AN EXOTIC SPHERE WITH POSITIVE SECTIONAL CURVATURE

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1 AN EXOTIC SPHERE WITH POSITIVE SECTIONAL CURVATURE PETER PETERSEN AND FREDERICK WILHELM In memory of Detlef Gromoll Dring the 950s, a famos theorem in geometry and some perplexing examples in topology were discovered that trned ot to have nexpected connections. In geometry, the development was the Qarter Pinched Sphere Theorem. ([Berg], [Kling], and [Ra]) Theorem (Rach-Berger-Klingenberg, 95-96) If a simply connected, complete manifold has sectional crvatre between 4 and, i.e., 4 < sec ; then the manifold is homeomorphic to a sphere. The topological examples were [Miln] Theorem (Milnor, 956) There are 7-manifolds that are homeomorphic to, bt not di eomorphic to, the 7-sphere. The latter reslt raised the qestion as to whether or not the conclsion in the former is optimal. After a long history of partial soltions, this problem has been nally solved. Theorem (Brendle-Schoen, 007) Let M be a complete, Riemannian manifold and f : M! (0; ) a C fnction so that at each point x of M the sectional crvatre satis es f (x) < sec x f (x) : 4 Then M is di eomorphic to a spherical space form. Prior to this major breakthrogh, there were many partial reslts. Starting with Gromoll and Shikata ([Grom] and [Shik]) and more recently Syama ([Sy]) it was shown that if one allows for a stronger pinching hypothesis sec for some close to ; then, in the simply connected case, the manifold is di eomorphic to a sphere. In the opposite direction, Weiss showed that not all exotic spheres admit qarter pinched metrics [Weis]. Unfortnately, this body of technically di clt geometry and topology might have been abot a vacos sbject. Until now there has not been a single example of an exotic sphere with positive sectional crvatre. To some extent this problem was alleviated in 974 by Gromoll and Meyer [GromMey]. Date: May 6, Mathematics Sbject Classi cation. Primary 53C0.

2 PETER PETERSEN AND FREDERICK WILHELM Theorem (Gromoll-Meyer, 974) There is an exotic 7 sphere with nonnegative sectional crvatre and positive sectional crvatre at a point. A metric with this type of crvatre is called qasi-positively crved, and positive crvatre almost everywhere is referred to as almost positive crvatre. In 970 Abin showed the following. (See [Ab] and also [Ehrl] for a similar reslt for scalar crvatre.) Theorem (Abin, 970) Any complete metric with qasi-positive Ricci crvatre can be pertrbed to one with positive Ricci crvatre. Copled with the Gromoll-Meyer example, this raised the qestion of whether one cold obtain a positively crved exotic sphere via a pertrbation argment. Some partial jsti cation for this came with Hamilton s Ricci ow and his observation that a metric with qasi-positive crvatre operator can be pertrbed to one with positive crvatre operator (see [Ham]). This did not change the sitation for sectional crvatre. For a long time, it was not clear whether the appropriate context for this problem was the Gromoll- Meyer sphere itself or more generally an arbitrary qasi-positively crved manifold. The mystery was de to an appalling lack of examples. For a 5 year period the Gromoll-Meyer sphere and the ag type example in [Esch] were the only known examples with qasi-positive crvatre that were not known to also admit positive crvatre. This changed arond the year 000 with the body of work [PetWilh], [Tapp], [Wilh], and [Wilk] that gave s many examples of almost positive crvatre. In particlar, [Wilk] gives examples with almost positive sectional crvatre that do not admit positive sectional crvatre, the most dramatic being a metric on RP 3 RP : We also learned in [Wilh] that the Gromoll-Meyer sphere admits almost positive sectional crvatre. (See [EschKer] for a more recent and mch shorter proof.) Here we show that this space actally admits positive crvatre. Theorem The Gromoll-Meyer exotic sphere admits positive sectional crvatre. On the other hand, we know from the theorem of Brendle and Schoen that the Gromoll-Meyer sphere cannot carry pointwise, 4 pinched, positive crvatre. In addition, we know from [Weis] that it cannot carry sec and radis > and from [GrovWilh] that it also can not admit sec and for points at pairwise distance > : We still do not know whether any exotic sphere can admit sec and diameter > : The Diameter Sphere Theorem says that sch manifolds are topological spheres ([Berg3], [GrovShio]). We also do not know the di eomorphism classi cation of almost 4 pinched, positively crved manifolds. According to [AbrMey] and [Berg4] sch spaces are either di eomorphic to CROSSes or topological spheres. The class with sec and diameter > incldes the globally 4 pinched, simply connected, class, apparently as a tiny sbset. Indeed, globally 4 pinched spheres

3 AN EXOTIC SPHERE WITH POSITIVE CURVATURE 3 have niform lower injectivity radis bonds, whereas manifolds with sec and diameter > can be Gromov-Hasdor close to intervals. In contrast to the sitation for sectional crvatre, qite a bit is known abot manifolds with positive scalar crvatre, Ricci crvatre, and crvatre operator. Starting with the work of Hitchin, it became clear that not all exotic spheres can admit positive scalar crvatre. In fact, the class of simply connected manifolds that admit positive scalar crvatre is pretty well nderstood, thanks to work of Lichnerowicz, Hitchin, Schoen-Ya, Gromov-Lawson and most recently Stolz [Stol]. Since it is sally hard to nderstand metrics withot any symmetries, it is also interesting to note that Lawson-Ya have shown that any manifold admitting a nontrivial S 3 action carries a metric of positive scalar crvatre. In particlar, exotic spheres that admit nontrivial S 3 actions carry metrics of positive scalar crvatre. Poor and Wraith have also fond a lot of exotic spheres that admit positive Ricci crvatre ([Poor] and [Wrai]). By contrast Böhm-Wilking in [BohmWilk] showed that manifolds with positive crvatre operator all admit metrics with constant crvatre and hence no exotic spheres occr. This reslt is also a key ingredient in the di erentiable sphere theorem by Brendle-Schoen mentioned above. We constrct or example as a deformation of a metric with nonnegative sectional crvatre, so it is interesting to ponder the possible di erence between the classes of manifolds with positive crvatre and those with merely nonnegative crvatre. For the three tensorial crvatres, mch is known. For sectional crvatre, the grim fact remains that there are no known di erences between nonnegative and positive crvatre for simply connected manifolds. Probably the most promising conjectred obstrction for passing from nonnegative to positive crvatre is admitting a free tors action. Ths Lie grops of higher rank, starting with S 3 S 3, might be the simplest nonnegatively crved spaces that do not carry metrics with positive crvatre. The Hopf conjectre abot the Eler characteristic being positive for even dimensional positively crved manifolds is another possible obstrction to S 3 S 3 having positive sectional crvatre. The other Hopf problem abot whether or not S S admits positive sectional crvatre is probably mch more sbtle. Althogh or argment is very long, we will qickly establish that there is a good chance to have positive crvatre on the Gromoll-Meyer sphere, 7. Indeed, in the rst section, we start with the metric from [Wilh] and show that by scaling the bers of the sbmersion 7! S 4 ; we get integrally positive crvatre over the sections that have zero crvatre in [Wilh]. More precisely, the zero locs in [Wilh] consists of a (large) family of totally geodesic dimensional tori. We will show that after scaling the bers of 7! S 4 ; the integral of the crvatre over any of these tori becomes positive. The comptation is fairly abstract, and the argment is made in these abstract terms, so no knowledge of the metric of [Wilh] is reqired. The di clties of obtaining positive crvatre after the pertrbation of section cannot be over stated. After scaling the bers, the crvatre is no longer nonnegative, and althogh the integral is positive, this positivity is to a higher order than the size of the pertrbation. This higher order positivity is the best that we can hope for. De to the presence of totally geodesic tori, there can be no pertrbation of the metric that is positive to rst order on sectional crvatre [Stra]. The technical signi cance of this can be observed by assming that one has a C family of

