolic space. For n = 3 and n = 7 , we compute now a pseudo-Riemannian invariant of  n 𝕋 (involving its periodic geodesics) that will be useful to show that 𝕋n and G (Hn ) are not isometrically equivalent, provided that the latter is endowed with any of the metrics which are invariant by the canonical action of the identity component of the isometry group of H .

2000 Mathematics Subject Classification. 53B30, 53C22, 22F30

Key words and phrases. oriented lines, minitwistor, pseudo-Riemannian, quaternions, octonions, pitch

Partially supported by Conicet, Secyt-UNC, Foncyt and Antorchas.

THE SPACE OF ORIENTED LINES OF  n ℝ . 

We begin by recalling the definitions and some notation and results from [4]. An oriented line in  n ℝ is a pair ℓ (u,v) := ({tu + v | t ∈ ℝ },u) for some u,v ∈ ℝ, |u| = 1 , where u is the direction (orientation) of the oriented line. Let  n 𝕋 denote the set of all oriented lines of  n ℝ and

 n-1 n n TS = {(u,v ) ∈ ℝ × ℝ | |u| = 1,⟨u,v⟩ = 0}

the tangent space of the (n - 1) -dimensional sphere. Then ℓ : T Sn- 1 → 𝕋n is a bijection whose inverse is given by

F : 𝕋n → T Sn- 1, F (ℓ(u,v)) = (u,v - ⟨v,u⟩ u)

(1)

(here v - ⟨v,u⟩u is the point on the line which is closest to the origin). This correspondence is called in [2] the minitwistor construction. By abuse of notation we sometimes identify 𝕋n with T Sn- 1 .

The group SO ⋉ ℝn n of Euclidean motions of ℝn , with multiplication given by  ′ ′ ′ ′ (k,a)(k ,a ) = (kk ,a + ka ) , acts transitively on  n 𝕋 in the canonical way (k,a) ⋅ (ℝu + v,u) = (ℝku + a + kv,ku ) .

Two pseudo-Riemannian metrics g1,g2 on a smooth manifold M are said to be equivalent if there exists a diffeomorphism f and a constant c ⁄= 0 such that f : (M, g1) → (M, cg2) is an isometry. Given an inner product ⟨,⟩ we denote ∥x∥ = ⟨x,x ⟩ and |x| = ∘ |⟨x,x⟩| . Let 𝔸 denote either of the normed division algebras ℍ or 𝕆 (quaternions and octonions, respectively) and let × denote the cross product in Im 𝔸 , the vector space of purely imaginary elements of 𝔸 . Let K𝔸 be the group of automorphisms of × , that is, K ℍ = SO3 and K 𝕆 = G2 .

INVARIANT METRICS ON 𝕋n FOR n = 3  AND n = 7 .

For n = 3 or n = 7 we identify ℝn with Im ℍ or Im 𝕆 , respectively. For μ ∈ ℝ we defined in [4] the split pseudo-Riemannian metric gμ on 𝕋n as the one whose associated norm is given by

 2 ∥(x,y)∥μ = ⟨x,u × y⟩ + μ |x|

(2)

for any (x,y ) ∈ T (u,v)T Sn-1 = T ℓ(u,v)𝕋n . The metric gμ is of type (2,2) or (6, 6) and is invariant by the induced action of H = SO3 ⋉ ℝ3 or H = G2 ⋉ ℝ7 on 𝕋n , depending on whether n = 3 or n = 7 .

We proved in the same article that only for those dimensions there exists a pseudo-Riemannian metric which is invariant by the induced transitive action of a connected closed subgroup of SOn ⋉ ℝn (as usual, we consider Riemannian metrics as a particular case of pseudo-Riemannian ones). The metrics gμ are not isometric to each other. Moreover, for μ ⁄= 0 , gμ is equivalent to g1 and not equivalent to g 0 .

We recall some further notation from [4].

Notation. In the following we set m = n - 1 and consider the canonical orthonormal basis {e0,e1,...,em} of ℝn . We take o := ℓ(e0,0) as origin in 𝕋n .

