Table Of ContentFAKE 13-PROJECTIVE SPACES
WITH COHOMOGENEITY ONE ACTIONS
6
1
CHENXUHEANDPRIYANKARAJAN
0
2
n
a
J
Abstract. We show that someembedded standard 13-spheres inShimada’s
4 exotic15-sphereshaveZ2quotientspaces,P13s,thatarefakereal13-dimensional
1 projectivespaces,i.e.,theyarehomotopyequivalent,butnotdiffeomorphicto
thestandardRP13. AsobservedbyF.Wilhelmandthesecondnamedauthor
] in[RW],theDavisSO(2)×G2actionsonShimada’sexotic15-spheresdescend
G to the cohomogeneity one actions on the P13s. We prove that the P13s are
D diffeomorphic to well-known Z2 quotients of certain Brieskorn varieties, and
thattheDavisSO(2)×G2 actionsontheP13sareequivariantlydiffeomorphic
h. to well-known actions on these Brieskornquotients. The P13s areoctonionic
t analogues of the Hirsch-Milnorfake5-dimensional projective spaces, P5s. K.
a GroveandW.ZillershowedthattheP5sadmitmetricsofnon-negativecurva-
m
turethatareinvariantwithrespecttotheDavisSO(2)×SO(3)-cohomogeneity
[ one actions. In contrast, we show that the P13s do not support SO(2)×G2-
invariantmetricswithnon-negativesectional curvature.
1
v
3 Contents
2
7 1. Introduction 2
3 2. Preliminaries 5
0 2.1. Shimada’s exotic 15-spheres Σ15s, the embedded 13- and 14-spheres
k
.
1 and the Davis action 5
0 2.2. Brieskorn varieties, Kervaire spheres and homotopy projective spaces 8
6 3. The cohomogeneity one actions of G=SO(2) G on S13 and P13 11
1 4. The G-invariant metrics on M13 × 2 k k 17
: k
v 5. Rigidities of non-negatively curved metrics 23
i 6. Proof of Theorem 1.8 26
X
Appendix A. The computations of Riemann curvature tensors 33
r
a A.1. The Riemann curvature tensors in Proposition 5.3 34
A.2. The curvature formula in Lemma 6.2 35
A.3. The Riemann curvature tensors R ,...,R in Lemma 6.5 36
1 10
References 37
2000 Mathematics Subject Classification. 53C20,53C30.
1
2 CHENXUHEANDPRIYANKARAJAN
1. Introduction
A fake real projective space is a manifold homotopy equivalent, but not diffeo-
morphic, to the standard real projective space. Equivalently, it is the orbit space
ofa free exotic involutionon a sphere. A free involutionis calledexotic, if it is not
conjugate by a diffeomorphism to the standard antipodal map on the sphere. The
firstexamples ofsuch exotic involutions were constructed by Hirsch andMilnor on
S5 and S6, see [HM]. They are restrictions of certain free involutions on the im-
ages of embedded standard 5- and 6-spheres in Milnor’s exotic spheres [Mi]. Thus
the quotient spaces of such embedded S5 and S6 are homotopy equivalent, but not
diffeomorphic, to the standard real projective spaces.
Theanalogousexotic15-spheresΣ15swereconstructedbyN.Shimadain[Sh]as
certain7-spherebundlesoverthe8-sphere. Theantipodalmaponthe7-spherefiber
defines a natural involution T on the Σ15s. In [RW], F. Wilhelm and the second
named author observed that the images of certain embedded standard 13- and 14-
spheresinΣ15sareinvariantundertheinvolution,andthusthequotientspacesare
homotopy equivalent to the standard 13- and 14-real projective spaces. Our first
main result is the diffeomorphism classification of the quotients. In particular we
show the following
Theorem 1.1. The quotient spaces of the embedded 13-spheres in certain Shi-
mada’s spheres Σ15s are fake real projective spaces, i.e., they are homotopy equiv-
alent, but not diffeomorphic to the standard 13-projective space.
