We show how to specify preferred parameterisations on a homogeneous curve in
an arbitrary homogeneous space. We apply these results to limit the natural
parameters on distinguished curves in parabolic geometries.Comment: 10 page

Eastwood, Michael

Slovak, Jan

Commentationes Mathematicae Universitatis Carolinae

English

arXiv

summary:We show how to specify preferred parameterisations on a homogeneous curve in an arbitrary homogeneous space. We apply these results to limit the natural parameters on distinguished curves in parabolic geometries

Slovák, Jan

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeMichael Eastwood; Jan SlovákPreferred parameterisations on homogeneous curvesCommentationes Mathematicae Universitatis Carolinae, Vol. 45 (2004), No. 4, 597--606Persistent URL: http://dml.cz/dmlcz/119488Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2004Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 45,4 (2004)597–606 597Preferred parameterisations on homogeneous curvesMichael Eastwood, Jan SlovákAbstract. We show how to specify preferred parameterisations on a homogeneous curvein an arbitrary homogeneous space. We apply these results to limit the natural param-eters on distinguished curves in parabolic geometries.Keywords: homogeneous space, parabolic geometry, distinguished curvesClassification: Primary 22F30; Secondary 53A40, 53C301. IntroductionThis article is motivated by the theory of distinguished curves in parabolicgeometries , as developed in [2]. A parabolic geometry is, by definition, mod-elled on a homogeneous space of the form G/P where G is a real semisimple Liegroup and P is a parabolic subgroup. (There is also a complex theory whichcorresponds to the choices of complex G’s and P ’s with specific curvature re-strictions for the holomorphic cases.) The notion of Cartan connection replacesthe Maurer-Cartan form on G, viewed as a principal fibre bundle over G/P withstructure group P , and much of the geometry of G/P automatically carries overto parabolic geometries in general (see also [4]). In particular, the curves on G/Pobtained by exponentiating elements in the Lie algebra g of G have counterpartsin general obtained by development under the Cartan connection. These mat-ters are thoroughly discussed in [2] and will not be repeated here. Suffice it tosay that results concerning distinguished curves on G/P have immediate conse-quences for the corresponding general parabolic geometry. Here, we shall discussonly the homogeneous setting G/P . We shall also not touch problems related tohigher dimensional analogues of curves since curvature obstructs the developmentin general.2. Generalities on G/PFirstly, let us discuss a general homogeneous space, namely a smooth manifoldM equipped with the smooth transitive action of a real Lie group G. Each X ∈ ggives a 1-parameter Lie subgroup t 7→ exp(tX) of G and hence a parameterisedSupport from the Australian Research Council is gratefully acknowledged. The second authorwas partially supported by GAČR 201/02/1390.598 M.Eastwood, J. Slovákcurve t 7→ exp(tX)m throughm ∈ M , which we shall suppose to be non-constant.Conversely, without the parameterisation, such a curve is homogeneous , namelyit is the orbit of a Lie subgroup of the symmetry group G.To investigate homogeneous curves on M we may as well choose a basepointm◦ ∈ M and consider only curves passing through m◦. All other homogeneouscurves are obtained by translation under the action