The isotropy subgroup at o of the action of H on 𝕋n is Ho := Ko × ℝe0 , where Ko = {k ∈ K | ke0 = e0} , the isotropy subgroup at e0 of the action of K on Sm , that is, Ko = SO2 or Ko = SU3 for m = 2 or m = 6 , respectively. The infinitesimal isotropy action of Ho is given by

(d(k,ce )) (x,y ) = (kx, k (y - cx )) 0 o

(3)

for any  m m n (x,y ) ∈ ℝ × ℝ = To 𝕋 .

Let 𝔥 , 𝔥o , 𝔨 , 𝔨o be the Lie algebras of H , Ho , K and Ko , respectively. We have the following direct sum decompositions: ℝn = ℝe0 + ℝm , 𝔥o = 𝔨o + ℝe0 and also, since K acts transitively on Sm , 𝔨 = 𝔨o + 𝔪 , where 𝔪 = {^x | x ∈ ℝm } , with  ( 0 - xt ) ^x = ∈ 𝔨 x 0m . Hence 𝔥 decomposes as 𝔥 = 𝔥o ⊕ 𝔭 , with 𝔭 = 𝔪 ⊕ ℝm (by abuse of notation we denote the subgroup  n {1} × ℝ of H by  n ℝ , and use the same notation for its subgroups).

NULL, TIME - AND SPACE - LIKE GEODESICS OF 𝕋n .

We obtained in [4] the complete description of the geodesics of  n (𝕋 ,gμ) for n = 3 and n = 7 :

Proposition 1. For n = 3 or n = 7 , the geodesics in (𝕋n, g ) μ through o are exactly the curves s ↦→ expH (sX ) ⋅ o , for X ∈ 𝔭 . In particular they are defined on the whole real line and do not depend on μ .

In this short note we present the geometric meaning of a geodesic being null, time- or space-like. We begin by stating a relationship with the ruled (parametrized) surface associated to it. The following proposition, which holds for all n ∈ ℕ , is elementary and well-known; we include it and its proof for the sake of completeness.

Proposition 2. If σ (s) = ℓ(us,vs) is a curve in  n 𝕋 with  ′ u s ⁄= 0 for all s , then there exists a unique curve

ασ (s) = vs - τ (s)us

in the parametrized (possible singular) ruled surface φ (s,t) = v + tu σ s s in ℝn , satisfying  ′ ′ ⟨u ,α σ⟩ = 0 . This curve is called the striction line of φ σ .

Moreover, if σ (0) = o and  ′ (F ∘ σ) (0) = (x,y) , then

α (0) = 0 ⇐ ⇒ ⟨x,y⟩ = 0 ⇐ ⇒ |J | takes its minimum at t = 0, σ

where J is the Jacobi field along the parametrization t ↦→ te0 of σ (0) associated to the variation by geodesics determined by σ .

Proof. Take τ = ⟨u′,v′⟩ ∕|u′|2 and use that |u| = 1 implies ⟨u,u ′⟩ = 0 . Uniqueness is clear. The first equivalence of the second assertion is a consequence of  ′ ′ ′ ′ (F ∘ σ ) (0 ) = (u 0,v 0 - ⟨v0,e0⟩e0) , which follows from (1) since  ′ u0 = e0⊥u 0 and v0 = 0. Finally, the Jacobi field along the given parametrization of σ (0) is  d-|| J (t) = ds0 vs + tus and satisfies ( ) |J |2′(t) = 2⟨u ′0,v′0⟩ + 2t|u′0|2 . □

Let now again n = 3 or n = 7 and suppose as before that ℝn = Im 𝔸 , with 𝔸 = ℍ or 𝔸 = 𝕆 . If σ is a curve in 𝕋n as in the Proposition above, the × -pitch of σ is the function  ′ ′ ′2 ρ = ⟨u × u ,v ⟩∕|u | , which is well-defined, since the expression does not change if one substitutes v with v + τu , where τ is any smooth function.

For example, if σ describes a helicoid passing through the origin, that is, φ σ (s,t) = sv + tus , where u describes, with unit angular speed, a unit circle in a plane orthogonal to v , then its striction line is ασ (s) = sv . (By abuse of notation we admit degenerate helicoids, in the case v = 0 .) For n = 3 its × -pitch is the constant ρ such that 2πρ is the (signed) length travelled along the striction line whilst u gives one complete positive turn around it. For n = 7 , one has to consider instead the (signed) length travelled along the projection of the striction line onto the × -normal to the oriented plane determined by the oriented circle u (here, the × -normal to the oriented plane determined by an orthonormal set {x, y} is x × y ).