Remark 1.2. (a) In [RW], they showed that the quotients of the embedded 14-
spheres in some Σ15s are not diffeomorphic to the standard RP14 following the
Hirsch-Milnor argument.
(b) They also observed that the Hirsch-Milnor’s argument breaks down in the
case of the embedded 13-spheres as there is an exotic 14-sphere in contrast to the
6-sphere.
Our proof of diffeomorphism classification is through the study of the so called
Davis action of G = SO(2) G on Shimada’s exotic 15-spheres, where G is the
2 2
simple exceptional Lie group×as the automorphism group of the octonions O. For
each odd integer k, denote Σ15 the total space of the 7-sphere bundle over the
k
8-sphere, with the Euler class [S8] and the second Pontrjagin class 6k[S8] where
[S8] is the standard generator of the cohomology group H8(S8). Shimada showed
thateachΣ15 is homeomorphicto the standard15-sphere,but notdiffeomorphic if
k
k2 1 mod 127, see [Sh]. In [Da](or see Section 2.1), using the octonion algebra,
M.6≡DavisintroducedtheactionsofGonΣ15ssuchthatG actsdiagonallyonthe7-
k 2
sphere fiber andthe 8-sphere base,whereas SO(2) acts via Mo¨bius transformation.
It is observed in [RW], that the Davis action on Σ15 leaves the image S13 of the
k k
embedded 13-sphere invariant and commutes with the involution T. Thus the
restricted action on S13 descends to the quotient space P13 = S13/T. They also
k k k
observed that the G-actions on S13 and P13 are cohomogeneity one, i.e., the orbit
k k
spaces are one dimensional. On the other hand, for the cohomogeneityone actions
on the homotopy spheres, aside from linear actions on the standard spheres, there
are families of non-linear actions [St]. They are examples given by the 2n 1
−
dimensional Brieskornvarieties Md2n−1, which are defined by the equations
zd+z2+...+z2 =0 and z 2+ z 2+...+ z 2 =1.
0 1 n | 0| | 1| | n|
FAKE RP13 WITH COHOMOGENEITY ONE ACTIONS 3
The Brieskorn varieties carry cohomogeneity one actions by SO(2) SO(n) via
×
(eiθ,A)(z0,z1,...,zn)= e2iθz0,e−idθA(z1,...,zn)t
withA SO(n). Anaturalinvolution,de(cid:0)notedbyI,isdefinedbyI((cid:1)z0,z1,...,zn)=
∈
(z , z ,..., z ). It is clear that the involution has no fixed point and commutes
0 1 n
awditmh−ittsheaScOoh(−o2m)×ogSenOe(inty)-aocnteioanc;tiaonndbtyhuSsOt(h2e)quSotOie(nnt).spNacoeteNtdh2na−t1w=henMdn2n−=1/7I,
the actions on M13 and N13 restricted to the×group G = SO(2) G are also
d d × 2
cohomogeneity one. We have the following
Theorem 1.3. For each odd integer k, the G-manifolds: the 13-sphere S13 and the
k
Brieskorn variety M13, with G = SO(2) G are equivariantly diffeomorphic, and
so are the quotient spkaces P13 =S13/T a×nd 2N13 =M13/I.
k k k k
Remark 1.4. Theorem1.1followsfromTheorem1.3aboveandthe diffeomorphism
classification of Nd2n−1 in [AB] and [Gi] (or see Section 2.2).
Remark 1.5. The space P13, i.e., k = 1, is diffeomorphic to the standard RP13
1
from the construction in [Sh] and [RW]. From Theorem 1.3 above, the known
diffeomorphism classification of N13 implies that there are 64 different oriented
k
diffeomorphism types of P13s.
k
Remark 1.6. (a) The Davis actions of SO(2) G on Shimada’s exotic spheres
2
Σ15s can be viewed as the octonionic analogs×of the SO(2) SO(3) actions on
Mkilnor’s exotic spheres Σ7s found in the same paper [Da]. No×te that SO(3) is the
automorphism group of the quaternions, and a special case of the SO(2) SO(3)
×
actions on a certain Σ7 was found in [GM].
