HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC
HYPERSURFACES IN RANK ONE SYMMETRIC SPACES
JOS´
E CARLOS D´
IAZ-RAMOS, MIGUEL DOM´
INGUEZ-V´
AZQUEZ,
AND ALBERTO RODR´
IGUEZ-V´
AZQUEZ
Abs ac . We conclude he classi ica ion o cohomogenei y one ac ions on symme ic
spaces o ank one by classi ying cohomogenei y one ac ions on qua e nionic hype bolic
spaces up o o bi equi alence. As a by-p oduc o ou p oo , we p oduce uncoun ably
many examples o inhomogeneous isopa ame ic amilies o hype su aces wi h cons an
p incipal cu a u es in qua e nionic hype bolic spaces.
1. In oduc ion
Riemannian geome y, in a e y b oad sense, can be unde s ood as he s udy o hose
p ope ies o a smoo h mani old ha a e in a ian unde isome ies. Among Riemannian
mani olds wi h la ge isome y g oups, Riemannian symme ic spaces s and ou as a class
o hei own, no only in Riemannian Geome y, bu also in Lie g oup heo y o Global
Analysis. In his class, Euclidean spaces and symme ic spaces o ank one a e he mos
popula in Riemannian geome y. I has been an in e es ing p oblem o s udy isome ic
ac ions on mani olds wi h la ge isome y g oups, and se e al ypes o hem ha e been in-
es iga ed o e he yea s. One o he mos impo an amilies o isome ic ac ions is ha o
cohomogenei y one, ha is, p ope isome ic ac ions whose o bi space is one-dimensional,
o in o he wo ds, whose p incipal o bi s a e hype su aces. Cohomogenei y one ac ions
ha e ecen ly been o g ea in e es o he cons uc ion o geome ic s uc u es, such as
Eins ein me ics, Ricci soli ons, special holonomy, o minimal hype su aces, among o he s.
Howe e , i is also a na u al and impo an p oblem o ind all cohomogenei y one
ac ions on a gi en Riemannian mani old, usually jus up o o bi equi alence. This is a
classical p oblem in submani old geome y ha aces back o he ime o ´
E. Ca an, and
which u ns ou o be equi alen o he classi ica ion o homogeneous hype su aces up o
isome ic cong uence. By bo h his o ical and ma hema ical easons, i has equen ly been
linked o he in es iga ion o he so-called isopa ame ic hype su aces.
2020 Ma hema ics Subjec Classi ica ion. P ima y 53C35, Seconda y 57S20, 53C40.
Key wo ds and ph ases. Isopa ame ic hype su ace, cohomogenei y one ac ion, homogeneous subman-
i old, cons an p incipal cu a u es, symme ic space, qua e nionic hype bolic space, K¨ahle angle.
The au ho s ha e been suppo ed by he p ojec s PID2019-105138GB-C21 (AEI/FEDER, Spain) and
ED431C 2019/10, ED431F 2020/04 (Xun a de Galicia, Spain). The second and hi d au ho s acknowledge
suppo o he Ram´on y Cajal p og am (Agencia Es a al de In es igaci´on, Spain) and he FPU p og am
(Minis y o Educa ion, Spain), espec i ely.
1
2 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
A hype su ace is called isopa ame ic i i s nea by equidis an hype su aces ha e con-
s an mean cu a u e. Thus, e e y homogeneous hype su ace is isopa ame ic. In he
30s, Ca an himsel s udied he con e se implica ion. Since all he examples known o him
(which included all isopa ame ic hype su aces in Euclidean and eal hype bolic spaces,
and all isopa ame ic hype su aces wi h up o h ee p incipal cu a u es in sphe es) we e
homogeneous, he posed he ques ion: is i ue ha an isopa ame ic hype su ace is ho-
mogeneous? A su p ising nega i e answe o his ques ion a i ed almos o y yea s la e ,
wi h he cons uc ion o he i s examples o inhomogeneous isopa ame ic hype su aces in
sphe es by Ozeki and Takeuchi [32], soon gene alized by Fe us, Ka che and M¨unzne [22].
