Shirley Bromberg

Alberto Medina
email: medina@math.univ-montp2.fr

Geodesically Complete Lorentzian Metrics on Some Homogeneous 3 Manifolds

Keywords: Lorentzian metrics, complete geodesics, 3-dimensional Lie groups, Euler equation 2000 Mathematics Subject Classification: 53C22, 53C50, 57M50, 22E30

In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete.

Introduction

The aim of this paper is to characterize the geodesically complete left invariant Lorentzian metrics on unimodular 3-dimensional Lie groups. The main result (Theorem 2) improve and complete Proposition 3.2 given in [START_REF] Bromberg | Complétude de l'équation d'Euler[END_REF]. It is well known that an unimodular Lie group of dimension 3 contains an infinity of cocompact discret subgroups. Consequently our results provide a rich class of compact and complete (or incomplete) Lorentzian 3-manifolds. In particular, we obtain the geodesically complete and compact locally homogeneous Lorentz 3-manifolds with non compact (local) isotropy group as described in [START_REF] Dumitrescu | Géométries Lorentziennes de dimension 3: classification et complétude[END_REF]. Moreover, our results give a correct description of the complete (or incomplete) Lorentzian metrics on M = Γ\PSL(2, R), where Γ is a Fuchsian group, obtained from a left invariant Lorentzian metric on PSL(2, R).

To characterize the complete left invariant pseudo-Riemannian metrics which are not definite on arbitrary Lie groups is a difficult problem. On one side of the spectrum we find groups where all such metrics are complete, such as the Abelian groups, the compact groups, the 2-nilpotent groups [START_REF] Guediri | Sur la complétude des pseudo-métriques invariantes a gauche sur les groupes de Lie nilpotents[END_REF] or the group of isometries of the Euclidean plane, and on the other side, groups such as the group of the affine transformations of the line in which no Lorentzian metric is complete.

The structure of this work is as follows: In Section 2 the definitions and some general results are given, particularly the algebraic counterparts of the geometric properties of left invariant pseudo-Riemannian metrics defined on Lie groups. In Section 3 we characterize the left invariant metrics on 3-dimensional unimodular Lie groups that are geodesically complete. The following result is proved:

A left invariant metric on an unimodular 3-dimensional Lie group is geodesically complete if and only if all geodesics that run on 1-parameter subgroups are complete.

In Section 4 we give an example that shows that Theorem 2 is not valid without the hypothesis of unimodularity. We also show that previous results in the literature [START_REF] Guediri | Sur la complétude des varietés pseudo-Rimanniennes[END_REF] are not accurate.

Preliminaries and general results

Let G = T ǫ (G) be the Lie algebra of a Lie group G, where ǫ stands for the identity of G. Every left invariant pseudo-Riemannian metric q on G induces a left invariant connection ∇ on G, the Levi-Civita connection; a non degenerate quadratic form •, • on G, the polar form of q ǫ , the restriction of q to G; and a product x • y on G defined by

(x • y) + = ∇ x + y + ,
where x + is the left invariant vector field on G determined by x ∈ G. These objects are related by the Koszul formula:

x • y, z = 1 2 ( [x, y], z -[y, z], x + [z, x], y ) . If c(t) is a curve in G and x(t) = L c(t) -1 * ,c(t) c ′ (t),
where L σ denotes the left multiplication by σ in G, it follows that c(t) is a geodesic if and only if x(t) is a solution of the equation

ẋ = -x • x. ( 1
)
That ∇ is geodesically complete is equivalent to the completeness of the vector field given by equation [START_REF] Bromberg | Complétude de l'équation d'Euler[END_REF]. Clearly the energy, that is q ǫ (x) = x, x , is a first integral of equation [START_REF] Bromberg | Complétude de l'équation d'Euler[END_REF].

Notice that if G is Abelian, then x • y ≡ 0 for every left invariant pseudo-Riemannian metric. Hence, every left invariant pseudo-Riemannian metric on an Abelian Lie group is complete.

If φ : G → G * is the canonical linear isomorphism associated to •, • then the equation ( 1) can be written, in terms of the co-adjoint representation as

ξ = -ad * φ -1 ξ ξ (2)
with ξ ∈ G * . Equation ( 2) is called the Euler equation associated to the pseudo-Riemannian metric, and the corresponding vector field is called the Euler vector field. Consequently the solutions of equation ( 2) are curves in the orbits of the co-adjoint representation of G. This implies that (2) is complete if G is compact. The energy on G * is given by q ǫ (φ -1 ξ). Suppose that (G, k) is orthogonal or quadratic, i.e. k is a bi-invariant pseudo-Riemannian metric. Then x•x = 0 and equation ( 1) is obviously complete and the geodesics through ǫ are the 1-parameter subgroups of G. Note that every left invariant pseudo-Riemannian metric •, • on a orthogonal Lie group (G, k) is given by the formula x, y = k(u(x), y) where u is a k-symmetric linear isomorphism of G. Equation ( 1) is given in this case by

u( ẋ) = [x, u(x)]
which is equivalent by a linear change of coordinates to the Lax pair

ẋ = [x, u -1 x]. (3)
Equation ( 3) has two quadratic first integrals k(x, x), k(x, u -1 x).

The vector fields induced by a left invariant pseudo-Riemannian metric are special cases of homogeneous quadratic vector fields. Definition 1. A homogeneous quadratic vector field on the n-dimensional affine space R n is a vector field F : R n → R n , F = (F 1 , . . . , F n ), where F i is a homogeneous polynomial of degree 2, i = 1, . . . , n. If the solutions of the differential equation Ẋ = F (X) are defined for all t ∈ R, the vector field F is called complete.

The following definition provides the simplest example of complete homogeneous quadratic vector fields: Definition 2. An affine-quadratic (or linearizable) vector field is a homogeneous quadratic vector field such that Ẋ = F (X) is equivalent by a linear change of coordinates to

U = A(V )U + B(V ), V = 0,
where A is linear and the coordinate functions of B are homogeneous polynomials of degree 2.

It is easily checked that every affine-quadratic vector field is complete.

Definition 3. Let F be a homogeneous quadratic vector field. We define

I := {X = 0 ; X, F (X) are linearly dependent}. If X ∈ I and F (X) = 0, X is called a zero of F. If X ∈ I and F (X) = 0 then X is called a relaxed idempotent of F.
In fact a strict idempotent is a non trivial solution of F (X) = X. If X ∈ I and F (X) = 0, then there exists λ = 0 such that F (λX) = λX, i.e. for every relaxed idempotent there is a strict idempotent in the same direction (the line through X is invariant by F ). In what follows we make no distinction between relaxed or strict idempotents. Remark 1. A simple calculation shows that any idempotent of F gives an incomplete solution for Ẋ = F (X).

In fact the solution of Ẋ = F (X) with initial condition an idempotent X 0 is given by t → α(t)X 0 where α satisfies α = α 2 . Moreover if X 0 ∈ I for the Levi-Civita product, the geodesic through ǫ with velocity X 0 is given by t → exp β(t)X 0 , where β is such that β(0) = 0, β(t) = α(t) and α = α 2 . A result by Kaplan and Yorke (see [START_REF] Kaplan | Non associative real algebras and quadratic differential equations[END_REF]) states that for every quadratic homogeneous vector field F , I = ∅, hence Proposition 1. Any left invariant pseudo-Riemannian metric on G has a geodesic that runs on a 1-parameter subgroup. Furthermore if the pseudo-Riemannian metric is complete then the corresponding Levi-Civita product has a zero. Hence for such metrics, there is a 1-parameter subgroup of G which is a geodesic. Lemma 1. Let X be an idempotent of a quadratic vector field F and Q a quadratic form which is a first integral of Ẋ = F (X). Then Q(X) = 0.

Proof . Let P be the bilinear polar form associated to Q. Then for any curve t → X(t)

d dt Q(X) = 2P X, d dt X . If t → X(t) is a solution of Ẋ = F (X) and Q is a first integral, then 0 = d dt Q(X) = 2P X, d dt X = 2P (X, F (X)) = -2Q(X)
as was asserted.

The following theorem is a general result of completeness for homogenous quadratic vector fields in dimension 2.

Theorem 1 ([2]

). Let F be a homogeneous quadratic vector field on the plane. The following assertions are equivalent:

(i) Ẋ = F (X) is complete;
(ii) F is quadratic-affine or the differential equation Ẋ = F (X) is equivalent, by a linear change of coordinates, to a system ẋ = y(ax + by), ẏ = y(cx + dy),

where (a + d) 2 -4(ad -bc) < 0.