According to the definition, if two curves in  n 𝕋 are H -congruent, then they have the same × -pitch, but if n = 7 , they might have different pitches if they are just congruent by an element of SO7 ⋉ ℝ7 .

Next we make explicit the identification of ℝn with Im ℍ and Im 𝕆 , if n = 3 or n = 7 , respectively. Let {1,i,j,k} be the standard orthonormal basis of ℍ . Let i⊥ denote the orthogonal complement of ℝi in Im 𝔸 . Given any unit element e ∈ 𝕆 orthogonal to ℍ ⊂ 𝕆 , we consider the orthonormal bases B = {j,k} 2 or B6 = {j,e,je,k,ie,ke} of  ⊥ i  m = TiS and use them to identify this vector space with  m ℝ . Let  ⊥ ⊥ Li : i → i be defined by Li(z) = iz = i × z . We identify as usual (ℝm, Li) = ℂm ∕2 .

In the following Lemma we consider on ℂ3 the canonical real inner product of the underlying six-dimensional Euclidean space.

Lemma 3. Let  3 x, y ∈ ℂ , with x ⁄= 0 and ⟨x,y⟩ = 0 . Then there exist g ∈ SU3 and a,b,c ∈ ℝ , b,c > 0, such that g(x ) = cj and g (y) = ak + be .

Proof. Let c = |x| and write  ′ a ′ y = a x + cix + y , with  ′ y ⊥ ℂx . Clearly  ′ a = 0 since ⟨x,y⟩ = 0 . Since SU3 acts transitively on  5 S , there exists g1 ∈ SU3 such that g1 (x ) = cj . Hence g1(ix) = ck and g1(y′) ∈ (ℂj )⊥ ~= ℂ2 (with the induced orientation). Since SU2 acts transitively on S3 , there exists g2 ∈ SU3 fixing j (and hence also k ) such that g (g (y′)) = be 2 1 for some b ≥ 0 . Thus, g = g ∘ g 2 1 satisfies the requirements. □

Proposition 4. Let n = 3 or n = 7 . Any nonconstant geodesic in 𝕋n is congruent by the action of H (up to orientation preserving reparametrization) to exactly one of the following geodesics

σ0 (s) = ℓ(i,sk), σ (s ) = ℓ ((cos s)i + (sin s)j,s(ak + be))

for some a,b ∈ ℝ, b ≥ 0 . Moreover, σ0 is a null geodesic for any μ and its corresponding ruled surface is a plane. The number a is the × -pitch of the ruled surface determined by σ (a helicoid) and ∥σ ′∥μ = μ - a . That is, σ is a space-like, time-like or null geodesic if and only if the × -pitch of the corresponding ruled surface if smaller, bigger or equal to μ , respectively.

Proof. First we show that σ0 and σ are geodesics. We call y = ak + be , consider (0,k) and (^j,y) as elements of 𝔭 ⊂ 𝔥 and observe that

σ0 (s ) = ℓ (i,sk ) = (1, sk) ⋅ ℓ (i,0) = expH (s(0,k)) ⋅ o.

We also have that expK (s^j)i = icos s + j sin s . Moreover, by definition of the multiplication on H , expH s (^j,y) = (expK (s^j),sy) , since ⟨y,j⟩ = 0 . Hence,

σ (s) = ℓ(icos s + j sin s,sy) = exp s (^j,y) ⋅ o. H

Therefore, σ0 and σ are geodesics by Proposition 1.

Given a nonconstant geodesic γ in 𝕋n , since the action of H on 𝕋n is transitive, we may suppose that γ (0) = o . Hence (F ∘ γ )′(0 ) = (x, y) ∈ ℝm × ℝm . If x = 0, there exists g ∈ Ko (which acts transitively on  m- 1 S ) with g(y) = ck . Hence, (g ∘ γ) (s) = σ0 (cs) for all s . If x ⁄= 0, by looking at the action (3) of Ho on To 𝕋n , one may suppose additionally that ⟨x,y ⟩ = 0 (see the geometric meaning of this condition in Proposition 2). If n = 7 , by Lemma 3, there exists g ∈ Ko = G ~= SU3 such that g ∘ γ and σ differ in an orientation preserving reparametrization. The case n = 3 , where  ~ Ko = SO2 = U1 , is clear. The curve σ0 is not H -congruent to a reparametrization of σ , since by (3) the Ho -orbit of  ′ σ0(0) consists of the elements (0, y) with y in a sphere. On the other hand, one has

