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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 or let be the space of oriented lines in . In a previous article we characterized up to equivalence the metrics on 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 is diffeomorphic to , the space of all oriented geodesics of the -dimensional hyperbolic space. For and , we compute now a pseudo-Riemannian invariant of (involving its periodic geodesics) that will be useful to show that and 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 .
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 .
We begin by recalling the definitions and some notation and results from [4]. An oriented line in is a pair for some , where is the direction (orientation) of the oriented line. Let denote the set of all oriented lines of and
the tangent space of the -dimensional sphere. Then is a bijection whose inverse is given by
(1) |
(here 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 with .
The group of Euclidean motions of , with multiplication given by , acts transitively on in the canonical way .
Two pseudo-Riemannian metrics on a smooth manifold are said to be equivalent if there exists a diffeomorphism and a constant such that is an isometry. Given an inner product we denote and . Let denote either of the normed division algebras or (quaternions and octonions, respectively) and let denote the cross product in Im , the vector space of purely imaginary elements of . Let be the group of automorphisms of , that is, and .
INVARIANT METRICS ON FOR AND .
For or we identify with Im or Im , respectively. For we defined in [4] the split pseudo-Riemannian metric on as the one whose associated norm is given by
(2) |
for any . The metric is of type or and is invariant by the induced action of or on , depending on whether or .
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 (as usual, we consider Riemannian metrics as a particular case of pseudo-Riemannian ones). The metrics are not isometric to each other. Moreover, for , is equivalent to and not equivalent to .
We recall some further notation from [4].
Notation. In the following we set and consider the canonical orthonormal basis of . We take as origin in .
The isotropy subgroup at of the action of on is , where , the isotropy subgroup at of the action of on , that is, or for or , respectively. The infinitesimal isotropy action of is given by
(3) |
for any .
Let , , , be the Lie algebras of , , and , respectively. We have the following direct sum decompositions: , and also, since acts transitively on , , where , with . Hence decomposes as , with (by abuse of notation we denote the subgroup of by , and use the same notation for its subgroups).
NULL, TIME - AND SPACE - LIKE GEODESICS OF .
We obtained in [4] the complete description of the geodesics of for and :
Proposition 1. For or , the geodesics in through are exactly the curves , for . In particular they are defined on the whole real line and do not depend on .
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 , is elementary and well-known; we include it and its proof for the sake of completeness.
Proposition 2. If is a curve in with for all , then there exists a unique curve
in the parametrized (possible singular) ruled surface in , satisfying . This curve is called the striction line of .
Moreover, if and , then
where is the Jacobi field along the parametrization of associated to the variation by geodesics determined by .
Proof. Take and use that implies . Uniqueness is clear. The first equivalence of the second assertion is a consequence of , which follows from (1) since and Finally, the Jacobi field along the given parametrization of is and satisfies .
Let now again or and suppose as before that Im , with or . If is a curve in as in the Proposition above, the -pitch of is the function , which is well-defined, since the expression does not change if one substitutes with , where is any smooth function.
For example, if describes a helicoid passing through the origin, that is, , where describes, with unit angular speed, a unit circle in a plane orthogonal to , then its striction line is . (By abuse of notation we admit degenerate helicoids, in the case .) For its -pitch is the constant such that is the (signed) length travelled along the striction line whilst gives one complete positive turn around it. For , 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 (here, the -normal to the oriented plane determined by an orthonormal set is ).
According to the definition, if two curves in are -congruent, then they have the same -pitch, but if , they might have different pitches if they are just congruent by an element of .
Next we make explicit the identification of with Im and Im , if or , respectively. Let be the standard orthonormal basis of . Let denote the orthogonal complement of in Im . Given any unit element orthogonal to , we consider the orthonormal bases or of and use them to identify this vector space with . Let be defined by . We identify as usual .
In the following Lemma we consider on the canonical real inner product of the underlying six-dimensional Euclidean space.
Lemma 3. Let , with and . Then there exist and , such that and .
Proof. Let and write , with . Clearly since . Since acts transitively on , there exists such that . Hence and (with the induced orientation). Since acts transitively on , there exists fixing (and hence also ) such that for some . Thus, satisfies the requirements.
Proposition 4. Let or . Any nonconstant geodesic in is congruent by the action of (up to orientation preserving reparametrization) to exactly one of the following geodesics
for some . Moreover, is a null geodesic for any and its corresponding ruled surface is a plane. The number is the -pitch of the ruled surface determined by (a helicoid) and . That is, 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 , respectively.
Proof. First we show that and are geodesics. We call , consider and as elements of and observe that
We also have that . Moreover, by definition of the multiplication on , , since . Hence,
Therefore, and are geodesics by Proposition 1.
Given a nonconstant geodesic in , since the action of on is transitive, we may suppose that . Hence . If there exists (which acts transitively on ) with . Hence, for all . If by looking at the action (3) of on , one may suppose additionally that (see the geometric meaning of this condition in Proposition 2). If , by Lemma 3, there exists such that and differ in an orientation preserving reparametrization. The case , where , is clear. The curve is not -congruent to a reparametrization of , since by (3) the -orbit of consists of the elements with in a sphere. On the other hand, one has
and the -pitch of is . Hence, the last assertion is true.
Remark. For and , the geometric interpretation given above of a geodesic in being null, time- or space-like is of course a rephrasing of that given in [1] involving angular momentum.
A GEOMETRIC INVARIANT OF
It is well-known that is diffeomorphic to , the space of all oriented geodesics of , for any Hadamard manifold of dimension (see [3]). For and , we compute now a pseudo-Riemannian invariant of (involving its periodic geodesics) that will be useful in [5] to show that if is the -dimensional hyperbolic space, then and 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 .
We remark that in [4] we obtained the geodesics of 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 or and let denote the subset of consisting of the initial velocities of periodic geodesics of through .
Proposition 5. The frontier of in is a subspace of dimension .
Proof. Since is homogeneous we may suppose that . Clearly the geodesic in Proposition 4 is periodic if and only if , while is not periodic. By that proposition, is the orbit of the isotropy action (3) of the multiples of the initial velocity of . Under the identification one has . Therefore , since acts transitively on the unit sphere in .
We show that the frontier of equals . Since clearly if and for all , we have that is contained in the frontier of . Next we verify the other inclusion. Suppose that . If we are done. If , we have . Hence . Therefore , which belongs to the interior of . This completes the proof of the proposition.
[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