These examples led o an added di icul y in he classi ica ion p oblem o isopa ame ic
hype su aces in sphe es, which has gi en ise o ou s anding esul s o e he las ew yea s
[12, 24, 30, 34, 13, 14]. O he inhomogeneous isopa ame ic hype su aces ha e been ound
in symme ic spaces such as complex and qua e nionic p ojec i e spaces [20, 21] and in
ce ain symme ic spaces o non-compac ype, such as complex hype bolic spaces [17],
o mo e gene ally, he symme ic spaces wi h Dynkin diag am o (BC )- ype [15, 19].
Howe e , none o hese examples, unlike he ones in sphe es, ha e cons an p incipal cu -
a u es, wi h only one ema kable excep ion: one inhomogeneous amily o isopa ame ic
hype su aces wi h cons an p incipal cu a u es in he Cayley hype bolic plane [15].
The classi ica ion o cohomogenei y one ac ions up o o bi equi alence in Euclidean
spaces ollows om he classi ica ion o isopa ame ic hype su aces in Rnob ained by
Seg e [33]. In symme ic spaces o compac ype and ank one, he co esponding classi i-
ca ion ollows om se e al wo ks. In sphe es i was ob ained by Hsiang and Lawson [23],
in complex p ojec i e spaces by Takagi [36], and in qua e nionic p ojec i e spaces and he
Cayley plane by Iwa a [25, 26]. The e is also a classi ica ion o cohomogenei y one ac ions
on i educible symme ic spaces o compac ype due o Koll oss [28].
The p oblem is mo e di icul in he non-compac case. The main eason is ha , unlike
in he compac se ing, he e a e wo main ypes (namely, educ i e and pa abolic) o
maximal subg oups o he isome y g oup o a symme ic space o non-compac ype, and
pa abolic subg oups con ain many subg oups ha ac ansi i ely on he space. Thus, he
in es iga ion o o bi s o subg oups o a pa abolic subg oup equen ly leads o complica ed
linea algeb a o combina o ial p oblems (in a ce ain sense simila , o example, o he
ones a ising in he ou s anding classi ica ion p oblem o o ally geodesic submani olds [27]),
o which e y ew ideas ha e been de eloped (c . [4, 6, 16]). The i s classi ica ion esul
o cohomogenei y one ac ions on a symme ic space o non-compac ype was gi en by
Ca an [11] o eal hype bolic spaces, while he was s udying isopa ame ic hype su aces
in spaces o cons an cu a u e. Howe e , he classi ica ion in complex hype bolic spaces
and he Cayley hype bolic plane, due o Be nd and Tama u [5], only a i ed se en y yea s
la e . The e a e se e al s uc u al esul s o symme ic spaces o non-compac ype [3, 6],
bu a ull classi ica ion is s ill no a ailable, no e en in qua e nionic hype bolic spaces.
This is p ecisely he poin whe e we s a ou s udy. The main aim o his a icle
is o classi y cohomogenei y one ac ions on qua e nionic hype bolic spaces up o o bi
equi alence. Ou me hod elies pa ially on he ideas de eloped in [5], whe e i is p o ed
ha his classi ica ion can be educed o a ce ain p oblem ha we sol e in his pape .
HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 3
The i s main esul o his a icle can be s a ed in e ms o qua e nionic algeb a. We
deno e by H he eal di ision algeb a o he qua e nions, endowed wi h i s s anda d complex
s uc u es i,jand k. Le Hnbe a igh qua e nionic ec o space o dimension n. The
compac symplec ic g oup Sp(n) is he g oup o qua e nionic ma ices (ac ing on he le
on Hn) ha p ese e he s anda d qua e nionic bilinea o m Pn
i=1 ¯ iwi, whe e , w ∈Hn,
and ba deno es conjuga ion. This bilinea o m na u ally induces an inne p oduc in Hn
ha makes i isome ic wi h R4n. By Jwe will deno e he qua e nionic s uc u e o Hn,
ha is, he subspace o eal endomo phisms o Hngene a ed by he igh mul iplica ions
by i, j and k, which can he e o e be seen as he Lie algeb a o Sp(1).