of G. Let P denote thestabiliser subgroup of m◦ so that M = G/P . We shall now suppose that G issemisimple and P is parabolic. In this case, there is a splittingg = p ⊕ ninto subalgebras with n nilpotent (as in [2]). This splitting is not canonical. It is,however, well-defined up to the Adjoint action of P and we obtain, therefore, apreferred subset(1){Adp X s.t. p ∈ P and X ∈ n}⊂ g,which we may use to generate homogeneous curves. Such curves are evidently non-constant but not all non-constant homogeneous curves arise in this way. Thesespecial curves are said to be distinguished . Equivalently, distinguished curvesthrough the basepoint m◦ ∈ M are those of the form t 7→ p exp(tX)m◦ for somep ∈ P and X ∈ n.As an example, consider G = SL(3, R) with P the upper triangular matrices.We may take(2) n =0 0 0∗ 0 0∗ ∗ 0.Thent 7→1 1 00 1 00 0 1 expt0 0 01 0 00 0 0modP =1 + t 1 0t 1 00 0 0modPandt 7→ expt0 0 01 0 01 1 0modP =1 0 0t 1 0t+ 12 t2 t 1modPare typical distinguished curves whereast 7→ expt0 −1 01 0 00 0 0modP =cos t − sin t 0sin t cos t 00 0 1modPPreferred parameterisations on homogeneous curves 599is homogeneous but (with this parameterisation) not distinguished.Suppose t 7→ γ(t) ∈ M is a distinguished curve with γ(0) = m◦ and let Cdenote its unparameterised image. In this article, we shall answer the question‘what are the possible reparameterisations of C as a distinguished curve?’ A directapproach to this question is given in [2, §3]. Here, we shall reason indirectly byfirstly establishing the following on general grounds.Theorem 1. Let C be an unparameterised distinguished curve passing throughm◦ ∈ M = G/P . The freedom in reparameterising C with origin at m◦ is of twopossible types: –affine t 7→ at for a 6= 0projective t 7→ at/(ct+ 1) for a 6= 0 and c arbitrary.If we drop the requirement that the parameter be zero at m◦, then translation isalso allowed so the freedom becomest 7→ at+ b or t 7→at+ bct+ d,respectively. The proof of Theorem 1 will be given in §4. Once this theorem isestablished, it is a matter of elementary computation to decide, for a given C,which type of freedom pertains. Examples will be given in §4. For the proof ofTheorem 1 we shall need some general considerations as in the following section.3. Lie algebras of vector fields in one dimensionThe following is a classical topic and Theorem 3 is due to Lie [3] (see also [5]).We are grateful to Ian Anderson for pointing out to us the translation and com-mentary on Lie’s article given by Ackerman and Hermann [1]. Nevertheless, webelieve that it is useful to give an independent, elementary, and self-containedtreatment.Theorem 2. Suppose g is a finite-dimensional subalgebra of the Lie algebra ofsmooth vector fields on R. Let x be the standard coördinate on R and supposeg ∋ ∂/∂x. Then g is one of the following: –g = span{∂∂x}g = span{∂∂x , eλx ∂∂x}g = span{∂∂x , x∂∂x}g = span{∂∂x , sin(λx)∂∂x , cos(λx)∂∂x}g = span{∂∂x , x∂∂x , x2 ∂∂x}g = span{∂∂x, sinh(λx) ∂∂x, cosh(λx) ∂∂x}.600 M.Eastwood, J. SlovákProof: If dim g = 1, then g = span{∂/∂x} and we are done. Next, if dim g = 2,then g = span{∂/∂x, g(x)∂/∂x} for some smooth non-constant function g(x).Now,[∂∂x, g(x) ∂∂x]= g′(x) ∂∂xso closure under Lie bracket implies g′(x) = µ + λg(x). This is a differentialequation we may solve: –g(x) = Ceλx +D if λ 6= 0or g(x) = µx+ C if λ = 0.These are the two-dimensional subalgebras stated in the theorem.Now suppose dim g = k + 1 ≥ 3 and choose a basis∂∂x , g1(x)∂∂x , . . . , gk(x)∂∂xof g. From closure under Lie bracket of ∂/∂x with the other basis vectors, weimmediately encounter a system of ordinary differential equations with constantcoefficientsg′i(x) = µi +k∑j=1λijgj(x), for i = 1, . . . , k.We may conclude that the functions gi(x) and, therefore, all vector fields in g arereal-analytic.Since dim g ≥ 3, there is a vector field g(x)∂/∂x ∈ g withg(x) = xN + · · · for some N ≥ 2.Because g is finite-dimensional, we may choose g(x) with N maximal. But theng ∋[[∂∂x, g(x) ∂∂x], g(x) ∂∂x]=[g′(x) ∂∂x, g(x) ∂∂x]=((g′(x))2 − g(x)g′′(x))∂∂x=(Nx2(N−1) + · · ·)∂∂x ,contradicting maximality of N unless N = 2. Therefore, dim g = 3 and(3) g = span{∂∂x, g(x) ∂∂x, g′(x) ∂∂x},where(4) g(x) = x2 + ax3 + · · · .Preferred parameterisations on homogeneous curves 601But then g contains the vector field[g′(x) ∂∂x, g(x) ∂∂x]− 2g(x) ∂∂x=((g′(x))2 − g(x)g′′(x) − 2g(x))∂∂x=(2ax3 + · · ·)∂∂x,again contradicting maximality of N unless a = 0. Now, in order for (3) to beclosed under Lie bracket we must haveg′′(x) =[∂∂x, g′(x) ∂∂x]∈ span{∂∂x, g(x) ∂∂x, g′(x) ∂∂x}and to be, in addition, consistent with a = 0 in (4), we conclude thatg′′(x) = 2 + νg(x), for some constant ν.This differential equation, with initial conditions imposed by (4), has solutionsg(x) = (2/λ2)(cos(λx) − 1) if ν < 0or g(x) = x2 if ν = 0or g(x) = (2/λ2)(cosh(λx) − 1) if ν > 0.It remains to observe that (3) is, indeed, closed under Lie bracket in these cases.Notice that this proof is local: the same conclusion holds for vector fields onany open interval (a, b) ⊆ R. As subalgebras of Vect(R), the various possibilitieslisted in the statement of Theorem 2 for different values of λ are distinct. Locally,however, the parameter λ may be eliminated and more besides. Precisely, thereis a coördinate change near the origin: –y =1− e−λxλ⇒ eλx ∂∂x =∂∂y and∂∂x = (1− λy)∂∂ywhencespan{∂∂x, eλx ∂∂x}∼= span{y ∂∂y, ∂∂y}∼= span{∂∂x, x ∂∂x}whilst the coördinate change y = tan((λx)/2) gives∂∂x =λ2(1 + y2) ∂∂y , sin(λx)∂∂x = λy∂∂y , cos(λx)∂∂x =λ2(1− y2) ∂∂ywhencespan{∂∂x, sin(λx) ∂∂x, cos(λx) ∂∂x}∼= span{∂∂x, x ∂∂x, x2 ∂∂x}and y = tanh((λx)/2) gives∂∂x =λ2(1− y2) ∂∂y , sinh(λx)∂∂x = λy∂∂y , cosh(λx)∂∂x =λ2(1 + y2) ∂∂ywhencespan{∂∂x , sinh(λx)∂∂x , cosh(λx)∂∂x}∼= span{∂∂x , x∂∂x , x2 ∂∂x}.We have proved the following.602 M.Eastwood, J. SlovákTheorem 3. Suppose s is a finite-dimensional subalgebra of the Lie algebra ofvector fields in a neighbourhood of the origin in R. Suppose s contains a vectorfield that does not vanish at the origin. Then there is a neighbourhood U of theorigin and a change of coördinates such that one of the following three possibilitiesholds.(5)s|U ∼= span{∂∂x}s|U ∼= span{∂∂x , x∂∂x}s|U ∼= span{∂∂x , x∂∂x , x2 ∂∂x}.4. ReparameterisationsLet C be an arbitrary smooth connected curve in a smooth manifoldM homo-geneous under the action ρ : G × M → M of a connected Lie group G. There isa homomorphism of Lie algebras ρ̇ : g → Vect(M) given byρ̇(X)(m) = ∂∂t(exp(−tX)m)|t=0and the symmetry algebra of C is defined bys = {X ∈ g s.t. ρ̇(X)(m) is tangent to C for all m ∈ C}.Clearly, s is a subalgebra of g and C is homogeneous if and only if ρ̇(s)|C containsnon-trivial vector fields at each point of C. In this case, we may invoke Theorem 3to conclude that ρ̇(s)|C is at most three-dimensional and locally has one of thethree forms listed in (5).Now suppose that C is homogeneous and pick a basepoint m◦ ∈ C. Supposethat X ∈ s ⊂ g is nilpotent in