4 4 PETER PETERSEN AND FREDERICK WILHELM metrics fg t g tr with g 0 a metric of nonnegative crvatre. If, sec g t P > 0 t0 for all planes P so that sec g0 P 0; then g t has positive crvatre for all s ciently small t > 0: Since no sch pertrbation of the metric in [Wilh] is possible, it will not be enogh for s to consider the e ect of or deformation on the set, Z; of zero planes of the metric in [Wilh]. Instead we will have to check that the crvatre becomes positive in an entire neighborhood of Z: This will involve nderstanding the change of the fll crvatre tensor. According to recent work of Tapp, any zero plane in a Riemannian sbmersion of a biinvariant metric on a compact Lie grop exponentiates to a at. Ths any attempt at pertrbing any of the known qasipositively crved examples to positive crvatre wold have to tackle this isse [Tapp]. In contrast to the metric of [EschKer], the metric in [Wilh] does not come from a left (or right) invariant metric on Sp () : So althogh the Gromoll Meyer sphere is a qotient of the Lie grop Sp () ; we do not se Lie theory for any of or crvatre comptations or even for the de nition of or metric. Or choice here is perhaps a matter of taste. The overriding idea is that althogh none of the metrics considered lift to left invariant ones on Sp () ; there is still a lot of strctre. Or goal is to exploit this strctre to simplify the exposition as mch as we can. Or sbstitte for Lie theory is the pll-back constrction of [Wilh]. In fact, the crrent paper is a contination of [PetWilh], [Wilh], and [Wilh]. The reader who wants a thorogh nderstanding of or argment will ltimately want to read these earlier papers. We have, nevertheless, endeavored to make this paper as selfcontained as possible by reviewing the basic de nitions, notations, and reslts of [PetWilh], [Wilh], and [Wilh] in sections, 3, and 4. It shold be possible to skip the earlier papers on a rst read, recognizing that althogh most of the relevant reslts have been restated, the proofs and comptations are not reviewed here. On the other hand, Riemannian sbmersions play a central role throghot the paper; so the reader will need a working knowledge of [On]. After establishing the existence of integrally positive crvatre and reviewing the reqired backgrond, we give a detailed and technical smmary of the remainder of the argment in section 5. Unfortnately, aspects of the speci c geometry of the Gromoll-Meyer sphere are scattered throghot the paper, starting with section ; so it was not possible to write section 5 in a way that was completely independent of the review sections. Instead we o er the following less detailed smmary with the hope that it will s ce for the moment. Starting from the Gromoll-Meyer metric the deformations to get positive crvatre are (): The (h h ) Cheeger deformation, described in section 3 (): The redistribtion, described in section 6. (3): The (U D) Cheeger deformation, described in section 3 (4): The scaling of the bers, described in section (5): The partial conformal change, described in section 0 (6): The (U; D) Cheeger deformation and a frther h deformation. We let g ; g ; ; g ;;3 ; ect. be the metrics obtained after doing deformations (), () and (), or (), (), and (3) respectively.

5 AN EXOTIC SPHERE WITH POSITIVE CURVATURE 5 It also makes sense to talk abot metrics like g ;3 ; i.e. the metric obtained from doing jst deformations () and (3) withot deformation (). All of the deformations occr on Sp () : So at each stage we verify invariance of the metric nder the varios grop actions that we need. For the prpose of this discssion we let g ; g ; ; g ;;3 ; ect. stand for the indicated metric on both Sp () and 7 : g ;3 is the metric of [Wilh] that has almost positive crvatre on 7. g ;;3 is also almost positively crvatre on 7, and has precisely the same zero planes as g ;3 : Some speci c positive crvatres of g ;3 are redistribted in g ;;3. The reasons for this are technical, bt as far as we can tell withot deformation () or methods will not prodce positive crvatre. It does not seem likely that either g ; or g ;;3 are nonnegatively crved on Sp () ; bt we have not veri ed this. Deformation (4), scaling the bers of Sp ()! S 4 ; is the raison d être of this paper. g ;;3;4 has some negative crvatres, bt has the redeeming featre that the integral of the crvatres of the zero planes of g ;3 is positive. In fact this integral is positive over any of the at tori of g ;3 : The role of deformation (5) is to even ot the positive integral. The crvatres of the at tori of g ;3 are pointwise positive with respect to g ;;3;4;5 : To nderstand the role of deformation (6); recall that we have to check that we have positive crvatre not only on the 0 planes of g ;3 ; bt in an entire neighborhood (of niform size) of the zero planes of g ;3 : To do this sppose that or zero planes have the form P span f; W g : We have to nderstand what happens when the plane is pertrbed by moving its foot point, and also what happens when the plane moves within the bers of the Grassmannian. To deal with the foot points, we extend and W to families of vectors F and F W on Sp () : These families can be mltivaled and F W contains some vectors that are not horizontal for the Gromoll-Meyer sbmersion. All pairs f; W g that contain zero planes of 7 ; g ;3 are contained in these families, and the families are de ned in a xed neighborhood of the 0 locs of g ;3 : All of or argments are valid for all pairs fz; V g with z F and V F W, provided z and V have the same foot point. In this manner, we can focs or attention on berwise deformations of the zero planes. To do this we consider planes of the form P span f + z; W + V g where ; are real nmbers and z and V are tangent vectors. Ultimately we show that all vales of all crvatre polynomials P (; ) crv ( + z; W + V ) are positive. Allowing ;, z and V to range throgh all possible vales describes an open dense sbset in the Grassmannian ber. The complement of this open dense set consists of planes that have either no z component or no W component. These crvatres can be compted as combinations of qartic, cbic, and qadratic terms in sitable polynomials P (; ) : In sections and 3 we show that these combinations/crvatres do not decrease mch nder or deformations (in a proportional sense); so the entire Grassmannian is positively crved.