We also conside he Lie g oup Sp(1)Sp(n) = Sp(1) ×Sp(n)/Z2, which ac s on Hnas
(q, A)· =A q−1. This is an impo an g oup in Di e en ial Geome y, as i a ises in
Be ge ’s holonomy lis , ha is, he lis o Lie g oups which can be ealized as he holonomy
o i educible, simply connec ed and non-locally symme ic Riemannian mani olds. Thus,
a Riemannian mani old is called qua e nionic K¨ahle i i has dimension 4n, is no Ricci-
la , and i s holonomy is isomo phic o a subg oup o Sp(1)Sp(n), n≥2. The simples
examples o symme ic, qua e nionic K¨ahle spaces a e he qua e nionic p ojec i e spaces,
and hei non-compac duals, he qua e nionic hype bolic spaces. In any case, unde s and-
ing algeb aic p ope ies linked o holonomy g oups is a i s undamen al s ep owa ds he
s udy o mo e geome ic ques ions, such as hose ela ed o cu a u e (e.g. he celeb a ed
LeB un-Salamon conjec u e [29]) o submani olds (e.g. he heo y o calib a ions [10]).
Simila ly, he p oblem o submani old geome y ha we add ess in his pape elies on a
linea algeb aic p oblem ha we desc ibe below.
We say ha a eal subspace Vo Hnis p o ohomogeneous i he e exis s a connec ed Lie
subg oup o Sp(1)Sp(n) ha ac s ansi i ely on he uni sphe e o V. A p o ohomogeneous
subspace o Hnhas cons an qua e nionic K¨ahle angle, a concep ha is cen al in ou
s udy and ha we ecall now. Le πVdeno e he o hogonal p ojec ion on o a ec o
subspace V, and de ine
PJ=πV◦J, whe e J∈J.
We say ha Vhas cons an qua e nionic K¨ahle angle (ϕ1, ϕ2, ϕ3), wi h ϕ1≤ϕ2≤ϕ3, i
o any ∈V he symme ic bilinea o m
L :J×J→R, L (J, J0) = hPJ , PJ0 i,
has eigen alues cos2(ϕi)h , i,i∈ {1,2,3}. We poin ou he e he ac ha he bilinea
o ms L , ∈V, desc ibed abo e do no necessa ily diagonalize simul aneously (al hough
we can p o e a p io i ha hey do so o p o ohomogeneous subspaces o dimension g ea e
o equal han 5, see Co olla y 4.2, and by classi ica ion esul s o dimension di e en
om 3).
The i s main esul o his a icle is o classi y, up o cong uence by elemen s in
Sp(1)Sp(n), p o ohomogeneous subspaces o Hn. We s a e he e he moduli space o such
subspaces o dimension kin Hnby p esen ing hei possible qua e nionic K¨ahle angles.
In Theo em A, and in wha ollows, deno es disjoin union.
4 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
Theo em A. The moduli space Mk,n o non-ze o p o ohomogeneous subspaces o dimen-
sion kin Hn, up o cong uence in Sp(1)Sp(n), is desc ibed in he ollowing able:
Mk,n k≤n n < k ≤4n
3
4n
3< k ≤2n k > 2n
k≡0 (mod 4) (R+
4 R−
4) (R−
4×Z2)S{(0, ϕ, ϕ)}ϕ∈[0,π
2]{(0,0,0)}
k≡2 (mod 4) {(ϕ, π
2,π
2)}ϕ∈[0,π
2]{(0,π
2,π
2)} {(0,π
2,π
2)} ∅
k6= 3 odd {(π
2,π
2,π
2)} ∅ ∅ ∅
k= 3 (R+
3 R−
3) (R−
3×Z2)∅ {(ϕ, ϕ, π
2)}ϕ∈{0,π
3}{(0,0,π
2)}
whe e Λ = {(ϕ1, ϕ2, ϕ3)∈[0, π/2]3:ϕ1≤ϕ2≤ϕ3}, and
R+
3={(ϕ, ϕ, π/2) ∈Λ : ϕ∈[0, π/2]},
R−
3={(ϕ, ϕ, π/2) ∈Λ : ϕ∈[π/3, π/2)},
R+
4={(ϕ1, ϕ2, ϕ3)∈Λ : cos(ϕ1) + cos(ϕ2)−cos(ϕ3)≤1},
R−
4={(ϕ1, ϕ2, ϕ3)∈Λ : cos(ϕ1) + cos(ϕ2) + cos(ϕ3)≤1, ϕ36=π/2},
S={(ϕ1, ϕ2, ϕ3)∈Λ : cos(ϕ1) + cos(ϕ2) + εcos(ϕ3)=1, o ε= 1 o ε=−1}.