(b) The DavisactionsofSO(2) SO(3)onMilnor’sexotic spheresalsoleavethe
×
images of the embedded 5-sphere invariant, and hence induce cohomogeneity one
actions on the Hirsch-Milnor’s fake 5-projectivespaces as observedin [RW]. These
actions are equivariantly diffeomorphic to those on the Brieskorn varieties N5’s,
d
which was first discovered by E. Calabi(unpublished, cf. [HH, p. 368])
Remark 1.7. In [ADPR], U. Abresch,C.Dura´n, T. Pu¨ttmannandA. Rigasgavea
geometricconstructionof free exotic involutions on the Euclideansphere S13 using
the wiedersehen metric on the Euclidean sphere S14. Thus the quotient spaces are
fake 13-projective spaces. Moreover, in [DP], Dura´n and Pu¨ttmann provided an
explicitnonlinearactionofO(2) G ontheEuclideansphereS13,andshowedthat
2
×
it is equivariantly diffeomorphic to the Brieskornvariety M13.
3
The second part of this paper is the study of the curvature properties of the
invariantmetrics on S13 and P13 with G=SO(2) G . Since any invariant metric
on the quotient spacekP13 cankbe lifted to an inv×aria2nt metric on S13, we restrict
k k
ourselves to the spheres S13s, or equivalently M13s. Note that M13 and M13 are
k k k k
equivariantly diffeomorphic, and so we assume that k 1. −
≥
On a Riemannian manifold with cohomogeneity one action, the principal or-
bits are hypersurfaces, and there are precisely two non-principal orbits that have
codimensions strictly bigger than one if the manifold is simply-connected. They
are called singular orbits. In [GZ1], K. Grove and W. Ziller constructed invariant
metrics with non-negative sectional curvature on cohomogeneity one manifolds for
which both singular orbits have codimension two. Particularly, their construction
yields non-negativelycurvedmetrics on10 of 14 (unoriented) Milnor’s spheres and
4 CHENXUHEANDPRIYANKARAJAN
allHirsch-Milnor’sfake5-projectivespaces. However,noteverycohomogeneityone
manifold admits an invariant metric with non-negative curvature. The first exam-
ples were found by K. Grove, L. Verdiani, B. Wilking and W. Ziller in [GVWZ],
and then generalized to a larger class in [He] by the first named author. The most
interesting class in [GVWZ] is the Brieskorn varieties Md2n−1. The Brieskornvari-
ety Md2n−1 is homeomorphic to the sphere, if and only if, both n and d are odd.
SInO([G2)VWSZO],(nit)iisnvsahroiwanetdmtheatrticfowrinth≥no4n-annedgadti≥ve3c,urMvad2tnu−r1e.dIonespnarotticsuulpapr,orthtearne
isnon×on-negativelycurvedSO(2) SO(7)invariantmetriconM13,ifd 3. Since
Gis apropersubgroupinSO(2) ×SO(7), there aremoreinvariandtmetric≥s onM13.
× k
One may suspect that there might be a chance to find an invariant metric with
non-negative curvature. Nevertheless we show that the obstruction does appear
even though the metric has a smaller symmetry group.
Theorem 1.8. For any odd integer k 3, the Brieskorn variety M13 does not
support an SO(2) G invariant metric≥with non-negative curvature. k
2
×
Remark 1.9. The techniques used to prove Theorem 1.8 are similar to those in
[GVWZ]and[He]. Howeverthe specialfeature ofthe LiegroupG andthe strictly
2
larger class of invariant metrics make the argument more involved.
Remark 1.10. For the Brieskorn variety M13 with d 4 an even integer, the
d ≥
principal isotropy subgroup has a simpler form than the one in the odd case, see
Remark2.11. This leadstoa muchmorecomplicatedformofthe invariantmetrics
inthe evencase,seeRemark 4.4,whichis notcoveredby ourproof. So foraneven
integer d 4, the question whether M13 admits an SO(2) G -invariant metric
≥ d × 2
with non-negative curvature remains open.