6 6 PETER PETERSEN AND FREDERICK WILHELM The role of the Cheeger deformations in (6) is that any xed plane with a nondegenerate projection to the vertical space of 7! S 4 becomes positively crved, provided these deformations are carried ot for a s ciently long time. Althogh the zero planes P span f; W g all have degenerate projections to the vertical space of 7! S 4 ; there are of corse nearby planes whose projections are nondegenerate. Exploiting this idea we get Proposition 0.. If all crvatre polynomials whose corresponding planes have degenerate projection onto the vertical space of 7! S 4 are positive on 7 ; g ;;3;4;5 ; then 7 g ;;3;4;5;6 is positively crved, provided the Cheeger deformations in (6) are carried ot for a s ciently long time. Proof. The assmptions imply that a neighborhood N of the 0 locs of g ;3 is positively crved with respect to g ;;3;4;5 : The complement of this neighborhood is compact, so g ;;3;4;5;6 is positively crved on the whole complement, provided the Cheeger deformations in (6) are carried ot for enogh time. Since Cheeger deformations preserve positive crvatre g ;;3;4;5;6 is also positively crved on N. So g ;;3;4;5;6 is positively crved. Ths the deformations in (6) allow s the comptational convenience of assming that the vector z is in the horizontal space of 7! S 4 : In the seqel, we will not se the notation g ; g ; ; g ;;3, ect.. Rather we will se more sggestive notation for these metrics, which we will specify in Section 5. Acknowledgments: The athors are gratefl to the referee for nding a mistake in an earlier draft in Lemma 5.3, to Karsten Grove for listening to an extended otline of or proof and making a valable expository sggestion, to Kriss Tapp for helping s nd a mistake in an earlier proof, to Blkard Wilking for helping s nd a mistake in a related argment and for enlightening conversations abot this work, and to Pala Bergen for copy editing.. Integrally Positive Crvatre Here we show that it is possible to pertrb the metric from [Wilh] to one that has more positive crvatre bt also has some negative crvatres. The sense in which the crvatre has increased is speci ed in the theorem below. The idea is that if we integrate the crvatres of the planes that sed to have zero crvatre, then the answer is positive after the pertrbation. The theorem is not speci c to the Gromoll-Meyer sphere. Theorem.. Let (M; g 0 ) be a Riemannian manifold with nonnegative sectional crvatre and : (M; g 0 )! B a Riemannian sbmersion. Frther assme that G is an isometric grop action on M that is by symmetries of and that the intrinsic metrics on the principal orbits of G in B are homotheties of each other. Let T M be a totally geodesic, at tors spanned by geodesic elds X and W sch that X is horizontal for and D (W ) H w is a Killing eld for the G action on B: We sppose frther that X is invariant nder G; D (X) is orthogonal to the orbits of G; and the normal distribtion to the orbits of G on B is integrable. Let g s be the metric obtained from g 0 by scaling the lengths of the bers of by p s :

7 AN EXOTIC SPHERE WITH POSITIVE CURVATURE 7 Let c be an integral crve of d (X) from a zero of jh w j to a maximm of jh w j along c;whose interior passes throgh principle orbits. Then Z Z crv gs (X; W ) s 4 (D X (jh w j)) : c In particlar, the crvatre of spanfx; W g is integrally positive along c; provided H w is not identically 0 along c: Here and throghot the paper we set crv (X; W ) R (X; W; W; X) : The formlas for the crvatre tensor of metrics obtained by warping the bers of a Riemannian sbmersion by a fnction on the base were compted by Detlef Gromoll and his Stony Brook stdents in varios classes over the years. We were made aware of them via lectre notes by Carlos Dran [GromDr]. They will appear shortly in the textbook [GromWals]. In the case when the fnction is constant, these formlas are necessarily mch simpler and can also be fond in [Bes], where scaling the bers by a constant is referred to as the canonical variation. To ltimately get positive crvatre on the Gromoll-Meyer sphere, we have to control the crvatre tensor in an entire neighborhood in the Grassmannian, so we will need several of these formlas. In fact, since the particlar W that we have in mind is neither horizontal nor vertical for ; we need mltiple formlas jst to nd crv(x; W ) : For vertical vectors U; V V and horizontal vectors X; Y; Z H; for : M! B we have (R gs (X; V ) U) H s (R (X; V ) U) H + s s A AX U V R gs (V; X)Y s R(V; X)Y + s (R(V; X)Y ) V + s A X A Y V (.) R gs (X; Y ) Z s R (X; Y ) Z + s (R (X; Y ) Z) V + s R B (X; Y ) Z The sperscripts H and V denote the horizontal and vertical parts of the vectors, R and A are the crvatre and A-tensors for the npertrbed metric g; R gs denotes the new crvatre tensor of g s ; and R B is the crvatre tensor of the base. To eventally nderstand the crvatre in a neighborhood of the Gromoll-Meyer 0-locs, we will need formlas for R gs (W; X) X and (R gs (X; W ) W ) H where X is as above and W is an arbitrary vector in T M: Lemma.3. Let : (M; g 0 ) c! B be as above. Let X be a horizontal vector for and let W be an arbitrary vector in T M: Then R gs (W; X) X s R(W; X)X + s (R(W; X)X) V +s R B W H ; X X + s A X A X W V (R gs (X; W ) W ) H s (R (X; W ) W ) H + s s A AX W V W V + s R B X; W H W H