Complete left invariant pseudo Riemannian metrics def ined on 3-dimensional unimodular Lie groups

The Lie bracket of a 3-dimensional Lie algebra can be described using the vector product × corresponding to the positive definite scalar product •, • and to an orientation as follows [START_REF] Milnor | Curvatures of left invariant metrics on Lie groups[END_REF]:

[X, Y ] = L(X × Y ),
where

L ∈ End (G).
If the group is unimodular, i.e. tr(ad

X ) = 0 for all X ∈ G, then L is •, • -symmetric. Hence G has a •, • -orthonormal basis of eigenvectors of L, B = (E 1 , E 2 , E 3 ), that is [E 2 , E 3 ] = α 1 E 1 , [E 3 , E 1 ] = α 2 E 2 , [E 1 , E 2 ] = α 3 E 3 .
The signature of the eigenvalues characterizes the Lie algebra as is shown in the following diagram signature of α 1 , α 2 , α 3 corresponding Lie group type (a) 0, 0, 0

R ⊕ R ⊕ R Abelian (b) +, 0, 0 Heisenberg nilpotent (c) +, +, + O(3) compact (d) +, +, 0 E(2) solvable (e) +, -, 0 E(1, 1) solvable (f) +, +, - Sl(2, R) semisimple Theorem 2.
The following assertions are equivalent for left invariant pseudo-Riemannian metrics defined on 3-dimensional unimodular Lie groups (i) the Euler equation is complete;

(ii) all geodesics that run on 1-parameter subgroups are complete;

(iii) the Euler field has no idempotents.

Notice that, by Remark 1, (ii) ⇒ (iii) since an idempotent of the Euler equation produces an incomplete geodesic that runs on a 1-parameter subgroup. Hence the only implication that requires proof is (iii) ⇒ (i).

In the first four cases, all pseudo-Riemannian metrics are complete, the first three being widely known:

(a) the Euler equation for a left invariant pseudo-Riemannian metric on an Abelian Lie group is complete;

(b) the Euler equation for a left invariant pseudo-Riemannian metric on Heisenberg group is linearizable;

(c) the Euler equation for a left invariant pseudo-Riemannian metric on a compact Lie group has a definite quadratic first integral: the Killing form.

The proof of Theorem 2 consists on considerations of case by case situations.

Proposition 2. Every left invariant pseudo-Riemannian metric on E(2) is complete.

Proof . In a suitable basis, the Lie bracket is given by the relations:

[E 2 , E 3 ] = 0, [E 3 , E 1 ] = E 2 , [E 1 , E 2 ] = E 3 (4)
and the Euler equation is

ξ1 = x 3 ξ 2 -x 2 ξ 3 , ξ2 = x 1 ξ 3 , ξ3 = -x 1 ξ 2 .
Since the semi-definite quadratic ξ 2 2 + ξ 2 3 is a first integral of the Euler vector field, by Lemma 1 the Euler vector field has no idempotents.

We split the proof of the completeness of Euler vector field in two cases: i) If the pseudo-Riemannian metric •, • is degenerate on the vector space spanned by E 2 , E 3 , then x 1 does not depend on ξ 1 and the two last equations of the Euler equation lie in the ξ 2 ξ 3 plane. This planar system of equations has a positive definite quadratic first integral, hence it is complete. The first equation is of the form

ξ1 = f (t)ξ 1 + g(t)
and its solution is defined for all values of t since f and g are so.

ii) If the pseudo-Riemannian metric •, • is non degenerate on the vector space spanned by E 2 , E 3 , then there is a basis of E( 2) with E 1 ⊥ •,• Vect{E 2 , E 3 } that satisfies the same relations for the bracket stated in equation [START_REF] Guediri | Sur la complétude des varietés pseudo-Rimanniennes[END_REF]. Then the first integral ξ 2 2 + ξ 2 3 and the energy, that in this basis has the form ξ 2 1 + aξ 2 2 + bξ 2 ξ 3 + cξ 2 3 , can be combined to get a positive definite first integral. Hence the solutions of the Euler equation are complete.

In [START_REF] Guediri | On completeness of left-invariant Lorentz metrics on solvable Lie groups[END_REF] another proof of Proposition 2 and an alternative statement and proof of Proposition 3 are provided. Proposition 3. If the Euler product of a pseudo-Riemannian metric defined on E(1, 1) has no idempotents then the Euler equation has a definite quadratic first integral, and hence such a metric is complete.

Proof . In a suitable basis, the Lie bracket is given by the relations

[E 1 , E 2 ] = E 2 , [E 1 , E 3 ] = -E 3 , [E 2 , E 3 ] = 0 (5)
and the Euler equation is

ξ1 = -x 2 ξ 2 + x 3 ξ 3 , ξ2 = x 1 ξ 2 , ξ3 = -x 1 ξ 3 .
Besides the energy: Φ -1 ξ, Φ -1 ξ = x 1 ξ 1 + x 2 ξ 2 + x 3 ξ 3 , the Euler vector field has ξ 2 ξ 3 as a first integral. If the Euler vector field has no idempotents, the pseudo-Riemannian metric is non degenerate on [G, G] = Vect{E 2 , E 3 }. In fact, for a pseudo-Riemannian metric which is degenerate on [G, G], x 1 does not depend on ξ 1 and any ξ 2 such that x 1 = 1 for ξ 3 = 0 gives an idempotent of the Euler vector field.

Hence, if the Euler vector field has no idempotents, there exists e 1 ⊥ •,• [G, G] such that the bracket relations given by equation ( 5) are still satisfied, and the energy in these coordinates is given by:

λξ 2 1 + µξ 2 2 + 2cξ 2 ξ 3 + νξ 2 3 .
Taking ξ 2 = 0, the equation for idempotents becomes

ξ 1 = x 3 ξ 3 = νξ 2 3 , ξ 3 = -x 1 ξ 3 = -λξ 1 ξ 3 .
which has a solution if and only if λµ < 0. Hence, if the Euler equation has no idempotents λµ > 0. Analogously λν > 0. Since ξ 2 ξ 3 is a first integral

λξ 2 1 + µξ 2 2 + νξ 2 3
is a first integral which is a definite quadratic form.

Proposition 4. If the Lax pair vector field for a pseudo-Riemannian metric on Sl(2, R) has no idempotents then it is complete.

In order to deal with Sl(2, R) some previous considerations are needed. The Killing form k of sl (2, R) is a quadratic form of index (2, 1), and induces a bi-invariant Lorentz metric on Sl(2,R). Thus the completeness of a left invariant pseudo-Riemannian metric is equivalent to the completeness of the Lax pair. We express this equation in two adapted basis (all non stated products are obtained either by symmetry or by antisymmetry):

(A) given a k-orthonormal basis, i.e. an ordered basis (E 1 , E 2 , E 3 ), such that

1 = k(E 1 , E 1 ) = k(E 2 , E 2 ) = -k(E 3 , E 3 ),
the Lie bracket is given by the table

[ , ] E 1 E 2 E 3 E 1 0 -κ E 3 -κ E 2 E 2 0 κ E 1 E 3 0 and matrix of ad X is ad X = κ   0 -x 3 x 2 x 3 0 -x 1 x 2 -x 1 0   ;
(B) given a k-hyperbolic basis, i.e. an ordered basis (E 1 , E 2 , E 3 ), such that

k(E 1 , E 1 ) = k(E 2 , E 3 ) = 1,
the Lie bracket is given by

[ , ] E 1 E 2 E 3 E 1 0 κ E 2 -κ E 3 E 2 0 κ E 1 E 3 0
and ad X has the matrix

ad X = κ   0 -x 3 x 2 -x 2 x 1 0 x 3 0 -x 1   , where κ = k([E 1 , E 2 ], E 3 ).
Since it is possible to take care of a constant by a linear change of coordinates, it will be omitted in what follows.

As was said before, the proof will consist in dealing with each case of the Jordan form of u.

Lemma 2. If u has exactly one eigendirection then the Lax pair vector field has always idempotents.

Proof . Let E be an eigenvector. There are two possibilities: either k(E, E) = 0 or k(E, E) = 0. Assume with no loss of generality that the eigenvalue is 1. Suppose first that E is an eigenvector such that k(E, E) = 0. Then k(E, E) > 0. In fact, under the hypothesis, the subspace E ⊥ is a 2-dimensional non degenerate subspace which is u invariant. If k(E, E) < 0, k is positive definite on E ⊥ . Thus u is diagonalizable on E ⊥ , which contradicts the assumption.

Let E 1 be an eingenvector such that k(E 1 , E 1 ) = 1, and

E 2 , E 3 ∈ E ⊥ such that k(E 2 , E 2 ) = -k(E 3 , E 3 ) = 1, k(E 2 , E 3 ) = 0.
The matrix of u and of u -1 are:

  1 0 0 0 α β 0 -β δ   , 1 ∆   ∆ 0 0 0 δ -β 0 β α   ,
where ∆ = β 2 + αδ. Notice that in this case the characteristic polynomial of u, p u (x) = (x -1) (x -(α + δ)x + ∆), has one simple real root. Hence ∆ > 0 and (α -δ) 2 -4β 2 < 0. The Lax pair is given by:

ẋ1 = 1 ∆ (βx 2 2 + (α -δ)x 2 x 3 + βx 2 3 ), ẋ2 = x 1 ∆ (-βx 2 + (∆ -α)x 3 ), ẋ3 = x 1 ∆ ((∆ -δ)x 2 + βx 3 ). If (x 1 , x 2 , x 3
) is an idempotent of the equation then x 1 = 0 and ∆/x 1 is an eigenvalue for

-β ∆ -α ∆ -δ β .
Since the characteristic polynomial of this matrix is

x 2 -(β 2 + (∆ -α)(∆ -δ))
, then the planar system has idempotents if and only if

0 < β 2 + (∆ -α)(∆ -δ) = ∆(β 2 + αδ + 1 -(α + δ))
or if and only if β 2 + αδ + 1 -(α + δ) > 0, because ∆ > 0. This inequality holds since