From Theorems 1.3 and 1.8, we have the following
Corollary 1.11. For any odd integer k 3, the fake 13-projective space P13 does
not support an SO(2) G invariant met≥ric with non-negative curvature. k
2
×
Remark 1.12. In contrastto the P13s, it is observedby O. Dearricottthat, follow-
k
ing Grove-Ziller’s construction, all fake Hirsch-Milnor’s 5-projective spaces admit
SO(2) SO(3) invariant metrics with non-negative curvature, see [GZ1, p. 334].
×
Remark 1.13. As observed in [ST], all P13s and S13s support even SO(2) SO(7)
k k ×
invariantmetricsthatsimultaneouslyhavepositiveRiccicurvatureandalmostnon-
negativesectionalcurvature. FortheinvariantmetricswithpositiveRiccicurvature
alone, it also follows from the result in [GZ2]. A Riemannian manifold admits an
almost non-negative sectional curvature if it collapses to a point with a uniform
lower curvature bound.
From the classification of cohomogeneity one actions on homotopy spheres in
[St] by E. Straume, M13s with G = SO(2) G are the only nonlinear actions
where the symmetry grokupdoes not have the×form2 SO(2) SO(n). Combining the
×
classification in [St], the obstructions in [GVWZ] and Theorem 1.8, we have the
following
Corollary 1.14. For n 2, let Σn be a homotopy sphere. Suppose that Σnadmits
≥
a non-negatively curved metric that is invariant under a cohomogeneity one action.
Then either
FAKE RP13 WITH COHOMOGENEITY ONE ACTIONS 5
(1) Σn is equivariantly diffeomorphic to the standard sphere and the action is
linear, or
(2) n = 5, Σ5 is the standard 5-sphere and the non-linear action is given by
SO(2) SO(3) on the Brieskorn variety M5, with k 3 odd.
× k ≥
We refer to the Table of Contents for the organization of the paper. Theorem
1.3 is proved in Section 3, and Section 6 is the proof of Theorem 1.8.
Acknowledgement. It is a great pleasure to thank Frederick Wilhelm who
has brought this problem to our attention, and we had numerous discussions with
him on this paper. We also thank Wolfgang Ziller for useful communications, and
Karsten Grove for his interest.
2. Preliminaries
Inthis section,werecallthe Davisactiononthe exotic 15-spheresΣ15s,andthe
k
Brieskorn varieties with cohomogeneity one action. We refer to [Ba] and [Mu] for
the basics of the algebra of the Cayley numbers (i.e., the octonions) and the Lie
group G .
2
2.1. Shimada’s exotic 15-spheres Σ15s, the embedded 13- and 14-spheres
k
and the Davis action. Consider the Cayley numbers O and let u u¯ be the
standardconjugation. ArealinnerproductonOisdefinedbyu v =1/7→2(uv¯+vu¯).
Let e ,e ,...,e beanorthonormalbasisofOoverRwithe =· 1. Wefollowthe
0 1 7 0
mult{iplications of}elements in O given by [Mu], for example, e e = e , e e = e
1 2 3 1 4 5
and e e =e . Any v O has the following form
1 7 6
∈
v =v e +v e + +v e .
0 0 1 1 7 7
···
Denote v = v the real part and v = v e +...+v e the imaginary part. We
0 1 1 7 7
ℜ ℑ
have
v¯=v e v e ... v e
0 0 1 1 7 7
− − −
and
v 2 =v2+v2+ +v2 =vv¯.
| | 0 1 ··· 7
The unit 7-sphere consists of all unit octonions:
S7 = v O: v =1 .
{ ∈ | | }
We write S8 = O O as the union of two copies of O which are glued together
φ
along O 0 via⊔the following map
−{ }
(2.1) φ:O 0 O 0
−{ } → −{ }
u
u φ(u)= .