g and ρ̇(X)(m◦) 6= 0. Then we shall say thatt 7→ exp(tX)m◦ ∈ Cis a preferred parameterisation of C.Theorem 4. The freedom in reparameterising a homogeneous curve with a pre-ferred parameter is one of two possible types: –affine t 7→ at for a 6= 0projective t 7→ at/(ct+ 1) for a 6= 0 and c arbitrary.Proof: Since X is nilpotent in g, certainly ρ̇(X) is nilpotent in ρ̇(s)|C . Byinspection, we may find the nilpotent elements in each of the local forms (3): –span{∂∂x}∋ a ∂∂xspan{∂∂x , x∂∂x}∋ a ∂∂xspan{∂∂x, x ∂∂x, x2 ∂∂x}∋ (p − qx)2 ∂∂x.Preferred parameterisations on homogeneous curves 603In the first two cases,a ∂∂x= ∂∂t⇐⇒ x = at,which gives affine freedom, whilst in the third case(p − qx)2 ∂∂x =∂∂t ⇐⇒ x =p2t1 + pqt,which gives projective freedom. Proof of Theorem 1: The parameterisations on a distinguished curve have theform t 7→ exp(tY )m◦ where Y is P -conjugate to an element of n in accordancewith (1). Certainly, there is affine freedom in such a parameterisation because Ycan be replaced by aY . But the allowed Y are, in particular, nilpotent. There-fore, the parameterisations on C as a distinguished curve are ipso facto preferredparameterisations on C as a homogeneous curve. Theorem 4 now implies that,if there is any additional freedom, it must be projective. But just one projectivetransformation, together with affine freedom, generates all projective freedom andthe proof is complete. Theorem 1 is useful in practice. Consider the general distinguished curve t 7→p exp(tX)m◦ for fixed p ∈ P and X ∈ n. The dichotomy offered by Theorem 1implies that if there are reparameterisations other than affine, then the specificprojective freedom t 7→ t/(t + 1) occurs. In this case, we may find q ∈ P andY ∈ n such thatp exp(tX) = q exp( tt+ 1Y)modP, ∀ tor, equivalently,(6) exp(−tt+ 1Y)r exp(tX) ∈ P, ∀ twhere r = q−1p ∈ P . The existence of suitable r ∈ P and Y ∈ n is a restrictionon X . Furthermore, if we adopt the Levi decomposition P = LU correspondingto our choice of n, then the L-component of r may be absorbed into Y . Hence,Theorem 1 has the following corollary.Corollary 5. Suppose P = LU is a Levi decomposition of a parabolic subgroupP of a semisimple Lie group G. Let g = p⊕ n be the associated decomposition ofthe Lie algebra of G. Then the distinguished curve t 7→ p exp(tX)modP admitsa projective reparameterisation if and only if there are r ∈ U and Y ∈ n such that(6) holds.We close this article with a complete analysis of the distinguished curves in thereal flag manifold SL(3, R)/P where P is the subgroup of upper triangular matri-ces. As already remarked in §2, we may take n to be the strictly lower triangular604 M.Eastwood, J. Slovákmatrices (2). We shall use Corollary 5 with U taken to be the upper triangularmatrices with 1’s along the diagonal. Consider, for example, the distinguishedcurve(7) t 7→1 1 00 1 00 0 1 expt0 0 01 0 00 0 0modP.According to Corollary 5, it admits a projective reparameterisation if and only ifwe can find a, b, c, u, v, w such thatexp−tt+ 10 0 0u 0 0v w 01 a b0 1 c0 0 11 0 0t 1 00 0 1 =∗ ∗ ∗0 ∗ ∗0 0 ∗ .Multiplying through by (t+ 1)2 yields(t+ 1)2 0 0−t(t+ 1)u (t+ 1)2 0−t(t+ 1)v + 12 t2uw −t(t+ 1)w (t+ 1)21 a b0 1 c0 0 11 0 0t 1 00 0 1for the left hand side. Expanding and equating coefficients of t to zero in the sub-diagonal entries gives algebraic equations for a, b, c, u, v, w whose general solutionsare0 0 0u 0 0v w 0 =0 0 01 0 00 0 01 a b0 1 c0 0 1 =1 1 b0 1 c0 0 1 .The existence of