β 2 + αδ > 1 4 (α + δ) 2 ≥ (α + δ) -1.
Hence the planar system has two eigendirections corresponding to two eigenvalues of different sign. Finally to produce an idempotent for the Lax pair, it is necessary to solve the equation

∆ x 1 = ∆ 2 βx 2 2 + (α -δ)x 2 x 3 + βx 2 3
, where ∆/x 1 is an eigenvalue and (x 2 , x 3 ) a corresponding eigenvector. Since the quadratic form βx 2 2 + (α -δ)x 2 x 3 + βx 2 3 is definite, this equation has always a solution. Now suppose that k(E, E) = 0. Let E 3 be an eigenvector. By hypothesis, k(E 3 , E 3 ) = 0. Let E 2 be such that k(E 3 , E 2 ) = 1 and k(E 2 , E 2 ) = 0. Choose E 1 k-orthogonal to E 2 and E 3 , with k(E 1 , E 1 ) = 1, which is possible because of the signature of k. Hence we obtain a khyperbolic basis (E 1 , E 2 , E 3 ). Since u is k-symmetric, the matrix of u in the basis constructed above is

  γ a 0 0 1 0 a b 1   , whose characteristic polynomial is p u (x) = (x -γ)(x -1) 2 .
Since u has only one eigendirection, γ = 1 and the matrix of

u -1 is   1 -a 0 0 1 0 -a a 2 -b 1   .
The Lax pair is given by:

ẋ1 = -ax 1 x 2 + (a 2 -b)x 2 2 + (b -1)x 2 x 3 , ẋ2 = ax 2 2 , ẋ3 = ax 2 1 -(a 2 -b)x 1 x 2 -ax 2 x 3 .
Notice that the hypothesis imply that a = 0. An idempotent is produced by setting x 2 = 1/a and solving the other two equations, with this value for x 2 .

Lemma 3. Let u be diagonalizable. Then the pseudo-Riemannian metric on Sl(2, R) defined by u has a definite quadratic first integral (and hence is complete) provided that the Lax pair vector field has no idempotents.

Proof . If u is diagonalizable then there is a k-orthonormal basis E 1 , E 2 , E 3 with E 1 , E 2 , E 3 eigenvectors of u. In fact, if u has one eigenvalue with eigenspace of dimension 3, then u is a multiple of the identity and the assertion follows trivially. If u has three different eigenvalues, then the proof is straightforward. Assume u has two eigenvalues and that one of them has eigenspace of dimension 2, i.e. there exists a basis

{E 1 , E 2 , E 3 } such that u(E 1 ) = λE 1 , u(E 2 ) = λE 2 , u(E 3 ) = µE 3 , λ = µ.
Then the restriction of u to the eigenspace corresponding to λ is a multiple of the identity. Since

k(E 1 , E 3 ) = k(E 2 , E 3 ) = 0, then k(E 3 , E 3 ) = 0. Hence the k-non degenerate subspace Vect{E 3 } is invariant by u. This implies that the eigenspace Vect{E 1 , E 2 } = Vect{E 3 } ⊥ is k-non degenerate.
And the assertion follows.

Hence the matrix of u -1 in the k-orthonormal basis of eigenvectors is given by

  λ 1 0 0 0 λ 2 0 0 0 λ 3   ,
where λ -1 1 , λ -1 2 and λ -1 3 are the eigenvalues of u. The Lax pair is given by ẋ1

= (λ 3 -λ 2 ) x 2 x 3 = ax 2 x 3 , ẋ2 = (λ 1 -λ 3 ) x 1 x 3 = bx 1 x 3 , ẋ3 = (λ 1 -λ 2 ) x 1 x 2 = (a + b)x 1 x 2 ,
where

a = λ 3 -λ 2 , b = λ 1 -λ 3 .
The equation for idempotents is:

x 1 = a x 2 x 3 , x 2 = b x 1 x 3 , x 3 = (a + b) x 1 x 2 ,
which is equivalent to:

b(a + b) x 2 1 = a(a + b) x 2 2 = ab x 2 3 . (6)
Equation ( 6) has a non trivial solution if and only if a, b and a + b have the same signature, that is, if and only if ab > 0. Hence the Euler product has no non trivial idempotents if and only if ab < 0 or ab(a + b) = 0. If ab(a + b) = 0 the Lax pair is linearizable. Assume ab < 0. Clearly

bx 2 1 -ax 2 2
is a first integral. This quadratic, which is a semi-definite quadratic form, and the quadratic first integral x 2 1 + x 2 2 -x 2 3 can be combined into a quadratic positive definite first integral.

Lemma 4. If u has two linearly independent eigenvectors corresponding to two different eigenvalues then the left invariant pseudo-Riemannian metric on Sl(2, R) defined by u is complete whenever the Lax pair vector field has no idempotents.

Proof . Let p u (x) = (x -α) 2 (x -β) be the characteristic polynomial of u, E an eigenvector of u corresponding to the simple eigenvalue β. Denote by V the subspace {x ∈ G : (u -α Id) 2 x = 0}. Then V and the subspace generated by E are u invariant and E ⊥ k V. The last assertion follows from the following calculation: when x ∈ V,

u 2 (x) -2αu(x) + α 2 x = 0 and α 2 k(E, x) = -k(E, u 2 (x) -2αu(x)) = -k(E, u 2 (x)) + 2αk(E, u(x)) = -β 2 k(E, x) + 2αβk(E, x) since u is k-symmetric. Hence (α 2 + β 2 -2αβ)k(E, x) = 0 and k(E, x) = 0. (By hypothesis, α = β.) For every E ′ ∈ V eigenvector corresponding to α, k(E ′ , E ′ ) = 0 because if k(E ′ , E ′ ) = 0, then Vect{E, E ′ } is a u invariant, k -non degenerate subspace. Hence Vect{E, E ′ } ⊥ k is u invariant of dimension 1.
This implies the existence of an eigenvector E ′′ / ∈ Vect{E, E ′ } which by hypothesis is not possible.

The subspace V is k-non degenerate, of dimension 2 and has an isotropic vector. Hence it is a hyperbolic plane. Since

k is of index (2, 1), k(E, E) > 0. Let E 1 = E/ k(E, E). Let E 2 , E 3 ∈ V such that k(E 2 , E 2 ) = 0 = k(E 3 , E 3 ), k(E 2 , E 3 ) = 1 and u(E 2 ) = αE 2 , u(E 3 ) = γE 2 + αE 3 ,
where 0 = γ = E 3 , E 3 . In this k-hyperbolic basis, the matrix of u -1 is

  b 0 0 0 a -γa 2 0 0 a   ,
where a = α -1 and b = β -1 .

The Lax pair associated to the pseudo-Riemannian metric is

ẋ1 = γa 2 x 2 3 , ẋ2 = -γa 2 x 1 x 3 + (a -b) x 1 x 2 , ẋ3 = (b -a) x 1 x 3 .
The first and third equations conform a planar system which has idempotents if and only if γ(b -a) > 0.

The idempotent obtained produces an idempotent for the Lax pair vector field. Then, if the Lax pair vector field has no idempotent we get a semi-definite quadratic first integral for the whole system and a definite quadratic first integral for the planar system:

(a -b)x 2 1 + γa 2 x 2 3 .
Thus the planar system is complete. The second equation is of the type

ẋ2 = f (t) + g(t)x 2 .
where f , g are defined for all t, therefore it is also complete.

Lemma 5. If u has two linearly independent eigenvectors corresponding to the same eigenvalue, then the Lax pair associated to u on Sl(2, R) is linearizable.

Proof . Assuming that u has two eigendirections and exactly one eigenvalue, there is a basis of

G, E 1 , E 2 , E 3 , such that M B (u) =   α 1 0 0 α 0 0 0 α   .
We have that

E ⊥ 1 = Vect{E 1 , E 3 }. In fact, since u is k-symmetric, k(E 1 , u(E 2 )) = k(u(E 1 ), E 2 ) k(E 1 , E 1 ) + ak(E 1 , E 2 ) = ak(E 1 , E 2 )
which implies that k(E 1 , E 1 ) = 0, and

k(E 3 , u(E 2 )) = k(u(E 3 ), E 2 ) k(E 3 , E 1 ) + ak(E 3 , E 2 ) = ak(E 3 , E 2 )
which implies that k(E 1 , E 3 ) = 0. Hence

k(E 1 , E 2 ) = 0.
The subspace V = Vect{E 1 , E 2 } is non degenerate and u invariant. Thus V ⊥ k is a onedimensional subspace, non degenerate and u invariant. The Lax pair associated to the pseudo-Riemannian metric is ẋ1 = a 2 x 2 3 , ẋ2 = -a 2 x 1 x 3 , ẋ3 = 0, which is linearizable.

We can summarize the situation in Sl(2, R) with the following proposition Proposition 5. A left invariant pseudo-Riemannian metric in Sl(2, R) with associated isomorphism u and minimal polynomial q u (x), is complete if and only if (i) q u (x) has degree less than or equal to 2. In this case the Euler equation is linearizable;

(ii) q u (x) = (x -α 1 )(x -α 2 )(x -α 3 ) where α i = α j if i = j and

1 α 3 - 1 α 2 1 α 3 - 1 α 1 > 0,
where α 3 is the eigenvalue of u corresponding to an eigenvector E such that k(E, E) < 0.