8 8 PETER PETERSEN AND FREDERICK WILHELM Remark.4. Notice that the rst crvatre terms vanish in both formlas on the totally geodesic tors. Proof. We split W W V + W H and get R gs (W; X) X R gs W V ; X X + R gs W H ; X X s R(W V ; X)X + s R(W V ; X)X V + s A X A X W V + s R W H ; X X + s R W H ; X X V + s R B W H ; X X s R(W; X)X + s (R(W; X)X) V + s R B W H ; X X + s A X A X W V To nd the other crvatre we se R gs (X; W ) W R gs X; W V W V + R gs X; W H W V +R gs X; W V W H + R gs X; W H W H Since A X A W HW V and A W HA X W V are vertical the above crvatre formlas imply In addition we have R gs X; W H W V H s R X; W H W V H R gs X; W V W H H s R X; W V W H H : R gs X; W V W V H s R X; W V W V H + s s A AX W V W V R gs X; W H W H H s R X; W H W H H + s R B X; W H W H : Therefore (R gs (X; W ) W ) H s (R (X; W ) W ) H + s s A AX W V W V +s R B X; W H W H as claimed. Now let X and W be as in the theorem. We set H w D W H and V W V. To prove the theorem we need to nd crv B (X; H w ) and A X V: Lemma.5. R B (H w ; X) X DX D X jh w j jh w j H w Proof. Since X is invariant nder G; [X; H w ] 0: Since X is also a geodesic eld R B (H w ; X) X r X r Hw X: Similarly, since the normal distribtion to the orbits of G on B is integrable we can extend any normal vector z to a G invariant normal eld Z, and get that all terms of the Koszl formla for hr Hw X; Zi vanish. In particlar, r Hw X is tangent to the orbits of G: If K is another Killing eld we have that X commtes with K as well as H w ; and [K; H w ] is perpendiclar to X as it is again a Killing eld. Combining this with or hypothesis that the intrinsic metrics on the principal orbits of G in B are

9 AN EXOTIC SPHERE WITH POSITIVE CURVATURE 9 homotheties of each other, we see from Koszl s formla that r Hw X is proportional to H w and can be calclated by Ths R B (H w ; X) X r X DX jh w j jh w j Lemma.6. hr Hw X; H w i hr X H w ; H w i D X jh w j jh w j D X jh w j ; so r Hw X D X jh w j H w : jh w j H w H w DX jh w j DX jh w j D X r X H w jh w j jh w j! jh w j D X D X jh w j (D X jh w j) DX jh w j jh w j H w H w jh w j DX D X jh w j H w : jh w j R B (X; H w ) H w jh w j r X (grad jh w j) : Proof. Let Z be any vector eld. Using that H w is a Killing eld we get showing that Ths hr Hw H w ; Zi hr Z H w ; H w i D Z hh w ; H w i D Z jh w j jh w j D Z jh w j hjh w j grad jh w j ; Zi r Hw H w jh w j grad jh w j : R B (X; H w ) H w r X r Hw H w r Hw r X H w DX jh w j r X (jh w j grad jh w j) r Hw H w jh w j D X jh w j (D X jh w j) grad jh w j (jh w j r X grad jh w j) r Hw H w jh w j (D X jh w j) grad jh w j (jh w j r X grad jh w j) + D X jh w j jh w j grad jh w j jh w j (jh w j r X grad jh w j)

10 0 PETER PETERSEN AND FREDERICK WILHELM (.7) It follows that crv B (X; H w ) R B (H w ; X) X; H w DX D X jh w j hh w ; H w i jh w j jh w j (D X D X jh w j) : Next we focs on ja X V j : Lemma.8. A X V D X jh w j H w : jh w j Proof. Since X and W are commting geodesic elds on a totally geodesic at tors, r X W 0: So A X V (r X V ) H (r X W r X H w ) H (r X H w ) H r B H w X D X jh w j H w jh w j Combining this A tensor formla with eqation.7 and Lemma.3 yields crv gs (X; W ) s crv (X; W ) + s crv B (X; H w ) s ja X V j + s 4 ja X V j s crv (X; W ) s (jh w j (D X D X jh w j)) s (D X jh w j) + s 4 (D X jh w j) Since crv (X; W ) 0; this frther simpli es to (.9) crv gs (X; W ) s (D X (jh w j D X jh w j)) + s 4 (D X jh w j) : If c is an integral crve of X from a zero of H w to a maximm of jh w j along c; then the rst term integrates to 0 along c; yielding Z Z crv gs (X; W ) s 4 (D X jh w j) c as desired. As we ve mentioned, to get positive crvatre on the Gromoll-Meyer sphere we will have to nderstand the fll crvatre tensor. Combining the calclations above we have Lemma.0. Let X and W be as in Theorem.. Then H w R gs (W; X) X s DX D X jh w j jh w j c s D X jh w j A X H w jh w j (R gs (X; W ) W ) H s s D X jh w j A Hw W V s jh w j r X (grad jh w j) : jh w j

11 AN EXOTIC SPHERE WITH POSITIVE CURVATURE Remark.. The two A tensors A X H w and A Hw W V involve derivatives of vectors that are not tangent or normal to the totally geodesic tori. They cannot be determined abstractly, and are in fact dependent on the particlar geometry. We give estimates for them in the case of the Gromoll-Meyer sphere in Lemma 9. below.. Review of the geometry of Sp () The next three sections are a review of [PetWilh], [Wilh], and [Wilh]. We let h : S 7! S 4 and h ~ : S 7! S 4 be the Hopf brations corresponding to the right A h and left A ~h actions of S 3 on S 7. Points on S 7 are denoted by pairs of qaternions written as colmn vectors. The qotient map for action on the right is a h : 7! (ac; c (jaj jcj )); and the qotient map for action on the left is a ~h : 7! (ac; c (jaj jcj )): The image is S 4 ( ) H R [Wilh]. Proposition.. (The Pllback Identi cation) Sp() is di eomorphic to the total space of the pllback of the Hopf bration S 7 h! S 4 via S 7 Ih! S 4, where S 4 I! S 4 is the antipodal map. In fact, the biinvariant metric on Sp() is isometric (p to rescaling) to the sbspace metric on the pllback ( I h) S 7 S 7 () S 7 () ; where S 7 () is the nit 7-sphere and S 7 () S 7 () has the prodct metric. In [GromMey] it was shown that 7 is the qotient of the S 3 -action on Sp() given by a b qaq qb A ; q; : c d qcq qd We let q ; : Sp()! 7 denote the qotient map. It was observed by Gromoll and Meyer that 7 is the S 3 bndle over S 4 of type (; ), sing the classi cation convention of [Miln]. The sbmersion p ; : 7! S 4 is indced by e h p j Sp() : Sp ()! S 4 ; where p : S 7 S 7! S 7 is projection onto the second factor. The Gromoll-Meyer metric on 7 is indced by the biinvariant metric via q ;. The metric stdied in [Wilh], g, is indced via q ; d ; by the pertrbation of the biinvariant metric that was stdied in [PetWilh]. We will review the de nition of this metric in the next section. The isometry grop of the metric discovered by Gromoll and Meyer is O() SO (3) : The O()-action is indced on 7 by the action A O() on Sp () de ned as O() Sp()! Sp() (A; U) 7! AU:

12 PETER PETERSEN AND FREDERICK WILHELM The SO(3)-action is indced on 7 by the S 3 action A h on Sp () de ned as As in [Wilh] we have S 3 Sp ()! Sp () a b a bq q; 7! c d c dq : Proposition.. Every point in 7 has a point in its orbit nder A SO() A h that can be represented in Sp() by a point of the form cos t sin t p; sin t cos t with t 0; 4 ; p; S 3 H; and Re () 0. Since only A h acts by isometries with respect to the metrics we stdy, the points in the previos proposition have to be mltiplied by SO () to get Proposition.3. Every point in 7 has a representative point (N p; N ) in its orbit nder A h that in Sp() has the form cos sin cos t sin t (N p; N ) p; sin cos sin t cos t cos cos t + sin sin t sin cos t + cos sin t p; sin cos t + cos sin t cos cos t sin sin t with t 0; 4 ; [0; ] ; p; S 3 ; and Re () 0. We have a similar representation in S 7 : Corollary.4. Every point in S 7 has a point in its orbit nder A ~h A h of the form cos sin cos t cos cos t + sin sin t N sin cos sin t sin cos t + cos sin t with t 0; 4 ; [0; ] ; S 3 ; and Re () 0. The h ber of N consists of the points Np : p S 3 : We need a basis for the tangent space of Sp () that is well adapted to the Gromoll-Meyer sphere and its symmetry grop. It trns ot that a left invariant framing is ill sited for this prpose; rather we se a basis that comes from S 7 via the embedding Sp () S 7 S 7 : To get the correct basis we point ot Proposition.5. SO () A h acts on S 7 by symmetries of ~ h: The action indced on S 4 has Z kernel and indces an e ective SO () SO (3) action that respects the join decomposition S 4 S S : The SO () factor acts in the standard way on S and as the identity on S : The SO (3) action is standard on the S factor and the identity on the S factor. (See [GlWarZil], cf also the proof of Proposition. in [Wilh].)

13 AN EXOTIC SPHERE WITH POSITIVE CURVATURE 3 Remark.6. At a representative point cos sin (N p; N ) sin cos cos t sin t cos cos t + sin sin t sin cos t + cos sin t sin t p; cos t sin cos t + cos sin t p; cos cos t sin sin t ; the parameter ; is the S coordinate in S S ; is the S coordinate, t is the distance to the singlar S in S S and p parameterizes the bers of p ; : 7! S 4 ; giving s a partial coordinate system (t; ; ; p) for 7. We denote the singlar S in S S by SR and we denote the singlar S by SIm : The points in SR are represented in Sp () by the points with t 0; and S Im corresponds to the set where t 4 : Ths SR h ~ cos sin p j Sp() p; Sp () : [0; ] ; p S 3 and sin cos SIm h ~ cos + sin sin + cos p j Sp() p p; p Sp () : sin + cos cos sin [0; ] ; ; p S 3 ; and Re () 0 : Throghot the paper, and will be prely imaginary nit qaternions that satisfy. Using sch a choice for and gets s a basis for the vertical space of h at N S 7 by setting v Np; # N p; # N p: The bers of h and h ~ have a one-dimensional intersection when t > 0 and coincide when t 0: v is tangent to this intersection. We get a basis for the horizontal space of h by selecting a sitable vector perpendiclar to N: When 0 a natral choice is ^N sin t cos t For general we jst mltiply by an element in SO () and get cos sin sin t ^N sin cos cos t cos sin t + sin cos t : sin sin t + cos cos t : With this choice we de ne the basis for the horizontal space as x ^Np; y ^N p ^N p; ^N p: These vectors are well-adapted to the Gromoll-Meyer sphere since x is normal to the S S s in S S S 4 ; y is tangent to the S s in S S S S S 4 ; and the s are tangent to the S s in S S S S S 4 :

14 4 PETER PETERSEN AND FREDERICK WILHELM We call x; y; and v, vectors, and we call ; ; # ; and #, vectors. When t 0, or formla for Np becomes cos Np p sin which has no \. So the vectors v; # ; # become indistingishable. This re ects the fact that the bers of h and ~ h coincide when t 0: Similarly or formlas for the vectors x; ; become indistingishable at t 0: This re ects the fact that the set where t 0 in S 4 is the singlar S S S S 4 ; i.e. the place where the S s are collapsed. On the other, hand at t 0, y becomes sin cos p sin cos and hence is well de ned, re ecting the fact that y is tangent to the circles of the join decomposition. Proposition.7. On S 7 the combined Hopf action A ~h A h leaves the splitting invariant and leaves the splitting p span fx; ; g span fyg span fv; # ; # g span fxg span fyg span f ; g span fvg span f# ; # g invariant when t > 0: Proof. Since A ~h acts by symmetries of h; it at least preserves the horizontal and vertical splitting of h. Bt it also leaves its own horizontal and vertical spaces invariant. The A ~h invariance of span fvg span f# ; # g when t > 0 follows from the fact that span fvg is the intersection of the two vertical spaces and span f# ; # g its orthogonal complement in the vertical space of h: The A ~h invariance of span fxg span fyg span f ; g when t > 0 follows from the fact that at the level of S 4 ; A ~h preserves or join decomposition. Finally, span fyg is A ~h invariant when t 0 since on S 4 ; the set where t 0 is the xed point set of A ~h ; and span fyg is the tangent space to this xed point set. A similar argment gives s the statement for A h : As observed in [PetWilh], T Sp () has a splitting T Sp () V V H; where V and V are the vertical spaces for the Hopf brations that describe Sp () ( I h) S 7 S 7 () S 7 () ; and H is the orthogonal complement of V V with respect to the biinvariant metric.

15 AN EXOTIC SPHERE WITH POSITIVE CURVATURE 5 The vectors (v; 0) (N p; 0) ; (# ; 0) (N p; 0) ; (# ; 0) (N p; 0) form an orthogonal basis for V : Similarly, (0; v) (0; N ) ; (0; # ) (0; N ) ; (0; # ) (0; N ) form a orthogonal basis for V : To get a basis for H at representative points we de ne cos sin sin t ^N ; sin cos cos t cos sin cos t ^N sin cos sin t and x ;0 y ;0 ; ; ^N p; ^N ; ^N p; ^N ^N p; ^N ^N p; ^N We refer the reader to [Wilh] for the comptations that show that x ;0 ; y ;0 ; ; ; and ; are tangent to Sp (). A corollary of the previos proposition is Corollary.8. The Gromoll-Meyer action A ; A h leaves span x ;0 ; ( ; ) ; ( ; ) span y ;0 span f(v; 0) ; (# ; 0) ; (# ; 0)g span f(0; v) ; (0; # ) ; (0; # )g invariant and leaves the splitting span x ;0 span y ;0 span f( ; ) ; ( ; )g span f(v; 0)g span f(# ; 0) ; (# ; 0)g span f(0; v)g span f(0; # ) ; (0; # )g invariant when t > 0: 3. Cheeger Deformations The metric stdied in [Wilh] is indced via q ; by the pertrbation of the biinvariant metric that was stdied in [PetWilh]. We start by reviewing its constrction. In [Cheeg] a general method for pertrbing the metric g on a manifold M of nonnegative sectional crvatre was proposed. Varios special cases of this method were rst stdied in [Berg], [BorDesSent], and [Wal]. If G is a compact grop of isometries of (M; g), then we let G act on G M by q (p; m) (pq ; qm):