7→ u2
| |
For any two integers m and n, let E be the manifold formed by gluing the two
m,n
copies of O S7 via the following diffeomorphism on (O 0 ) S7:
× −{ } ×
u um un
(2.2) Φ :(u,v) (u,v )= , v .
m,n 7→ ′ ′ u2 um un!
| | | | | |
The natural projection p : E S8 sends (u,v) to u and (u,v ) to u. It
m,n m,n ′ ′ ′
gives E the structure of an S7-bu→ndle over S8 with the transition map Φ .
m,n m,n
The total space E is homeomorphic to S15, if and only if, m+n= 1; see [Sh,
m,n
±
Section 2].
6 CHENXUHEANDPRIYANKARAJAN
Using the fact that G is the automorphismgroup of O, in [Da], Davis observed
2
that G acts on E as follows:
2 m,n
g(u,v)=(g(u),g(v))
and
g(u,v )=(g(u),g(v )).
′ ′ ′ ′
From[Da,Remark1.13],theG2-manifoldsEm,nandEm′,n′ areequivariantlydiffeo-
morphic, whenever (m,n) = (m,n) or (n,m). Furthermore, the bundles E
m,n
admit another SO(2) symmet±ry via Mo¨b±ius transformations that commutes with
the G -action. Write an element γ SO(2) as
2
∈
a b
(2.3) γ =γ(a,b)= and a2+b2 =1.
b a
(cid:18)− (cid:19)
In terms of the coordinate charts, the action on the sphere bundle E is defined
m,n
by
(2.4) γ⋆u = (au+b)( bu+a)−1
−
γ⋆u = ( b+au)(a+bu) 1
′ ′ ′ −
−
and
( bu+a)mv( bu+a)n
(2.5) γ⋆v = − −
bu+am+n
|− |
(a+bu¯)mv (a+bu¯)n
′ ′ ′
γ⋆v = .
′ a+bu¯ m+n
| ′|
The formulas above are compatible with the transition map Φ . Davis showed
m,n
the following
Lemma 2.1 (Davis). The formulas (2.4) and (2.5) give a well-defined action of
SO(2) on E . Furthermore the action is G -equivariant, and for any v O(not
m,n 2
∈
necessarily unit) we have
γ⋆v = v and γ⋆v = v .
′ ′
| | | | | | | |
Suppose now that m+n=1 and k =m n. So k is an odd number and
−
k+1 k+1
(2.6) m= and n= − .
2 2
We set Σ15 = E , and note that it is homeomorphic to the 15-sphere. A Morse
k m,n
function on Σ15 in [Sh] is given by
k
v (u(v ) 1)
′ ′ −
(2.7) f (x)= ℜ = ℜ .
1
1+ u2 1+ u(v ) 1 2
| | | ′ ′ − |
q q
Note that f has only two critical points as (u,v)=(0, 1). Set
1
±
(2.8) S1k4 =f1−1(0)= x∈Σ1k5 :ℜv =ℜ(u′(v′)−1)=0
anditisdiffeomorphictothestan(cid:8)dardS14 forallk. Considerthef(cid:9)ollowingfunction
on S14:
k
(uv) v
′
(2.9) f (x)= ℜ = ℜ .
2
1+ u2 1+ u 2
| | | ′|
q q
FAKE RP13 WITH COHOMOGENEITY ONE ACTIONS 7
It is straightforward to verify that on S14, the function f has precisely two non-
k 2
degenerate critical points as (u,v )=(0, 1). It follows that
′ ′
±
S1k3 = f2−1(0)∩S1k4
(2.10) = x Σ : (uv)= v = v = (u(v ) 1)=0 Σ15
∈ k ℜ ℜ ℜ ′ ℜ ′ ′ − ⊂ k
is diffeomorphic to(cid:8)the standard 13-sphere for all k. Let (cid:9)
(2.11) T : E E
m,n m,n
→
(u,v) (u, v) and (u,v ) (u, v )
′ ′ ′ ′
7→ − 7→ −
be the antipodal map on the fiber S7. The two spheres S14 and S13 are invariant
k k
under this involution T. Denote
P14 =S14/T and P13 =S13/T
k k k k
the quotient spaces.