In this case, the Euler equation of a complete left invariant pseudo-Riemannian metric has a definite quadratic first integral;

(iii) q u (x) = (x -α 1 )(x -α 2 ) 2 and E 3 , E 3 1 α 2 - 1 α 1 > 0,
where E 3 is a cyclic vector associated to α 2 .

Some examples and remarks

It is worth noting that, except for the case E(2) and the case in Proposition 5(iii) we have shown that a left invariant pseudo-Riemannian metric on a non unimodular Lie group is complete if and only if either the Euler equation is linearizable or has a definite quadratic first integral.

The following example shows that this is not the case for E(2) (and an example can be found in the same way in Sl(2, R)). However in these cases the differential equation is of the type:

ẋ1 = X 1 (x 1 , x 2 ), ẋ2 = X 2 (x 1 , x 2 ), ẋ3 = X 3 (x 1 , x 2 , x 3 ) (X 1 , X 2 , X 3 quadratic polynomials)
where the two first equations have a definite quadratic first integral.

Example 1. Consider on E(2) the pseudo-Riemannian metric given by

-x 2 1 + x 2 2 + 2x 2 x 3 .
in the basis considered in Proposition 2. The corresponding Euler equation is

ξ1 = ξ 1 ξ 2 , ξ2 = ξ 2 3 , ξ3 = -ξ 2 ξ 3 .
This equation is complete and is not linearizable. However ξ 1 is not bounded, hence no first integral can have bounded level surfaces.

In [START_REF] Guediri | Sur la complétude des varietés pseudo-Rimanniennes[END_REF] a characterization of the complete pseudo-Riemannian metrics on Sl(2, R) is given:

Theorem 1 .
The Lorentzian left invariant pseudo-Riemannian metric on Sl(2, R) is defined by the isomorphism φ is complete only in the two following cases:

1) φ has an eigenspace of dimension at least two;

2) φ is diagonalizable over the real numbers, has two eigenvalues of different sign and the eigenvectors associated to these eigenvalues generate a Lorentzian plane.

However, the following examples show that this characterization is not correct.

Example 2. Consider on Sl(2, R) the left invariant Lorentzian metric given in a k-orthonormal basis B = (E 1 , E 2 , E 3 ) by

x 2 1 + x 2 2 + x 2 x 3 . In the k-hyperbolic basis deduced from B, B ′ = (E 1 , (1/ √ 2)(E 2 + E 3 ), (1/ √ 2)(E 2 -E 3
)), the k-symmetric isomorphism u that corresponds to the Lorentzian metric is given by the matrix

M B ′ (u) =   1 0 0 0 1/2 1 0 0 1/2   .
The characteristic polynomial of u is p u (x) = (x -1/2) 2 (x -1), with eigendirections E 1 and E 2 + E 3 . The Lax pair equation corresponding to this Lorentzian metric in B ′ coordinates is ẋ1 = 4x 2 3 , ẋ2 = x 1 (x 2 -4x 3 ), ẋ3 = -x 1 x 3 .

The planar system given by the first and third equations is complete. Hence the system is complete. Notice that u has neither an eigenspace of dimension 2 nor is it diagonalizable. The pseudo-Riemannian metric is given by:

x, x = k(u(x), x) = 1 2 x 2 1 + 1 3 x 2 2 -x 2 3 which is Lorentzian. Notice that all eigenvalues of u are positive The Lax pair corresponding to this pseudo-Riemannian metric is ẋ1 = -2x 2 x 3 , ẋ2 = x 1 x 3 , ẋ3 = -x 1 x 2 .

This equation has the following first integrals Example 4. Consider the pseudo-Riemannian metric in the same basis as before given by x 2 1 -x 2 2 + 2x 2 3 . The isomorphism u associated to this pseudo-Riemannian metric has E 1 , E 2 , and E 3 as eigenvectors with (respectively) eigenvalues 1, -1 and 2. Hence we have two opposite eigenvalues whose corresponding eigenvectors generate a Lorentzian plane. The Lax pair equation in this case is: ẋ1 = 1 2 x 2 x 3 , ẋ2 = 3 2 x 1 x 3 , ẋ3 = 2x 1 x 2 .

x 2 1 + x 2 2 -x 2
Since 1 3 , 1, 2 1 3 is an idempotent for the Lax pair vector field, the pseudo-Riemannian metric is not complete.

We have shown that for unimodular Lie groups of dimension 3 the simplest case for incompleteness occur. This is not the case in general as the following example shows.

Example 5. Let G be a non unimodular Lie algebra de Lie of dimension 3. Then there is a basis (e 1 , e 2 , e 3 ), of G such that the Lie bracket is given by [ , ] e 1 e 2 e 3 e 1 0 αe 2 + βe 3 γe 2 + δe 3 e 2 0 0 e 3 0 where α + δ = 2 [START_REF] Milnor | Curvatures of left invariant metrics on Lie groups[END_REF]. For α = 1/2, δ = 3/2, and energy 2ξ 2 1 -ξ 2 ξ 3 , the Euler equation is:

ξ1 = ξ 2 ξ 3 , ξ2 = ξ 1 ξ 2 , ξ3 = 3ξ 1 ξ 3 .
Clearly the Euler vector field has no idempotents and the curve

ξ(t) = 1 √ 2 1 1 - √ 2t , 1 (1 - √ 2t) 1/2 , 1 (1 - √ 2t) 3/2
is an incomplete solution with initial condition ξ(0) = (1/ √ 2, 1, 1).

Example 3 .

 3 Let u be the isomorphism with matrixM B (u) =

3 and x 2 2 + x 2 3 .

 323 Hence the solutions are bounded and the equation is complete.

Theorem. Si la métrique invariante à gauche sur Sl(2, R) définie par l'endomorphisme φ est lorentzienne, elle n'est complète que dans les deux cas suivants:

Acknowledgements

The authors wish to thank the anonymous referee for the careful reading and the pertinent observations that led, in particular, to a revision of the proof of Lemma 4 and to reformulate accordingly Proposition 5.


Theophanis Tsandilas
email: theophanis.tsandilas@inria.fr

Emmanuel Dubois

Mathieu Raynal
email: mathieu.raynal@irit.fr

Modeless Pointing with Low-Precision Wrist Movements

Keywords: Pointing techniques, constrained wrist movement, elastic devices, rate control, clutching

Wrist movements are physically constrained and take place within a small range around the hand's rest position. We explore pointing techniques that deal with the physical constraints of the wrist and extend the range of its input without making use of explicit mode-switching mechanisms. Taking into account elastic properties of the human joints, we investigate designs based on rate control. In addition to pure rate control, we examine a hybrid technique that combines position and rate-control and a technique that applies non-uniform position-control mappings. Our experimental results suggest that rate control is particularly effective under low-precision input and long target distances. Hybrid and non-uniform position-control mappings, on the other hand, result in higher precision and become more effective as input precision increases.

Introduction

Technology becomes more and more ubiquitous, and a variety of handheld devices such as Wii Remotes start becoming widely available. Researchers and designers have been envisioning scenarios that move user interaction beyond desktop computers. A user interacts with a public display in a museum to get more information about an exhibit or with a wall display in an airport to learn more about a flight. Driven by such scenarios, research in Human-Computer Interaction has been exploring new pointing techniques that go beyond the use of a mouse. For example, Vogel and Balakrishnan [START_REF] Vogel | Distant Freehand Pointing and Clicking on Very Large, High Resolution Displays[END_REF] proposed techniques for freehand pointing in front of highresolution displays. Other work [START_REF] Raynal | Tactile Camera Vs. Tangible Camera: Taking Advantage of Small Physical Artefacts to Navigate into Large Data Collections[END_REF] has explored the use of mobile phones as pointing devices for interaction with public displays. In such scenarios, user interaction does not rely on the presence of specialized input devices and sophisticated motion tracking systems. Pointing precision becomes an important issue. Factors that affect pointing precision include natural hand tremor and limited hand precision [START_REF] Bérard | On the Limits of the Human Motor Control Precision: The Search for a Device's Human Resolution[END_REF][START_REF] Rahman | Tilt Techniques: Investigating the Dexterity of Wrist-Based Input[END_REF] or the use of low-resolution input devices.

Several solutions have tried to improve pointing precision by proposing mechanisms that balance between absolute and relative pointing. "Clutching" is a common input recalibration mechanism that allows users to reposition the frame of reference of absolute pointing. Despite its simplicity, it requires explicit mode transitions between pointing and clutching. When using a mouse, clutching is activated naturally, by lifting the mouse and translating it over the table. However, when the input device is positioned in free space, mode switching relies on the use of a button [START_REF] Zhai | Human Performance in Six Degree of Freedom Input Control[END_REF] or a special hand gesture [START_REF] Vogel | Distant Freehand Pointing and Clicking on Very Large, High Resolution Displays[END_REF]. Such mechanisms may not be available in certain situations (e.g., when the user manipulates a physical object with no additional input channels), rely on error-prone hand-gesture recognition mechanisms or compete with the activation of other actions, such as the selection of a target.

