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arXiv:0805.0812v3 [math.DG] 23 Oct 2008

AN EXOTIC SPHERE WITH POSITIVE SECTIONAL

CURVATURE

PETER PETERSEN AND FREDERICK WILHELM

In memory of Detlef Gromoll

During the 1950s, a famous theorem in geometry and some perplexing examples

in topology were discovered that turned out to have unexpected connections. In

geometry, the development was the Quarter Pinched Sphere Theorem. ([Berg1],

[Kling], and [Rau])

Theorem (Rauch-Berger-Klingenberg, 1952-1961) If a simply connected, complete

manifold has sectional curvature between 1/4 and 1, i.e.,

1/4 < sec ≤ 1,

then the manifold is homeomorphic to a sphere.

The topological examples were [Miln]

Theorem (Milnor, 1956) There are 7-manifolds that are homeomorphic to, but

not diffeomorphic to, the 7-sphere.

The latter result raised the question as to whether or not the conclusion in the

former is optimal. After a long history of partial solutions, this problem has been

finally solved.

Theorem (Brendle-Schoen, 2007) Let M be a complete, Riemannian manifold

and f : M −→ (0,∞) a C∞–function so that at each point x of M the sectional

curvature satisfies

f (x)

4

Then M is diffeomorphic to a spherical space form.

< secx≤ f (x).

Prior to this major breakthrough, there were many partial results. Starting with

Gromoll and Shikata ([Grom] and [Shik]) and more recently Suyama ([Suy]) it was

shown that if one allows for a stronger pinching hypothesis δ ≤ sec ≤ 1 for some

δ close to 1, then, in the simply connected case, the manifold is diffeomorphic to a

sphere. In the opposite direction, Weiss showed that not all exotic spheres admit

quarter pinched metrics [Weis].

Unfortunately, this body of technically difficult geometry and topology might

have been about a vacuous subject. Until now there has not been a single example

of an exotic sphere with positive sectional curvature.

To some extent this problem was alleviated in 1974 by Gromoll and Meyer

[GromMey].

Date: May 6, 2008.

2000 Mathematics Subject Classification. Primary 53C20.

1

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2 PETER PETERSEN AND FREDERICK WILHELM

Theorem (Gromoll-Meyer, 1974) There is an exotic 7–sphere with nonnegative

sectional curvature and positive sectional curvature at a point.

A metric with this type of curvature is called quasi-positively curved, and positive

curvature almost everywhere is referred to as almost positive curvature. In 1970

Aubin showed the following. (See [Aub] and also [Ehrl] for a similar result for

scalar curvature.)

Theorem (Aubin, 1970) Any complete metric with quasi-positive Ricci curvature

can be perturbed to one with positive Ricci curvature.

Coupled with the Gromoll-Meyer example, this raised the question of whether

one could obtain a positively curved exotic sphere via a perturbation argument.

Some partial justification for this came with Hamilton’s Ricci flow and his observa-

tion that a metric with quasi-positive curvature operator can be perturbed to one

with positive curvature operator (see [Ham]).

This did not change the situation for sectional curvature. 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 quasi-positively curved manifold.

The mystery was due to an appalling lack of examples. For a 25–year period the

Gromoll-Meyer sphere and the flag type example in [Esch1] were the only known

examples with quasi-positive curvature that were not known to also admit positive

curvature.

This changed around the year 2000 with the body of work [PetWilh], [Tapp1],

[Wilh2], and [Wilk] that gave us many examples of almost positive curvature. In

particular, [Wilk] gives examples with almost positive sectional curvature that do

not admit positive sectional curvature, the most dramatic being a metric on RP3×

RP2. We also learned in [Wilh2] that the Gromoll-Meyer sphere admits almost

positive sectional curvature. (See [EschKer] for a more recent and much shorter

proof.) Here we show that this space actually admits positive curvature.

Theorem The Gromoll-Meyer exotic sphere admits positive sectional curvature.

On the other hand, we know from the theorem of Brendle and Schoen that the

Gromoll-Meyer sphere cannot carry pointwise,

addition, we know from [Weis] that it cannot carry

sec ≥ 1 and radius >π

and from [GrovWilh] that it also can not admit

sec ≥ 1 and four points at pairwise distance >π

We still do not know whether any exotic sphere can admit

sec ≥ 1 and diameter >π

The Diameter Sphere Theorem says that such manifolds are topological spheres

([Berg3], [GrovShio]). We also do not know the diffeomorphism classification of “al-

most1

4–pinched”, positively curved manifolds. According to [AbrMey] and [Berg4]

such spaces are either diffeomorphic to CROSSes or topological spheres.

The class with sec ≥ 1 and diameter >π

connected, class, apparently as a tiny subset. Indeed, globally1

1

4–pinched, positive curvature. In

2

2.

2.

