4. Homogeneity and holonomy

Transcription

1 4 Homogeneity and holonomy Objekttyp: Chapter Zeitschrift: L'Enseignement Mathématiue Band (Jahr): 48 (2002) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften Die Rechte liegen in der Regel bei den Herausgebern Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz, wwwlibraryethzch

2 Let Gp =M, dim(m) >2, be an irreducible submanifold of the Euclidean space with rank(m) > 1 andfull homogeneous Then M is contained in a sphère ([o4]) THEOREM 311 We summarize ail the above facts in the following theorem G-p =M, dim(m) >2, be an irreducible andfull of the Euclidean space Then, and only if M is contained in a sphère; if if and only if M is an orbit of an s -représentation Let THEOREM 312 homogeneous submanifold (i) rank(m) > 1 (ii) rank(m) > 2 The next corollary uses the fact that the minimal homogeneous submani foldsof Euclidean spaces must be totally géodésie (see [D]) Let G-p =M, dim(m) > 2, be an irreducible and homogeneous submanifold of the Euclidean space with parallel mean 0 and Mis either minimal in a sphère or it curvature vector H Then H COROLLARY 31 full is an orbit of an s -représentation Homogeneity and holonomy 4 In this section we briefly relate homogeneity and holonomy In particular, we are interested in the computation of the holonomy group in homogeneous situations We shall put spécial emphasis on the tangent bundle of a ho mogeneousriemannian manifolds and the normal bundle of a homogeneous submanifold of Euclidean space But, in the first part, we will work in the framework of arbitrary homogeneous (pseudo)metric vector bundles with a con nectionthis is because, in our opinion, the main ideas are better understood in this context Another reason is that one can prove, without extra efforts, very gênerai results which could hâve applications to the pseudoriemannian case Let E A M be a finite dimensional real vector bundle over M with a covariant derivative operator V (also called a connection), which corresponds, as usual, to a connection in the sensé of distributions More precisely, there exists a distribution TL on TE such that (1) H V = TE, where V is the vertical distribution; (2) (/i c )*(^) = 7~(>tJ, c ()i for ail cgr, where fi c is multiplication by c

3 Let (, ) be a C metric on the fibres and let g be a Riemannian metric on M (in fact, (, ) and g need not be positive definite) We assume that there is a Lie group G which acts on E by bundle morphisms, whose induced action on M is by isometries and is transitive Moreover, we assume that the action on E préserves both the metric on the fibres and the connection A vector X in the Lie algebra Q of G induces, in a natural way, a Killing vector field X both on E and M, Le, if Çp G E (resp p G M) then X(ÇP ) '= X-tp := /=oexp(tx)4, (resp X(p) := Xp := \t=oexp(tx)p), where exp(tx) is the one parameter subgroup associated with X We will always keep in mind, as remarked above, the following two important cases : (a) M = G/H is a homogeneous Riemannian manifold, where G is a Lie subgroup of the isometry group /(M), E = TM is the tangent bundle and V is the usual Levi-Civita connection (b) M=G-v, where vekn and G is a Lie subgroup of the isometry n group I(R ) Hère, E= v{m) is the normal bundle endowed with the usual normal connection V-1 The bundle E is endowed with the so-called Sasaki (Riemannian) metric g Namely, (i) 7ï is perpendicular to the vertical distribution V, defined by the tangent l space to the fibres E = ir~ () (ii) The restriction of g to V coincides with the metric on the fibres (iii) 7T is a Riemannian submersion The Sasaki metric may be regarded as follows A curve c(t) in E may be viewed as a section along the curve c(t) = 7v(c(t)) In this way, (c / (0),c / = <! 0 c(0, t \oc) +^(c / (0),c / Observe that G acts by isometries, with respect to the Sasaki metric, on E As is well known, the fibres E, em, are totally géodésie submanifolds of E In fact, if c(t) is a curve in M starting at, then the parallel transport rf along c(t) defines an isometry from E into Ec^ Let j(s) be a curve in E and consider f(s, t) = rrtc (j(s)) We hâve that (rf^c)), t/xt'cs ))) does j 1 not dépend on t and so, where A dénotes the shape operator of E totally géodésie as a submanifold of E Then E is