This paper explores solutions that achieve pointing precision without requiring the user to switch between modes. We focus on movements of the wrist, which are important for the manipulation of handheld devices and can be easily detected by accelerometer and gyroscope sensors of common mobile devices. We investigate three solutions. Based on the observation that wrist movements occur within a limited range, beyond which natural self-centering forces develop, we investigate rate (velocity) control as a first solution. Inspired by the work of Casiez et al. [START_REF] Casiez | The Effect of Spring Stiffness and Control Gain with an Elastic Rate Control Pointing Device[END_REF], we explore a second technique that combines position and rate control to extend the input range of the hand. We claim that the wrist can be viewed as a hybrid device: isotonic when movements take place around its neutral position, and elastic when its position extends beyond a certain range. Finally, we examine a third technique influenced by previous research on multi-scale pointing [START_REF] Guiard | Navigation as Multiscale Pointing: Extending Fitts' Model to Very High Precision Tasks[END_REF] and non-uniform position-control mappings [START_REF] Appert | High-Precision Magnification Lenses[END_REF][START_REF] Poupyrev | The Go-Go Interaction Technique: Non-Linear Mapping for Direct Manipulation in VR[END_REF]. We show that rate control is insensitive to input precision, which means that it is particularly valuable when input precision is low. We measure and control input precision by dividing the input range into discrete units of input. We use input precision as an independent variable to explore the design space and evaluate the three techniques.

Related Work

In indirect pointing, the pointing device is decoupled from the display. This separation results in two different spaces: [START_REF] Appert | High-Precision Magnification Lenses[END_REF] the display space, which is the space of the pointer's movement, and (2) the motor (or control) space, where the manipulation of the input device takes place. In relative-pointing devices, the mapping between the two spaces can change dynamically, for example, by using recalibration mechanisms that change the frame of reference of pointing on the display space. Clutching is the most common recalibration mechanism [START_REF] Hinckley | A Survey of Design Issues in Spatial Input[END_REF]. When using a mouse, clutching can be performed by lifting and moving the mouse off the table. When pointing occurs in free space, clutching requires the activation of an explicit mode-switching mechanism, for example, by pressing a button [START_REF] Zhai | Human Performance in Six Degree of Freedom Input Control[END_REF] or by changing the hand's posture [START_REF] Vogel | Distant Freehand Pointing and Clicking on Very Large, High Resolution Displays[END_REF].

The Control-Display (C-D) gain [START_REF] Gibbs | Controller Design: Interactions of Controlling Limbs, Time-Lags and Gain in Positional and Velocity Systems[END_REF] maps the movement of the input device (motor space) to the movement of the display pointer (display space). It can be calculated as the ratio of the pointer velocity to the velocity of the input device. Casiez et al. [START_REF] Casiez | The Impact of Control-Display Gain on User Performance in Pointing Tasks[END_REF] examined the impact of C-D gains on pointing performance with a high-resolution mouse. Their results indicate that pointing time follows an L curve as a function of C-D gain. When the gain is low, performance slows down because clutching becomes frequent. When it is high enough, the C-D has no effect on user performance, i.e., increasing its value has no cost. Problems due to the accuracy of hands and fingers appear in tiny movements of about 0.2 mm. When the hand moves in free space, however, problems appear in larger movements in the range of 3 -5 mm [START_REF] Bérard | On the Limits of the Human Motor Control Precision: The Search for a Device's Human Resolution[END_REF]. In this case, the use of high C-D gains is problematic. For example, ray casting, which involves high C-D gains when used at a distance, is extremely sensitive to hand tremor and results in high error rates [START_REF] Myers | Interacting at a Distance: Measuring the Performance of Laser Pointers and Other Devices[END_REF][START_REF] Oh | Laser Pointers as Collaborative Pointing Devices[END_REF][START_REF] Vogel | Distant Freehand Pointing and Clicking on Very Large, High Resolution Displays[END_REF].

A pointing device can control either the position (position control) or the velocity (rate control) of the pointer. Position control is most often used with isotonic devices, e.g., a mouse, while rate control is used with isometric and elastic devices, e.g., joysticks. Rate control eliminates the need for clutching but relies on a self-centering mechanism, not present in isotonic devices. Zhai [START_REF] Zhai | Human Performance in Six Degree of Freedom Input Control[END_REF] compared all the possible mappings between input types (isotonic and isometric) and types of control (position and rate) and found that isotonic rate-control input was about 50% slower than isometric rate-control input and isotonic position-control input. More recently, Casiez and Vogel [START_REF] Casiez | The Effect of Spring Stiffness and Control Gain with an Elastic Rate Control Pointing Device[END_REF] examined the effect of the stiffness of elastic devices on the performance of rate control. They found that rate control performed well even for very low stiffness values, as long as a self-centering mechanism was present and velocity-control functions were carefully selected. They also found that pointing performance was only 15% slower when stiffness was zero, i.e., when pointer control was purely isotonic.

Research in Virtual Reality has proposed hybrid movement mappings to facilitate the manipulation of virtual objects, avoiding the use of high C-D gains, explicit interaction modes, and clutching mechanisms. The Go-Go technique introduced by Poupyrev et al. [START_REF] Poupyrev | The Go-Go Interaction Technique: Non-Linear Mapping for Direct Manipulation in VR[END_REF] extends the reach of a user's hand in the virtual world by applying non-uniform C-D gains. The technique uses a one-to-one movement mapping between the real and a virtual hand as long as the hand stays within a fixed area around the user. This ensures that users can manipulate nearby objects with precision. However, when the arm of the user extends beyond the proximity area, the C-D gain grows through a non-linear (parabolic) function. Variable C-D gains can help users reach remote objects without having to "clutch". A major drawback of the technique is that high gains develop towards extreme input positions, requiring users to bring remote objects closer so that they can effectively interact with them. Bowman and Hodges [START_REF] Bowman | An Evaluation of Techniques for Grabbing and Manipulating Remote Objects in Immersive Virtual Environments[END_REF] explored variations of the Go-Go technique, including the stretch go-go technique, which controls the velocity of the virtual arm rather than its absolute position. Unfortunately, little evidence about the effectiveness of the above techniques exists. More recently, Appert et al. [START_REF] Appert | High-Precision Magnification Lenses[END_REF] introduced the Ring lens, a high-precision magnification lens that applies position control at two scales. The cursor moves with a low C-D gain within a Ring lens, supporting high-precision control in the magnified area. As the cursor reaches the border (ring) of the lens, the lens follows the movement of the cursor. The technique outperformed both speed-depended precision control and mode-switching precision control.

A few approaches have examined hybrid designs. Dominjon et al. [START_REF] Dominjon | A Comparison of Three Techniques to Interact in Large Virtual Environments Using Haptic Devices with Limited Workspace[END_REF] proposed an interaction technique for haptic devices that combines position and rate control. The technique is based on the visualization of the input space as a three-dimensional bubble. Movement is isotonic and position controlled inside the bubble, but it becomes elastic and rate controlled as the cursor crosses the boundaries of the bubble to the outside. Dominjon et al. [START_REF] Dominjon | A Comparison of Three Techniques to Interact in Large Virtual Environments Using Haptic Devices with Limited Workspace[END_REF] implemented the bubble technique with a PHANTOM haptic device and evaluated it on 3D-model painting tasks. The technique was more efficient than both absolute positioning and clutching and received higher subjective rankings. RubberEdge [START_REF] Casiez | Reducing Clutching by Combining Position and Rate Control with Elastic Feedback[END_REF] is a hybrid device that applies a similar approach to 2D pointing tasks. It resembles to a regular touch pad, but its boundaries are elastic. When the finger of the user moves from the central isotonic zone of the pad to its elastic boundaries, cursor movement becomes rate controlled, allowing the user to traverse long distances without clutching.

3

The Wrist as a Constrained Pointing Device

Wrist movements are constrained by the hand's joints. Joint constraints decrease the operating range of input, hindering user performance. This is a major problem when users need to interact with large visual spaces without losing in pointing precision.

Clutching techniques as well as techniques that let users switch between relative and absolute pointing can increase the range of movement but require an additional input channel. Unfortunately, such mechanisms may not be available in certain situations, for example, when users interact with non-specialized devices or physical objects. Physical constraints may not pose a problem if movements of the hand are highly accurate, when pointing distances are relatively short or targets are large. Problems arise when hand movements are imprecise, distances are large, and targets are small. To better describe this problem we study user performance in relation to the number of discrete units of movement. Fig. 1 presents how the input and output space can be partitioned into units of movement. U is the range of the display measured in units, and y is the size (in mm, dots or pixels) of the minimum unit of output movement. Clearly, targets on the display have to be larger than y to be selectable. Chapuis and Dragicevic [START_REF] Chapuis | Effects of Motor Scale, Visual Scale and Quantization on Small Target Acquisition Difficulty[END_REF] found that the discretization of the output had practically no effect on pointing performance as long as targets were selectable. This implies that performance only depends on motor-space control and not on the visual representation of the cursor. In most real applications, the minimum output movement y is never larger than 1 pixel, which allows for smooth cursor transitions and the selection of tiny tar-gets. In this case, U represents the display size in pixels. Similarly to output range, we can view input range as a discretized entity, where u is the range of detectable discrete input units, and x is the size of each unit in mm. The minimum input unit x repre- sents the minimum distance that can be recognized by an input device and controlled by the user. When the number of output units is greater than the number of input units (U > u), the user cannot reach all the output units with an absolute mapping between input and output. Given an input-output configuration (u, U), where U is much greater than u (U >> u), our goal is to examine pointing techniques that maximize pointing performance. The output-to-input ratio:

(

determines the minimum number of clutches required to traverse the whole output range. An alternative definition of the C-D gain in terms of input and output units is:

CDU gain = U m u m (2)
where U m is the number of output units covered when the input device moves u m units. Notice that, in contrast to the C-D gain, the above measure is not unit-free.

