CURVATURE AND SYMMETRY OF MILNOR SPHERES
KARSTEN GROVE AND WOLFGANG ZILLER
Dedicated to Detlef Gromoll on his 60th birthday
Since Milnor's discovery of exotic spheres [Mi], one of the most intriguing p*
*roblems in Rie-
mannian geometry has been whether there are exotic spheres with positive curvat*
*ure. It is
well known that there are exotic spheres that do not even admit metrics with po*
*sitive scalar
curvature [Hi] . On the other hand, there are many examples of exotic spheres w*
*ith positive
Ricci curvature (cf. [Ch1], [He],[Po], [Na]) and this work recently culminated *
*in [Wr ] where it
is shown that every exotic sphere that bounds a parallelizable manifold has a m*
*etric of positive
Ricci curvature. This includes all exotic spheres in dimension 7. So far, howev*
*er, no example
of an exotic sphere with positive sectional curvature has been found. In fact, *
*until now, only
one example of an exotic sphere with non-negative sectional curvature was known*
*, the so-called
Gromoll-Meyer sphere [GM ] in dimension 7. As one of our main results we prove:
Theorem A. Fifteen of the 27 exotic spheres in dimension 7 admit metrics of non*
*-negative
sectional curvature.
The exotic spheres that occur in this theorem are exactly those that can be e*
*xhibited as
3-sphere bundles over the 4-sphere , the so-called Milnor spheres . Each such *
*exotic sphere
can be written as an S3 bundle in infinitely many distinct ways, cf. [EK ]. O*
*ur metrics are
submersion metrics on these sphere bundles and we will obtain infinitely many n*
*on-isometric
metrics on each of these exotic spheres, see Proposition 4.8. We do not know i*
*f any of the
remaining exotic spheres in dimension 7 admit metrics with non-negative curvatu*
*re, or if the
metrics on the Milnor spheres above can be deformed to positive curvature. But *
*no obstructions
are known either.
Another central question in Riemannian geometry is, to what extent the conver*
*se to the cele-
brated Cheeger-Gromoll soul theorem holds [CG ]. The soul theorem implies that *
*every complete,
non-compact manifold with non-negative sectional curvature is diffeomorphic to *
*a vector bundle
over a compact manifold with non-negative curvature. The converse is the questi*
*on which total
spaces of vector bundles over compact non-negatively curved manifolds admit (co*
*mplete) met-
rics with non-negative curvature. In one extreme case, where the base manifold *
*is a flat torus,
there are counterexamples [OW ],[Ta]. In another extreme case, where the base m*
*anifold is a
sphere (the original question asked by Cheeger and Gromoll) no counter examples*
* are known.
But there are also very few known examples, all of them coming from vector bund*
*les whose
principal bundles are Lie groups or homogeneous spaces (cf. [CG ], [GM ], [Ri1]*
*, and [Ri2] ). It
is easy to see that the total space of any vector bundle over Sn with n 3 admi*
*ts a complete
non-negatively curved metric. Another of our main results addresses the first n*
*on-trivial case.
Theorem B. The total space of every vector bundle and every sphere bundle over *
*S4 admits a
complete metric of non-negative sectional curvature.
__________
The first named author was supported in part by the Danish National Research *
*Council and both authors
were supported by a RIP (Research in Pairs) from the Forschungsinstitut Oberwol*
*fach and by grants from the
National Science Foundation.
1
2 KARSTEN GROVE AND WOLFGANG ZILLER
The special case of S2 bundles over S4 will give rise to infinitely many non-*
*negatively curved
6-manifolds with the same homology groups as CP3, but whose cohomology rings ar*
*e all distinct,
see (3.9).
From a purely topological relationship between bundles with base S4 and S7 (c*
*f. Section 3
and [Ri3]) it will follow that most of the vector bundles and sphere bundles ov*
*er S7 admit a
complete metric of non-negative curvature, see (3.13).
In [GZ2 ] we will use the constructions of this paper to also analyze bundles*
* with base CP2,
CP2# CP2, and S2 x S2.
From representation theory it is well known that any linear action of the rot*
*ation group SO(3)
has points whose isotropy group contains SO(2). A proof of this assertion for g*
*eneral smooth
actions of SO(3) on spheres was offered in [MS ]. However, this turned out to b*
*e false. In fact,
among other things, Oliver [Ol] was able to construct a smooth SO(3) action on *
*the 8-disc
D8, whose restriction to the boundary 7-sphere S7 is almost free, i.e., has onl*
*y finite isotropy
groups. Explicitly, the isotropy groups of the example in [Ol] are equal to 1; *
*Z2; D2; D3 and D4.
By completely different methods we exhibit infinitely many such actions on the *
*7-sphere.
Theorem C (Exotic Symmetries of the Hopf Fibration).For each n 1 there exists *
*an almost
free action of SO(3) on S7 which preserves the Hopf fibration S7 ! S4 and whose*
* only isotropy
groups, besides the principal isotropy group 1, are the dihedral groups D1 = Z2*
*, D2 = Z2x Z2,
Dn, Dn+1, Dn+2, and Dn+3. Furthermore these actions do not extend to the disc D*
*8 if n 4.
In the case of the exotic 7-spheres we produce the first such examples.
Theorem D. Let 7 be any (exotic) Milnor sphere. Then there exist infinitely man*
*y inequiv-
alent almost free actions of SO(3) on 7, one or more for each fibration of 7 by*
* 3-spheres,
preserving this fibration.
Since the SO(3) actions in Theorem C and D take fibers to fibers, they induce*
* an action of
SO(3) on the base S4. This action of SO(3) on S4 is a fixed action, which yield*
*s the well known
decomposition of S4 into isoparametric hypersurfaces [Ca], [HL ]. Hence our act*
*ions on S7 and
7 can be viewed as lifts of this action of SO(3) on S4 to the total space of th*
*e S3 fibrations.
All of the above results follow from investigations and constructions related*
* to manifolds of
cohomogeneity one, i.e., manifolds with group actions whose orbit spaces are on*
*e-dimensional.
For closed manifolds this means that the orbit space is either a circle (and al*
*l orbits are principal)
or an interval. In the first case it is easy to see that the manifold supports *
*an invariant metric
with non-negative curvature. In the second more interesting case, the interior*
* points of the
interval correspond to principal orbits and the endpoints to non-principal orbi*
*ts. Although very
difficult, it is tempting to make the following
Conjecture. Any cohomogeneity one manifold supports an invariant metric of non-*
*negative
sectional curvature.
If true, this would imply in particular that the Kervaire spheres, which carr*
*y a cohomogeneity
one metric [HH ], and are exotic spheres in dimension 8k +1, support an invaria*
*nt metric of non-
negative curvature. In [BH ] it was shown that the Kervaire spheres do not admi*
*t a metric with
positive sectional curvature, invariant under the group action.
One of our key results is a small step in the direction of this conjecture.
Theorem E. Any cohomogeneity one manifold with codimension two singular orbits *
*admits a
non-negatively curved invariant metric.
CURVATURE AND SYMMETRY OF MILNOR SPHERES 3
The importance of Theorem E is due to the surprising fact, that the class of *
*cohomogeneity
one manifolds with singular orbits of codimension two is extremely rich. This i*
*s illustrated by
our other key result.
Theorem F. Every principal L bundle over S4 with L = SO(3) or SO(4) supports a *
*cohomo-
geneity one L x SO(3) structure with singular orbits of codimension two.
Theorems A and B are now easy consequences of E and F in conjunction with the*
* Gray-
O'Neill curvature formula for submersions. The SO(3) actions in Theorem C and D*
* arise from
this construction as well, since the group SO(3) commutes with the principal bu*
*ndle action and
hence induces an action on every associated bundle.
Another consequence of Theorem E is the following:
Theorem G. On each of the four diffeomorphism types homotopy equivalent to RP5 *
*there exists
infinitely many non-isometric metrics with non-negative sectional curvature.
The paper is organized as follows. Section 1 is devoted to general properties*
* of cohomogeneity
one manifolds and to an important construction of principal bundles in this fra*
*mework. In
Section 2 we prove Theorem E and Theorem G. The constructions in Section 1 are *
*used to
prove Theorem F in Section 3. In Section 4 we analyze induced SO(3) actions on *
*associated
bundles and derive Theorem C and D. Finally, in Section 5 we examine the geomet*
*ry of our
examples in more detail and raise some open questions.
It is our pleasure to thank J. Shaneson for general help concerning topologic*
*al questions, and
R. Oliver for sharing his insight about SO(3)-actions on discs.
1.Principal Bundles And Cohomogeneity One Manifolds
We first recall some basic facts about manifolds of cohomogeneity one and est*
*ablish some
notation.
Let M be a closed, connected smooth manifold with a smooth action of a compac*
*t Lie group
G. We say that the action G x M ! M is of cohomogeneity one if the orbit space*
* M=G is
1-dimensional. A cohomogeneity one manifold is a manifold with an action of coh*
*omogeneity
one.
Consider the quotient map ss : M ! M=G. When M=G is one dimensional, it is ei*
*ther a
circle S1, or an interval I. In the first case all G orbits are principal and s*
*s is a bundle map. It
then follows from the homotopy sequence of this bundle that the fundamental gro*
*up ss1(M) of
M is infinite. In the second case there are precisely two non-principal G-orbit*
*s corresponding
to the endpoints of I, and M is decomposed as the union of two tubular neighbor*
*hoods of the
non-principal orbits, with common boundary a principal orbit. All of this actua*
*lly holds in the
topological category (cf. [Mo ]).
Unless otherwise stated, we will only consider the most interesting case, whe*
*re M=G = I.
For this we will make the description above more explicit in terms of an arbitr*
*ary but fixed
G-invariant Riemannian metric on M, normalized so that with the induced metric,*
* M=G =
[-1; 1]. Fix a point x0 2 ss-1(0) and let c : [-1; 1] ! M be the unique minima*
*l geodesic
with c(0) = x0 and ss O c = id[-1;1]. Note that c : R ! M intersects all orbits*
* orthogonally,
and c : [2n - 1; 2n + 1] ! M, n 2 Z are minimal geodesics between the two non-p*
*rincipal
orbits, B = ss-1(1) = G . x , x = c(1). Let K = Gx be the isotropy groups *
*at
x and H = Gx0 = Gc(t); -1 < t < 1, the principal isotropy group. By the slice*
* theorem,
we have the following description of the tubular neighborhoods D(B-) = ss-1([-1*
*; 0]) and
D(B+) = ss-1([0; 1]) of the non-principal orbits B = G=K :
(1.1) D(B ) = G xK D` +1
4 KARSTEN GROVE AND WOLFGANG ZILLER
where D` +1is the normal (unit) disk to B at x . Hence we have the decompositi*
*on
(1.2) M = D(B-) [ED(B+) ;
where E = ss-1(0) = G . x0 = G=H is canonically identified with the boundaries *
*@D(B ) =
G xK S` , via the maps G ! G x S` , g ! (g; _c(1)). Note also that @D` +1= S`*
* =
K =H. All in all we see that we can recover M from G and the subgroups H and K .
In general, suppose G is a compact Lie group and H K G are closed subgroup*
*s such
that K =H = S` are spheres. It is well known (cf. [Bes, p.195]) that a transit*
*ive action of a
compact Lie group K on a sphere S` is linear and is determined by its isotropy *
*group H K.
