gromoll meyer an exoctic sphere with non negative sectional curvature

8/2/2019 Gromoll Meyer an Exoctic Sphere With Non Negative Sectional Curvature

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Annals of Mathematics

An Exotic Sphere With Nonnegative Sectional CurvatureAuthor(s): Detlef Gromoll and Wolfgang MeyerReviewed work(s):Source: The Annals of Mathematics, Second Series, Vol. 100, No. 2 (Sep., 1974), pp. 401-406Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1971078 .

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An exotic pherewithnonnegativesectional urvature

By DETLEF GROMOLL AND WOLFGANG MEYERIn this notewe construct n exotic 7-sphere ' with a metricof non-negative sectional curvatureK. It is obtained as the quotientof a certainisometric ction of Sp(1) on Sp(2), and hence as a riemannian ubmersionfrom p(2). By a formula fO'Neill, ' automatically nheritsnonnegative

sectional curvature. It turnsout that K is even (strictly) positiveon anopen dense set of points. It is not knownyetwhether or not this metriccan be deformed nto one withpositivecurvatureeverywhere. However,there is a conjecturethat on any manifold, metricwithnonnegative ec-tionalcurvature which s positiveat somepoint, an be deformed nto onewithpositivecurvatureeverywhere.ByAubin'sTheorem, similarresultholds forRicci curvature [1]. We shall see that 7 has naturallymanysymmetries. In particular, 0(2) x SO(3) acts as an isometry group on7

1. The construction f 7Let Sp(n) denotethe groupof symplecticn x n quaternionmatrices;i.e., Q E Sp(n) if and only if QQ* = Q*Q = Id, whereQ* is the transposedconjugatematrixofQ. Stm ill denote the standardrn-sphere.The field fquaternionswill be identifiedwithR'.We consider n action of S3 x S3 Sp(1) x Sp(1) on Sp(2) given by

(q, x q2Q) q,02 ?Q [q2 ?)whereq2denotesthe conjugate ofq2. This action is clearly free, and thequotientmanifoldSp(2)/S3x S3 is diffeomorphico S4. A diffeomorphismSp(2)/S3 x S3 S4 is given by

orbts~s3a b) 2~~ff2 - 1fd12)rbitS3XS3( d) (2bdyilb -lid yas one can check easily. In particular,the diagonal A in S3 x S3 actsfreelyon Sp(2). The quotientmanifold 7 = Sp(2)/A s an S3-bundleover


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402 D. GROMOLL AND W. MEYERS3 X S3/A S3

Sp(2)/A =7ISp(2)/S3x S3 S4.S3-bundlesverS4 with tructure roupSO(4) are classified yr3(SO(4))Z (0Z. For (m,n) e Z0D one can construct hecorrespondingundleoftype(m,n) byglueingtwo trivialbundlesR4 x S3 togetheron (R4 0) x S3 byidentifyingu, q) inthe firstopywith u I u I-', (u/! IU I )mq(u/Iu II)"') inthe

second copy. Accordingto Milnor [2], whenever m + n = 1, the totalspace is homeomorphico S7, and the differentiabletructure s exotic if(m-n)2 1 mod7. We will identify with the total space of the bundlecorrespondingo (2, -1) andhencewithan exotic7-sphere. It actuallywasMilnor'sdescription f this sphere which suggested consideration f theabove action.THEOREM 1. 7 is the exotic7-sphere f type 2, -1).Proof. Consider hemaps h1,h2:R4 x S3 17

h (u, q) = orbit, p(u)( q )-uq1h2(v,r) = orbitAp(v) ,1)- r v

wherep(u) = (1 + IunI)-"2. LettingQ= (a ) Sp(2),wehavehj(R4x S3) = {orbit Q id #0}h2(R4 S3) = {orbit,Q Ib #0} .

Hence,hj(R4x S3) Uh2(R4 S3) = 17. Furthermore,hemapshl,h2 re dif-ferentiablembeddings; he inverses are givenbyhT'(orbit Q) = IId!K2(bd,dad Ila 11-1)2h-1(orbit,) = IIl 1l-2(WY-6bob cI -1)

Finally, h-lh((u,q) = (u 1f11-2, U/f1u 1ff)2q(u/fu f)`), which completes theargument. 2. The action of 0(2) x SO(3) on 7

On Sp(2) we considerthe standardbi-invariantmetricgiven by theKillingform. The actionofS3 x S3 described boveis isometric. Therefore,


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AN EXOTICSPHERE 4037 inherits natural metricfromSp(2) such that the projectionSp(2) I'becomes riemannian ubmersion. Now observe that the action of A com-mutes with the actionof the group 0(2) x S3 on Sp(2) givenby (A x q)Q =AQQ1?),forA E 0(2), q e Sp(1)= S3. Thus, 0(2) x S3 acts on ' byisometriesvia

(A x q)orbit,Q = orbitA Q( 0This action on 7/ has a kernelZ2 _ d x Z2,since

orbit Q( I ?) orbit'(-1 0)Q(1 0)(-1 ) orbit QOne can see that 0(2) x SO(3) = 0(2) x S3/Idx Z2 acts effectively n 7.