Research in biomechanics has proposed several models to describe the mechanics of joints and muscles. Lemay and Crago [START_REF] Lemay | A Dynamic Model for Simulating Movements of the Elbow, Forearm, and Wrist[END_REF] used a simple mathematical model to express passive elastic torques M in human limbs, according to which the joint's stiffness is composed of both a linear and an exponential component. The exponential component dominates close to the limits of a limb's operating range, but movement is highly isotonic around the rest position. Lehman and Calhoun [START_REF] Lehman | An Identified Model for Human Wrist Movements[END_REF] studied extensions and flexions of the wrist. They found that wrist movements are isotonic in a range of 40 degrees to the left and the right of the rotational axis. Passive torques were less than 0.1 Nm within this movement range. Elastic torques started appearing rapidly as movement extended towards its extreme positions. Fig. 2 presents the angular range of wrist rotations and our hypothetical model of the wrist as a hybrid device. The precision of such wrist movements can be particularly low. Rahman et al. [START_REF] Rahman | Tilt Techniques: Investigating the Dexterity of Wrist-Based Input[END_REF] found that errors inflated considerably as users tried to control more than 12 levels of purely flexion-based tilts when interacting with a mobile phone.

Techniques

We examined three techniques that allow the user to point by moving the wrist without making use of explicit mode-switching mechanisms. Motivated by previous work on hybrid movement mappings [START_REF] Casiez | Reducing Clutching by Combining Position and Rate Control with Elastic Feedback[END_REF][START_REF] Dominjon | A Comparison of Three Techniques to Interact in Large Virtual Environments Using Haptic Devices with Limited Workspace[END_REF][START_REF] Poupyrev | The Go-Go Interaction Technique: Non-Linear Mapping for Direct Manipulation in VR[END_REF], we reconsider the use of rate control in conjunction with wrist movements. Rate control depends on the existence of a selfcentering mechanism. Previous results [START_REF] Casiez | The Effect of Spring Stiffness and Control Gain with an Elastic Rate Control Pointing Device[END_REF] suggest that the stiffness of an elastic device has no effect on the effectiveness of rate control. Rate control can work equally well with low stiffness values provided that a self-centering mechanism is present. Their results also suggest that when movement occurs within a small range, the removal of external self-centering mechanisms only slightly affects user performance. Given that the wrist has a natural resting position and movement takes place within a relatively small range around it, we expected that rate control would be a viable solution even if we did not externally reinforce the wrist's centering mechanism.

Pure Rate Control

Rate control is based on the application of a transfer function that maps the displacement d from a neutral input position to velocity units. Previous work [START_REF] Casiez | The Effect of Spring Stiffness and Control Gain with an Elastic Rate Control Pointing Device[END_REF][START_REF] Zhai | Human Performance in Six Degree of Freedom Input Control[END_REF] has made use of linear transfer functions. Our informal tests, however, showed that non-linear transfer functions result in better motor control. This can be explained by the fact that the neutral position of the wrist is not strict but expands within a certain range where self-centering is absent and velocity control is difficult. The form of polynomial functions (see Fig. 3) that we tested is the following:

v(x) = sign(x)•|x| b , b 1 (3)
where x = 0 is the hand's resting position. Both the input range and velocity are normalized in [-1, 1]. We observed that transfer functions that grow rapidly around the zero position, i.e., when b is low, are sensitive to input precision. Velocities must be low enough so that small targets can be easily selected. If the width of the smallest target is W min , the minimum cursor velocity v min must be as follows:

(4)

where T react is the delay for the user to react and stop the movement when the cursor enters the area of the target. Based on previous experimental results [START_REF] Maury | Rhythmic Menus: Toward Interaction Based on Rhythm[END_REF], we estimated this delay to be around 150 ms.

Hybrid Control

We explored mixed-control designs, balancing between high precision afforded by position control and smooth long-distance movement afforded by rate control. Our approach is based on existing techniques [START_REF] Casiez | Reducing Clutching by Combining Position and Rate Control with Elastic Feedback[END_REF][START_REF] Dominjon | A Comparison of Three Techniques to Interact in Large Virtual Environments Using Haptic Devices with Limited Workspace[END_REF] but does not assume the availability of specialized elastic devices. As shown in Fig. 4, we divide the input range into three zones. The central zone is reserved for position control. The portion of the display that corresponds to movements within this zone is communicated to the user as a framed window. The side input zones let the user reposition this window by controlling its velocity. As a result, pointing takes place at two stages. First, the user turns the hand out of its central zone to bring the window around the target. Then, the hand returns towards the central zone to point to the target within the window's boundaries. We follow the approach of Dominjon et al. [START_REF] Dominjon | A Comparison of Three Techniques to Interact in Large Virtual Environments Using Haptic Devices with Limited Workspace[END_REF], who visually communicate the range of position control, rather than the approach of Casiez et al. [START_REF] Casiez | Reducing Clutching by Combining Position and Rate Control with Elastic Feedback[END_REF], who use a single cursor representation. In our case, there is no haptic feedback to communicate transitions to the user, and hence, visual feedback is essential. We also found that breaking the pointing task into two different scales, i.e., rough pointing with rate control, and precise pointing with position control, was more appropriate than assuming that rate and position control constitute symmetric ways of pointing.

Clearly, the range of input units determines the maximum size of the window, but how to divide this range is not straightforward. Larger central zones increase the active width of position-controlled pointing but do not necessarily result in faster movements because (1) a shorter range is reserved for rate control, and (2) transitions from position to rate control, and inversely, require longer hand movements. The model proposed by Casiez et al. [START_REF] Casiez | Reducing Clutching by Combining Position and Rate Control with Elastic Feedback[END_REF] assumes that movement has two parts. Movement first occurs in the isotonic area until the user's finger reaches the elastic boundary of the device. Then, movement continues in the elastic zone until the target is finally selected. In contrast, our model assumes that movement always returns to the central isotonic zone to complete the pointing task. Our early tests have shown that direct target selection with rate control is particularly hard when a hybrid design is used. When movement takes place in a single side (i.e., elastic) zone of the input, the cursor can only move towards a single direction. If the cursor overshoots the target, movement has to return to the isotonic zone. Fig. 4 demonstrates the steps of a selection task based on this model.

Non-Uniform Position Control

We have designed a window-based technique that applies position control to move the window of high-precision pointing. As shown in Fig. 5, a movement of the hand out of its central zone translates the window to the left or to the right by using a high C-D gain. The position of the window freezes when the wrist starts moving towards its central zone. This mechanism allows the user to recalibrate the movements, keeping high-precision pointing around the wrist's rest position. During recalibration, the user may have to repeat multiple forward and backward movements before bringing a distant target within the window. More precisely, the number of such movements depends on the target's distance, the input resolution, the size of the central zone, and the selected C-D gains. This recalibration mechanism is analogous to regular clutching mechanisms but does not require the use of additional input channels. Also, it makes use of two distinct gains (CDU low , CDU high ), which allows for minimizing the number of recalibration actions. The steps required by the third technique to point to distant targets are similar to the ones required by the hybrid technique except that now position control is used to position the window. The cursor moves to the boundaries of the window. Then, the window moves to the target, following a series of recalibration actions. Finally, the cursor points to the target. If w units of a total of u input units are reserved for absolute pointing, the maximum output distance d traversed without recalibration can be computed as follows:

d = w CDU low + u w 2 CDU high (5)
To maximize this distance and, at the same time, ensure that all the pixels can be visited, we can select CDU low to be equal to 1 pixel/unit and CDU high to be equal to w:

d max = w 2 2 + 1+ u 2 w (6)
The value w o that maximizes this expression can be calculated as follows:

"d "w w o = 0 # $w o + 1+ u 2 % & ' ( ) * = 0 # w o = 1+ u 2 (7)
This equation shows that the distance is maximized when the zone of absolute pointing is approximately half of the input range. This estimation has been based on the highest possible value for CDU high . As we have already discussed, previous results [START_REF] Casiez | The Impact of Control-Display Gain on User Performance in Pointing Tasks[END_REF] show that C-D gains do not impact pointing performance as long as no clutching and input resolution problems arise. Besides, other results [START_REF] Chapuis | Effects of Motor Scale, Visual Scale and Quantization on Small Target Acquisition Difficulty[END_REF] show that discretization of the cursor's movement does not hurt user performance. Yet, our own tests showed that high gains could hinder motor control. Our explanation is that nonuniform control requires users to switch from a low to a high gain, adapting accordingly their movement strategies. It seems that the higher the difference between the two gains, the higher becomes the cost of such movement adaptations. Therefore, we select CDU high to be equal to w/a, where a>1. Then, based again on Equation 5, we find that the value w o that maximizes the distance d is as follows:

w o = a + u 2 (8)
In practice, the best values for a and w must be empirically selected.

Experiment

We conducted an experiment to compare the performance of the three techniques. We focused on low-precision wrist input with up to 241 input units.