Thus the diagram of inclusions
G B
B- = G=K- j-____ BBj+BB B+ = G=K+
____ BBB
(1.3) K- K+
BB __
` BBB ___ `
S -= K-=H i- BBB ___i+ S += K+=H
H
determines a manifold
(1.4) M = G xK- D`-+1[G=H G xK+ D`++1
on which G acts by cohomogeneity one via the standard G a*
*ction on
G xK D` +1 in the first coordinate. Thus the diagram (1.3)defines a cohomogen*
*eity one
manifold, and we will refer to it as a cohomogeneity one group diagram, which w*
*e sometimes
denote by H {K-; K+} G. We also denote the common homomorphism j+ O i+ = j- O*
* i-
by j0:H ! G.
We are now ready for the main construction in this section: Principal bundles*
* over cohomo-
geneity one manifolds.
Let L be any compact Lie group, and M any cohomogeneity one manifold with gro*
*up diagram
H {K-; K+} G. It is important to allow the G-action on M to be non-effective,*
* i.e. G and
H have a common normal subgroup, since this will produce more principal bundles*
* over M, see
e.g. (3.1),(3.2).
For any Lie group homomorphisms OE : K ! L, OE0 : H ! L with OE+ O i+ = OE-*
* O i- = OE0,
let P be the cohomogeneity one L x G-manifold with diagram
L x GG
(OE-;j-)wwwGG(OE+;j+)GGGww
wwww GGG
(1.5) K- K+
HH ww
HHH www
i-HHHH wwwi+w
H
Clearly the subaction of L x G by L = L x {e} on P is free since L \ (l; g)K (l*
*; g)-1 =
(l; g)(L x {e} \ K )(l; g)-1 as well as L \ (l; g)H(l; g)-1 = (l; g)(L x {e} \ *
*H)(l; g)-1 is the
trivial group for all (l; g) 2 L x G. Moreover, P=L = M since it has a cohomog*
*eneity one
description H {K-; K+} G. It is also apparent that the non-principal orbits i*
*n P have
the same codimension as the non-principal orbits in M, as well as the same slic*
*e representation,
since the normal bundles in M pull back to the normal bundles in P under the pr*
*incipal bundle
projection P ! M. In summary:
CURVATURE AND SYMMETRY OF MILNOR SPHERES 5
Proposition 1.6.For every cohomogeneity one manifold M as in (1.3)and every cho*
*ice of
homomorphisms OE : K ! L with OE+ O i+ = OE- O i-, the diagram (1.5)defines a*
* principal L
bundle over M.
Note, moreover, that the L x G-action on P may well be effective even if the *
*G-action on M
is not.
We now move on to discuss induced actions on associated bundles:
Let F be a smooth manifold on which L acts, L x F ! F. Consider the total spa*
*ce of the
associated bundle V = P xL F. Observe that the product of the trivial G-action *
*on F with the
sub-action of G = {e} x G L x G on P induces a natural G-action on V .
Lemma 1.7 (Isotropy Lemma).The natural G-action on V = P xL F has exactly the f*
*ollowing
types of isotropy groups
OE-1(Lu) and OE-10(Lu)
where Lu , u 2 F are the isotropy groups of L x F ! F.
Proof.Consider the L-orbit, L(x; u) = {(`x; `u) | ` 2 L} of a point (x; u) 2 P *
*x F. Then
GL(x;u)= {g 2 G | gL(x; u) = L(gx; u) = L(x; u)}
= {g 2 G | 9` 2 L : (gx; u) = (`x; `u)}
= {g 2 G | 9` 2 Lu : (`-1; g) 2 (L x G)x}
However, (L x G)x is some (^`; ^g)-conjugate of one of (OE+; j+)(K+); (OE-; j-)*
*(K-)_or (OE0; j0)(H),
and the claim follows. |_*
*_|
2. Non-Negative Curvature On Homogeneous Bundles
The primary purpose of this section is to prove Theorem E of the Introduction.
As in [Ch1] we will construct non-negative curvature metrics on M = D(B-) [ED*
*(B+) (cf.
1.2) with the additional property that the common boundary E = @D(B-) = @D(B+) *
*is totally
geodesic in M. This is a very strong restriction, which, by the soul theorem [C*
*G ], implies that
also B are totally geodesic . With this in mind, all we have to do is to const*
*ruct G-invariant
non-negative curvature metrics on the bundles D(B ) = G xK D` +1(cf. (1.1)), t*
*hat agree
on the common boundary E = G=H = G xK S` and are product metrics near the bou*
*ndary.
From the Gray-O'Neill curvature submersion formula (cf. [ON ] or [Gr]), we kn*
*ow that the
product metric of a left invariant, Ad(K)-invariant metric of non-negative curv*
*ature on G with a
K-invariant non-negative curvature metric on D`+1(which is product near S`= @D`*
*+1) induces
a G-invariant non-negative curvature metric on the quotient G xK D`+1(which is *
*product near
G=H = G xK S` = @(G xK D`+1)). The difficulty in the above strategy is therefo*
*re, that
in general the restriction of such metrics on G xK- D`-+1and on G xK+ D`++1to G*
*=H =
G xK- S`-= G xK+ S`+ are different.
Consider any closed Lie subgroups H K G of a compact Lie group G, with Lie *
*algebras
h k g. Fix any left invariant, Ad(K)-invariant Riemannian metric, < ; > on G*
* and let
m = k? and p = h? \ k relative to this metric. On G=H and K=H we get induced (s*
*ubmersed)
G-, respectively K-invariant metrics which are also denoted by < ; > . As usual*
* we make the
identifications p + m ' TH G=H and p ' TH K=H via action fields, i.e. , X + A !*
* (X + A)*H
and X ! X*Hrespectively.
The homogeneous space G=H can be identified with the orbit space G xK K=H of *
*G x K=H
by the K-action (k; (g; kH))p!_(gk-1; kkH). The identification is given by gH !*
* K(g; H) with
inverse K(g; kH) ! gkH. By K=H we mean K=H endowed with the metric < ; >, whe*
*re
> 0. In this terminology we have:
6 KARSTEN GROVE AND WOLFGANG ZILLER
Lemmap2.1.The_G-invariant metric < ; > on G=H induced from the product metric *
*on G x
K=H via G xK K=H ' G=H is determined by
< ; >|m = < ; >|m and < ; >|p = ___+1< ; >|p
Proof.The vertical space (= tangent space to K-orbit) at (1; H) 2 G x K=H is gi*
*ven by
Tv(1;H)= h x {0} + {(-X; X*H) | X 2 p}
Thus (U; YH*) 2 T(1;H)GxK=H, U 2 g; Y 2 p is horizontal if and only if U = Z +A*
* 2 p+m = h?
satisfies - + = 0 for all X 2 p, i.e.
Th(1;H)= m x {0} + {(Y; YH*) | Y 2 p}:
Now (A; 0) projects to A 2 m TH G=H and (Y; YH*) projects to ( + 1)Y 2 p TH G*
*=H.
In particular, the horizontal lift of A 2 TH G=H to (1; H) is (A; 0), and Y 2 *
*p TH G=H
lifts to _1_+1(Y; YH*). This proves the claim since the norms of these vectors*
* are given by
*
* __
k(A; 0)k2 = kAk2 and k_1_+1(Y; YH*)k2 = (_1_+1)2(2kY k2+ kY k2) = ___+1kY k2. *
* |__|
As an immediate consequence of this lemma, we see that if Q is a fixed biinva*
*riant metric on
G, and we choose < ; > above as
(2.2) < ; >|m= Q|m and < ; >|k= +1_Q|k
p__
then the metric on G=H induced via G xK K=H as above, is the same as the one*
* induced
directly via Q. This is essentially the method that Cheeger used in [Ch1] to co*
*nstruct a non-
negatively curved metric on the connected sum of two projective spaces. The pr*
*oblem now,
however, is that in general a metric like (2.2)on G has some negative sectional*
* curvature, as we
will see, since a = +1_> 1.
We need to work in a slightly more general context. As before G is a compact *
*Lie group and
k g a subalgebra. Let K G be the (immersed) Lie sugbroup of G with Lie algebr*
*a k, i.e. K
need not be compact. As before let Q be a fixed biinvariant metric on g and a >*
* 0. Define
(2.3) Qa|m= Q|m and Qa|k= aQ|k
and denote again by Qa also the corresponding left and Ad(K) invariant metric o*
*n G. We need
the following curvature formulas for this left invariant metric (see e.g [Es],[*
*DZ ] for special cases).
Proposition 2.4.For any a > 0 let Ra be the curvature tensor of the metric Qa d*
*efined in
(2.3). Then for any A; B 2 m and X; Y 2 k we have
Qa(Ra(A+X; B + Y )(B + Y ); A + X) =
1_ 2 1_flfl 2 2 flfl
4k[A; B]m + a[X; B] + a[A;QY+]k4[A; B]k+ a [X; YQ]+
1_ 3 2 3_ 2
4a(1 - a) k[X; Y ]kQ+ 4(1 - a) k[A; B]k+ a[X;QY ]k
where subscripts denote components. In particular, (G; Qa) has non-negative cur*
*vature whenever
0 < a 1, or if k is abelian and a 4_3.
Proof.For a = 1 this is the well known formula for the sectional curvature of a*
* biinvariant
metric. For a 6= 1, we claim that Qa is a submersed metric. Indeed, on G x K co*
*nsider the
biinvariant (semi-) Riemannian metric induced from < ; > = Q x bQ|k( b negative*
* allowed) on
g x k. When b = _a_1-awe claim that the map G x K ! G, (g; k) 7! gk is a (semi-*
*) Riemannian
CURVATURE AND SYMMETRY OF MILNOR SPHERES 7
submersion. In fact this can be viewed as a special case of (2.1) above, when H*
* is trivial, by
noticing that in this case the vertical space given by
Tv(1;1)= {(-X; X) | X 2 k} T(1;1)G x K
is non-degenerate since b 6= -1 (this would not be true in the general case whe*
*re h 6= {0}). The
rest of the argument in (2.1)carries over verbatim and we see that the submerse*
*d metric on G
is scaled by _b_b+1= a in the k-direction.
To compute the sectional curvature, we use the Gray-O'Neill formula. Conside*
*r a 2-plane
in T1G = g spanned by A + X and B + Y as in (2.4). The corresponding horizontal*
* lifts to
T(1;1)GxK are (A+ b__b+1X; _1_b+1X) and (B+ b__b+1Y; _1_b+1Y ) respectively. Mo*
*reover, when extending
the G-coordinate to left invariant vector fields and the K-coordinate to right *
*invariant vector
fields, the resulting fields are easily seen to be horizontal. The Gray-O'Neill*
* formula then yields:
Qa(Ra(A + X; B + Y )(B + Y ); A + X) = ff + 3_4fi
where i j
ff = gxk
and flh i
vflfl2
fi = flfl(A + _b_b+1X; _1_b+1X); (B +f_b_b+1Y;l_1_b+1Y:)
gxk
Now
fl 2 fl fl 2 fl
ff =1_4flfl[A + _b_b+1X; B++1_b_b+1Y_]flfl4b flfl[_1_b+1X; _1_b+1Y ]flfl
Q Q
fl 2 fl fl 2 fl
=1_4flfl[A; B]m + _b_b+1([X; B]++1[A;_Y4])flflflfl[A; B]k++(_b_b+1)2[X;1Y_]f*
*lfl4b(_1_b+1)4k[X; Y ]k2Q
Q Q
where we have used [m; k] m and [k; k] k. In terms of a = _b_b+1(and hence 1-*
*a = _1_b+1andb =
_a_
1-a) we have
fl 2 fl *
* 2
ff = 1_4k[A; B]m+a[X; B] + a[A;2YQ]k+ 1_4fl[A; B]k+ a2[X;QY+]fl1_4a(1 - a)3k*
*[X; Y ]kQ:
Using the fact that for right invariant vector fields X*; Y *, [X*; Y *] = -[*X*
*;*Y ] = -[X; Y ] in
terms of left invariant vector fields, we get:
fli vfl2 *
* j
fi= flfl[A; B]m + _b_b+1[A; Y ] + _b_b+1[X; B] + [A; B]k+ (_b_b+1)2[X;fYl];*
*f-(_1_b+1)2[X;lY ]
gxk
fli vfl2 j
= flfl[A; B]k+ (_b_b+1)2[X; Y ]; -(_1_b+1)2[X;fYl]fl
gxk
since m x 0 Th.