4 We now describe how the holonomy algebra (ie, the Lie algebra of the holonomy group of the connection V of the bundle E A M) is linked with the group G As we saw above the fibres E of the bundle E are totally of géodésie Then the projection on E a XGQ, Killing field X oï E, induced the fibre E Observe gives a Killing field (X) of to so(e ), the Lie that this projection vanishes at, so (X) belongs algebra of SO(E ) The Lie algebra spanned by thèse (X) is included in the Lie algebra of the normalizer N(Rol ) of the holonomy group Hol^ in by some B(X)B o,0 B(X)B B(X)B SO(E ) This is due to the following géométrie reason: for any curve cin x g 'c rrt = grtc g~, since G préserves the connection (and l where Hol dénotes the holonomy group of the so grolp g~ = Uolg p, connection on the bundle E) M and geg, be the flow on E associated to the horizontal component [X] n of Killing field X (ie if p G Ep, then t( p ) is the parallel transport of Çp along the curve exp(sx) -p from 0 to t) Let F be the flow of the Killing Let rf the field X on E, ie, F( p ) := exp(tx)^p Then the fact that isometries and parallel transport are géométrie objects implies that off =Ff orf Taking into account this identity, one finds that <p t := r* t off defines a one parameter group of isometries of E with the following properties : (i) 4>t(E ) = E, (ii) 4>t\E belongs to A^(Hol^), the normalizer in SO( "^) of the holonomy group is the claimed projection of the HOI4 and (iii) cj) t E = e B^X) where (X) v dénotes vertical field (X)Ç (ie Killing XtoE = [X^], where []v projection) Note that (iii) is a simple conséuence of the gênerai fact that if two flows FfFf commute then FfoFj = Ff +r rf \ \ B(X)B B(X)B The following theorem makes précise the above description and establishes, using the transitivity of G on M, the inclusion of the holonomy algebra into the Lie algebra generated by the (X) B(X)B THEOREM 41 ([OSv]) The Lie algebra C generated by {B {X) :XeÇ} contains the Lie algebra of the holonomy group Hol^ and is contained in the Lie algebra TV(Hol^) of its normalizer in SO(E ) Proof In order to illustrate better the main ideas we will only prove a simplified version of the theorem The inclusion in the normalizer was observed before Let L dénote the Lie group associated to C and let G E Let us consider S^ := GL CE Note that either Sçp nsrt Sçp = S^, for ail rjp, ee =oor a It is standard to show that Sç is a subbundle of E over M (of course not vector subbundle) Observe that the connected component of the fibre at

5 of Sçg is L-^, since the connectée! component of the isotropy subgroup G is contained in L So, the restrictions X\ s^ and [X]K are both tangent to Sç g and hence G acts the horizontal component [X\U transitively on M, {[X] n ( ) X : space Hç (note that c7^ G is also tangent to S^ Since G} coincides with the horizontal tt*(x) =X) for ail î]g^ Then 7-^ This implies that Hol* -^ C L Ç, where Hol* is the connected component of Hol^ (ie, the restricted holonomy group) In other words, any orbit of Hol* is contained in an orbit of L To get the inclusion Hol* CL one has to carry out a similar argument but replacing Eby the principal bundle over Mof orthonormal basis of E - Applications E= (VX), TM, the tangent bundle: in this case we will show that where X(p) =X-p, pem (cf [N]) Indeed, B(X)B (X) = where 7^ is the géodésie of M with initial condition G Ep If M is locally irreducible and the scalar curvature is not (identically) zéro, then the restricted holonomy group O* of M is non exceptional, ie it acts on T M as an s -représentation (see [Sim, p 229]) Then O* coincides with the connected component of its normalizer in SO(T M) So, the Lie algebra of O* is algebraically generated by (X) :Xe G} More generally, if M is not Ricci flat the same conclusion holds due to [K] and is now a {B(X)B conséuence of next proposition But Alekseevsky-Kimel'feld [AK] proved that a homogeneous Riemannian manifold cannot be Ricci flat, unless it is flat (a conceptual proof is due to Heintze and appeared in [88, p 553]) Then the holonomy algebra can always be calculated in this way for a locally irreducible M (the so-called Kostant's method) The following resuit is essentially due to Lichnerowicz and it is a conséuence of Berger's list [Bl] Since it is difficult to find in the literature we include a simple proof Let M be a Riemannian manifold which is irreducible and let g be the Lie algebra of the local holonomy group O^oc at Let n be the normalizer of g in so(t M) Then n contains g properly if and only if M is Kàhler and Ricci flat near PROPOSITION 41 at GM