16 6 PETER PETERSEN AND FREDERICK WILHELM If b is a biinvariant metric on G, then for each l > 0 we get a prodct metric l b + g on GM: The qotient of this action then indces a new metric, g l ; of nonnegative sectional crvatre on M. It was observed in [Cheeg] that we may expect the new metric to have fewer 0 crvatres and symmetries than the original metric, g g. The qotient map of this action is denoted by (3.) q GM : G M! M: In [PetWilh] we stdied the e ect of pertrbing the biinvariant metric on Sp() sing Cheeger s method and the S 3 S 3 S 3 S 3 action indced by the commting S 3 -actions a b A p p a p ; b ; c d c d a b a b A p d ; ; c d p c p d a b aq b A q h ; ; c d cq d a b a bq A q h ; : c d c dq If T M, then ^ T (G M) denotes the horizontal vector, with respect to q GM ; satisfying dp ^ ; where p : G M! M is the projection onto the second factor. Similarly if P T M is a tangent plane, then ^P T (G M) is the horizontal plane satisfying dp ( ^P ) P. Cheeger s observation was that ([Cheeg], cf [PetWilh], Proposition.0) Proposition 3.. (i): If the crvatre of P is positive with respect to g, then the crvatre of dq GM ( ^P ) is positive with respect g l. (ii): The crvatre of dq GM ( ^P ) is positive with respect to g l if the A-tensor of q GM is nonzero on ^P. (iii): If G S 3, then the crvatre of dq GM ( ^P ) is positive if the projection of P onto T O G is nondegenerate. (iv): If the crvatre of ^P is 0 and A q GM vanishes on ^P, then the crvatre of dq GM ( ^P ) is 0. Remark 3.3. According to [Tapp], no new positive crvatre can be created via (ii) if M is a Lie grop with a biinvariant metric. Following [PetWilh] and [Wilh], or comptations will be based on deformations of the biinvariant metric on Sp () : The biinvariant metric indced by Sp () S 7 () S 7 () is called b: The biinvariant metric we se is scaled so that the vectors x ;0 etc. have nit length. Ths we se b; also called b p in [PetWilh] and [Wilh], which is indced by Sp () S 7 ( p )S 7 ( p ), where S 7 ( p ) is the sphere of radis p. The e ect of the Cheeger pertrbation A h A h is to scale V and V and to preserve the splitting V V H and bj H. The amont of the scaling is < and converges to as the scale on the S 3 -factor in S 3 S 3 Sp() converges to and converges to 0 when the S 3 S 3 factor is scaled to a point. We will call

17 AN EXOTIC SPHERE WITH POSITIVE CURVATURE 7 the reslting scales on V and V, and. To simplify the exposition, we set and call the reslting metric g. It follows that g is the restriction to Sp() of the prodct metric S 7 S 7 where S 7 denotes the Berger metric obtained from S 7 ( p ) by scaling the bers of h by p. The following reslts can be fond in [PetWilh]. Proposition 3.4. Let g denote a metric obtained from the biinvariant metric on Sp() via Cheeger s method sing the S 3 S 3 S 3 S 3 -action, A A d A h A h. Then A A d A h A h is by isometries with respect to g. In particlar, A ; is by isometries with respect to g, and hence g indces a metric of nonnegative crvatre on the Gromoll-Meyer sphere, 7. Proposition 3.5. Let A H : H M! M be an action that is by isometries with respect to both g and g l. Let H AH denote the distribtion of vectors that are perpendiclar to the orbits of A H. P is in H AH with respect to g if and only if dq GM ( ^P ) is in H AH with respect to g l. In fact, g (; w) g l (; dq GM ( ^w)) for all ; w T M: Notational Convention: Let q GM : G (M; g )! (M; g l ) be a Cheeger sbmersion. Sppose that : M! B is a Riemannian sbmersion with respect to both g and g l. It follows that z is horizontal for : M! B with respect to g if and only if dq GM (^z) is horizontal for with respect to g l : To keep the notation simpler, we can think of this correspondence as a parameterization of the horizontal space, H ;gl ; of with respect to g l by the horizontal space, H ; g of with respect to g. We can then denote vectors and planes in H ; gl by the corresponding vectors and planes in H ; g. We will do this for the A A d Cheeger deformation, bt not for the A h A h Cheeger deformation. Note that if t [0; 4 ) then the orthogonal projection p V h ;V ~h : V h! V ~h with respect to the nit metric on S 7 is an isomorphism. In fact the matrix of p Vh ;V ~h with respect to the ordered bases v; # ; # and v; # ~ ; # ~ is cos(t) 0 A : 0 0 cos(t) The horizontal space of q ; with respect to g is given by Proposition 3.6. [Wilh] For t [0; 4 ) the horizontal space of q ; with respect to g at the representative point (N p; N ) is spanned by x ;0 ; y ;0 ; ; + tan(t) # ; ; + tan(t) # ; v ; v # ; # p p V h ;V ~h (ppn )! V h ;V ~h (p pn ) ;!) # ; # p V h ;V ~h (p pn )! ;