Remark 2.2. Note thatMilnor’sexotic 7-spheresΣ7sarediffeomorphic to 3-sphere
bundles overthe 4-sphere. The involutionT on Σ15s is the analogueof the natural
involution on Σ7s given by the antipodal map of the 3-sphere fiber, see [Mi] and
[HM].
In [RW], Wilhelm and the second named author observedthat the Davis action
of G = SO(2) G on Σ15 leaves both S14 and S13 invariant and commutes with
× 2 k k k
the involution T.
Lemma 2.3. The SO(2) G action on Σ15 restricts to an action on the spheres
S14, S13 and descends to t×he q2uotient spaceskP14, P13.
k k k k
Proof. It is easy to see that the action commutes with the involution T. So it is
sufficient to show that the defining conditions of S13 and S14 in Σ15 are preserved
k k k
bytheSO(2) G action. InthefollowingwegiveaproofforS13,andtheargument
for S14 is sim×ilar2. k
k
Since G is the automorphism group of O, it is easy to see that the defining
2
conditions are preserved. Next we consider the action by SO(2). Let γ =γ(a,b) in
equation (2.3). Note that (xy)= (yx) for any x,y O. We have
ℜ ℜ ∈
1
(γ⋆v) = (a bu)mv(a bu)n
ℜ a bu ℜ{ − − }
| − |
1
= (a bu)m+nv
a bu ℜ −
| − |
1 (cid:8) (cid:9)
= (a v b (uv))
a bu ℜ − ℜ
| − |
= 0,
and
1
((γ⋆u)(γ⋆v)) = (au+b)(a bu)−1(a bu)mv(a bu)n
ℜ a bu ℜ − − −
| − |
1 (cid:8) (cid:9)
= (au+b)v
a bu ℜ
| − |
= 0.
8 CHENXUHEANDPRIYANKARAJAN
For the coordinates (u,v ), since u(v ) 1 = uv¯/ v 2 and u(v ) 1 = 0; it
′ ′ ′ ′ − ′ ′ ′ ′ ′ −
| | ℜ
follows that (u¯v )=0. Similar to the case of (u,v), we have
′ ′
ℜ (cid:0) (cid:1)
1
(γ⋆v ) = (a+bu¯)mv (a+bu¯)n
′ ′ ′ ′
ℜ a+bu¯ ℜ{ }
| ′|
1
= (a+bu¯)v
′ ′
a+bu¯ ℜ{ }
| ′|
= 0
and
(γ⋆u)(γ⋆v ) 1 = a+bu¯ ( b+au)(a+bu) 1(a+bu¯) n(v ) 1(a+bu¯) m
′ ′ − ′ ′ ′ − ′ − ′ − ′ −
ℜ | |ℜ −
(cid:0) (cid:1) = a+bu¯′ (cid:8)( b+au′)(a+bu′)−1(a+bu¯′)−1(v′)−1 (cid:9)
| |ℜ −
= a+bu¯ a(cid:8)2+b2 u 2+ab(u +u¯) ( b+au)((cid:9)v ) 1
′ ′ ′ ′ ′ ′ −
| | | | ℜ −
= 0. (cid:16) (cid:17) (cid:8) (cid:9)
This shows that S13 is invariant under the SO(2) action, which finishes the proof.
k
(cid:3)
Remark 2.4. In [RW], following the Hirsch-Milnor argument in [HM], they also
showedthatP14 andP13 arehomotopyequivalenttothe standardRP14 andRP13
k k
for all k; and P14 is not diffeomorphic to RP14, when k 3,5 mod 8.
k ≡
2.2. Brieskornvarieties,Kervairespheresandhomotopyprojectivespaces.
F(2onr an1y)-idnitmegeenrssionna≥l su3bmanadnidfol≥d o1f,Cthn+e1B,rdieefisknoerdnbvyartiheetyeqMuad2tni−on1sis the smooth
−
zd+z2+ +z2 = 0
0 1 ··· n
z 2+ z 2+ + z 2 = 1.
(cid:26) | 0| | 1| ··· | n|
When d = 1, M12n−1 is diffeomorphic to the standard sphere S2n−1; and when
d=2, M22n−1 is diffeomorphic to the unit tangent bundle of Sn.