6 Proof Let us endow tr(ab) Assume that n <$o(t M) with the usual scalar product (A, B) = n=goî, then décompose orthogonally is Then J G ï are ideals of n and so [g, *] =0 Now choose 07^ /<! 5 with commutes J So, a symmetric endomorphism which commutes with g & c and then each eigenspace of J\ defines a parallel distribution near Since M is locally irreducible at we conclude, by de Rham's Décomposition 2 2 id We may assume, by rescaling J, that J = Theorem, that JJ = - and g If we -id -c / Extending J^ by parallelism we obtain a parallel almost complex structure on M Thus, M is Kàhler near It is well-known [KN, Proposition 45, satisfies : p 149, vol ll] that the Ricci curvature Ric M of a Kàhler manifold M If 7 is any curve in a small neighbourhood the parallel transport along 7, then of joining to p, and r7r 7 is _L g So, M is Ricci flat near (7 The above two formulas, together with the Ambrose-Singer holonomy theorem also show the converse since J E= v(m), the normal bundle of a submanifold of Rn Recall that this in case the non trivial part of the normal holonomy représentation is an s -représentation Hence, the semisimple part of the normal holonomy group coincides with the connected component of its own normalizer (in the orthogonal group) If M is an irreducible submanifold which is not a curve, (the maximal parallel then the group G gives the parallel transport in and flat subbundle of i/(m) (see [03]) So, in this case, the Lie algebra of the normal holonomy group is algebraically generated by X G G} (X) be the the affine hâve that as to can regarded (X) Moreover, we projection subspace + v (M) of the Killing field of R n (restricted to this normal space) induced by So, the normal holonomy group measures how far G is from acting polarly and how far M is from being a principal orbit (in which case this projection would be trivial from the définition of polarity) {B(X)B : B(X)B IgÇ Polar actions on the tangent bundle and symmetry We recall briefly the gênerai notions of polar and hyperpolar actions on Riemannian manifolds; for more détails we refer to [Da, PT2, PTI, HPTT] Let M be a complète Riemannian manifold and let G be a closed subgroup

7 of the full group of isometries of M A complète embedded and closed submanifold of M is called a section if does intersect any orbit of G in M and is perpendicular to orbits at intersection points If there exists a section in M then the action of G is called a polar action Observe that from a section we can obtain, by means of the group, sections which contain any given point An action is called hyperpolar if it is polar and a section is in addition flat Of course in the case of R n thèse two concepts coincide Let now M be a complète simply connected Riemannian manifold and let TM be its tangent bundle endowed with the Sasaki metric We will regard M as the (Riemannian) embedded submanifold of TM which consists of the zéro vectors We hâve the following characterization of symmetric spaces in terms of polar (or euivalently, hyperpolar) actions on TM The following resuit was obtained by J Eschenburg and the third author when writing the article [EO] THEOREM 42 Let M be a simply connected complète Riemannian manifold Then the tangent bundle TM admits a polar action having M as an orbit if and only if M is symmetric Proofi Assume Mis irreducible Let G act polarly on TM and G0 =M T M, since horizontal and If isa section for this action with G then vertical distributions are perpendicular with respect to the Sasaki metric Since meets G -orbits perpendicularly, we hâve that the horizontal distribution of TM is tangent to the G -orbits Then the parallel transport of any v G T M belongs to G-v If the codimension of Gv is greater than 1, then the holonomy group does not act transitively on the (unit) sphère of T M Hence M is symmetric by the theorem of Berger [81, Sim] If G v has codimension 1 then M must be two point homogeneous* and hence x Mk symmetric by [Wa] (for a conceptual proof see [Sz]) If M = M\ x is reducible, by projecting Killing vector fields to the factors we obtain a x G& and such that G/ acts polarly bigger group, let us say G = G\ x c on Mt Let us show the converse As we noted in Section 2, the transvection group Tr(AO acts transitively on any holonomy bundle Then the polarity follows from the fact the holonomy représentation acts polarly It follows from the above results that an irreducible homogeneous space in which holonomy agrées with isotropy must be symmetric