solutions shows that the distinguished curve (7) admits projectivereparameterisations. On the other hand, this same exercise for the curvet 7→ expt0 0 01 0 01 1 0modPgives an inconsistent set of equations for a, b, c, u, v, w. According to Theorem 1and Corollary 5, it admits only affine reparameterisations.Notice that the criterion (6) of Corollary 5 depends only on the L-conjugacyclass of X ∈ n. Therefore, to say which distinguished curves admit projectivereparameterisations it suffices to say whether (6) is satisfied for X ∈ n normalisedunder the Adjoint action of L. We obtain the following table of normal forms.Preferred parameterisations on homogeneous curves 605Normal form Reparameterisation0 0 01 0 00 0 0 projective0 0 00 0 00 1 0 projective0 0 01 0 00 1 0 projective0 0 01 0 01 0 0 projective0 0 00 0 01 1 0 projective0 0 00 0 01 0 0 projective0 0 01 0 0x 1 0 affine if x 6= 0We have to be careful, however, with the decision which of the above normal formsgive rise to different distinguished curves. In our case, the lines four through sixin the table are in the same orbit of the Adjoint action of the entire P and sothe distinguished curves indicated by these lines coincide. Indeed, a simple checkreveals(expZ)−10 0 00 0 01 0 0 expZ =0 0 01 0 01 0 0 , Z =0 0 00 0 −10 0 0while the other case is symmetric. We should also like to remark, that the latterobservation yields a sufficient condition for coincidences of classes of distinguishedcurves. There are also examples of such a coincidence where the correspondingL-orbits are not in the same orbit of P . In our case, however, the first three linesand the last two lines in the table obviously produce different curves.This completes our analysis of distinguished curves in this real flag manifold.It is more efficient than the direct approach of [2] because Theorem 1 tells us, inadvance, what sort of reparameterisation we may expect on a distinguished curve.606 M.Eastwood, J. SlovákReferences[1] Ackerman M., Hermann R., Sophus Lie’s 1880 Transformation Group Paper, Math SciPress, Brookline, Mass., 1975.[2] Čap A., Slovák J., Žádńık, V., On distinguished curves in parabolic geometries, Transfor-mation Groups 9 (2004), no. 2, 143–166.[3] Lie S., Theorie der Transformationsgruppen, Math. Ann. 16 (1880), 441–528.[4] Sharpe R.W., Differential Geometry, Springer, New York, 1997.[5] Strigunova M.S., Finite-dimensional subalgebras of the Lie algebra of vector fields on thecircle (in Russian), Trudy Mat. Inst. Steklova 236 (2002), 338–342.Department of Pure Mathematics, University of Adelaide, South Australia 5005E-mail : meastwoo@maths.adelaide.edu.auDepartment of Algebra and Geometry, Masaryk University, Janáčkovo nám. 2a,662 95 Brno, Czech RepublicE-mail : slovak@math.muni.cz(Received November 24, 2003, revised February 23, 2004)

Preferred parameterisations on homogeneous curves

We show how to specify preferred parameterisations on a homogeneous curve in an arbitrary homogeneous space. We apply these results to limit the natural parameters on distinguished curves in parabolic geometries.Michael Eastwood, Jan Slovakhttp://www.karlin.mff.cuni.cz/cmuc/cmucemis/cmucemis.htm

Eastwood, Michael George

Slovak, J.

Adelaide Research & Scholarship

Czech digital mathematics library

http://dml.cz/bitstream/handle/10338.dmlcz/119488/CommentatMathUnivCarolRetro_45-2004-4_4.pdf

Preferred Parameterisations on Homogeneous Curves

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