 ′ 2 ∥ σ (0)∥μ = ∥ (j,ak + be )∥ μ = ⟨j,i × (ak + be)⟩ + μ|j| = ⟨j,- aj + bie⟩ + μ = μ - a,

and the × -pitch of σ is ρ (s) = ⟨(icos s + j sin s) × (j coss - isin s),ak + be⟩ = a . Hence, the last assertion is true. □

Remark. For n = 3 and μ = 0 , the geometric interpretation given above of a geodesic in (𝕋n,gμ) being null, time- or space-like is of course a rephrasing of that given in [1] involving angular momentum.

A GEOMETRIC INVARIANT OF 𝕋n

It is well-known that 𝕋n is diffeomorphic to G (H ) , the space of all oriented geodesics of H , for any Hadamard manifold of dimension n (see [3]). For n = 3 and n = 7 , we compute now a pseudo-Riemannian invariant of  n 𝕋 (involving its periodic geodesics) that will be useful in [5] to show that if H is the n -dimensional hyperbolic space, then  n 𝕋 and G (H ) are not isometrically equivalent, provided that the latter is endowed with any of the metrics which are invariant by the canonical action of the identity component of the isometry group of H .

We remark that in [4] we obtained the geodesics of 𝕋n without needing to compute explicitly the Levi-Civita connection. That is why we give this pseudo-Riemannian invariant instead of a more standard one, like the curvature, since the computation of the latter would have been probably rather cumbersome.

For n = 3 or n = 7 and ℓ ∈ 𝕋n let A denote the subset of Tℓ𝕋n consisting of the initial velocities of periodic geodesics of 𝕋n through ℓ .

Proposition 5. The frontier of A in  n T ℓ𝕋 is a subspace of dimension m .

Proof. Since 𝕋n is homogeneous we may suppose that ℓ = o . Clearly the geodesic σ in Proposition 4 is periodic if and only if a = b = 0 , while σ0 is not periodic. By that proposition, A is the orbit of the isotropy action (3) of the multiples of the initial velocity of σ (s) = ℓ((coss) i + (sin s)j,0) . Under the identification  n m m To𝕋 ~= ℝ × ℝ one has σ ′(0 ) = (j,0) . Therefore A = {(x,cx) ∈ ℝm × ℝm | c ∈ ℝ} , since Ko acts transitively on the unit sphere in ℝm .

We show that the frontier of A equals {0} × ℝm . Since clearly (0, y)∈∕A if y ⁄= 0 and (0,y) = limn → ∞ (y∕n,ny ∕n) for all  m y ∈ ℝ , we have that  m {0} × ℝ is contained in the frontier of A . Next we verify the other inclusion. Suppose that limn →∞ (xn, cnxn) = (x,y) . If x = 0 we are done. If x ⁄= 0 , we have  2 cn |xn | = ⟨cnxn, xn⟩ . Hence limn→ ∞ cn = ⟨y,x⟩ ∕|x|2 := c . Therefore (x,y) = (x,cx) , which belongs to the interior of A . This completes the proof of the proposition. □

References

[1]    B. Guilfoyle & W. Klingenberg, An indefinite Kähler metric on the space of oriented lines, J. London Math. Soc. 72 (2005), 497-509.        [ Links ]

[2]    N. J. Hitchin, Monopoles and geodesics, Comm. Math. Phys. 83 (1982), 579-602.        [ Links ]

[3]    G. Keilhauer, A note on the space of geodesics. Rev. Unión Mat. Argent. 36 (1990), 164-173.        [ Links ]

[4]    M. Salvai, On the geometry of the space of oriented geodesics of Euclidean space, Manuscr. Math. 118 (2005), 181-189.        [ Links ]

[5]    M. Salvai, On the geometry of the space of oriented lines of the hyperbolic space, preprint.        [ Links ]

Marcos Salvai
FaMAF-CIEM, Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
salvai@mate.uncor.edu

Recibido: 3 de noviembre de 2005
Aceptado: 19 de septiembre de 2006

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