This classi ica ion includes ypical examples such as o ally eal subspaces (p ecisely
hose wi h qua e nionic K¨ahle angle (π/2, π/2, π/2)), o ally complex subspaces (wi h
qua e nionic K¨ahle angle (0, π/2, π/2)), qua e nionic subspaces (wi h qua e nionic K¨ahle
angle (0,0,0)), subspaces o cons an K¨ahle angle ϕ∈(0, π/2) inside a o ally complex
ec o subspace (wi h qua e nionic K¨ahle angle (ϕ, π/2, π/2)), complexi ica ions o sub-
spaces o cons an K¨ahle angle ϕ∈(0, π/2) in a o ally complex subspace (wi h qua e -
nionic K¨ahle angle (0, ϕ, ϕ)), and (Im H) , ∈Hn, 6= 0 (wi h qua e nionic K¨ahle angle
(0,0, π/2)). Howe e , he e a e some o he non-classical examples. Some o hem we e
in oduced in [15], bu he e a e some o he s, which a e basically p esen ed and classi ied
in Sec ion 5. A basis o hese subspaces can be calcula ed explici ly, bu o R±
3and R±
4
i s exp ession is a he long. See P oposi ion 5.3 o R±
3and P oposi ions 5.10 and 6.1 o
R±
4 o ge u he de ails. Fu he mo e, he e a e non-cong uen subspaces o Hnwi h he
same K¨ahle angles. These co espond p ecisely o he in e sec ions R+
3∩R−
3=R−
3and
R+
4∩R−
4=R−
4.
We poin ou he e h ee main ools ha ha e been essen ial o ob ain his classi ica ion.
Fi s we use he classical gene aliza ion o he hai y ball heo em ega ding he possi-
ble ank o con inuous dis ibu ions on sphe es [35] in o de o educe he classi ica ion
p oblem o eal subspaces o Hnwi h cons an qua e nionic K¨ahle angle o subspaces o
dimensions 3 and mul iples o 4 (Sec ion 3). Secondly, we p o ide a Lie heo e ic a gu-
men elying on esul s by Bo el [9] and Mon gome y and Samelson [31] on g oups ac ing
e ec i ely and ansi i ely on sphe es, o p o e ha , o subspaces o dimension g ea e o
equal han 5, he maps L ha a e used o de ine qua e nionic K¨ahle angle diagonalize
simul aneously (Co olla y 4.2). In hi d place, using he p e ious esul s, we can show ha
a p o ohomogeneous subspace o dimension 4lis he sum o p o ohomogeneous subspaces
HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 5
o dimension 4 wi h he same qua e nionic K¨ahle angle (Sec ion 4.2). All his educes
he classi ica ion o p o ohomogeneous subspaces o dimensions 3 and 4. A his s age,
we ac ually ob ain he mo e gene al classi ica ion o eal subspaces o dimensions 3 and 4
wi h cons an qua e nionic K¨ahle angle. This is a (ha d) p oblem o linea algeb a ha
is sol ed in Sec ion 5.
The i s consequence o Theo em A is he classi ica ion o cohomogenei y one ac ions
on qua e nionic hype bolic spaces HHn+1 up o o bi equi alence. In ac , Be nd and
Tama u explained in [5] how o ob ain his classi ica ion. Conside he symme ic pai
(G, K) = (Sp(1, n + 1),Sp(1) ×Sp(n+ 1)) ep esen ing he symme ic space HHn+1. We
deno e by g=k⊕p he co esponding Ca an decomposi ion, and le abe a maximal
abelian subspace o p, which is one-dimensional because HHn+1 is o ank one. Le
g=g−2α⊕g−α⊕g0⊕gα⊕g2αbe he es ic ed oo space decomposi ion o gwi h
espec o a. Then, gαis isomo phic o a qua e nionic ec o space Hnendowed wi h he
s anda d qua e nionic bilinea o m, and K0∼
=Sp(1) ×Sp(n), he connec ed Lie subg oup
o Gwhose Lie algeb a is k0=g0∩k=Nk(a), no malizes gαand ac s on gαin he canonical
way. The classi ica ion o cohomogenei y one ac ions on HHn+1 can be ob ained i we de-
e mine he p o ohomogeneous subspaces Vo gα∼
=Hn. I Vis such a p o ohomogeneous
subspace, we de ine he Lie subalgeb a sV=a⊕(gαV)⊕g2αo g, and deno e by SV
he connec ed Lie subg oup o Gwi h Lie algeb a sV( h oughou his a icle deno es
he o hogonal complemen o a ec o subspace). Then N0
K0(SV)SV=N0
K0(V)SVac s on
HHn+1 wi h cohomogenei y one, whe e N0
K0(·) deno es he connec ed componen o he
iden i y o he no malize in K0. Knowing all such subspaces Vup o cong uence by an
elemen o Sp(1)Sp(n) de e mines all cohomogenei y one ac ions on HHn+1 up o o bi
equi alence.