For U; V 2 k we have
(U; V ) = _1_b+1(b(U + V ); U + V ) + _1_b+1(-(-U + bV ); -U + bV )
and hence (U; V )v = _1_b+1(U - bV; -U + bV ). Moreover,
2 2 2
k(U; V )vk2gxk= (_1_b+1)2 kU - bV kQ + bk - U + bV=kQ1_b+1kU - bV kQ:
This yields
fl 2 fl
fi= _1_b+1flfl[A; B]k+ (_b_b+1)2[X; Y ] +=b(_1_b+1)2[X;(Y1]flfl- a) k[A; B*
*]k+ a[X;2YQ]k
Q
which completes the proof of the curvature formula.
8 KARSTEN GROVE AND WOLFGANG ZILLER
If a 1 all terms are non-negative. For a > 1 the latter two terms will be ne*
*gative in general.
However, if k is abelian the formula reduces to
Qa(Ra(A+X; B + Y )(B + Y ); A + X) = 1_4k[A; B]m + a[X; B] + a[A;2YQ]k+ (1 - 3_*
*4a) k[A; B]kk2Q
which is non-negative for a 4_3as claimed. __
|__|
Remark 2.5.In general there are two planes with strictly negative curvature on *
*(G; Qa) for any
a > 1 arbitrarily close to 1. Indeed, one can usually easily find two planes sp*
*anned by A + X
and B + Y with [A; B] = -a2[X; Y ] and [X; B] + [A; Y ] = 0 which will have neg*
*ative sectional
curvature if [X; Y ] 6= 0.
We are now ready to prove the main result of this section.
Theorem 2.6.Suppose G is a connected, compact Lie group and H K G are closed
subgroups with K=H = S1. Then for any biinvariant metric on G, there is a G-inv*
*ariant non-
negatively curved metric on G xK D2 which is a product near the boundary G=H = *
*G xK S1,
and so that the metric restricted to G=H is induced from the given biinvariant *
*metric on G.
Proof.Fix a biinvariant metric Q on g and let m = k?; p = h? \ k as before. By *
*assumption, p
is 1-dimensional and is an abelian subalgebra of g. Moreover, if H H is the in*
*effective kernel
of the K-action on S1 = K=H = (K=H )=(H=H ) we have h= h since the isotropy gro*
*up of an
effective action on S1 is finite. Since H is normal in K, h and hence p is pres*
*erved by Ad(K).
This implies that the metric Qa defined by
Qa|m= Q|m; Qa|p= aQ|p and Qa|h= Q|h
is Ad(K)-invariant. Since p is also a subalgebra, this metric can also be viewe*
*d as in (2.3)(with
k = p ) and hence (2.4)implies that its curvature is non-negative if a 4=3.
Let K=H = S1 be equipped with the metric induced from Qa|k. By Lemma (2.1), t*
*he metric on
p__
GxK S1 ' G=H is then given by Q on m and ___+1aQ on p. Now pick e.g. a = 4=3;*
* = 3 and
a K-invariant metric on D2 with non-negativepcurvature,_whichpis_product near t*
*he boundary
circle @D2 = S1, and on S1 is the metric 3S1 = 3K=H from above.
The quotient metric on G xK D2 induced from the product metric on G and on D2*
* has_all_
the desired properties claimed in (2.6). *
* |__|
It follows immediately from (2.6)and the discussion in the beginning of this se*
*ction, that we
can construct non-negatively curved metrics on each half G xK D2, matching smo*
*othly near
@(GxK- D2) ' G=H ' @(GxK+ D2) to yield G-invariant metrics on M = GxK-D2[EGxK+D2
with non-negative curvature. This finishes the proof of Theorem E.
Remark 2.7.For a metric on D2 we can choose a rotationally symmetric metric dt2*
*+ f(t)2d2,
where f is a concave function which is odd with f0(0) = 1 in order to guarantee*
* smoothness
of the metric. Suppose K=H = (S1; Q) is a circle of length 2ssr. Then the induc*
*ed metric on
the principal orbit G=H at c(t) (where t = 0 corresponds to the singular orbit *
*G=K) can be
2a
described as GxK f(t)_rK=H which, using (2.1), is then given by Q on m and _f__*
*f2+r2Q on p. Hence
2
we need to choose a t0 such that f2(t) = _r_a-1, for t t0. Notice that the lar*
*ger the radius r is,
or if we choose 1 < a 4=3 close to 1, the larger t0 needs to be, and hence the*
* diameter of M
will be large.
Remark 2.8.In the case where a non-regular orbit is exceptional, i.e., is a hyp*
*ersurface, one can
just choose the biinvariant metric on G itself to induce a metric on the disc b*
*undle G xK D1,
CURVATURE AND SYMMETRY OF MILNOR SPHERES 9
which then has the same properties as in Theorem 2.6. Hence one obtains a non-*
*negatively
curved metric on every cohomogeneity one manifold with non-regular orbits of co*
*dimension
2.
We point out that there are many cohomogeneity one manifolds with non-negativ*
*e curvature,
whose singular orbits have codimension bigger than 2. One large class is the li*
*near cohomo-
geneity one actions on round spheres Sn(1), classified in [HL ], and characteri*
*zed as the isotropy
representations of compact rank two symmetric spaces. There are also many isome*
*tric coho-
mogeneity one actions on compact symmetric spaces with their natural metric of *
*non-negative
curvature, recently classified in [Ko ] in the irreducible case. In almost all *
*of these examples,
none of the principal orbits are totally geodesic. The difficulty in proving th*
*e conjecture that
every cohomogeneity one manifold carries a metric with non-negative curvature m*
*ay lie in that
one needs a better understanding of how to glue the two halfs together without *
*making the
middle totally geodesic.
One particularly intriguing class of cohomogeneity one manifolds are the Kerv*
*aire spheres,
which are the 2n - 1 dimensional manifolds defined by the equations
zd0+ z21+ . .z.2n= 0 ; |z0|2+ . .|.zn|2 = 1:
For d odd, they are homeomorphic to spheres, and if 2n-1 1 mod 8 they are not *
*diffeomorphic
to spheres. As discovered in [HH ], they carry a cohomogeneity one action by S*
*O(2)SO (n)
defined by (ei; A)(z0; . .;.zn) = (e2iz0; eiA(z1; . .;.zn)t). This action was e*
*xamined in detail
in [BH ], where they showed that the group picture is given by K- = SO(2) x SO(*
*n - 2); K+ =
O(n - 1); H = Z2x SO(n - 2), with embeddings given by (ei; A) 2 K- SO(2) SO(2)*
* SO(n -
2) ! (ei; R(d); A) with R(d) a rotation by angle d, A 2 K+ ! (det(A); (det(A); *
*A)), and
(ffl; A) 2 Z2x SO(n - 2) = H ! (ffl; (ffl; ffl; A)). In particular, one obtains*
* a different action for
each odd d, and the non-principal orbits have codimension 2 and n - 1.
In the special case n = 3, where these actions define a cohomogeneity one act*
*ion on S5, they
were first discovered by E.Calabi, who also observed that they descend to cohom*
*ogeneity one
actions on the homotopy projective spaces S5=Z2, where Z2 is the element -id2 S*
*O(2). In [Lo]
it was shown that this homotopy projective space contains four diffeomorphism t*
*ypes, according
to d 1; 3; 5; 7 mod 8, and two homeomorphism types, according to d 1; 3 mod 8*
*. Hence
each of the four possible differentiable structures on RP5 carries infinitely m*
*any cohomogeneity
one actions by SO(2) SO(3), and since the codimension of the singular orbits in*
* this case are
both equal to two, they all admit an invariant metric with non-negative section*
*al curvature by
Theorem E. One easily shows that the effective group picture is given by G = SO*
*(2) SO(3); K- =
SO(2) with embedding ei ! (e2i; (R(d); id)) , K+ = O(2) with embedding A ! (1; *
*(1; A))
and H = Z2 = (1; diag(-1; -1; 1).
To finish the proof of Theorem G, we need to show that these metrics are neve*
*r isometric to
each other. For this we first note that if the action of SO(2)SO (3) extends to*
* a transitive action,
then it must be linear and hence corresponds to the case d = 1 which is the wel*
*l known tensor
product action. If d > 1, we will argue that SO(2)SO (3) is the id component of*
* the isometry
group, and since the group actions are never conjugate to each other, the corre*
*sponding metrics
cannot be isometric either. Notice that any isometries, besides the elements of*
* SO(2)SO (3), must
preserve the G orbits and hence induce isometries of the homogeneous metrics on*
* the principal
orbits SO(2)(SO (3)=Z2). One easily shows that for any invariant metric on this*
* homogeneous
space, any further isometries in the id component come from right translations *
*by NSO(3)(Z2)=Z2.
But these right translations do not extend to G=K+ and hence are not well defin*
*ed on M. This
finishes the proof of Theorem G.
10 KARSTEN GROVE AND WOLFGANG ZILLER
Using the same methods as in [Se], one shows that there do not exist any SO(2*
*)SO (3) invariant
metrics with positive curvature on these 5-dimensional cohomogeneity one manifo*
*lds. This
implies that if we apply Hamilton's flow to our metrics of non-negative curvatu*
*re, one cannot
obtain a metric of positive curvature since Hamilton's flow preserves isometrie*
*s.
3. Topology of Principal Bundles
In this section we discuss the Proof of Theorem F from the Introduction.
Notice that over S4, every principal SO(2) bundle is trivial and well known o*
*bstruction theory
implies that every k-dimensional vector bundle with k > 4 is the direct sum of *
*a 4-dimensional
bundle and a trivial bundle. Hence we only need to examine principal SO(3) and *
*SO(4) bundles.
This is also why Theorem E and F, together with the O'Neill submersion formula,*
* implies
Theorem B.
To employ the methods of Section 1 we begin by describing the well known coho*
*mogeneity
one action by SO(3) on S4 in a language that will be needed for our constructio*
*n of principal
bundles. Let V = {A | A a3 x 3 real matrix withA = At; tr(A) = 0}. Then V is*
* a five
dimensional vector space with inner product = trAB. SO(3) acts on V via *
*conjugation
g . A = gAg-1 and this action preserves the inner product and hencePactsPon S4(*
*1) V . Every
point in S4(1) is conjugate to a matrix in F = {diag(1; 2; 3) | i= 0; 2i= 1*
*} and hence
the quotient space is one dimensional. The singular orbits B consist of those *
*matrices A with
two eigenvalues ithe same, negative for B- and positive for B+. Clearly,pF_is a*
* greatpcircle_p_
in S4(1) thatpis_orthogonalpto_allporbits_and we can choose x- = diag(2= 6; -1*
*= 6; -1= 6),
x+ = diag(1= 6; 1= 6; -2= 6) and hence K- = S(O(1)O(2)); K+ = S(O(2)O(1)) SO*
*(3). As
long as 1 > 2 > 3 we obtain the principal isotropy group H = S(O(1)O(1)O(1))= Z*
*2x Z2.
Notice that B- and B+ are both Veronese surfaces in S4(1) which are antipodal t*
*o each other
at distance ss=3.