2includes the globally1

4–pinched, simply

4–pinched spheres

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AN EXOTIC SPHERE WITH POSITIVE CURVATURE3

have uniform lower injectivity radius bounds, whereas manifolds with sec ≥ 1 and

diameter >π

2can be Gromov-Hausdorff close to intervals.

In contrast to the situation for sectional curvature, quite a bit is known about

manifolds with positive scalar curvature, Ricci curvature, and curvature operator.

Starting with the work of Hitchin, it became clear that not all exotic spheres can

admit positive scalar curvature. In fact, the class of simply connected manifolds

that admit positive scalar curvature is pretty well understood, thanks to work of

Lichnerowicz, Hitchin, Schoen-Yau, Gromov-Lawson and most recently Stolz [Stol].

Since it is usually hard to understand metrics without any symmetries, it is also

interesting to note that Lawson-Yau have shown that any manifold admitting a non-

trivial S3action carries a metric of positive scalar curvature. In particular, exotic

spheres that admit nontrivial S3actions carry metrics of positive scalar curvature.

Poor and Wraith have also found a lot of exotic spheres that admit positive Ricci

curvature ([Poor] and [Wrai]). By contrast B¨ ohm-Wilking in [BohmWilk] showed

that manifolds with positive curvature operator all admit metrics with constant

curvature and hence no exotic spheres occur. This result is also a key ingredient in

the differentiable sphere theorem by Brendle-Schoen mentioned above.

We construct our example as a deformation of a metric with nonnegative sec-

tional curvature, so it is interesting to ponder the possible difference between the

classes of manifolds with positive curvature and those with merely nonnegative cur-

vature. For the three tensorial curvatures, much is known. For sectional curvature,

the grim fact remains that there are no known differences between nonnegative and

positive curvature for simply connected manifolds. Probably the most promising

conjectured obstruction for passing from nonnegative to positive curvature is admit-

ting a free torus action. Thus Lie groups of higher rank, starting with S3×S3, might

be the simplest nonnegatively curved spaces that do not carry metrics with posi-

tive curvature. The Hopf conjecture about the Euler characteristic being positive

for even dimensional positively curved manifolds is another possible obstruction to

S3×S3having positive sectional curvature. The other Hopf problem about whether

or not S2× S2admits positive sectional curvature is probably much more subtle.

Although our argument is very long, we will quickly establish that there is a

good chance to have positive curvature on the Gromoll-Meyer sphere, Σ7. Indeed,

in the first section, we start with the metric from [Wilh2] and show that by scaling

the fibers of the submersion Σ7−→ S4, we get integrally positive curvature over

the sections that have zero curvature in [Wilh2]. More precisely, the zero locus in

[Wilh2] consists of a (large) family of totally geodesic 2–dimensional tori. We will

show that after scaling the fibers of Σ7−→ S4, the integral of the curvature over

any of these tori becomes positive. The computation is fairly abstract, and the

argument is made in these abstract terms, so no knowledge of the metric of [Wilh2]

is required.

The difficulties of obtaining positive curvature after the perturbation of section

1 cannot be over stated. After scaling the fibers, the curvature is no longer nonneg-

ative, and although the integral is positive, this positivity is to a higher order than

the size of the perturbation. This higher order positivity is the best that we can

hope for. Due to the presence of totally geodesic tori, there can be no perturbation

of the metric that is positive to first order on sectional curvature [Stra]. The tech-

nical significance of this can be observed by assuming that one has a C∞family of

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4 PETER PETERSEN AND FREDERICK WILHELM

metrics {gt}t∈Rwith g0a metric of nonnegative curvature. If, in addition,

∂

∂tsecgtP

????

t=0

> 0

for all planes P so that secg0P = 0, then gthas positive curvature for all sufficiently

small t > 0. Since no such perturbation of the metric in [Wilh2] is possible, it will

not be enough for us to consider the effect of our deformation on the set, Z, of zero

planes of the metric in [Wilh2]. Instead we will have to check that the curvature

becomes positive in an entire neighborhood of Z. This will involve understanding

the change of the full curvature tensor.

According to recent work of Tapp, any zero plane in a Riemannian submersion

of a biinvariant metric on a compact Lie group exponentiates to a flat. Thus any

attempt at perturbing any of the known quasipositively curved examples to positive

curvature would have to tackle this issue [Tapp2].

In contrast to the metric of [EschKer], the metric in [Wilh2] does not come from

a left (or right) invariant metric on Sp(2). So although the Gromoll–Meyer sphere

is a quotient of the Lie group Sp(2), we do not use Lie theory for any of our

curvature computations or even for the definition of our metric. Our choice here is

perhaps a matter of taste. The overriding idea is that although none of the metrics

considered lift to left invariant ones on Sp(2), there is still a lot of structure. Our

goal is to exploit this structure to simplify the exposition as much as we can.