It is knownthat 4 = dim0(2) x SO(3) is themaximaldimension f com-pact groups that can act effectively n any exotic 7-sphere.* 0(2) x SO(3)has been realized as an isometry roup na different escription f ' givenbyBrieskorn,wherethe natural metric,however,has sectional curvaturesof either sign.3. The curvature of 'To fix notations we briefly eview some facts about riemannian ub-mersions. ConsiderriemannianmanifoldsM, M with dim M> dimM, anda submersion : M-) M; i.e., w s surjective and of maximal rank. For eachp e M, we have a submanifold -'(p) ofM, the fiber f the submersion verp. The tangent space Mq of M at q splits into an orthogonalsum Mq=

Aq'0 Al, whereAl is the tangent space of the fiber '-1(w(q)) nd A' is theorthogonal omplement.w s calleda riemannian ubmersionfw*:AI Mi(q)is isometricor ll qe M. AT andA' are calledthe vertical nd horizontaldistributions f the riemannian ubmersion. For a vectorfieldZ on M, letZT denote ts verticalcomponent. Any vector field X on M has a uniquehorizontal iftZ onM; .e., T =0 and r*X =Xor. The sectional curvaturesK of M and k of M are relatedby O'Neill's formula cf. [3]): If X, Y areorthonormal ector fields n an open subset of M, then(1) K(X, Y)o c = K(X, Y) ? 3f[X, Y]T I2I4Weshalluse thisformula o compute hecurvature f7, butfirst e needanother elementary act.

* Communicatedo us by W. C. Hsiang, unpublished.


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404 D. GROMOLL AND W. MEYERLet G be a Lie group withLie algebra . L and R denote eftand righttranslations,Ad = RI'L* theadjoint representation.For u E D consider heleft nvariantvector field ou andthe right nvariantvectorfieldRau with

L*u Ig Lg*u,Rau Ig Rg*u. For a bi-invariantmetric , > on G and a, bE I,define function : G R byf(g) = . Then(2) (Lg*u)f - .Now we can establisha formula orthecurvature of 7 which nvolves onlydata of the Lie algebras inquestionand the adjointrepresentation.

THEOREM. Let QE Sp(2). For a quaternion , set a+ = (0 ?) anda =( a)(a) The tangent paceof the iberwc-'(wu(Q))f the ubmersionwc:p(2)-7is givenbyAT = {RQ*(a+ + a-) - LQ*a+ Re a = 0} .

(b) Let u, v be orthonormalvectors n theLie algebra of Sp(2) suchthat u = LQ*u E A- and v = LQ*VE AQ-. Then(3) K(w7?i, Y) = ~L- in, v]ff2V]?max112+l


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AN EXOTIC SPHERE 405Computation fw42(M,]Q):We have(4) 2d)a(2x, Y) = O)a0?() YQa(X)-()a(L, Th)=W"a(L9, II),sinceX, Y are horizontal,nd

dcojgXs ) IQ= dca&(?t ).Onthe otherhand,(5) 2d)a(iU, V = 2d0)a(LQ*u,LQ*v)= (LQ*u)oa(L*v) - (LQ*v)oca(L*u) (oa(LQ*[Th,])Usingthe definitionfco,and (2) we get

(LQ*u)oa(L*v) = (LQ*u) = ,and

(LQ*v)oa(L*u) = Finally,

(Oa(LQ*[u,v]) = Combining hiswith (4) and (5) yields

w(0., i])Q=I-


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406 D. GROMOLL AND W. MEYERSTATE UNIVERSITY OF NEW YORK AT STONY BROOK

REFERENCES[1 ] T. AUBIN, Metriquesriemanniennes t courbure,J. Diff.Geom.4 (1970),383-424.[2] J. W. MILNOR, On manifolds omeomorphico the 7-sphere,Ann. of Math. 64 (1956),399-405.[3] B. O'NEILL, The fundamental quations of a submersion,Mich. Math. J. 13 (1966),459-469.

(ReceivedFebruary6, 1973)

gromoll meyer an exoctic sphere with non negative sectional curvature

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