Next, we lift these groups into S3 under the two-fold cover S3 = Sp(1) ! SO(3*
*) which
sends q 2 Sp(1) into a rotation in the 2-plane Im(q)? Im(H) with angle 2, wher*
*e is
the angle between q and 1 in S3(1). After renumbering the coordinates, the grou*
*p K- lifts to
Pin(2) = {ei}[{jei} which we abbreviate to ei [jei. Similarly, K+ lifts to Pin(*
*2) = ej [iej ,
and H =S(O(1)O(1)O(1)) SO(3) lifts to the quaternion group Q = {1; i; j; k}. Th*
*us
the group diagram for S4 is
S3II
vvvv II
vv III
vvv III
(3.1) ei [ jeiH ej [ iej
HHH vvvv
HHH vvv
HHH vvv
Q
We are now in a position to construct principal SO(3) bundles over S4. Since th*
*e second Stiefel
Whitney class w2 of the principal bundle SO(3) ! P* ! S4 is zero, there exists *
*a two fold cover
P of P* such that P ! S4 is a principal S3 bundle. We first construct a cohomog*
*eneity one
action by G on P, with S3 G, which then induces a cohomogeneity one action on *
*P* since
P* = P=oe, with oe = -1 central in S3, as long as oe is also central in G.
Principal bundles S3 ! P ! S4 are classified by an element in ss3(S3) = Z and*
* hence by an
integer k. Equivalently, we can consider the classifying map of the bundle f :S*
*4 ! BS3= HP1
and then k = f*(x)[S4] where x 2 H4(HP1 ; Z) = Z is the generator corresponding*
* to HP1
HP1 . Hence we can also consider k as the Euler class of the principal S3 bundl*
*e, regarded as a
CURVATURE AND SYMMETRY OF MILNOR SPHERES 11
sphere bundle over S4, and evaluated on the fundamental class. Indeed the latte*
*r follows from
the fact that the universal principal S3 bundle over HP1 is the Hopf bundle wi*
*th Euler class
x. Throughout the rest of the paper we denote by Pk ! S4 the principal S3 bundl*
*e with Euler
class k.
We can now use the S3 cohomogeneity one action on S4 in (3.1)and the main con*
*struction
in (1.6)to arrive at the following group diagram:
S3 x S3T
jjjjjj TTTT
jjjjj TTTTT
jjjj TTTTT
(3.2) (eip-; ei) [ (j; j)(eip-; ei) (ejp+; ej ) [ (i; i)(ejp+; ej )
TT jjj
TTTTT jjjjj
TTTTT jjjjj
TTTTT jjjj
4Q
where 4Q = {(1; 1); (i; i); (j; j); (k; k); }. In order for H to be a subgroup*
* of K , we
need that p 1 mod 4 and then we get K =H = S1. Hence (3.2)defines a cohomogen*
*eity one
manifold Pp-;p+. Notice that the action of S3xS3 is again ineffective, the effe*
*ctive version being
S3x S3= (1; 1) = SO(4). As in (1.6), it now follows that S3 = S3x 1 acts freel*
*y on Pp-;p+and
that P=S3 is a cohomogeneity one manifold as in (3.1)and hence equivariantly di*
*ffeomorphic
to S4. Thus we obtain a principal bundle
S3 ! Pp-;p+! S4
Since oe = (-1; 1) is central in S3 x S3, we also obtain a cohomogeneity one ac*
*tion by SO(3) x
SO(3) on the principal SO(3) bundle P* = P=(-1; 1) ! S4.
To identify the principal bundle, we prove:
Proposition 3.3.The principal S3 bundle Pp-;p+! S4 is classified by k = (p2-- p*
*2+)=8.
Proof.The Gysin sequence of the sphere bundle S3 ! Pk ! S4 yields that the non-*
*zero
cohomology groups of Pk are: H0 = H7 = Z and H4(Pk; Z) = Z=kZ if k 6= 0 and H3 *
*= H4 = Z
if k = 0. Hence we can recognize |k| by computing the cohomology groups of Pp-;*
*p+.
To do this in general for a cohomogeneity one manifold M :H {K-; K+} G, we *
*use
the Meyer-Vietoris sequence, where U = D(B ) = G xK D` +1 deformation retract*
*s to
B = G=K and U- \ U+ = G=H. Hence we get a long exact sequence
ss*--ss*+i-1 i i i
(3.4) ! Hi-1(B-) Hi-1(B+) ----! H (G=H) ! H (M) ! H (B-) H (B+) !
where ss are the projections of the sphere bundles G=H = G xK S` = @D(B ) ! *
*B =
G=K . Notice that in our case of (3.2)above, the restriction of the principal S*
*3 bundle Pp-;p+!
S4 to the S3 orbits S3=ei [ jei ' RP2 ' S3=ej [ iej and S3=Q in S4 are all triv*
*ial, since
the classifying space HP1 for principal S3 bundles has no 1-, 2- or 3-skeleton*
*. Thus B =
G=K = S3 x RP2 and G=H = S3 x (S3=Q) up to diffeomorphism. In particular we ob*
*tain:
H3(B ; Z) = Z; H4(B ; Z) = 0, and H3(G=H; Z) = Z + Z, and the Meyer-Vietoris se*
*quence
(3.4)for P = Pp-;p+becomes:
ss*--ss*+3 4
H3(P) ! H3(B-) H3(B+) = Z + Z ----! H (G=H) = Z + Z ! H (P) ! 0
In order to compute H4(P), we need to compute the cokernel of ss*-- ss*+. In ou*
*r case, this
cokernel is determined by the determinant of ss*-- ss*+:Z2 ! Z2. If the determi*
*nant is equal to
12 KARSTEN GROVE AND WOLFGANG ZILLER
0, then H4(P) = Z, and if it is non-zero then H4(P) is a cyclic group with orde*
*r the absolute
value of the determinant. Consider the commutative diagram:
o 3 3 o
S3 x S3_____//_S x S =K
(3.5) |j| ||
fflffl|ss fflffl|
S3 x S3=H____//S3 x S3=K
where Ko are the identity components of K . The maps are two fold covers and *
*as before
it follows that S3 x S3=Ko = S3 x (S3=ei) = S3 x S2 and since S3 x S3=K = S3 x*
* RP2, it
follows that * :H3(G=K ) ! H3(G=Ko) is an isomorphism. The map j is an 8-fold c*
*over and
if we write S3 x S3=H = S3 x (S3=Q), then j* in H3 is an isomorphism on the fir*
*st factor and
multiplication by 8 on the second. Hence j*:H3(S3 x S3=H; Z) = Z2 ! H3(S3 x S3;*
* Z) = Z2
has determinant 8 and therefore det(ss*-- ss*+) = det(o*-- o*+)=8. It remains t*
*o determine the
induced map in H3 for the S1 bundle o . For this purpose we consider the follow*
*ing commutative
diagram of fibrations, where we drop the index for the moment (see e.g. [WZ , *
*p.228]).
o =o 3 3 oae1
S1______//S3 x S3__//S x S =K______//BS1
(3.6) | | | |
| id| | r|
fflffl| fflffl|h fflffl|ae2 fflffl|
S1 x S1___//S3 x S3____//S2 x S2___//_BS1x BS1
coming from the S1 bundle o and the S1 x S1 bundle h (product of Hopf bundles).*
* If we let
H*(BS1) = Z[s] and H*(BS1x BS1) = Z[t1; t2] then r*(t1) = ps; r*(t2) = s since *
*the inclusion
S1 ! S1 x S1 is given by ei ! (eip; ei). If we set H*(S3 x S3) = (u; v), then *
*the only
non-zero differentials in the spectral sequence for ae2 are d2(u) = t21; d2(v) *
*= t22. By naturality
the differentials in the spectral sequence for ae1 are given by d2(u) = p2s2; d*
*2(v) = s2 and hence
a generator 1 in H3(S3x S3=Ko) goes to (-u; p2v) under o*. Thus o-(1) = (-u; p2*
*-v); o+(1) =
(-u; p2+v) and the matrix of o*-- o*+is given by:
-1 1
p2- -p2+
which implies that |k| = |p2-- p2+|=8.
Next we will show that k = (p2-- p2+)=8 with a fixed choice of sign, i.e., th*
*e sign does
not depend on p-; p+. For this, consider the manifolds P7p-;p+and P7p+;p-. We c*
*laim that the
Euler class of the corresponding S3 bundles differ by a sign. First note that *
*the antipodal
map -id: S4 ! S4 interchanges the two halves of S4 relative to the decompositio*
*n (3.1).
Since it is orientation reversing the Euler class of the pull back bundle (-id)*
**Pp-;p+is the
negative of Pp-;p+. Moreover, (-id)*Pp-;p+is a cohomogeneity one manifold with*
* diagram
as for Pp-;p+, except the roles of i and j are switched. Precomposing the S3 x *
*S3 action by
(A; A): S3xS3 ! S3xS3 where A is the inner automorphism of S3 given by A(i) = j*
*; A(j) = i
and A(k) = k-1 we see that Pp+;p-and (-id)*Pp-;p+are equivariantly diffeomorphi*
*c. In
particular, the Euler class of Pp+;p-and Pp-;p+have opposite signs.
To see which sign is the correct one (although this is not important for our *
*main results), we
need to compute the Euler class in one particular case. For this one can take t*
*he well known
cohomogeneity one action by SO(4) on S7 (see e.g. [TT ] ) which is given by the*
* representation
c^3c1(the isotropy reperesentation of the rank 2 symmetric space G2= SO(4)). *
*This action
preserves the Hopf fibration S3 ! S7 ! S4 with Euler class 1. By computing the*
* isotropy
CURVATURE AND SYMMETRY OF MILNOR SPHERES 13
groups of this SO(4) action, one shows (cf. [GZ1 ]) that they are the same as t*
*he_ones_for P-3;1
and hence k = (p2-- p2+)=8 as claimed. *
* |__|
Corollary 3.7.Every principal S3, respectively SO(3), bundle over S4 has a coho*
*mogeneity
one action by G = SO(4), respectively G = SO(3)xSO (3), in fact in general seve*
*ral inequivalent
ones.
Proof.We only need to convince ourselves that every integer k can be written as*
* (p2-- p2+)=8,
where p 1 mod 4. Set p- = 4r + 1; p+ = 4s + 1 and hence k = (r - s)(2r + 2s +*
* 1). Then
if we let r = -s, we get k = -2s, for r = s + 1 we get k = 4s + 3 and for r = s*
* - 1 we
get k = -4s + 1. These solutions can also be written in the following more conv*
*enient form:
(p-; p+) = (2k + 1; -2k + 1) if k 0 mod 2, (p-; p+) = (-k - 2; -k + 2) if k 1*
* mod 4 and
(p-; p+) = (k + 2; k - 2) if k 3 mod 4. Hence every integer k can be achieved,*
*_in_general in
several different ways. |*
*__|
Remark 3.8.For each value of k 6= 0 there exist only finitely many solutions of*
* k = (p2--
p2+)=8 = (r - s)(2r + 2s + 1), which can all be described as follows: Set m = *
*r - s and
n = 2r + 2s + 1. Then k = nm with n odd, and r = (2m + n - 1)=4; s = (-2m + n -
1)=4. Hence for each way of writing k as a product nm with n odd (including si*
*gn changes
for both n and m), we get a solution for p- and p+, if r and s are integers. N*
*otice that
if k = 2t, then m = 2t; n = 1 and hence one only gets one solution: p- = 2k + *
*1; p+ =
-2k + 1 and it is not hard to see that in all other cases, one obtains several *
*solutions. Hence
all principal S3 bundles with k 6= 2t have several inequivalent cohomogeneity o*
*ne actions by
G = SO(4). If, e.g., k = 105, the following is the complete set of 8 solutions*
*: (p-; p+) =
(29; 1); (-31; -11); (37; -23); (41; 29); (-47; 37); (73; -67); (-107; -103); (*
*-211; 209).