Roughly wen y yea s a e Be nd and B ¨uck [2] announced he i s examples o coho-
mogenei y one ac ions using his p ocedu e, we ob ain he ull classi ica ion o cohomogene-
i y one ac ions on qua e nionic hype bolic spaces up o o bi equi alence as a consequence
o Theo em A. Toge he wi h he esul s by Be nd and Tama u [5], his inishes he
classi ica ion o cohomogenei y one ac ions on non-compac symme ic spaces o ank one:
Theo em B. The moduli space o cohomogenei y one ac ions on HHn+1 up o o bi equi -
alence is gi en by he disjoin union
{N, K, SU(1, n + 1)}
4n
G
k=1 Mk,n.
The ac ions e e enced he e a e:
(1) N: he ac ion ha p oduces a ho osphe e olia ion.
(2) K: he ac ion ha p oduces a amily o geodesic sphe es cen e ed a a poin .
(3) SU(1, n+1): he ac ion ha p oduces a amily o ubes a ound a o ally geodesic CHn+1.
(4) Mk,n: he cohomogenei y one ac ions o he connec ed Lie subg oups o Sp(1, n + 1)
wi h Lie algeb as Nk0(V)⊕a⊕(gαV)⊕g2α, whe e Vis a p o ohomogeneous subspace
o dimension ko gα∼
=Hn.

6 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
We no e ha , in his classi ica ion, he ac ion o Sp(1, `)×Sp(n+ 1 −`)⊂Sp(1, n + 1)
which gi es ubes a ound a o ally geodesic lowe dimensional qua e nionic hype bolic
space HH`,`∈ {1, . . . , n}, in HHn+1 a e included in i em (4), whe e in his case Vis a
qua e nionic subspace o gα∼
=Hn(hence, o qua e nionic K¨ahle angle (0,0,0)) o eal
dimension k= 4(n−`+ 1). Mo eo e , i we ake Va line in gα(i.e. k= 1), hen N0
K0(V)
is i ial and we eco e he ac ion ha gi es ise o he so-called sol able olia ion [4].
In ou s udy o p o ohomogeneous subspaces o Hnwe ha e also encoun e ed non-
cong uen pai s o subspaces wi h he same cons an qua e nionic K¨ahle angles. Mo eo e ,
we p o e in Sec ion 6 ha an H-o hogonal di ec sum o subspaces o dimension 4 wi h he
same cons an qua e nionic K¨ahle angle is p o ohomogeneous i and only i any wo ac o s
a e cong uen unde an elemen o Sp(n). Howe e , e en i ha di ec sum is no p o o-
homogeneous, i has cons an qua e nionic K¨ahle angle in some cases. Thus, i we ake
Va non-p o ohomogeneous subspace wi h cons an qua e nionic K¨ahle angle as abo e,
and deno e by SV he subg oup o Gwhose Lie algeb a is sV=a⊕(gαV)⊕g2α, hen:
(1) since Vhas cons an qua e nionic K¨ahle angle, ubes a ound SV·oa e isopa ame ic
and ha e cons an p incipal cu a u es by [15, Theo em 4.5], and (2) hese ubes a e no
homogeneous by [5, Theo em 4.1]. Hence, we ha e he ollowing ema kable consequence:
Theo em C. The e exis uncoun ably many inhomogeneous isopa ame ic amilies o hy-
pe su aces wi h cons an p incipal cu a u es in HHn+1 wi h n≥7, up o cong uence.
We ecall ha he only examples o inhomogeneous isopa ame ic amilies o hype su -
aces wi h cons an p incipal cu a u es known so a in any i educible Riemannian sym-
me ic space a e he celeb a ed examples in sphe es by Fe us, Ka che and M¨unzne [22]
and a single example ound in he Cayley hype bolic plane [15]. Thus, his is he i s ime
an uncoun able collec ion o such examples is p oduced in some symme ic space.