Participants and Apparatus

Twelve volunteers (three women and nine men), 21 to 40 years old (the median age was 27) participated. Two participants used their left hand to perform the tasks. One participant had also participated in the pilot experiments (see below).

We used an Ascension Flock of Birds 6-DOF motion tracker to detect extensions and flexions of the wrist around a vertical axis. Participants were seated and moved a sensor cube (25.4mm 25.4mm 20.3mm) within a range of about 50 -80 cm from the transmitter and a distance of about 70 -80 cm from the monitor. We attached the sensor cube on a solid pen-like extension (see Fig. 6c), allowing participants to grab it more comfortably. We used a 22-inches monitor with a 1680 1050 resolution. We detected only rotations parallel to the ground and used a simple Kalman filter to remove noise. To control for input precision, we discretized the rotation values measured by the magnetic tracker by diving the effective rotational range into discrete input units. The experimental software was written in Java 1.6.

Task and Stimuli

Participants performed a series of reciprocal 1D pointing tasks by selecting two targets forward and then backward in succession. Targets were rendered as solid vertical bars and were selected by pressing a key with the non-dominant hand. Participants were required to successfully select a target before moving to the next. They were told to perform tasks "as fast as possible, trying to avoid errors". For the rate technique, the cursor was rendered as a one-pixel-thick vertical line. For the hybrid and the non-uniform position-control technique, we used two cursor representations to communicate the two different levels of input control, as shown in Fig. 6. When the wrist of the user moved out of the central zone, the cursor turned into a small rectangle and passed control to the window, which started moving. Its position relative to the boundaries of the window slightly changed, providing direct feedback about the position of the wrist with respect to the central zone of input.

Optimizing the Techniques: Summary of Two Pilot Experiments

For each technique, we had to choose several parameters. What transfer function to use? How to split the input range? What C-D gains to choose? To reduce the design space, we conducted two pilot experiments. Six volunteers participated in each pilot.

Pilot 1. The first experiment tested pure rate control under three transfer functions (Fig. 7a) and two levels of low input precision: 21 and 61 input units. We selected the functions f 2 and f 3 to approximate an optimal transfer function for both levels of precision. We expected that the transfer function f 1 would be the slowest one, especially under the low input precision. Under a range of 21 units, the lowest velocity allowed by f 1 is 20.1 1.5 pixels/ms or 63 pixels/sec. For targets of 8 pixels and minimum response times of 150 ms, this value is higher than 53 pixels/sec, which is the velocity limit calculated by Equation 2. The transfer function has a significant effect on selection time (F1.68, 8.38=15.47, p=0.002) and error ( 2df=2,N=6 =10.17, p=0.006). The results are summarized in Fig. 8. The error rate is considerably high for f 1 , especially when input precision is low. The best function for errors is f 3 . The results suggest that as long as an optimal transfer function is selected, there is no clear penalty for reducing the input precision: errors rates are not hurt and the cost in speed is minimal, less than 5%.

Fig. 8. Overall results for Pilot 1

Pilot 2. The second pilot explored issues related to the design of the hybrid technique. We compared three assignments of input range to isotonic and elastic zones. More specifically, we tested a single level of input precision of 61 input units under the three configurations shown in Table 1. Presentation order was counterbalanced among participants. The CDU gain for cursor's movement was set to 4 pixels/units. As a result, the size S of the area cursor for the three configurations was 44, 124, and 204 pixels, respectively. For each configuration, we empirically selected two transfer functions (see Fig. 7): a linear function f 1 , and a parabolic function f 2 . Results are summarized in Fig. 9. An ANOVA repeated-measures analysis indicates a significant main effect of input configuration on selection time (F 1.67,8.33 =37.84, p<0.0001). Pairwise comparisons with Bonferroni's adjustment show that C1 is significantly slower than both C2 (p=0.001) and C3 (p=0.004). There is no significant difference between C2 and C3 (p=1.0). The mean error rate is also higher for C1 but with no consistency among participants. The pointing time is not significantly different for the two transfer functions (F 1,5 =1.4, p=0.29), but the mean error is lower for f 1 under C2 and C3. In conclusion, we can select a configuration with f 1 as transfer function, where 50% to 84% of the input range is dedicated to the central zone.

Conditions

The experiment tested three pointing techniques: rate control (Rate), hybrid control (Hybrid), and non-uniform position control (Position). For each one, the experiment tested two levels of input precision: 61 (low) and 241 (high) discrete input units. These values are higher than the ones tested by Pilot 1 and 2, but they still correspond to a low input precision. We made sure that participants could visit every pixel on a window of 61 or 241 pixels with our configuration when using a simple positioncontrol technique. We also verified that the Device's Human Resolution [START_REF] Bérard | On the Limits of the Human Motor Control Precision: The Search for a Device's Human Resolution[END_REF] was high enough so that participants could comfortably select the smallest 8-pixel targets. For each condition, we tried to optimize user performance by refining input control based on our theoretical analysis and the two pilot studies. Below, we justify how various parameters for each technique were selected.

Rate Control. F o r t h e l o w -precision condition, we used the transfer function f 3 (x)=sgn(x)2|x| 3.3 . Pilot 1 showed that this function provided the best tradeoff between time and errors. However, it was unclear whether this function would be optimal under larger numbers of discrete input units. Informal tests with 241 units showed optimal results for functions ranging between f 2 and f 3 . We chose f(x) =sgn(x)2|x| 2.5 .

Hybrid Control.

The CDU gain for position-control was set to 1 pixel/unit. This gain enables the selection of every pixel and makes optimal use of the available input range. Table 2 shows how we divided the input range for the two levels of input precision. For both, we used a linear transfer function (see f 1 in Fig. 7b). Notice that we avoided using large central zones, as they would result in longer sub-movements for the final target selection. Position Control. Equation 8 calculates the size (in input units) of the zone of absolute pointing that minimizes the number of recalibration actions. This zone is approximately half of the input range provided that a << u. Yet, a minimal number of recalibration actions do not necessarily result in optimal performance. We simplified our analysis by keeping the design of the technique as similar as possible to the design of Hybrid. We used the input configuration shown in Table 2 and set CDU low to 1 pixel/unit. Finally, we empirically selected CDU high to 20 pixels/unit for the low input precision (61 units) and to 8 pixels/unit for the high input precision (241 pixels).

Design and Procedure

A full-factorial repeated-measures within-participants design was used. The presentation order of techniques (Rate, Hybrid, and Position) and input precision (21 and 241 units) within each technique were counterbalanced among participants. For each combination of technique and input precision, participants completed three blocks of trials. We tested three target distances D (250, 500, and 1000 pixels) and three target widths (8, 16, and 32 pixels). Each block consisted of nine randomly sorted trials, which correspond to the nine combinations of target distances and widths. For each technique, participants also completed a total of nine practice trials. To reduce fatigue, we allowed participants to take brief breaks between blocks and techniques. Experimental sessions lasted approximately 40 to 50 minutes.

Measures and Hypotheses

We measured selection time as the total time spent to complete a reciprocal pointing task divided by two. We also measured error rates. We expected that the performance of Position and Hybrid would improve as input precision increases. Given the results of Pilot 1, however, our hypothesis was that input precision would have no effect on the performance of Rate. We also predicted that Position would be less effective under the low-precision condition due to the recalibration problem. We expected that Hybrid would provide the best balance between pointing precision and speed.

Results

We eliminated a total of 25 outliers (1.2% of total measurements) for values three standard deviations away from the within-cell mean. The main results are presented in Fig. 10. An ANOVA repeated-measures analysis indicates a significant main effect of input precision on time (F 1,11 =91.23, p=1.210 -6 ). This effect, however, is different for each technique. More specifically, there is a significant interaction between input precision a n d t e c h n i q u e ( F 1.74,19.16 =37.42, p=4.810 -7 ). Pairwise comparisons with Bonferroni's adjustment show that its effect is significant for Hybrid (p=2.610 -5 ), significant for Position (p=6.010 -7 ), but no significant for Rate (p=0.876). Consistent with our findings from Pilot 1, the results suggest that input resolution has minimal or practically no effect on the performance of pure rate control as long as an optimal transfer function is selected.

The effect of the technique on time is significant (F 1.87,20.55 =7.39, p=0.0044). Yet, pairwise comparisons show that differences between the three techniques are only significant under low precision. Position is significantly slower than both Rate (p=2.210 -4 ) and Hybrid (p=0.013), but no significant difference is found between the two latter techniques (p=0.11). Although rate control performs well in terms of pointing time, it results in high error rates. The error rate is approximately 5% for this technique, in contrast to a 1.1 -1.7% error rate observed for Hybrid and Position. Freedman's non-parametric test shows that the effect of the technique on errors is statistically significant ( 2df=2,N=12 =9.14, p=0.01). Overall, our results indicate an advantage of Hybrid over Position for 61 input units and an advantage of both these techniques over Rate for 241 input units.

Fig. 10. Overall results of the main experiment

Typical error rates in Fitts' law experiments are in the range of 3 -4%. The error rates that we observe for Hybrid and Position are considerably lower, even though the instructions given to the participants were consistent all over the experimental session and rather neutral towards precision. Our explanation is that the two window-based techniques reduce the likelihood of errors by dividing the pointing task into two sequential subtasks. Rapid ballistic movement, which is rough and imprecise, takes place during the first subtask. Errors are possible, but the user must correct them before continuing with the final pointing movement. As the participants were given visual feedback about the success of the first movement in the form of an area cursor, errors during this step were eliminated. In addition, the second subtask was relatively slow and precise, as it took place in a limited area around the target.