18 8 PETER PETERSEN AND FREDERICK WILHELM Notation: We will call the seven vectors in Proposition 3.6, x ;0 ; y ;0 ; ;0 ; ;0, v;, # ; ; and # ; respectively. We will call the span of the rst for H ; and the span of the last three V ;. Althogh or partial framing of T Sp () is well adapted to stdy the Gromoll- Meyer sphere it is neither left nor right invariant. For example, the left invariant eld that eqals x ;0 at is (L Q ) 0 Q 0 cos t sin t x ;0 : sin t ; cos t cos t sin t sin t cos t sin t cos t cos t sin t 0 Since varies, x ;0 is not left invariant. Note also that one shold think of f ; g as de ning a global distribtion rather than as global vector elds. The fact that S is not parallelizable corresponds to the fact that is not canonically determined by : Conseqently, any statement that we make abot a single nit span f ; g is valid for any span f ; g : Similarly any statement abot a single nit span f ; g is valid for any span f ; g ; and any statement abot a single # span f# ; # g is valid for any # span f# ; # g : 4. Zero Crvatres of 7 In this nal review section we discss the zero crvatres of 7 ; g : The description that we give is more geometric than that of [Wilh]. We give a brief idea of why the zeros occr, bt for a fll jsti cation we combine [Wilh] and [Tapp] with new comptations of the zero crvatres when t 0: These were not given in [Wilh] becase they were not needed. We give them here to flly jstify or description and also becase they give a avor of some of the important isses of [Wilh]. From Proposition 3., we see that a 0 plane for g mst have a degenerate projection onto the tangent spaces to the orbits of all for S 3 actions, A ; A d ; A h ; and A h : There is a vector eld tangent to Sp () that is normal to the orbits of all for actions. We call this eld :When restricted to an S 7 factor, is the eld that is normal to the S 3 S 3 s in the join decomposition S 7 S 3 S 3 ; that corresponds to writing a point in S 7 as a with a; c H: c is of corse in span x ;0 ; y ;0 ; bt the combination is qite complicated. (sin t cos ) x;0 (sin ) y ;0 p : sin t cos + sin So does not have mch to do with or join decomposition S 4 S R S Im : Rather it is the geodesic eld that is the gradient of the distance from the point where 0

19 AN EXOTIC SPHERE WITH POSITIVE CURVATURE 9 (t; ) (0; 0) : In or coordinate system for S 4 ; the antipodal point to (t; ) (0; 0) is (t; ) 0; : So is the eld that is tangent to the meridians between these two points. Ths is mltivaled at the two poles. This corresponds to the fact that or formla for is 0 0 at these poles. Unfortnately is everywhere normal to the Gromoll-Meyer action. Fortnately the vectors ZV fu T Sp () jcrv b (; U) 0g are typically not horizontal for the Gromoll-Meyer sbmersion p ;. However, from [Tapp] we know that every time a vector U is horizontal for p ; ; we get a zero plane in 7 ; even with respect to g : The projection to S 4 of the points in 7 that have zero crvatre planes containing are Theorem 4.. The points in S 4 over which there is a horizontal vector for q ; : Sp ()! 7 that is in ZV are the meridians emmanating from (t; ) (0; 0) that make an angle that is 6 with the meridians that go from (t; ) (0; 0) throgh SIm : The set is therefore 4 dimensional with a for dimensional complement. In [Wilh] it is described as the sblevel set L (t; ) ; where L : S 4! R is ( cos(t) sin() p if (t; ) 6 (0; 0) or 0; L(t; ) sin +sin t cos : 0 if (t; ) (0; 0) or 0; Combining this with the main theorem of [Tapp] and Proposition 4.7 below gives s Theorem 4.. Of corse there can also be zero planes that do not contain : Since (generically) spans the orthogonal complement of the orbit of the S 3 S 3 S 3 S 3 action, sch planes necessarily have a nondegenerate projection onto the tangent space to the entire orbit of S 3 S 3 S 3 S 3 ; bt a degenerate projection onto the orbit of each individal S 3 action. In addition, the plane mst have zero crvatre for the biinvariant metric and be horizontal for the Gromoll-Meyer sbmersion, it is not srprising that sch planes are fairly rare. Theorem 4.. The set of points Z in S 4 over which there is a 0 plane in 7 is the nion of the points described in Theorem 4. with the points where cos 0: To get a qick idea of how these other zeros occr, we point ot that the horizontal vectors for q ; : Sp ()! 7 that are also perpendiclar to the orbits of A h A h are span x ;0 ; y ;0 when t > 0 and n o span x ;0 ; y ;0 ; ;0 ; ;0 when t 0: Since span x ;0 ; y ;0, the isse boils down to its complementary vector in span x ;0 ; y ;0 : Fortnately does have a projection onto the tangent space to the orbits of A A d : Combining this with the other reqirements for zero planes it is arged in [Wilh], that the points in S 4 over which there are 0 planes are those described in the previos theorem. The actal zero planes have the form

20 0 PETER PETERSEN AND FREDERICK WILHELM Theorem 4.3. If P is a plane with 0 crvatre at a point where (t; sin ) 6 (0; 0) and cos 6 0, then P has the form where If has the form then W has the form P span f; W g W V V : x ;0 cos ' + y ;0 sin '; v cos ; v # + sin ; # cos + # sin where '; # ; # span f# ; # g ; correspond to spherical combinations ; of f ; g that satisfy ; and (cos ; sin ) is the point in the rst qadrant of R that is on the nit circle and on the ellipse parameterized by cos (4.4) 7! ; sin : L (t; ) When cos 0; there are zero planes of the form described above. In addition there are zero planes of the form where W P span x ;0 ; W v + # p 3 ; v # p! 3 : Remark 4.5. There is a frther conjgacy condition for a vector of the form of W to actally be horizontal for q ; : 7! S 4 : Becase of this, in a given ber of 7! S 4 over a point in Z S 4 most points do not in fact have zero crvatres, and at most points where there is a zero crvatre, there is jst one zero crvatre. None of these isses will be important for s, so we will not review them. Remark 4.6. The nit circle and the ellipse in qestion do not intersect when L (t; ) > : When this happens the corresponding W s are not horizontal for the Gromoll-Meyer sbmersion. 4.. Zero Crvatres at t 0. When t 0; all points have positive crvatre except for certain points with cos sin 0: The lack of 0 planes in 7 is cased by the zero planes of Sp () not intersecting the horizontal distribtion of the Gromoll Meyer sbmersion. The reason for this is the fact that the nit circle and the ellipse in (4.4) do not intersect when L (t; ) > : So the corresponding W s are not horizontal for the Gromoll-Meyer sbmersion. For example, if t 0 and sin 6 0; then y ;0 and L (0; ) : For span y ;0 ; W to have 0 crvatre, with respect to g, W mst have the form for some prely imaginary S 3 H: cos (v; v) + sin (#; #) (N p; N )!