Our substitute for Lie theory is the pull-back construction of [Wilh1]. In fact,

the current paper is a continuation of [PetWilh], [Wilh1], and [Wilh2]. The reader

who wants a thorough understanding of our argument will ultimately want to read

these earlier papers. We have, nevertheless, endeavored to make this paper as self-

contained as possible by reviewing the basic definitions, notations, and results of

[PetWilh], [Wilh1], and [Wilh2] in sections 2, 3, and 4. It should be possible to skip

the earlier papers on a first read, recognizing that although most of the relevant

results have been restated, the proofs and computations are not reviewed here. On

the other hand, Riemannian submersions play a central role throughout the paper;

so the reader will need a working knowledge of [On].

After establishing the existence of integrally positive curvature and reviewing the

required background, we give a detailed and technical summary of the remainder

of the argument in section 5. Unfortunately, aspects of the specific geometry of the

Gromoll-Meyer sphere are scattered throughout the paper, starting with section 2;

so it was not possible to write section 5 in a way that was completely independent

of the review sections. Instead we offer the following less detailed summary with

the hope that it will suffice for the moment.

Starting from the Gromoll-Meyer metric the deformations to get positive curva-

ture are

(1): The (h1⊕ h2)–Cheeger deformation, described in section 3

(2): The redistribution, described in section 6.

(3): The (U ⊕ D)–Cheeger deformation, described in section 3

(4): The scaling of the fibers, described in section 1

(5): The partial conformal change, described in section 10

(6): The ∆(U,D) Cheeger deformation and a further h1–deformation.

We let g1, g1,2, g1,2,3, ect. be the metrics obtained after doing deformations (1),

(1) and (2), or (1), (2), and (3) respectively.

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AN EXOTIC SPHERE WITH POSITIVE CURVATURE5

It also makes sense to talk about metrics like g1,3, i.e. the metric obtained from

doing just deformations (1) and (3) without deformation (2).

All of the deformations occur on Sp(2). So at each stage we verify invariance of

the metric under the various group actions that we need. For the purpose of this

discussion we let g1, g1,2, g1,2,3, ect. stand for the indicated metric on both Sp(2)

and Σ7.

g1,3 is the metric of [Wilh2] that has almost positive curvature on Σ7. g1,2,3

is also almost positively curvature on Σ7, and has precisely the same zero planes

as g1,3. Some specific positive curvatures of g1,3 are redistributed in g1,2,3. The

reasons for this are technical, but as far as we can tell without deformation (2) our

methods will not produce positive curvature. It does not seem likely that either

g1,2or g1,2,3are nonnegatively curved on Sp(2), but we have not verified this.

Deformation (4), scaling the fibers of Sp(2) −→ S4, is the raison d’ˆ etre of this

paper. g1,2,3,4has some negative curvatures, but has the redeeming feature that the

integral of the curvatures of the zero planes of g1,3is positive. In fact this integral

is positive over any of the flat tori of g1,3.

The role of deformation (5) is to even out the positive integral. The curvatures

of the flat tori of g1,3are pointwise positive with respect to g1,2,3,4,5.

To understand the role of deformation (6), recall that we have to check that we

have positive curvature not only on the 0–planes of g1,3, but in an entire neighbor-

hood (of uniform size) of the zero planes of g1,3. To do this suppose that our zero

planes have the form

P = span{ζ,W}.

We have to understand what happens when the plane is perturbed by moving its

foot point, and also what happens when the plane moves within the fibers of the

Grassmannian.

To deal with the foot points, we extend ζ and W to families of vectors Fζ and

FW on Sp(2). These families can be multivalued and FW contains some vectors

that are not horizontal for the Gromoll-Meyer submersion. All pairs {ζ,W} that

contain zero planes of?Σ7,g1,3

valid for all pairs {z,V } with z ∈ Fζand V ∈ FW, provided z and V have the same

foot point. In this manner, we can focus our attention on fiberwise deformations of

the zero planes.

To do this we consider planes of the form

?

are contained in these families, and the families

are defined in a fixed neighborhood of the 0–locus of g1,3. All of our arguments are

P = span{ζ + σz,W + τV }

where σ,τ are real numbers and z and V are tangent vectors. Ultimately we show

that all values of all curvature polynomials

P (σ,τ) = curv(ζ + σz,W + τV )

are positive.

Allowing σ,τ, z and V to range through all possible values describes an open

dense subset in the Grassmannian fiber. The complement of this open dense set

consists of planes that have either no z component or no W component. These cur-

vatures can be computed as combinations of quartic, cubic, and quadratic terms in

suitable polynomials P (σ,τ). In sections 12 and 13 we show that these combina-

tions/curvatures do not decrease much under our deformations (in a proportional

sense); so the entire Grassmannian is positively curved.