If k = 0, i.e., on P7 = S4 x S3, we obtain infinitely many inequivalent cohom*
*ogeneity one
actions corresponding to p- = p+.
Using the principal S3 bundle Pk with Euler class k, we can consider the asso*
*ciated 2-sphere
bundle Mk = Pk xS3S2 ! S4, where S3 acts on S2 via the two fold cover S3 ! SO(3*
*). This
can also be described as Mk = Pk=S1 with S1 S3. We now observe the following i*
*nteresting
consequence of our results:
Corollary 3.9.The total space of the S2 bundles Mk ! S4, which admit a metric w*
*ith non-
negative sectional curvature, have the same integral cohomology groups as CP3, *
*but distinct
cohomology rings for k 2.
Proof.The Gysin sequence of the sphere bundle S2 ! Mk ! S4 yields that the non-*
*zero
cohomology groups H*(Mk; Z) are H0 = H2 = H4 = H6 = Z. From the Gysin sequence *
*of
the circle bundle S1 ! Pk ! Mk, and H4(Pk; Z) = Zk, we get that if x; y are the*
* generators
in H2(Mk; Z) and H4(Mk; Z) , then x2 = ky. Hence Mk all have the same cohomolog*
*y groups
as CP3 , but distinct cohomology rings, as long as k 2. Notice that Mk and M-*
*k are __
diffeomorphic, M1 is diffeomorphic to CP3, and M0 is diffeomorphic to S2 x S4.*
* |__|
Next, we consider the case of principal S3 x S3 bundles P over S4 and the cor*
*responding
principal SO(4) bundles P* ! S4 with P* = P=(-1; -1). These bundles are classi*
*fied by
elements of ss3(S3 x S3) = ss3(SO (4)) = Z Z and hence by pairs of integers (k*
*; l). For this
identification, we use the convention in [Mi]: To an element (k; l) we associat*
*e the element in
ss3(SO (4)) given by q 2 S3 ! (u ! qkuql) 2 SO(4). Under the two-fold cover S3x*
* S3 ! SO(4)
given by (q1; q2) ! (u ! q1uq-12) this hence corresponds to the element q ! (qk*
*; q-l) in
ss3(S3 x S3). Another way to describe these integers is as follows: If we start*
* with a principal
14 KARSTEN GROVE AND WOLFGANG ZILLER
S3x S3 bundle Pk;l! S4, then we obtain two principal S3 bundles P=S3x 1 and P=1*
* x S3 and
these are now classified by their Euler class -l and k.
To construct cohomogeneity one actions on these principal bundles, we start w*
*ith the group
diagram
S3ix S3 xUS3U
iiiiii UUUU
iiiii UUUUU
iiii UUUUU
(3.10) (eip-; eiq-; ei) [ (j; j; j)K0- (ejp+; ejq+; ej ) [ (i; i; i)K*
*0+
UUUU iiiii
UUUUU iiiii
UUUUU iiiii
UUUUU iiii
4Q
which defines a cohomogeneity one manifold P10p-;q-;p+;q+as long as p ; q 1 m*
*od 4. S3xS3x1
acts freely on it with quotient S4 and hence Pp-;q-;p+;q+is a principal S3 x S3*
* bundle over S4.
As such, it is classified by two integers k and l as above. The analogue of Pro*
*position 3.3 is now
Proposition 3.11.The principal S3 x S3 bundle Pp-;q-;p+;q+! S4 is classified by*
* k = (p2--
p2+)=8 and l = -(q2-- q2+)=8. Hence every principal S3x S3, respectively SO(4) *
*bundle over S4
has a cohomogeneity one action by G = S3xS3xS3=(1; 1; 1), respectively G = SO(4*
*)xSO (3).
Proof.The formula for k and l follows from (3.3)since the group diagram for P=S*
*3 x 1 x 1 is
the S3 x S3 cohomogeneity one picture for Pq-;q+and hence l = -(q2-- q2+)=8 and*
* similarly
k = (p2-- p2+)=8.
As before it follows that for each k; l , there exist solutions p ; q to k =*
* (p2-- p2+)=8 and
l = -(q2-- q2+)=8 with p ; q 1 mod 4. If k 6= 0; l 6= 0, there are only finit*
*ely many solutions,
and for k 6= 0; l = 0, i.e., on P7kx S3, there exist infinitely many different *
*cohomogeneity one
actions.
The ineffective kernel of the S3 x S3 x S3 action is (1; 1; 1), hence on P* =*
* P=(-1; -1; 1)_
the effective action is by S3 x S3 x S3=<(-1; -1; -1); (-1; -1; 1)> = SO(4) x S*
*O(3). |__|
We finally point out that among the linear cohomogeneity one actions on spher*
*es [HL ],
only S2; S3; S4; S5 and S7 admit cohomogeneity one actions where both singular *
*orbits have
codimension 2. Moreover in each case there is only one effective action, and t*
*he groups are
S1; T2; SO(3); SO(2) SO(3) and SO(4) respectively. Among the non-linear cohomog*
*eneity one
actions, there exist infinitely many such actions by SO(2)SO (3) on S5.
Since ss4(SO (3)) = Z2; ss4(SO (4)) = Z2 Z2 , and ss4(SO (5)) = Z2 there are*
* only 1,3, re-
spectively 1 non-trivial vector bundle among the 3,4, respectively 5-dimensiona*
*l vector bundles
over S5. One easily shows that the total space of each of the corresponding pri*
*ncipal bundles is
diffeomorphic to a Lie group and hence all vector bundles and sphere bundles ov*
*er S5 admit a
metric with non-negative curvature.
We now explore the consequences to the existence of the SO(4) action on S7 fo*
*r vector bundles
and sphere bundles over S7. Since ss6(SO (k)) = 0 for k = 2; 5; 6; 7 (see [Ja])*
* it follows that only
principal SO(3) and SO(4) bundles over S7 can be non-trivial, and both admit tw*
*o-fold covers
to principal S3 and S3x S3 bundles. We first consider the case of principal S3 *
*bundles. As was
mentioned in the proof of (3.3), the cohomogeneity one picture for the SO(4) ac*
*tion on S7 is
the same as that for P-3;1. Hence, if we apply the construction in Section 1 to*
* obtain principal
S3 bundles over S7, one is forced to consider the same cohomogeneity one pictur*
*e as that for
Pp-;-3;p+;1. By Proposition 3.11, Pp-;-3;p+;1= Pk;1is a principal S3xS3 bundle *
*over S4, where
k = (p2-- p2+)=8. One can of course also argue directly, that for every princip*
*al S3x S3 bundle
CURVATURE AND SYMMETRY OF MILNOR SPHERES 15
Pk;1over S4, we have Pk;1=S3 x 1 = P1 = S7 and hence Pk;1can be regarded as a p*
*rincipal
S3 bundle over S7. As such, it is classified by an element r 2 ss6(S3) = Z12(s*
*ee [Ja]), and
it was shown in [Ri3] that r = k(k + 1)=2 and hence each principal S3 bundle ov*
*er S7 with
r = 0; 1; 3; 4; 6; 7; 9; 10 can be written in the form Pk;1in infinitely many w*
*ays. We thus obtain:
Corollary 3.12.Eight of the 12 principal S3 bundle over S7, classified by r = 0*
*; 1; 3; 4; 6; 7; 9
and 10, admit infinitely many cohomogeneity one actions by S3 x S3 x S3= (1; 1*
*; 1).
As a consequence, the associated bundles over S7 with fiber S2 or R3also carr*
*y infinitely many
metrics with non-negative curvature. Note, however, as was done in [Ri3], that *
*the total space
of the principal S3 bundles over S7 not achieved by (3.12), i.e., r = 2; 5; 8; *
*11 are diffeomorphic
to the corresponding ones for r = 10; 7; 4; 1 -2; -5; -8; -11 mod 12. In fact,*
* they are simply
the pull back of these bundles via the reflection R in the equator S6 in S7. As*
* a consequence
the corresponding associated S2 and R3 bundles also have diffeomorphic total sp*
*aces. Thus all
3-dimensional vectorbundles and corresponding sphere bundles over S7 have compl*
*ete metrics
of non-negative curvature.
Similarly, principal S3xS3 bundles over S7are classified by (r; s) 2 Z12Z12, *
*and it follows as
in (3.11)that every such bundle, with r; s = 0; 1; 3; 4; 6; 7; 9; 10, admits in*
*finitely many cohomo-
geneity one actions by S3xS3xS3xS3. As before the principal bundles with r; s =*
* 2; 5; 8; 11 as
well as the corresponding associated bundles with fiber S3 or R4 have total spa*
*ces diffeomorphic
to the ones with r; s = 0; 1; 3; 4; 6; 7; 9; 10. In summary:
Corollary 3.13.All three dimensional and 80 of the 144 four dimensional vector *
*bundles over
S7, as well as the corresponding sphere bundles, have metrics with non-negative*
* sectional cur-
vature.
4. Almost Free SO(3) Actions
As we have seen in Section 3, there are typically many different ways of repr*
*esenting the
principal bundles discussed in this paper as cohomogeneity one manifolds. This *
*will in general
yield different induced actions on associated bundles, and will enable us, in p*
*articular, to prove
Theorem C and D in the Introduction.
Recall, that any S3 bundle over S4 is associated to a principal SO(4) bundle *
*over S4, which
in turn is determined by its two-fold universal cover, a principal S3 x S3 bund*
*le over S4. Each
of these bundles are thus determined by a pair of integers (k; l) 2 Z x Z = ss3*
*(S3 x S3) =
ss3(SO (4)), where we use the convention described in the previous section. For*
* (k; l) 2 Z x Z let
Mk;l! S4; P*k;l! S4; Pk;l! S4 denote the corresponding S3 bundle, principal SO(*
*4) bundle
and principal S3 x S3 bundle respectively. In Section 3 we saw that for any cho*
*ice of integers
p ; q 1 mod 4, satisfying k = (p2-- p2+)=8 and l = -(q2-- q2+)=8 there is a c*
*ohomogeneity
one action by S3 x S3 x S3 on Pk;lwith diagram (3.10), which induces an effecti*
*ve action of
SO(4)xSO (3) on P*k;l. The SO(4) subaction is the free principal action on P*k;*
*land the subaction
by SO(3) is a lift of the cohomogeneity one action of SO(3) on S4. In particula*
*r SO(3) acts on
the total space of every associated S3 bundle taking fibers to fibers.
Theorem 4.1.The SO(3) action on Mk;l, induced from (3.10)as described above, pr*
*eserves the
S3 fibration Mk;l! S4 and has exactly the following orbit types:
(1); (Z2); (D2); (D|p-+q-|_); (D|p--q-|_); (D|p++q+|_); and (D|p+-q*
*+|_)
2 2 2 2
where D0, in this context, should be interpreted as both SO(2) and O(2).
16 KARSTEN GROVE AND WOLFGANG ZILLER
Proof.To compute the isotropy groups of this action, we apply the Isotropy Lemm*
*a 1.7 to the
corresponding ineffective S3 action on Mk;l= P*k;lxSO(4)S3 = Pk;lxS3xS3S3. In t*
*he latter
description S3 x S3 acts on S3 via quaternion multiplication (Q1; Q2) . v = Q1v*
*Q-12. The
isotropy groups of this S3x S3 action on S3 are 4S3 S3x S3 and conjugates ther*
*eof, i.e. the
subgroups S3a= {(b; aba-1) | b 2 S3} for some a 2 S3.