21 AN EXOTIC SPHERE WITH POSITIVE CURVATURE When t 0; we have V h V ~h so none of the horizontal vectors N p; N + p V h ;V ~h (ppn ) can have the reqired form (N p; N ) : When (t; ) (0; 0) or 0;, the de nition of is ambigos: The de nition of x ;0 is also ambigos since the coordinate is nonexistent at t 0: In fact, the three vectors x ;0 ; ;0 ; and ;0 project nder p ; q ; to a basis for the normal space of SR S4 : Declaring that a particlar prely, imaginary nit qaternion is amonts to declaring that a particlar nit normal vector to SR is x;0. This choice is somewhat irrelevant since, on the level of S 4 ; the isometric action A h xes SR and acts transitively on the normal space. Ths, to nd 0 crvatres when (sin t; sin ) (0; 0) ; we only need to consider planes of the form P span fz; W g where z span x ;;0 ; y ;0 and W V ; : There will of corse be other 0 planes, bt they are the images of these nder A h : Since L (0; ) ; when 6 0; ; there are no 0 crvatres when t 0; provided is not 0; 4 ; ; or 3 4 : The details can be fond in [Wilh], bt the basic reason is contained in the remark above, when 6 0; ; then y;0 ; and the W s that together with y form 0 planes are not horizontal at t 0: The strctre of the 0 planes when (t; ) (0; 0) or 0; was claimed in [Wilh, p. 556] to be Proposition 4.7. Let ' x ;0 cos ' + y ;0 sin ' for some ' ; : If (t; ) (0; 0) or 0; and jsin 'j, then there are vales of p for which ' is in 0 planes of the form where and P span ' ; W W cos (v; v) + sin (# ; # cos + # sin ) ': Any other 0 plane is the image of one of these nder A h : The details of this were not given in [Wilh], since it was not crcial to the goal of that paper. Since we will need to se it, we will prove it here. The vale of cos is determined by '; the relationship can be inferred from or proof. Proof. As explained in [Wilh] it is enogh to consider planes of the form P span fz; W g where z is horizontal for p ; : 7! S 4 and W V V is horizontal for q ; : Sp ()! 7 : Since t 0; we can se the isometries A h to frther redce or consideration to planes with z span x ; ; y ; : In other words we may replace z with ' x ;0 cos ' + y ;0 sin ':

22 PETER PETERSEN AND FREDERICK WILHELM Using Proposition 4.6 in [Wilh], this then forces W to have the form listed in the statement. It only remains to check what W s are horizontal for q ; when t 0: We explained above that when ' y ;0 ; the reqired W is not horizontal. When ' x ;0 ; the reqired W is W v + p 3 #; v p! 3 # : We see that W can be realized in the form N p; N + p V h ;V ~h (ppn ) by choosing and p so that p! 3 + pp : Now we consider the general problem of realizing in the form The rst coordinate, of W forces s to set W cos (v; v) + sin (# ; # cos + # sin ) N p; N + p V h ;V ~h (ppn ) : v cos + sin # ; cos + sin : The qestion then becomes whether there is a choice of p that will achieve the desired second coordinate. The second coordinate of W can be written (4.8) v cos + (# cos + # sin ) sin N cos + N sin : for cos + sin : On the other hand if we set then pp cos + 0 sin N + p V h ;V ~h (ppn ) N ( cos + sin ) + (pp) N N cos N sin + ( cos + 0 sin ) N N ( cos + cos ) + N ( sin + 0 sin ) since N is real and therefore commtes with all qaternions. Eqating this with 4.8 gives s the eqations or cos cos + cos ; sin sin + 0 sin cos cos ; 0 sin ( + ) sin :

23 AN EXOTIC SPHERE WITH POSITIVE CURVATURE 3 We can always choose p so that 0 points in the direction of + : The isse is that since + has a variable length, sometimes there are soltions and sometimes there are not. In fact, if we set then or eqations become L j + j ; cos cos ; sin sin L : So the qestion becomes whether or not the nit circle (cos ; sin ) intersects the ellipse whose parametrization is cos 7! ; sin : L Ths when jlj there are soltions and when L > there are no soltions. So it remains to analyze how L depends on ': Since we have Ths cos + sin ; '; cos ( ') + sin ( ') cos ' + sin ': L j + j So or condition, L for 0 crvatre is or cos ' + : cos ' cos ' Keeping in mind that ' or cos ' ; ; we get 3 ' 3 sin ' :

24 4 PETER PETERSEN AND FREDERICK WILHELM 5. Frther Smmary Having scaled the bers of 7! S 4 ; we can get pointwise positive crvatre along any single (formerly) at tors via a conformal change. The idea is that the Hessian of the conformal factor needs to cancel the s term in Eqation.9, (5.) crv gs (X; W ) s (D X (jh w j D X jh w j)) + s 4 (D X jh w j) : Unfortnately, there is no conformal change that will prodce pointwise positive crvatre along all of the tori simltaneosly. The problem is that only the component of W that is horizontal for 7! S 4 appears in or crvatre formla. Along any one tors, the vector H w is a Killing eld for the SO (3) action on S 4 ; bt the precise Killing eld, and more importantly the ratio jhwj jw j varies from tors to tors, so the size of the reqired Hessian varies as well. This di clty is overcome by sing only a partial conformal change. The restriction of the metric to the distribtion span f(np; N)g is not modi ed. The metric only changes on the orthogonal complement of span f(np; N)g : The details are carried ot in Section 0. A frther di clty is created by the fact that the pieces of the zero locs with L (t; ) and cos 0 intersect at certain points over points in S 4 where t 6 and cos 0: A description of this intersection is given in [Wilh], Theorem E(iv,v). The di clty this creates is that the natral choices of conformal factors do not agree on this intersection. To circmvent this di clty, in Sections 7, 8, and the appendix we analyze the e ect on Eqation 5. of rnning the h Cheeger pertrbation for a long time. If is the parameter of this pertrbation, then it trns ot that making small has the e ect of concentrating all of the terms on the right hand side of eqation 5., s (D X (jh w j D X jh w j)) + s 4 (D X jh w j) ; arond t 0: We will make small enogh so that we can choose a (partial) conformal factor that is constant near the intersection of the two pieces of the zero locs, and hence not have to worry abot the con ict that the intersection creates. The intersection of the two pieces of the zero locs, also creates a notational con ict. To simplify the exposition we will henceforth write explicitly only abot the planes at points where L (t; ) : With the obvios modi cations in notation and a few simpli cations, the argment simltaneosly will give s positive crvatre near the planes where cos 0: Unfortnately, to really move the spport of the partial conformal change away from the intersection we have to make depend on s: In the end we will pick O s 67 : This means that or ltimate metric is not obtained as an in nitesimal pertrbation of any (known) metric with nonnegative crvatre. This fact will frther complicate or exposition. Before we can explain why, some frther clari cation is needed. Imagine that we have a deformation in which all of the former zero crvatre planes, span f; W g ; are positively crved. Next comes the danting challenge of establishing that an entire neighborhood (of niform size) of these planes in the Grassmannian is positively crved. We have to consider what happens when we move the base point of or plane and also when we move the plane with ot moving the base point.