As expected, the effect on pointing time is significant for both target width (F 1.73,19.03 =236.77, p=9.110 -14 ) and distance (F 1.90,20.84 =460.99, p=8.310 -18 ). We also find a significant interaction effect between technique and distance (F 3.62,39.87 =18.04, p=3.510 -8 ). As shown in Fig. 11, the performance of Position degrades considerably in long distances, as the cost of recalibration actions becomes higher. A pairwise analysis using Bonferroni's adjustment shows that Position is significantly slower than both Rate (p =2.710 -4 ) and Hybrid (p=4.410 -4 ) for distances of 1000 pixels and significantly slower than Rate (p =0.006) for distances of 250 pixels.

Predictive Models

We can attempt to predict user performance for other levels of input precision. For Rate, we use the regular formulation of Fitts' law (see Eq. 9 in Fig. 12), which is independent of the number of input units. For Hybrid and Position, we use Eq. 10. The parameter s (0 s1) represents the portion of the input range dedicated to the movement in the central zone. S is the size of the movable window, which can be calculated as S = CDU gain •s•u. To produce Eq. 10, we followed an analysis similar to that of Cao et al. [START_REF] Cao | Peephole Pointing: Modeling Acquisition of Dynamically Revealed Targets[END_REF]. Due to limited space, we do not provide the details here.

MT = a + b log 2 D W +1
Eq. 9 Based on the above models, we have extrapolated movement time with up to 961 input units and distances as high as 1500 pixels. Fig. 13 gives an overview of the trends. Position exhibits a poor performance under low input precision, but outperforms Rate and Hybrid as precision improves. The performance of Position is particularly sensitive to both input precision and target distances. Variations in the performance of Hybrid are less radical. The technique seems to be more appropriate in the range of 61 of 241 units. However, it can be a good alternative for higher levels of precision, particularly when pointing involves large distances. Clearly, prediction results for the position-control and the hybrid technique do not apply to short target distances. As long as movement takes place only within this window, movement will be purely position controlled and faster than the time predicted by the above models.

Limitations and Future Directions

We examined levels of input precision from 21 to 241 units and targets distances from 250 to 1000 pixels. Output configurations can vary greatly beyond the range of values reported here. For example, wall displays can support very high resolutions. In such environments, pointing precisely and moving quickly in space are equally important, so future work must verify how our techniques behave in such settings. Some previous work [START_REF] Frees | Prism Interaction for Enhancing Control in Immersive Virtual Environments[END_REF][START_REF] König | Adaptive Pointing -Design and Evaluation of a Precision Enhancing Technique for Absolute Pointing Devices[END_REF][START_REF] Vogel | Distant Freehand Pointing and Clicking on Very Large, High Resolution Displays[END_REF] has made use of pointer acceleration to effectively balance between precision and speed. To simplify our analysis, we did not consider pointer acceleration here. If used with our techniques, acceleration can be applied at two levels: (1) to the window of precise pointing, and (2) to the cursor within this window. The former solution could increase the range of the window's movement and hence reduce the number of recalibration actions. The latter solution, however, is problematic, as it can cause the de-calibration of the wrist movements away of the joint's neutral position. To extend our approach to two-dimensional tasks, we must first consider that hand movements are not perfectly symmetric. Wrist flexions and extensions have a wider angular range than ulnar and radial deviations (Fig. 2). Techniques must be adapted to accommodate these asymmetries. In addition, natural hand movement combines rotations of the wrist and movements of other joints (Fig. 14), which could be mapped to different levels of a pointing task, e.g., by using different CD gains. Exploring techniques that combine various joints is an interesting future direction. Another future goal is to test non-linear mappings between input and output [START_REF] Rahman | Tilt Techniques: Investigating the Dexterity of Wrist-Based Input[END_REF].

Conclusions

The paper explored pointing with low-precision input when the input device is controlled by movements of the wrist. We examined techniques that achieve pointing precision without making use of explicit mode-switching mechanisms. Wrist movements are physically constrained and take place within a small range around the hand's resting position. Based on this observation, we explored rate control as a possible solution. Our results show that rate control is effective under low levels of input precision. Interestingly, we found that as long as optimal transfer functions are selected, the performance of rate control remains practically constant for a range of input precision from 21 to 241 units. This suggests that rate control could be particularly valuable for low-precision input devices (e.g., low-fidelity camera-based capture of hand movements) and users with hand-tremor problems.

In addition to rate control, we examined two techniques that split the pointing task into two scales. Pointing at a macro scale is performed with an area cursor, which is visually communicated to the user as a framed window. Pointing at a micro scale is performed with a regular point cursor, which moves within the window. The first technique combines position and rate-control and derives from previous work [START_REF] Casiez | Reducing Clutching by Combining Position and Rate Control with Elastic Feedback[END_REF][START_REF] Dominjon | A Comparison of Three Techniques to Interact in Large Virtual Environments Using Haptic Devices with Limited Workspace[END_REF]. The second technique uses position control to move both the area and the point cursor and has been based on previous work on two-scale pointing [START_REF] Guiard | Navigation as Multiscale Pointing: Extending Fitts' Model to Very High Precision Tasks[END_REF] and early techniques in Virtual Reality that applied non-uniform position-control mappings [START_REF] Poupyrev | The Go-Go Interaction Technique: Non-Linear Mapping for Direct Manipulation in VR[END_REF]. According to our experimental results, these two techniques are more precise than pure rate control and their performance improves as input precision increases. The hybrid technique was particularly effective in balancing between precision and speed in the range of 61 to 241-input units. It outperformed the non-uniform position-control technique in long target distances over 1000 pixels. Nevertheless, our analysis suggests that the performance of the latter is faster as input precision improves. Based on these results, we derived predictive models of user performance, but future work must compare the techniques on different task scales. We are particularly interested in exploring their application to high-resolution displays where distances can be as high as 10k to 20k pixels. We also plan to study the proposed techniques on the movement of other human joints. For example, the thumb moves within a limited range. Enabling it to precisely point on large output surfaces in isolation of or in combination with hand movements is a challenging problem and certainly worth of future investigation.

Fig. 1 .

 1 Fig. 1. Input and output unit ranges.

Fig. 2 .

 2 Fig. 2. The wrist. (a) The range of its angular movements. (b) Modeled as a hybrid of an isotonic and an elastic device.

Fig. 3 .

 3 Fig. 3. Three different transfer functions: v(x) = sign(x)•|x| b , where b 1.

Fig. 4 .

 4 Fig. 4. Steps of a selection task with the hybrid technique when D > D 1 . The vertical line in the input space shows the input position. (a) The cursor travels a distance D 1 , which corresponds to a movement d 1 along the isotonic range of the input. (b) The window travels towards the target, which appears in a distance D 2 from the initial center of the window. The velocity v of the window increases as the input position extends towards its extreme sides. (c) The cursor moves to the target by covering a distance D 3 .

Fig. 5 .

 5 Fig. 5. Non-uniform position control. (a) Absolute pointing in the central zone of the input range with a low C-D gain. (b-d) Position recalibration with back and forth movements.

Fig. 6 .

 6 Fig. 6. Experimental task for the hybrid and the non-uniform position-control technique. (a) When the hand moves out of the central zone, the cursor becomes inactive and takes the form of a small rectangle. The short distance between this rectangle and the boundaries of the window provides feedback about input position out of the central zone. (b) A move of the hand back to the central zone activates the cursor, which takes its regular line form.

Fig. 7 .

 7 Fig. 7. Transfer functions tested by the two pilot studies, shown for the right half of the input range: (a) Pure rate control, and (b) Hybrid control.

Fig. 9 .

 9 Fig. 9. Overall results for Pilot 2

Fig. 11 .

 11 Fig. 11. Pointing time across distances.

Fig. 12 .

 12 Fig. 12. Models tested for the three techniques

Fig. 13 .

 13 Fig. 13. Extrapolated movement times for the three pointing techniques and three levels of input precision (u = 61, 241, and 961 units). Target width is constant (W = 20 pixels).

Fig. 14 .

 14 Fig. 14. Interaction mixing movements of various joints. A bracelet attached around the wrist allows for detecting its absolute position and measuring the rotations of the hand relative to the forearm. (a) Slight wrist rotations within a central zone. (b) Wrist rotations out of the central zone. (c) Movements of the whole arm. (d) Rotations of the forearm.

Table 1 .

 1 Input configurations tested by Pilot 2

	Config.	Left Zone % (units)	Right Zone % (units)	Central Zone % (units)	S (pixels)
	C1	40.98% (25)	40.98% (25)	18.03% (11)	44
	C2	24.59% (15)	24.59% (15)	50.82% (31)	124
	C3	8.20% (5)	8.20% (5)	83.61% (51)	204

Table 2 .

 2 Input configurations tested by the experiment

	Precision (units) Left Zone % (units)	Right Zone % (units) Central Zone % (units)	S (pixels)
	61	19.67% (12)	19.67% (12)	60.66% (37)	37
	241	24.90% (60)	24.90% (60)	50.21% (121)