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Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

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Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric.

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Page 1: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

Proceedings Ist International Meetingon Geometry and Topology

Braga �Portugal�

Public� Centro de Matem�atica

da Universidade do Minho

p� ����� ����

On Lie Groups with Left Invariant semi�Riemannian Metric

R� P� Albuquerque

� Introduction and General Results

J� Milnor in the well known ��� gave several results concerning curvatures ofleft invariant Riemannian metrics on Lie groups� Some of those results can bepartial or totally generalized to inde�nite metrics� We will �rst show three ofthose generalizations that we have obtained� These will serve our purposes lateron�

Let G be a real Lie group of dimension n and g its Lie algebra� Considering aleft invariant semi�Riemannian structure on G� let e�� � � � � en be an orthonormalbasis of left invariant vector �elds and �ijk their structure constants� that is�

�ei� ej� �nX

k��

�ijkek�

or equivalently� denoting �i ��ei� ei� � � i � n�

�ijk � �k � �ei� ej�� ek� �

Lemma � With structure constants �ijk as above� the sectional curvature satis��es the formula� for i �� j�

K�ei� ej � �i

nXk��

��jik��ikj � �j�k �jik � �j�i �kji� �k�i �kii�kjj

���jik � �k�j �ikj � �k�i �kji��ikj � �j�i �kji � �j�k �jik

��

Page 2: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

� R� P� Albuquerque

Proof� The Levi�Civita conexion given by Koszul formula veri�es

rei ej �Xk

���ijk � �k�i �jki � �k�j �kijek�

Hence� denoting by R the semi�Riemannian curvature tensor

R�ei� ejei � �rei rej ei �rej rei ei �r�ei�ej � ei�

we get the desired result by inspection on the right side of the identity

�i�jK�ei� ej ��R�ei� ejei� ej� �

As one can see� the curvatures depend continuously on the structure constants�

Lemma � If the transformation ad �ei is skew�adjoint� then

K�ei� ej ��i�j

nXk��

�k ��kji�

If ei is also orthogonal to �ej� g�� then necessarily K�ei� ej � � In the case ofRiemannian metric this condition is also su�cient�

Proof� If ad �ei is skew�adjoint� then

� only for indice i � �ei� ej�� ek�� � � �ei� ek�� ej��

that is� �ijk � ��j�k �ikj� It follows that

�kii � ��iki � �i�k �iik � �

These results make possible the simpli�cation of the formula of lemma ��

Recall that the Ricci curvature in a direction x � g and the scalar curvatureare� respectively�

r�x �nXi��

�i �R�x� eix� ei� and S �nXi��

�ir�ei � �Xi�j

K�ei� ej�

In a direction ej� the Ricci curvature becomes r�ej � �jP

i��j K�ej� ei�Let us denote by �p� n�p the signature ��� � � � ����� � � � �� of a metric with

p minus signs�

Page 3: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

On Lie Groups with Left Invariant semi�Riemannian Metric �

Theorem � If the Lie algebra g contains linearly independent vector �elds x� y� zso that

�x� y� � z�

then there exist left invariant metrics on G� of signature �p� n� p� such that��i� � p � n and r�x � � r�z or r�z � � r�x��ii� � p � n and S � ��iii� � p � n and S � �

Proof� We will take the metric induced by a scalar product in g� Fix a basisb�� � � � � bn of g such that b� � x� b� � y� b� � z� Let �ijk be the structure constantsof g for b�� � � � � bn� For any real number � � � consider an auxiliary basis e�� � � � � enand the Lie algebras g� whose structure constants for e�� � � � � en are those givenby the bracket product of g and the basis e� � �b�� e� � �b�� ei � ��bi �i � ��Computation shows

�e�� e�� � ��b�� �b�� � ��b� � e��

�ei� ej� � ��bi� ��bj� �

Xk

���ijkbk ����ij�e� � ���ij�e� � �Xk��

�ijkek

for i � � �� j � �� and

�ei� ej� � ���ij�e� � ���ij�e� � ��Xk��

�ijkek

for i� j � �� Clearly g� � g for � � as Lie algebras� since we only made achange of basis� Now� for any � p � n� de�ne the left invariant metric on g�with signature �p� n � p� which makes e�� � � � � en an orthonormal basis and sothat �� � �� � �� or �� � �� � ����

Once � � � we get a limit Lie algebra g� de�ned by �e�� e�� � ��e�� e�� � e�and �ei� ej� � otherwise� Using lemmas and �� one may check