Thohmeeoormemorp2hi.c5t(oBtrhieesksotarnnd).arSdusppphoesreenS2≥n 31,ainfdandd≥on2l.y iTf,hebomthannifaonldd Md da2nre−1odids
−
numbers. Assume that n and d are odd numbers, it is the Kervaire sphere, if and
only if, d 3 mod 8.
≡±
Remark 2.6. The Kervaire sphere is known to be exotic if n 1 mod 4.
≡
Denote I the following involution on Md2n−1:
(z ,z ,...,z ) (z , z ,..., z ).
0 1 n 0 1 n
7→ − −
Clearly it is fixed-point free. Atiyah and Bott showed the following result, see also
[Gi, Corollary 4.2].
Theorem 2.7 ([AB, Theorem 9.8]). If the involution I on the topological spheres
Md4m−3 and Mk4m−3 are isomorphic, then
d k mod 22m.
≡±
In particular the involution I acting on M34m−3 = S4m−3 is not isomorphic to the
standard antipodal map whenever m 2.
≥
Corollary 2.8. There are 64 smoothly distinct real projective spaces M13/I with
k
k =1,3,...,127.
FAKE RP13 WITH COHOMOGENEITY ONE ACTIONS 9
The group G˜ =SO(2)×SO(n) acts on Md2n−1 by
eiθ,A (z ,Z)= e2iθz ,e idθAZ , for (z ,Z) C Cn.
0 0 − 0
∈ ⊕
Note that o(cid:0)ur con(cid:1)vention is (cid:0)different from th(cid:1)e one in [GVWZ], as we have e idθ
−
for the action of eiθ on Z = (z ,...,z )t. The norm z is invariant under this
1 n 0
| |
action, and two points belong to the same orbit if and only if they have the same
valueof z . Lett bethe uniquepositivesolutionoftd+t2 =1,andthenwehave
| 0| 0 0 0
0 z t . It follows that the orbitspace is [0,t ]. The orbit types and isotropy
0 0 0
≤| |≤
subgroups of this action have been well-studied, see for example, [HH], [BH] and
[GVWZ].
In our case, we assume that d is odd. When n =7, the embedding G SO(7)
2
induces the action of G=SO(2) G on M13. To describe the isotropy su⊂bgroups
of the G-action we introduce the×follo2wing sdubgroups in G :
2
Denote O(6), the subgroup in SO(7) that maps e to e , SO(6) the sub-
1 1
• group that fixes e , and SU(3)=SO(6) G . ±
1 2
The other subgroup in G that fixes e∩ is denoted by SU(3) , and the
2 3 3
• complex structure on C3 = spanR e1,e2,e4,e7,e6,e5 is given by the left
{ }
multiplication of e . Note that
3
(SO(2) SO(5)) G =U(2) SU(3)
2 3
× ∩ ⊂
whereSO(2) SO(5) SO(7)hastheblock-diagonalform,andtheembed-
ding U(2) ×SU(3) is⊂given by h diag (deth) 1,h . To see this, take
3 −
A=diag ⊂A ,A (SO(2) SO(7→5)) G with
1 2 2
{ }∈ × ∩ (cid:8) (cid:9)
cost sint
A =
1 sint cost
(cid:18)− (cid:19)
for some t. Since e =e e , we have
3 1 2
A(e ) = A(e )A(e )
3 1 2
= (e cost+e sint)( e sint+e cost)
1 2 1 2
−
= e
3
and thus A SU(3) . Using the complex structure of SU(3) , A acts on
3 3 1
C=spanR e∈1,e2 byeit,andA2actsinvariantlyonC2 =spanR e4,e7,e6,e5 .
So the elem{ent A}embeds diagonally in SU(3) with (1,1)-entry{eit. }
3
The common subgroup SU(2) = SU(3) SU(3) and it is also given by
3
• SU(2) = SO(4) G where SO(4) SO(∩7) as A diag I ,A and I is
2 3 3
∩ ⊂ 7→ { }
the identity matrix.
