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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