This a icle is o ganized as ollows. We ecall some basic ac s abou symme ic spaces
in
§
2.1, and o cohomogenei y one ac ions in
§
2.2. The undamen al concep o qua e -
nionic K¨ahle angle is ecalled in Subsec ion 2.3 oge he wi h some impo an no a ion
ha will be used h oughou his a icle. In Sec ion 3 we use a gene aliza ion o he
hai y ball heo em o ule ou se e al possibili ies o qua e nionic K¨ahle angles. Then,
in Subsec ion 4.1 we p o e a simul aneous diagonaliza ion esul o subspaces o cons an
qua e nionic K¨ahle angle. This is used in
§
4.2 o p o e a ac o iza ion heo em o p o o-
homogeneous subspaces o dimension mul iple o 4. Al oge he , his educes ou s udy o
dimensions 3 (
§
5.1) and 4 (
§
5.2). The exis ence o inhomogeneous isopa ame ic hype su -
aces wi h cons an p incipal cu a u es in qua e nionic hype bolic spaces (Theo em C) is
es ablished in Sec ion 6. We inally p o e Theo ems A and B in Sec ion 7.
2. P elimina ies
We s a his sec ion by ecalling he main known esul s conce ning cohomogenei y
one ac ions on symme ic spaces o non-compac ype and ank one. Cohomogenei y one
ac ions wi h a non- o ally geodesic singula o bi a e buil using he concep o qua e nionic
K¨ahle angle, which we ecall in his sec ion. Also, we will p esen some p ope ies and
HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 7
suma ize all he examples o subspaces wi h cons an qua e nionic K¨ahle angle known up
o he p esen . The main e e ences o hese no ions and esul s a e [2], [5], and [15].
2.1. Symme ic spaces o non-compac ype and ank one.
Hu wi z’s heo em asse s ha any no med eal di ision algeb a Fis isomo phic o R,
C,Ho O. The hype bolic spaces o e hese algeb as cons i u e he symme ic spaces o
non-compac ype and ank one. In o he wo ds, i Mis a symme ic space o non-compac
ype and ank one, hen Mis ei he a eal hype bolic space RHn+1,n≥1, a complex
hype bolic space CHn+1,n≥1, a qua e nionic hype bolic space HHn+1,n≥1, o he
Cayley hype bolic plane OH2. As a symme ic space, any o hese mani olds Mcan be
iden i ied wi h a quo ien G/K o Lie g oups, whe e Gis he connec ed componen o he
iden i y o he isome y g oup o M, up o a ini e co e ing, and Kis he iso opy subg oup
o Gco esponding o a ce ain poin o∈M ha we ix om now on. Then one can ake
G=SO0(1, n + 1), SU(1, n + 1), Sp(1, n + 1), F−20
4and K=SO(n+ 1), S(U(1) ×U(n+ 1)),
Sp(1) ×Sp(n+ 1), Spin(9), depending on whe he F=R,C,H,O, espec i ely.
We deno e by gand k he Lie algeb as o Gand K, espec i ely, by B he Killing o m
o g, and by θ he Ca an in olu ion o gwi h espec o k. Le g=k⊕pbe he Ca an
decomposi ion o ginduced by θ. We ha e ha hX, Y i=−B(X, θY ) is an inne p oduc
ha es ic ed o pinduces a Riemannian me ic on G/K ha makes G/K isome ic o
M, up o homo he y.
Le abe a maximal abelian subspace o p, which is one-dimensional as Mhas ank
one, and le g=g−2α⊕g−α⊕g0⊕gα⊕g2αbe he co esponding es ic ed oo space
decomposi ion o g. He e, he oo space g0spli s as g0=k0⊕a, whe e k0is he Lie algeb a
o K0=NK(a), he no malize o ain K, which also no malizes gαand g2α. Mo eo e ,
g=k⊕a⊕n, whe e n=gα⊕g2α, is an Iwasawa decomposi ion o g. When F=R,
we ha e g−2α=g2α= 0 and nis abelian. O he wise, nis only wo-s ep nilpo en . In
ac , nis isomo phic o he (2n+ 1)-dimensional Heisenbe g algeb a when F=Cand o
a ce ain gene alized Heisenbe g algeb a i F∈ {H,O}(see [7]). Mo eo e , g2α, he cen e
o n, is equal o he de i ed algeb a o n, and has dimension 1, 3 o 7 o F=C,Ho O,
espec i ely. In addi ion o his, we can iden i y gαwi h Rn,Cn,Hn,O o F=R,C,H,
O, espec i ely. Indeed, gαis a Cli o d module o e Cl(m), whe e m= dim g2α, which is
he sum o equi alen Cli o d modules i m= 3, and is i educible i m= 7.