We can now read off the isotropy groups of the S3-action from (1.7). They ar*
*e OE-1(S3a)
and OE-10(S3a), where OE0:Q ! S3 x S3 is the diagonal embedding and OE :Pin(2)*
* ! S3 x S3
are the homomorphisms determined by OE-(ei) = (eip-; eiq-); OE-(j) = (j; j) and*
* OE+(ej ) =
(ejp+; ejq+); OE+(i) = (i; i).
Clearly OE-10(S3a) = <-1> = Z2 unless a 2 ; ; , and in these ca*
*ses OE-10(S3a) =
*; ; = Z4, except when a = 1, in which case OE-10(S3a) = Q.
Now consider those ei1; jei2 2 Pin(2) such that (eip-1; eiq-1) or j(eip-2; ei*
*q-2) 2 S3a.
If a = eit, then aeip-1a-1 = eiq-1 implies that ei(p--q-)1= 1 and ajeip-2a-1 = *
*jeiq-2
implies that ei(p--q-)2-2it= 1. Hence OE-1-(S3eit) = Pin(2) for
p- 6= q-. In the case of p- = q-, we get OE-1-(S3eit) = {ei} = S1 Pin(2) if a *
*= eit6= 1 and
OE-1-(S31) = Pin(2).
If a = jeit, then aeip-1a-1 = eiq-1implies that ei(p-+q-)1= 1 and ajeip-2a-1 *
*= jeiq-2
implies that ei(p-+q-)2-2it= 1. Hence OE-1-(S3jeit) = as the only
possibility, since p- 6= -q- when p-; q- 1 mod 4.
If a 62 {eit} [ j{eit}, then the only 1 with aeip-1a-1 = eiq-1is given by 1 =*
* 0; ss, i.e.
ei1 = 1 and there are precisely two values of 2 ( 2 and 2+ ss) with ajeip-2a-1 *
*= jeiq-2
and hence OE-1-(S3a) = = Z4
The groups OE-1+(S3a) are computed in exactly the same way. Finally, to obtai*
*n the isotropy
groups of the effective action by SO (3), we only need to observe that under th*
*e two-fold
cover S3 ! SO (3) the images of Z4; Q; (for p even), {ei}; P*
*in(2)_are_equal
to Z2; D2; Dp=2; SO(2) and O(2) respectively. *
* |__|
As pointed out in (3.7), for each (k; l) with k 6= 0; l 6= 0, there are only *
*finitely many solutions
(p ; q ) to the equations k = (p2-- p2+)=8; l = -(q2-- q2+)=8, when p ; q 1 m*
*od 4. As
explained there also, one of these solutions can be written as (p-; p+) = (2k +*
*1; -2k +1); (-k -
2; -k+2) or (k+2; k-2) when k 0 mod 2; k 1 mod 4 or k 3 mod 4 respectively. *
*Similarly
(q+; q-) = (2l + 1; -2l + 1); (-l - 2; -l + 2) or (l + 2; l - 2) when l 0 mod *
*2; l 1 mod 4 or
l 3 mod 4 respectively. If say l = 0, (q-; q+) = (4n + 1; 4n + 1) is obviously*
* a solution for all
n.
We exhibit the isotropy groups, other than (1); (Z2) and D2, of these particu*
*lar SO(3) actions
on Mk;lin the following table:
_________________________________________________
| | k even | k odd |
|_______|____________________|___________________|
| l even |D , D |D , D |
|__________|_|k+l|__|k-l1|___|_|k+2l1|=2__|k-2l3|=2|_
| l oddD | , D | D , D |
|________||2k+l1|=2_|2k-l3|=2|_|k-l4|=2__|k+l|=2|_
| l = 0 |D , D D| , D |
|_________|_|2n+1k|___|2nk|__||4n+3k|=2_|4n-1k|=2|_
Table 4.2. Isotropy Groups
In particular, we get:
CURVATURE AND SYMMETRY OF MILNOR SPHERES 17
Corollary 4.3.Each of the manifolds Mk;0admit infinitely many inequivalent almo*
*st free
SO(3) actions preserving the fibration Mk;0! S4 and inducing the same cohomogen*
*eity one
action on S4.
Remark 4.4.Notice that Mk;0are also precisely those Mk;lwhich can be regarded n*
*ot only as
S3 bundles over S4, but also as principal S3 bundles. Indeed, the glueing map q*
* ! {u ! qku}
for the bundle Mk;0commutes with the right action by S3 and hence S3 acts freel*
*y on Mk;0with
quotient S4. Thus Mk;0= Pk and one can therefore also directly lift the SO(3) a*
*ction on S4
using the cohomogeneity one action on Pk = Pp-;p+from Corollary 3.7. But this i*
*s an effective
action of S3 on Mk;0, instead of SO(3), and one easily sees that it also acts a*
*lmost freely with
isotropy groups the binary dihedral groups . These actions of S3, f*
*initely many for
each k, commute with the free principal action of S3, whereas the infinitely ma*
*ny almost free
actions of SO(3) in Corollary 4.3 do not.
A more detailed version of Corollary 4.3 in the special case of the Hopf fibr*
*ation S7 = M1;0!
S4 is included in the following result, which also implies Theorem C in the Int*
*roduction.
Theorem 4.5.For each n there is an action of SO(3) on S7 which preserves the Ho*
*pf fibration
S7 ! S4 and has exactly the following orbit types:
(1); (Z2); (D2); (D|2n-1|); (D|2n|); (D|2n+1|); (D|2n+2|)
where as before (D0) stands for (SO (2)) and (O (2)). In particular, for n 6= 0*
*; -1 this action is
almost free. Moreover the action does not extend to the disc D8 if n 6= 0; 1; 2.
Proof.The first part can just be read off from Theorem 4.1 and Table 4.2 when k*
* = 1; l = 0.
To prove that the actions do not extend to the disc, we use the work of Olive*
*r in [Ol] concerning
the structure of fixed point free SO(3) actions on discs. First, however, consi*
*der the case of an
SO(3) action on D8 with non-empty fixed point set DSO(3)6= ;. Since SO(3) has o*
*nly irreducible
representations in odd dimensions, it follows that the slice representation of *
*SO(3) at a fixed
point, restricted to SO(2) SO(3), has to have a fixed vector and hence dimDSO(*
*2)> 0. By
Smith theory DSO(2) DSO(3)has the integral cohomology of a point (cf. e.g. [Br,*
* Chapter
III]). In particular any component of DSO(2)with positive dimension has non-emp*
*ty boundary,
and @DSO(2)= DSO(2)\ @D = SSO(2). Thus , if an almost free action of SO(3) on S*
*7 extends
to D8, it cannot have fixed points.
Next, we will show, using [Ol], that any fixed point free action of SO(3) on *
*D8 has (Z3) or
(D3) among its orbit types on the boundary sphere S7, which then proves our the*
*orem. From
Corollary 1 of [Ol] we know in particular that D3 occurs as isotropy group for *
*any SO(3) action
on D8 without fixed points. In fact, it follows from Lemma 1 and Lemma 3 in [Ol*
*], that the
octahedral group O SO(3) has an isolated fixed point in the interior of D8 and*
* that D3 occurs
as an isotropy group of the linear representation of O at such a fixed point. I*
*n particular, D3 is
the isotropy of an interior point p 2 D8 and dimDD3 > 0. Again by Smith theory *
*DZ3 DD3
has the Z3-cohomology of a point. Thus each component of DZ3intersects @D = S n*
*on-trivially
in SZ3.
Relative to an SO(3)-invariant metric on D8, join p 2 DD3 DZ3 to a closest p*
*oint q 2
@DZ3 = SZ3 inside DZ3. In particular, Z3 SO(3)q. But SO(3)q also fixes the nor*
*mal vector
to the boundary at q, hence the minimal geodesic above, and therefore p. Hence*
* SO(3)q_
SO(3)p = D3, which implies that SO(3)q= Z3 or D3. *
* |__|
Remark 4.6.The group of linear symmetries of the Hopf fibration S7 ! S4 is give*
*n by (Sp(2)x
Sp(1))=Z2 where Sp(2) acts via matrix multiplication on S7 H2, and Sp(1) is th*
*e right Hopf
action. The cohomogeneity one action of SO(3) on S4 defines an embedding of a m*
*aximal SO(3)
18 KARSTEN GROVE AND WOLFGANG ZILLER
in SO(5), which under the two fold cover Sp(2) ! SO(5) lifts to a maximal Sp(1)*
*m Sp(2).
Hence (Sp(1)m x Sp(1))=Z2 = SO (4) (Sp(2) x Sp(1))=Z2, which also happens to b*
*e the
cohomogeneity one action of SO(4) on S7 with singular orbits of codimension two*
*, projects to
SO(3) SO(5) under the Hopf map. Thus there are two linear lifts of the cohomog*
*eneity one
action of SO(3) on S4. One is the almost free action by Sp(1)m on S7 with isotr*
*opy groups 1,
Z3, and D*3and the other is the action by SO(3) = 4 Sp(1)=Z2 SO(4), which has *
*isotropy
groups 1, Z2, D2, SO(2), and O(2). In particular, none of the actions in (4.5)a*
*re linear, except
n = 0, which is the latter one.
We now analyze the ramifications of (4.1)to the Milnor spheres. Recall that *
*Milnor [Mi]
showed that the Euler class of Mk;l! S4 is equal to e = k + l. Hence the Gysin *
*sequence and
Smale's solution of the Poincare conjecture implies that Mk;lis homeomorphic to*
* S7 if and only
if k + l = 1. By changing the orientation if necessary, we can assume that k + *
*l = 1. For
Theorem D, we also need the diffeomorphism classification of these homotopy sph*
*eres Mk;1-k
due to Eells and Kuiper [EK ]. According to [EK ], Mk;1-kis diffeomorphic to Mm*
*;1-m if and
only if k(k - 1) m(m - 1) mod 56, i.e. the diffeomorphism class is given by k(*
*k-1)_2mod 28 in
the group Z28of exotic 7-spheres.
Since k(k-1)_2mod 28 takes on precisely 16 different values [EK ], there are *
*16 different diffeo-
morphism types of topological 7-spheres which fiber over S4 with S3 as fiber. U*
*sing Theorem
E and F, this completes the proof of Theorem A in the introduction. In passing,*
* we note that
M2;-1generates the group Z28of all homotopy 7-spheres via connected sum.
It is now clear that the SO(3) actions considered in (4.1)(cf. Table 4.2) on *
*Mk;1-kand Mm;1-m
are, in general, different actions on the same homotopy sphere when k(k-1)_2 m(*
*m-1)_2mod28.
To make this more concrete, we exhibit the following special cases. As pointed*
* out in [EK ,
p.102], k(k - 1) m(m - 1) mod 56if and only if m k or 1 - k mod 7 , and m k *
*or 1 - k
mod 8 . Choosing the special case m k mod 56, we get:
Corollary 4.7.Let Mk;1-kbe any of the homotopy 7-spheres considered above. Then*
* for each
integer n, Mk;1-ksupports an SO(3) action with the following orbit types:
(1); (Z2); (D2); (D|k+56n+11|=2); (D|3(k+56n)-13|=2)
if k is even, and
(1); (Z2); (D2); (D|k+56n-21|=2); (D|3(k+56n)-23|=2)
if k is odd.
Of course, there are many more actions given by Theorem 4.1 on the exotic sph*
*eres Mk;1-k,
most of which are almost free. Even for the standard sphere, we get many additi*
*onal almost
free actions, besides the ones described in Theorem C, whenever Mk;1-kis diffeo*
*morphic to S7,
i.e for k 0; 1 mod 7 and k 0; 1 mod 8. They preserve a different fibration of*
* S7 by 3-spheres,
but in this case, we get only finitely many actions for each fibration.