Since G acts transitively on S6 = v O: v =0 and v =1 with SU(3) and
2
SU(3) as isotropy subgroups at e a{nd∈e resℜpectively, th|es|e two}groups are con-
3 1 3
jugate by an element in G .
2
We follow the notions in [GVWZ] to determine the isotropy subgroups. Denote
B the singular orbit with z = 0, and choose p = (0,1,i,0,...,0) B with
0
iso−tropy subgroup K . We|als|o denote B the sin−gular orbit with z ∈= t−, and
− + 0 0
| |
choose p =(t ,i td,0,...,0) with isotropy subgroupK+. Note that B and B
+ 0 0 +
−
have codimensions 2 and n 1 = 6 respectively. Let c(t) be a normal minimal
p −
geodesic connecting p = c(0) and p = c(L). The isotropy subgroup at c(t)(0 <
+
t<L) stays unchange−d that is the principal isotropy subgroup H. We have
10 CHENXUHEANDPRIYANKARAJAN
Theorem 2.9. The cohomogeneity one action of G = SO(2) G on M13 with d
× 2 d
odd has the following isotropy subgroups:
(1) The principal isotropy subgroup is
H=Z SU(2)=(ε,diag ε,ε,1,A )
2
· { }
where ε= 1 and A is a 4 4-matrix.
± ×
(2) At p , the isotropy subgroup is
−
cosdθ sindθ
K =SO(2)SU(2)= eiθ,diag ,1,A
− sindθ cosdθ
(cid:18) (cid:26)(cid:18)− (cid:19) (cid:27)(cid:19)
where A is a 4 4-matrix.
×
(3) At p , the isotropy subgroup is
+
K+ =O(6) G =(detB,diag detB,B )
2
∩ { }
where B O(6) G .
2
∈ ∩
Remark 2.10. Denote j, the complex structure given by the left multiplication of
e . For the group H, we have diag ε,ε,1,A (SO(2) SO(5)) G and A
3 2
U(2) SU(3) with detA=ε. For th{e group K} ∈, we have× ∩ ∈
3 −
⊂
cosdθ sindθ
diag ,1,A (SO(2) SO(5)) G
sindθ cosdθ ∈ × ∩ 2
(cid:26)(cid:18)− (cid:19) (cid:27)
and A U(2) SU(3) with detA=e jdθ.
3 −
∈ ⊂
Remark 2.11. If d is an even integer, then the isotropy subgroup K is the same
−
as in the case d odd. The other two isotropy subgroups are
H=Z SU(2)=(ε,diag I ,A )
2 3
× { }
K+ =Z SU(3)=(ε,diag 1,B )
2
× { }
where ε= 1, A SO(4) G =SU(2) and B SO(6) G =SU(3).
2 2
± ∈ ∩ ∈ ∩
ClearlytheG-actioncommuteswiththeinvolutionI andhenceinducesanaction
on N13 = M13/I. Write [z ,z ,...,z ] N13, the equivalent class under the
d d 0 1 7 ∈ d
involution I.
Corollary 2.12. The cohomogeneity one action of G = SO(2) G on N13 =
× 2 d
M13/I with d odd, has the following isotropy subgroups.
d
(1) The principal isotropy subgroup is
H¯ =Z (Z SU(2))=(ε ,diag ε ,ε ,1,A )
2 2 1 2 2
× · { }
where ε = 1 and A is a 4 4-matrix.
1,2
± ×
(2) The singular isotropy subgroup at [0,1,i,0,...,0] is
cosdθ sindθ
K¯ =Z SO(2)SU(2)= eiθ,diag ε ,1,A
− 2· sindθ cosdθ
(cid:18) (cid:26) (cid:18)− (cid:19) (cid:27)(cid:19)
where ε= 1 and A is a 4 4-matrix.
± ×
(3) The singular isotropy subgroup at [t ,i td,0,...,0] is
0 0
K¯+ =Z (O(6) G )=(εp,diag detB,B )
2 2
× ∩ { }
where ε= 1 and B O(6) G .
2
± ∈ ∩