The subalgeb a a⊕no gis sol able and nis i s de i ed subalgeb a. We deno e by Aand
by N he connec ed closed Lie subg oups o Gwi h Lie algeb as aand n, espec i ely. Then,
G=KAN is an Iwasawa decomposi ion o Gand AN is di eomo phic o M. Fu he mo e,
i we pull back he me ic on M o AN we ge a le -in a ian Riemannian me ic on AN.
Thus, Mis isome ic o he sol able Lie g oup AN endowed wi h a le -in a ian me ic.
As such, i is an example o a Damek-Ricci space (see [7]).
2.2. Cohomogenei y one ac ions on hype bolic spaces.
We can dis inguish h ee di e en classes o cohomogenei y one ac ions on symme ic
spaces o non-compac ype and ank one, up o o bi equi alence. I was shown in [2] ha
any such ac ion has a mos one singula o bi .
8 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
Ac ions wi h no singula o bi .
Be nd and Tama u [4] classi ied ac ions wi hou singula o bi s. They p o ed ha he e
a e exac ly wo such ac ions up o o bi equi alence.
(i) The ac ion o Non FHn+1 has cohomogenei y one. The o bi s o his ac ion a e
mu ually cong uen ho osphe es which o m a egula Riemannian olia ion on FHn+1,
called he ho osphe e olia ion.
(ii) Le Sbe he connec ed Lie subg oup o AN wi h Lie algeb a s=a⊕w⊕z, whe e wis
a ec o subspace o gαo codimension one. Di e en choices o wlead o conjuga e
ac ions. The ac ion o Son FHn+1 has cohomogenei y one and i s o bi s o m a
egula Riemannian olia ion on FHn+1, called he sol able olia ion.
Ac ions wi h a o ally geodesic singula o bi .
Be nd and B ¨uck [2] classi ied cohomogenei y one ac ions on FHn+1 wi h a o ally
geodesic singula o bi F. The emaining o bi s, which a e p incipal, a e ubes a ound he
o ally geodesic submani old F, whe e:
(i) F=R:F∈ {poin ,RH1,...,RHn−1};
(ii) F=C:F∈ {poin ,CH1,...,CHn,RHn+1};
(iii) F=H:F∈ {poin ,HH1,...,HHn,CHn+1};
(i ) F=O:F∈ {poin ,OH1,HH2}.
Ac ions wi h a non- o ally geodesic singula o bi .
Be nd and Tama u [5] ga e a cons uc ion me hod o all cohomogenei y one ac ions
wi h a non- o ally geodesic singula o bi in hype bolic spaces. Such ac ions only appea
i F6=R. We ecall ha K0ac s on gαby he adjoin ep esen a ion, and hence, i Vis
a eal subspace o gα,N0
K0(V) will deno e he connec ed componen o he iden i y o he
no malize o Vin K0.
Theo em 2.1. Le g=k⊕a⊕nbe an Iwasawa decomposi ion o he Lie algeb a o he
isome y g oup o he hype bolic space M=FHn+1,F∈ {C,H,O}.
(i) Le Vbe a non-ze o ec o subspace o gαsuch ha N0
K0(V)ac s ansi i ely on
he uni sphe e o V. Deno e by gαV he o hogonal complemen o Vin gα.
Then he connec ed subg oup o Gwi h Lie algeb a Nk0(V)⊕a⊕(gαV)⊕g2α
ac s on Mwi h cohomogenei y one, and he o bi h ough ois singula , p o ided ha
dim V≥2. Fu he mo e, e e y cohomogenei y one ac ion on Mwi h a non- o ally
geodesic singula o bi can be ob ained in his way up o o bi equi alence.