K��e�� e� � K��e�� e� � ��

��

K��e�� e� � ��� ��� K��ei� ej � � for fi� jg �� f� �� �g�

Hence

r��e� � ��

���

��

��� � �����

��

r��e� � ��

��

��� � ����

��

With the prescribed metric� r��e� and r��e� clearly have di�erent signs� Withrespect to scalar curvature it follows that

S� � ��K�e�� e� �K�e�� e� �K�e�� e� � �

����

Page 4: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

� R� P� Albuquerque

From the continuous dependence of the Ricci and scalar curvatures on the struc�ture constants� we get the desired results for the �xed values of p and for su��ciently small �� �

� The Special Class S

We will now study the Lie groups that do not satisfy the hypothesis of thelast theorem� With K� Nomizu� we consider a special class S of solvable Liegroups� A non�commutative Lie group G belongs to S if its Lie algebra g has theproperty that �x� y� is a linear combination of x and y� for any x� y � g�

In ��� it is shown that G � S if and only if there exists an abelian ideal nof codimension and an element b �� n such that �b� u� � u for every u � n�Furthermore� G � S if and only if every left invariant Riemannian metric on G

has sectional curvatures of constant sign�In order to prove our next theorem we deduced the following slight generalization

of a lemma from ��� The proof of this generalization is equal to the original�

Lemma � Let G be a Lie group with a left invariant semi�Riemannian metricand such that its Lie algebra can be decomposed as

g �� b � � n

where b is orthogonal to n and � b� b �� � � � n is an abelian ideal andL � ad �bjn � �Id � S� where S is the skew�adjoint part of L and � � R � Then

G has constant sectional curvature K � �����

The following is a generalization of ���Theorem �� Notice the new demonstrationof Case II�

Theorem � Let G be a Lie group of dimension n belonging to the special classS� Then

�i� Any left invariant semi�Riemannian structure on G� of signature �p� n�p�has constant sectional curvature K�

In particular� K is negative constant� if p � � or positive constant� if p � n��ii� Given any p � N � � p � n� and any K � R � we can construct a left

invariant metric of signature �p� n � p with K as constant sectional curvature�We may still conclude the same in the cases p � � K � and p � n� K � �

Proof� Let g �� b � � n be the Lie algebra of the Lie group G�

�b� u� � u �u� v� � � u� v � n�

Page 5: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

On Lie Groups with Left Invariant semi�Riemannian Metric �

�i Suppose g has a scalar product�

Case I� � � �jn�n is nondegenerate�There exists an unitary vector b� �� n such that �b�� n�� � Writing

b� � �b � u�� � � R n f g� u� � n

we have�b�� v� � �v v � n�

Applying lemma � to this case in which the operator S � � we get the desiredresult� with the obvious particullarities for signatures � � n and �n� �

Case II� � � �jn�n is degenerate�There exists e � n such that�e� ��� all over n� So� since � � � is nondegenerateon g� �e� b��� � Hence we may just suppose

�e� b�� �

De�ne two maps a and C by

a�x ��e� x�� C�x� y � a�xy � a�yx� x� y � g�

Immediately one recognizes the linearity and bilinearity� respectively� of a and C�From ker a � n� the skew�adjointness of C and

C�b� u � u � �b� u� C�u� v � � �u� v�� u� v � n�

we �nd that C � � � �� Now we can compute� for any x� y� z � g�

�rx y� z��

��� �x� y�� z� � � �y� z�� x� � � �z� x�� y� �

�a�x �y� z� �a�y �x� z� �a�y �z� x� �a�z �y� x�

�a�x �z� y� �a�z �x� y��� a�z �x� y� �a�y �x� z� �

Hencerx y ��x� y� e� a�yx

and thenR�x� yz � �rx ry z �ry rx z �r�x�y� z �

� � �y� z� rx e � a�zrx y � a�zry x� �x� z� ry e � a�xry z � a�yrx z �

� � �y� z��x� e� e� a�z �x� y� e� a�za�yx� a�z �y� x� e� a�za�xy

� �x� z��y� e� e� a�x �y� z� e� a�xa�zy � a�y �x� z� e � a�ya�zx �

Page 6: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

� R� P� Albuquerque

�ii If one wants sectional curvature K � choose the following metric� Takeb� �

pKb and de�ne a scalar product � � � on g satisfying

�b�� b��� �� �b�� n��

� � �jn�n of signature �p� � n� p � � p � n�

For what we have seen above� with the left invariant metric induced by this scalarproduct� G has constant sectional curvature K�

For K � � we do the same with b� �p�Kb and choosing a scalar product

on g satisfying�b�� b��� � �b�� n�� �

� � �jn�n of signature �p� n� � p � � p � n� �

Finaly� if one wants K � � it is su�cient to choose a left invariant metric onG that is degenerate on n� This must be an inde�nite metric� �

By Theorems and � we may conclude the following�

Proposition � Every non�abelian Lie group admits left invariant metrics of sig�nature �p� n� p such that p � n and S � � or � p and S � �

Let us denote by F�p the class of Lie groups such that every left invariantmetric of signature �p� n� p has sectional curvature of constant sign� As we saidbefore F� � S� Looking at the proof of Theorem we can establish

S � F� � F� � F�� � � � � � F�n�

In other words� it is useless to search for other Lie groups for which one has thesame nice results of Theorem ��

Notice that according to the well known Theorem of R�S� Kulkarni� whichsays that� for a connected manifold of dimension � � and inde�nite metric� if thesectional curvatures have an upper bound �or a lower then they are constant�one becomes aware that looking for Lie groups in F�p � � p � n� n � � is thesame as looking for those that have constant K for all such metrics�

� A Non�Complete semi�Riemannian Structure�

Let M be a simply connected semi�Riemannian manifold of dimension n andsignature �p� n�p� with constant sectional curvatureK and geodesically complete� in the usual concept� M is a simply connected space form� Consulting� forexample� ���� we note that a simply connected space form is di�eomorphic to theEuclidean space if and only if

Page 7: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

On Lie Groups with Left Invariant semi�Riemannian Metric �

�a p � n� n� and K � ��b K � ��c p � � and K � �

Otherwise� M is the product of the Euclidean space with a sphere�

Now suppose M is a simply connected Lie group eG� belonging to the specialclass S� provided with any metric given by part �ii of Theorem � such that pand K are out of cases �a� �b and �c above� General Lie group theory saysthat a simply connected and solvable Lie group is di�eomorphic to the Euclideanspace ����� Thus eG is di�eomorphic to the Euclidean space� We derive from thisthat with the prescribed semi�Riemannian structure� completeness must fail ineG�

� Example�

Fix n � �� A simply connected Lie group in the special class S is

G �

�� v sIn��

�� GL�n�R � v � R

n��� s �

As a manifold this is just M � Rn���R

�� Let e�� � � � � en be the canonical basis ofRn � T�v�sM for all �v� s �M �with the Lie product �v� s � �u� t � �v�su� st it

is easy to see that the Ui�v�s� � sei are left invariant vector �elds � in the abovenotation� n is the ideal spanned by U�� � � � � Un�� and b � Un�

We now de�ne a Lorentzian metric� on M giving its components relative tothe basis e�� � � � � en�

gij ��ij

s�� for i � n� gnn � �

s�

�this is just the left invariant metric which makes U�� � � � � Un an orthonormalbasis so that �Un� Un�� �� Now we can �nd the Christo�el symbols for theLevi�Civita connection� Calculations lead us to

�ni� � �h

ij � �hin � �n

in � �hnn � � �n

ii � �iin � �n

nn � �

s�

for all i� j� ��� h � n� �� �� i� Now let us �nd the geodesics of M� Suppose � ����� � � � � �n��� is such a curve so that �note �t �

�i� � vio� � � so � � and ��i� � io� �� � �o�

�notice the generalization� on both dimension and signature� of the �Poincar�e half space or

�Lobatchevski Plane�

Page 8: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

� R� P� Albuquerque

The system of ordinary di�erential equations of a geodesic gives�����i � ����i �

�� � �� �Pn��i�� �

��i � �

Denoting �i � ��i� from the �rst equation we get��

i

�i� ���

�� So �log j�ij� � �log ���

which gives� �i ��ios�o�� Henceforth� from the second equation we get

�� � �� �Q� � �

where Q � s��o

Pn��i��

�io� We have Q � if and only if all the io � � It is easy

to see that in this case

��t � �v�o� � � � � vn�� o� soe� �oso

t

is the desired geodesic� which happens to be the only complete one�Now suppose Q � � Let us start by consider �o �� � Making the substitution

� � z� and hence �� � z� � �z dzd�� the above di�erential equation sympli�es

to

zdz

d� Q � z� � Q� �D � �

pQ� �D

� ���

�� or � depending on the signal of �o� where

D ���os�o�Qs�o �

s�o���o �

n��Xi��

�io�

Equation ��� is easely integrable� giving us di�erent solutions for di�erent val�ues of D�

Case I� D � ��

��

pQ �

� �

pQ t�

so�

Thus

�t �so

� sopQt

��i�t � �i�t �io

�� sopQ t�

and

�i�t �io

sopQ�� so

pQ t

� io

sopQ

� vio�

Page 9: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

On Lie Groups with Left Invariant semi�Riemannian Metric �

Case II� D � � By integration of equation ���� we �nd

pD

log

pQ� �D �pD

� t � to�

where to ��pDlog

pQs�o�D�pD

so� �p

Dlog j�oj�so

pD

s�o� Solving for the implicit func�

tion� we get

�t ��pD e

pD�to�t

Q� e�pD�to�t

�i�t ��ioD e�

pD�to�t

s�o�Q� e�pD�to�t�

�i�t � �iopD

s�o�Q� e�pD�to�t

� �iopD

s�o�Q� e�pD to

� vio�

Case III� D � � Again� equation ��� gives

p�D arccos

�p�DpQ

�� t � to�

where to ��p�D arccos

�p�DsopQ

�� So

�t �

p�DpQ cos �

p�D�to t�

�i�t ��ioD

s�oQ cos��p�D�to t

�i�t � iop�Ds�oQ

tg�p�D�to t� io

p�Ds�oQ

ps�oQ�D � vio�

Finally� if one considers �o � � which is equivalent to s�oQ�D � � then thesolutions of case III will adapt perfectly�

We can conclude that� on our example of a semi�Riemannian homogeneousspace with constant sectional curvature � almost every geodesic is non complete�

There remains the question� Which is the condition on a Lie group with leftinvariant semi�Riemannian structure so that it is complete�

A theorem due to Marsden �cf���� says that any compact homogeneous semi�Riemannian space is complete� When trying to see what happens on Lie groupswe were led to the following interesting result which we have never heard about�Let G be a Lie group with left invariant metric�

Lemma � Right invariant vector �elds on G are Killing �elds�

Page 10: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

R� P� Albuquerque

Proof� In every manifold� the Lie derivative of a tensor A� with respect to adi�erentiable vector �eld X� veri�es

LXA � limt��

t� �t �A� A�

where f tg is the �local �ow of X ����� Now let X be a right invariant vector�eld on G and let us determine its �ow� A maximal integral curve � of X startingat e�

�� � e� ���t � X��t�

is precisely the �unique one�parameter subgroup of G associated to X �inducedby the Lie homomorphism t d

dt�� tX between the Lie algebras R and the one

consisting of right invariant vector �elds on G� Let f tg be the �ow of X� Wehave t�e � ��t� Given g � G��

dRg � �dt

��

� Rg� e

�d�

dt�

�� Rg� e

�Xe � Xg�

hence t�g � ��tg� i�e�� t � L��t� Thus t is an isometry� or� in other words� �t � � ��� � �� By the initial equality� LX � � �� as we wanted�

� Bi�invariant Metrics�

Recall that a Lie algebra g is compact if it is the Lie algebra of a compact Liegroup� We say that g is simple if it has no proper ideals other than � Recall alsoE� Cartan�s criterion for semisimplicity� g is semisimple if� and only if� its Killingform is nondegenerate�

Let us now recall some basic facts about the structure of a semisimple Liealgebra� that can be seen in ���� Every Lie algebra g admits a Cartan subalgebra�that is� a nilpotent subalgebra h which coincides with its normalizer in g� AllCartan subalgebras of g have the same dimension� This natural number is thencalled the rank of g� It is also well known that every complex semisimple Liealgebra g admits a root decomposition relative to one of its Cartan subalgebrash� that is� g can be decomposed as the direct sum

g �M���

g�

where

g� � fy � g � �x� y� � ��xy� x � hg

Page 11: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

On Lie Groups with Left Invariant semi�Riemannian Metric

and � is a subset of h�� complex dual of h� for which � � � i� g� �� � � n f gis called a root system�

Lie group theory tells us that h � g�� h is maximal abelian �even in the realcase and that the root subspaces g� �� �� have dimension �

When g is real and semisimple then the complexi�cation gc is semisimple�

and� if h is a Cartan subalgebra of g� then hc is a Cartan subalgebra of gc�

Lemma � Let g be a real semisimple Lie algebra and h one of its Cartan subal�gebras�

There exists a real linear form �� on h such that g� �� �g� de�ned asabove� if� and only if� there exists y � g n h and a nonzero root � of gc� relatve tohc� such that gc� � C y� In this a case g� � Ry�

The proof is immediate� for � � c�

Now we are able to present the only theorem of this section� Its proof wasmainly taken from ���lemma ����� the particular case when g is compact� Firstrecall� if a left invariant metric on a Lie group is bi�invariant� then all the adjointmorphisms ad �x� x � g� are skew�adjoint� In a Lie group G� the Killing form B

of its Lie algebra is Ad �G�invariant� so� when G is semisimple� the left invariantmetric induced by B is also right invariant�

Theorem � Let G be a Lie group with simple Lie algebra g of rank r� If g

satis�es one of the following conditions��i� dim�g or r are odd��ii� for some Cartan subalgebra h� in the root decomposition of �gc� hc there

is a root subspace of type C y with y � g��iii� g is compact�

then any bi�invariant metric on G is induced by a multiple of the Killing form�

Proof� Let � � � denote any bi�invariant metric on G and B the Killing form�There is a linear bijection S of g such that

�x� y�� B�S�x� y� x� y � g�

From this we can deduce that S commutes with all the ad �x� x � g�Now� if y � g is an eigenvector of S associated to a real eigenvalue � �� � then

S�x� y� � �x� S�y� � ��x� y�� so each eigenspace is an ideal� Since g is simple� thiseigenspace is all g and so � � �� �B on g� Thus we must assure the existenceof one real eigenvalue for S� If dim�g is odd this is trivial� Let h be a Cartansubalgebra of g� Since h is abelian and equals its normalizer� then �S�h� h� � h

and hence S�h � h� Thus odd rank implies� as above� an eigenvalue for S�

�note g simple �� gc simple

Page 12: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

� R� P� Albuquerque

If g satis�es �ii� then by the lemma there is a real linear form �� on h

with g� � Ry� that is� �x� y� � �xy� for all x � h� Then

�x� S�y� � S�x� y� � �xS�y� x � h

So S�y � �y for some nonzero � � R �

Finally� if g is compact� then �B is an inner product� S is symmetric andcertainly diagonalizable�

Theorem � may be generalized to Lie groups with reductive Lie algebra ��g� g�semisimple� since� with bi�invariant metric� the simple components of g and thecenter of g are all orthogonal to each other�

There is a large class of Lie algebras satisfying condition �ii of the theorem�the simple split Lie algebras� g is said to be split if any of its maximal R �diagonalizable subalgebras is a Cartan subalgebra� We have an R �diagonalizablesubalgebra a � g when there exists a basis in g with respect to which all operatorsad �x �x � a are expressed by diagonal matrices�

The following are examples of simple split Lie algebras ����� sln �n � ��sok�k�� �k � � sok�k �k � �� spn �n � ��

In ���remark �� we �nd a Lorentz metric on SL� with constant sectional curva�ture � and that this �metric is essentially the same as the Killing�Cartan form �We can now stablish� all bi�invariant metrics on SL� have constant sectional cur�vature�

� Remark on Complex Simple Lie Algebras�

Let g be a complex Lie algebra and � a faithfull representation of g in acomplex vector space� Notice that such representations exist by the well knowntheorem of Ado� We call the bilinear and symmetric form on g

��x� y � tr ���x��y�

a trace form�

With respect to � all endomorphisms ad �x are skew�adjoint and it is provenlike Cartan�s semisimplicity criterion that� if g is semisimple� then � is nondegen�erate �����

Proposition � Every trace form on a complex simple Lie algebra is a multipleof the Killing form�

Page 13: Albuquerque.-.On.Lie.Groups.with.Left.Invariant.semi-Riemannian.Metric

On Lie Groups with Left Invariant semi�Riemannian Metric �

The proof is obviously equal to the one of theorem �� This time there is noproblem with eigenvalues�

In other sense we have the following�

Corollary � If k is a simple Lie subalgebra of gn �n� C � then its Killing formis a multiple of the trace form

tr �XY � X� Y � k�

References

�� F� Barnet� On Lie groups that admit left�invariant Lorentz metrics of con�stant seccional curvature� Illinois J� Math� �� ����� ������

��� J� Milnor� Curvatures of left invariant metrics on Lie Groups� Advances inMathematics � ����� �������

��� K� Nomizu� Left invariant Lorentz metrics on Lie groups� Osaka J� Math�� ����� ����

��� B� O�Neill� Semi�Riemannian Geometry� Academic� ���

��� A� L� Onishchik and E� B� Vinberg� Lie groups and Lie algebras III� EMS�vol� �� Springer� ��

��� V� S� Varadarajan� Lie groups� Lie algebras and their Representations�Springer� ���

��� J� A� Wolf� Spaces of Constant Curvature� McGraw�Hill� ���

Rui Pedro AlbuquerqueDepartamento de Matem!aticaUniversidade de !EvoraPortugalEmail� rpa�dmat�uevora�pt