Not all of the actions in Table 4.2 and Corollary 4.7 are almost free. Indee*
*d, in the case
of Mk;lwith k = -l = 2r there exists no almost free lift preserving the fibrati*
*on, since (3.8)
implies that the only action obtained from (4.1)is the one described in Table 4*
*.2. For the
homotopy spheres Mk;1-k, the only actions in Table 4.2 which are not almost fre*
*e occur in the
case of k = 0; 1; -2; 3. Of course for k = 0; 1, which corresponds to the Hopf *
*fibration, (4.5)
gives rise to infinitely many almost free actions preserving the fibration. For*
* k = -2; 3, i.e. on
M-2;3= M3;-2, (4.1)implies that there exist one further action besides the one *
*described in
Table 4.2. It corresponds to (p-; p+) = (5; 1) , (q-; q+) = (-3; 5) and hence g*
*ives rise to an
almost free action with isotropy groups 1; Z2; D2; D3; D4. Hence in the case of*
* the homotopy
CURVATURE AND SYMMETRY OF MILNOR SPHERES 19
spheres Mk;1-kthere always exists at least one almost free action preserving th*
*e fibration. This
implies Thorem D in the Introduction.
We also observe that the SO(3) actions on Mk;lextend to an action of O(3). Fo*
*r this just
note that the element -id 2 SO(4) commutes with the structure group and the SO(*
*3) action
on P*k;land hence induces an action on the associated bundle.
All the actions in this section on Mk;lare isometric actions with respect to *
*the non-negatively
curved metrics we constructed in Section 2. We now consider the question whethe*
*r these metrics
can ever be isometric to each other, and show that, at least in the case of the*
* homotopy spheres
Mk;1-k, this can almost never be the case:
Proposition 4.8.If Mk;1-kand Mm;1-m are diffeomorphic, then the metrics of non-*
*negative
curvature constructed on them in Section 2 can only be isometric, if the corres*
*ponding isometric
SO(3) actions are conjugate. In particular, we obtain infinitely many such met*
*rics on each
Mk;1-kwhich are not isometric to each other.
Proof.To see this, we use the result by E.Straume that the degree of symmetry o*
*f any exotic 7-
sphere is at most 4, see [St1, Theorem C]. In other words the dimension of any *
*compact Lie group
G that acts effectively on an exotic 7-sphere is at most 4. Now fix a metric on*
* = Mk;1-ksuch
that one of the above SO(3) actions is isometric and let G SO(3) be the id-com*
*ponent of its
full isometry group. Following Straume, G is either SO(3) or a finite quotient *
*of SO(3) x SO(2).
In particular G contains only one subgroup SO(3), so if one of the other action*
*s of SO(3) on
is a subgroup of G, then the two actions must be conjugate. Hence, if the SO(3)*
* actions are
inequivalent, the corresponding metrics cannot be isometric and in fact have_di*
*fferent isometry
groups. |__|
We suspect that O(3) will always be the full isometry groups of the metrics w*
*e constructed
on Mk;l, see the next sections for some comments on this question.
5.Remarks and Open Problems
Recall the two steps in our approach to the Cheeger-Gromoll problem : (1) Any*
* principal
SO(k)-bundle over S4 has a cohomogeneity one G-structure with SO(k) G (and wit*
*h singular
orbits of codimension two). (2) Any cohomogeneity one G-manifold (with singula*
*r orbits of
codimension two) admits a G-invariant metric with non-negative curvature. As su*
*ggested in the
introduction, it is plausible that any cohomogeneity one manifold supports an i*
*nvariant metric
with non-negative curvature. This is just one of the reasons for the following *
*challenging:
Problem 5.1.Which principal SO(k)-bundles over Snwith k n support a cohomogene*
*ity one
G-structure with SO(k) G ?
Note that for each of the ways of writing Sn as a cohomogeneity one manifold *
*(cf. [HL ]), our
construction in Section 1 will in general yield several candidates for such bun*
*dles. In some cases
it will give rise to infinitely many such candidates, namely whenever S1 is a n*
*ormal subgroup
of K- or K+. One can further increase the flexibility of our construction by us*
*ing subactions of
the usual cohomogeneity one actions listed in [HL ], which are still cohomogene*
*ity one (see [St2]
for a complete list), and by making the actions ineffective. Of particular inte*
*rest here are of
course SO(8)-bundles over S8, since 4095 exotic 15-spheres can be presented as *
*(linear) 7-sphere
bundles over the 8-sphere (cf. [Sh] and [EK ]).
In view of our examples, it would be interesting to study in more detail the *
*topology of the
principal S3 x S3 bundles Pk;l! S4 and their associated 3-sphere bundles Mk;l! *
*S4. As we
observed before, in the case of the sphere bundles Mk;l, one can recover the Eu*
*ler class e = k + l
20 KARSTEN GROVE AND WOLFGANG ZILLER
from the torsion in H4. But for the principal bundles Pk;lthe only non-zero coh*
*omology groups
H*(Pk;l; Z) are H0 = H3 = H7 = H10= Z; H4 = Z(k;l). Indeed, in Section 3 we saw*
* that among
the total spaces Pk;1there are only 7 diffeomorphism types. In the special case*
* of the homotopy
spheres Mk;1-k, the total space has been classified up to diffeomorphism in [EK*
* ]. It would be
interesting to extend this classification:
Problem 5.2.Classify the manifolds Pk;land Mk;lup to homotopy, homeomorphism an*
*d dif-
feomorphism type.
See [Wh ] and [Tam ] for a partial classification of Mk;lup to homotopy and h*
*omeomorphism
type. Also notice that it was shown in [DW ] that for the corresponding vector *
*bundles Ek;l! S4,
the total spaces are diffeomorphic if and only if they are isomorphic as vector*
* bundles.
Since our manifolds Mk;lhave trivial ss2 and ss3 = Zk+l, one easily sees that*
* if one of the
manifolds Mk;lis homotopy equivalent to a homogeneous space G=H, then G=H = Sp(*
*2)= Sp(1),
where Sp(1) is one of the three possible embeddings of Sp(1) in Sp(2). Hence G=*
*H = Sp(2)= Sp(1)
x 1 = S7, G=H = Sp(2)=4 Sp(1) = T1S4 or G=H = Sp(2)= Sp(1) = SO (5)= SO(3) where
SO(3) SO(5) is the maximal embedding given by the cohomogeneity one action of *
*SO(3) on
S4. It was shown in [Be] that B7 = SO(5)= SO(3) carries a metric of positive se*
*ctional curvature
and that H4(B7; Z) = Z10. One easily shows that the first Pontrayagin class of*
* the tangent
bundle of B7 is equal to 6 times a generator in H4 = Z10. Furthermore, in [Tam *
*] it was shown
that the first Pontrayagin class of the tangent bundle of Mk;lis equal to 4l ti*
*mes a generator
in H4(Mk;l; Z) = Zk+l. Hence B7 cannot be homeomorphic to a principal S3 bundle*
* over S4,
i.e. l = 0. But it would be interesting to know if it can be homeomorphic or di*
*ffeomorphic to a
sphere bundle.
For the principal S3 bundles Pk over S4, we have that Pk is diffeomorphic to *
*P-k and that
P1 = S7. Furthermore, P2 = T1S4 since on T1S4 = Sp(2)=4 Sp(1) one has the fre*
*e action
by S3 given by left multiplication with diag(q; 1) and since H4(T1S4; Z) = Z2, *
*it follows that
this principal S3 bundle has k = 2. Hence P1 and P2 are diffeomorphic to homo*
*geneous
spaces. From the above, it follows that all other Pk are strongly inhomogeneous*
*, i.e. they do
not have the homotopy type of a homogeneous space, except possibly P10 which is*
* at least not
homeomorphic to a homogeneous space.
For the associated 2-sphere bundles Mk ! S4 considered in (3.9), it follows f*
*rom [On , Theorem
6] that the only homogeneous spaces that have the same integral cohomology grou*
*ps as CP3,
are CP3 = M1 itself and S2 x S4 = M0. Hence Mk with |k| 2 do not have the hom*
*otopy
type of a homogeneous space.
By construction, the total space P, respectively M, of any principal bundle, *
*respectively
associated sphere or vector bundle considered in this paper, as well as the bas*
*e S4, is the union
D- [ D+ of two disc bundles with common boundary S = @D- = @D+. Moreover, relat*
*ive to
the metrics of non-negative curvature on D- [ D+, S is totally geodesic. From t*
*he Cheeger-
Gromoll soul-construction [CG ] and Perelman's rigidity theorem [Pe] it follows*
* in particular that
there are two-planes with zero curvature at every point of D- [ D+.
All cohomogeneity one G-manifolds considered in this paper have G = S3; S3xS3*
* or S3xS3x
S3 (acting possibly ineffectively). For the metrics on D = G xK D2, note that*
* we can choose
a fixed biinvariant metric on G scaled by a fixed constant (e.g. a=4/3) in the *
*K direction, and
on D2 we choose a metric dt2+f2(t)d2 with a fixed convex function f. Although K*
* and hence
Qa depends on the particular example, it easily follows from (2.4)and the O'Nei*
*ll submersion
formula that there is a uniform bound C for the curvatures of all principal bun*
*dles considered
here, i.e. 0 sec(P) C. But, as explained in (2.7), there is no bound on the*
* diameter,
diam(P), since the length of the circles K goes to infinity. It is also appare*
*nt that all examples
CURVATURE AND SYMMETRY OF MILNOR SPHERES 21
have a uniform lower bound on their volumes, i.e., there is a v > 0 such that v*
*ol(D- [ D+) v
since this is true of vol(S). Similar bounds for curvature, volume and diamete*
*r hold for the
associated bundles and the base S4.
All these examples complement Cheegers classical finiteness theorem [Ch2] and*
* recent finite-
ness theorems by Petrunin-Tuschmann [PT ] and Tapp [Ta].
Since our examples of S3 bundles over S4 are 2-connected, they illustrate the*
* sharpness of
the following, even within the class of non-negatively curved manifolds.
Theorem 5.3 (Petrunin-Tuschmann).For each n and D, C > 0 there exist only finit*
*ely many
diffeomorphism types of simply connected compact Riemannian n-dimensional manif*
*olds M,
with | secM| C, diamM D and finite ss2(M).
In the special case where the lower curvature bound is a fixed positive numbe*
*r ffi > 0, the same
conclusion was obtained simultaneously by Fang and Rong in [FR] (in that case t*
*he bound on
diamM is automatic by the Bonnet-Myers theorem). Motivated by this, the followi*
*ng conjecture
was proposed in [FR]:
Conjecture (Rong).For each n, there are at most finitely many 2-connected posit*
*ively curved
n-manifolds.
Our examples show that this conjecture is false, if we replace positive with *
*non-negative
curvature.
The following result was first obtained in [GW ] in the special case where =*
* Sn(1). It was
then extended to arbitrary souls in [Ta].
Proposition 5.4.For each n and C; D; V > 0 and each metric on with diam D and
vol > V , there exist only finitely many vector bundles M over with a complete*
* metric of
non-negative curvature, such that is the soul and such that the sectional curv*
*atures of all
2-planes of M either tangent to or normal to are bounded above by C.
By the above remarks, in our examples of 3 and 4 dimensional vector bundles o*
*ver S4, we
have bounds on the curvatures of M and the volume of the soul , which is always*
* the zero
section of the vector bundle and isometric to the metric on the base S4. But th*
*e diameter of
the base necessarily goes to infinity since the length of the circles K =H = S1*
* goes to infinity.
We conclude our discussion with a few more remarks about the geometry and sym*
*metry of
our examples, in particular the principal S3x S3 bundles Pk;l! S4 and their ass*
*ociated sphere
bundles S3 ! Mk;l! S4 and vector bundles R4 ! Ek;l! S4. Much work has been done
previously on trying to construct metrics with non-negative or positive curvatu*
*re on the total
spaces Pk;l; Mk;land Ek;l. A natural approach is to consider Kaluza Klein type *
*metrics on the
principal L bundle P, where one chooses a principal connection to define the ho*
*rizontal space,
pulls back the metric from the base to the horizontal space, and defines the me*
*tric on the fiber
to be a biinvariant or left invariant metric on L. This metric then also induce*
*s a metric on the
associated sphere bundles and vector bundles. We call these metrics connection *
*type metrics.
In all three cases, the fibers of the projection onto the base are totally geod*
*esic and isometric to
each other. The metrics of positive Ricci curvature on Pk;land Mk;lconstructed *
*in [Na] and [Po]
are exactly of this type. But for the construction of non-negatively curved met*
*rics this approach
has been successfull only in the case where the principal bundle is a Lie group*
* or a homogeneous
space. In [DR ] it was shown that the only case in which the induced metric on *
*Mk;lhas positive
curvature, is when k = 0; l = 1 or k = 1; l = 0, i.e when Mk;l= S7. It would be*
* interesting to
know if non-negatively curved connection type metrics exist on the bundles Mk;l*
*with kl 6= 0; 1.
The metrics in our examples are not of this type. We will show that in our case*
* the metrics on
22 KARSTEN GROVE AND WOLFGANG ZILLER
the S3 fibers (as well as on the base S4) are cohomogeneity one metrics, and th*
*at the fibers are
not totally geodesic.
On Pk;lwe have the cohomogeneity one action by G = S31x S32x S33with the prin*
*cipal
bundle action given by S31x S32(but acting on the left on P). By construction, *
*the projection
Pk;l! S4 is a Riemannian submersion, with the horizontal distribution given by *
*a principal
connection, since the metric is S31x S32invariant. The same follows for the ass*
*ociated bundles
Mk;l= Pk;lxS31xS32S3 and Ek;l= Pk;lxS31xS32R4. In all three cases, the metric o*
*n the base
is given by the submersed metric on Pk;l=S31x S32, and as a cohomogeneity one m*
*etric on S4
under the action of S33, is described by three functions f1(t); f2(t); f3(t), t*
*he length of the three
action fields i*; j*; k* along c(t) with i; j; k in the Lie algebra of S33. Her*
*e c(t) is a fixed geodesic
perpendicular to all orbits, as in Section 1. Invariance of the metric under th*
*e isotropy action of
H = Q implies that these action fields must be orthogonal. It follows from Pere*
*lmann's rigidity
theorem that two of these functions are equal to 1, f2(t) = f3(t) = 1 on D- and*
* f1(t) = f3(t) = 1
on D+.
The metric on the fiber of Pk;l! S4 over the point c(t) in S4 is given by a l*
*eft invariant metric
Qton S31x S32. But this metric depends on t, and hence the fibers are not total*
*ly geodesic. This
completely describes the metric on Pk;l. It is interesting to observe that our *
*metrics are just
slightly more general than connection type metrics in that the metrics on the f*
*iber are allowed
to depend on a single parameter t.
Notice that the left invariant metric Qa on G is also right invariant under t*
*he maximal torus
T3 S31x S32x S33containing K0 which hence acts S1 ineffectively and by isometr*
*ies on each
half D = G xK D2 of Pk;l. But the intersection of T- and T+ acts trivially. F*
*urthermore, the
first two components of T3 act by isometries via right translation on the left *
*invariant metric
Qton S31x S32.
We now consider the geometry of the associated bundles Mk;l= Pk;lxS31xS32S3(r*
*) and Ek;l=
Pk;lxS31xS32R4, where we also allow ourselves the freedom of varying the radius*
* in S3(r). The
horizontal distribution and the metric on the base S4 is the same as before, an*
*d we only need to
describe the metric on the fibers. The fiber of the S3 bundle Mk;l! S4, over th*
*e point c(t) 2 S4
can be described as S31x S32xS31xS32S3(r) = S3 where the metric on S31x S32is g*
*iven by the left
invariant metric Qt. Only the right translations on S31x S32, that are still is*
*ometries of Qt, are
isometries of this metric on S3. These right translations consist of the action*
* by T2 = (ei; ei )
(in the case of D-) and this action of T2 on S3 is the standard cohomogeneity o*
*ne action on
S3. Hence all fibers of Mk;l! S4 are cohomogeneity one metrics on S3 (and in p*
*articular
not homogeneous). Choosing a basis of T2, the metric on S3 can be described by *
*the length
and inner product of the corresponding action fields along a normal geodesic in*
* the fiber. But,
unlike in the case of S4, since the principal isotropy group of the T2 action i*
*s trivial, the inner
product between these two action fields does not have to be 0. Hence the metric*
* on the fiber
is described by three functions h1(s; t); h2(s; t); h3(s; t), where s is the ar*
*c length parameter of
a normal geodesic in the cohomogeneity one metric on the fiber S3 over c(t). He*
*nce the metric
on Mk;lis completely described by (f1; f2; f3; h1; h2; h3): I x I ! R6.
Similarly, the metric on the fibers of Ek;l! S4 are cohomogeneity two metrics*
* dt2+ gtd2
with gta cohomogeneity one metric on S3 as above. In both cases, the fibers aga*
*in change from
point to point and hence are not totally geodesic.
Notice that on Mk;l(but not on Ek;l) one can describe a different metric usin*
*g the identification
Pk;lxS31xS32S3 = Pk;l=4S3 as a submersed metric from Pk;l. For this metric, the*
* horizontal
distribution and the metric on the base is the same, but the metric on the fibe*
*rs is now the
metric on S3 = 4S3OS31x S32induced from the left invariant metric Qt on S31x S3*
*2which
CURVATURE AND SYMMETRY OF MILNOR SPHERES 23
is invariant under right translations by T2 and hence again only a cohomogeneit*
*y one metric
on S3. To compare this metric with the previous metric, if we consider the metr*
*ic on Mk;l=
Pk;lxS31xS32S3(r) and let r go to infinity, then the limit is the new metric ju*
*st described.
Indeed, if we set L = S31x S32; H = 4S3, then one has the identification P xL L*
*=H ' P=H
given by [(p; `H)] ! H . `-1p. The fiber L xL L=H then gets identified with HO*
*L and if
Qt(X; Y ) = Q(AtX; Y ) it follows as in (2.1)that the metric is induced from th*
*e left invariant
2A
metric Q(_r__tAt+r2X; Y ), which as r goes to infinity, converges to Qt.
Next, we consider the isometry group of our examples. As was explained in Sec*
*tion 4, the
action of SO(3) on the principal bundle Pk;ldescends to an action of SO(3) on t*
*he total space
of Mk;lwhich acts by isometries in the metric of non-negative curvature that we*
* constructed.
Furthermore, the action can be extended to an isometric action by O(3). We susp*
*ect that this
group will always be the full isometry group of our metrics. In the special cas*
*e of exotic spheres
Mk;1-k, it follows from [St1, Theorem C], that the id component of the full iso*
*metry group can
be at most SO(3) x SO(2) or one of its finite quotients. An affirmative answer *
*to the following
question would of course rule out such an extension, and would imply that the g*
*roup SO(3) is
always the id-component of the full isometry group.
Problem 5.5.Does the SO(3) subaction of an (almost) effective SO(3) SO(2) actio*
*n on an
(exotic) 7-sphere have isotropy groups containing SO(2) ?
In [Da] it was observed that there are natural lifts of the cohomogeneity one*
* action of
SO(3)SO (2) on S4 to each of the manifolds Mk;l. Also, each exotic 7-sphere ca*
*n be exhib-
ited as a Brieskorn variety, and as such it again supports a natural action of *
*SO(3)SO (2). If the
exotic sphere is of the form Mk;1-k, this action is in general different from t*
*he previous ones. In
either case, however, the subaction of SO(3) is never almost free.
We also remark, that the action of SO(3)SO (2) on S4 lifts not only to Mk;las*
* in [Da], but to
the principal bundle Pk;las well. But this lift does not commute with the free *
*action of S3x S3
and hence one does not obtain a cohomogeneity one action on Pk;las we do in our*
* examples.
An essential difference between our examples and the Gromoll-Meyer metric [GM*
* ] on M2;-1,
is that it has a four dimensional isometry group, which agrees with the action *
*of SO(3)SO (2)
in [Da]. We finally rephrase the description of this metric on the Gromoll Meye*
*r sphere in our
context, thereby exhibiting similarities and differences.
Consider the following subgroup G = (S1 x S31) x (S32x S33) Sp(2) x Sp(2):
ae oe
cos - sin fifi
sin cos . diag(q1; q1); diag(q2;qq3)i2 Sp(1)
This subgroup acts on Sp(2) (via left and right multiplication) and any two com*
*binations of the
S3 factors act freely and one easily sees that in all cases the quotient is S4 *
*on which S1xS3 (the
S3 being the remaining S3 factor) acts by cohomogeneity one with the standard s*
*um action.
Hence G also acts by cohomogeneity one on Sp(2) with singular orbits of codimen*
*sion two and
three, and isometrically with respect to the biinvariant metric with non-negati*
*ve curvature. If
one chooses the free action by S32xS33, the principal S3xS3 bundle over S4 is t*
*he two fold cover
of the frame bundle of the tangent bundle of S4 and hence P1;1= Sp(2). If one c*
*hooses the free
action by S31x S32, it was shown in [GM ] that one obtains the principal bundle*
* P2;-1and hence
the associated sphere bundle P2;-1xS31xS32S3 = P2;-1=41;2S3 is the exotic spher*
*e M2;-1with a
submersed metric of non-negative curvature. As in our case, the action of S1xS3*
*3descends to an
action of the associated S3 bundle M2;-1which becomes an effective action by SO*
*(3) SO(2) and
24 KARSTEN GROVE AND WOLFGANG ZILLER
which, by [St1, Theorem C], is the id-component of the isometry group of the Gr*
*omoll-Meyer
metric. Notice that, as in our case, we also get a family of metrics with non-n*
*egative curvature
on the Gromoll-Meyer sphere by considering Sp(2) xS31xS32S3(r), and as r goes t*
*o infinity we
obtain the Gromoll-Meyer metric in the limit.
Of course, as a consequence of our results, it follows that P2;-1= Sp(2) also*
* has infinitely
many cohomogeneity one actions by S3x S3x S3, but with singular orbits of codim*
*ension two.
Another major difference between our metrics induced by these actions and the G*
*romoll-Meyer
metric, is that in their example there exists an open set of points in M2;-1on *
*which every two-
plane has positive curvature, whereas in our example, by construction, there ar*
*e always 2-planes
of 0 curvature at every point.
Motivated by Proposition 4.8 we conclude our discussion with the following na*
*tural question.
Problem 5.6.On each of the Milnor spheres (including the standard sphere), as w*
*ell as on the
homotopy RP5 in Theorem G, does the space of metrics with non-negative sectiona*
*l curvature
have infinitely many components ?
In a similar vein, in [KS ] it was shown that on some of the homogeneous spac*
*es SU(3)=S1 the
space of metrics with positive sectional curvature has at least two components.
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University of Maryland, College Park , MD 20742
E-mail address: kng@math.umd.edu
University of Pennsylvania, Philadelphia, PA 19104
E-mail address: wziller@math.upenn.edu
*