(ii) Le Vand V0be ec o subspaces o gαas in (i), and assume ha he co esponding
cohomogenei y one ac ions ha e non- o ally geodesic singula o bi s. Then, hese
ac ions a e o bi equi alen i and only i he e exis s k∈K0such ha Ad(k)V=V0.
2.3. Qua e nionic K¨ahle angle.
The me ic and he qua e nionic K¨ahle s uc u e on HHn+1 induce a posi i e de ini e
inne p oduc h·,·i on gαand a qua e nionic s uc u e Jon gα, espec i ely, such ha gα
is isomo phic o Hnas a ( igh ) qua e nionic Euclidean space. He e, by a qua e nionic
s uc u e Jwe unde s and a 3-dimensional ec o subspace o EndR(Hn), he space o eal
HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 9
endomo phisms o Hn∼
=R4n, admi ing a basis {J1, J2, J3}o o hogonal ans o ma ions
o Hn∼
=R4nsuch ha J2
i=−Id and JiJi+1 =Ji+2 =−Ji+1Ji, o each i∈ {1,2,3}
(indices modulo 3). Such a basis is called a canonical basis o he qua e nionic s uc u e J.
Some imes i is help ul o ega d Jas endowed wi h a posi i e de ini e inne p oduc ha
makes i isome ic o he Euclidean 3-space R3, and such ha he elemen s o J ha a e
o hogonal complex s uc u es o Hncons i u e he uni sphe e S2⊂Jwi h espec o such
inne p oduc . Th oughou his a icle, i is a ec o in Hnand Vis a eal subspace o Hn
(i.e. a ec o subspace o he eal ec o space R4nwi h he unde lying eal ec o space
s uc u e o Hn), we will deno e by H =R ⊕J and by HV=V+JV he qua e nionic
spans o ∈Hnand o V⊂Hn, espec i ely; some imes we will also w i e (Im H) o
e e o J .
Theo em 2.1 shows he c ucial ole played by eal subspaces Vo gα∼
=Hnand hei
beha io wi h espec o K0in he classi ica ion p oblem o cohomogenei y one ac ions on
HHn+1. No e ha he e ec i iza ion o K0on gα∼
=Hnis he Lie g oup Sp(1)Sp(n) =
(Sp(1) ×Sp(n))/{±(1,Id)}, which ac s in he s anda d way: (q, A)· =A q−1, whe e
q∈Sp(1) and A∈Sp(n). Thus, in his subsec ion we ga he some impo an e minology
and use ul ac s o s udy eal subspaces o a qua e nionic Euclidean space, up o cong uence
by elemen s o Sp(1)Sp(n).
Fi s ly, mo i a ed by Theo em 2.1, we will say ha a eal subspace V⊂Hnis p o-
ohomogeneous i he e is a connec ed subg oup o Sp(1)Sp(n) ha ac s ansi i ely on
he uni sphe e o V. Equi alen ly, Vis p o ohomogeneous i he connec ed Lie g oup
N0
Sp(1)Sp(n)(V) ac s ansi i ely on he uni sphe e o V. No e ha p o ohomogeneous sub-
spaces Vo gα∼
=Hna e p ecisely hose inducing cohomogenei y one ac ions on HHn+1
ia he cons uc ion in Theo em 2.1(i). We will also say ha wo eal subspaces Vand
Wo Hna e equi alen i he e exis s an elemen T∈Sp(1)Sp(n) such ha TV =W.
Obse e ha , by Theo em 2.1(ii), Vand Wa e equi alen p o ohomogeneous subspaces o
gα∼
=Hni and only i hey induce o bi equi alen cohomogenei y one ac ions on HHn+1.
Le us now ecall a use ul desc ip ion o he ac ion o Sp(1)Sp(n) on Hn. Le {X1, . . . , Xn}
and {Y1, . . . , Yn}be wo H-o hono mal bases o Hn, and le {J1, J2, J3}and {J0
1, J0
2, J0
3}
be wo canonical bases o he qua e nionic s uc u e o Hn. Then, he e exis s a unique
T∈Sp(1)Sp(n) such ha T(Xi) = Yiand TJj=J0
jT o all i∈ {1, . . . , n}and all j∈
{1,2,3}. Con e sely, any R-linea endomo phism o Hnwhich maps H-o hono mal bases
o Hn o H-o hono mal bases o Hnand in e wines canonical bases o he qua e nionic
s uc u e o Hnin he abo e desc ibed ashion lies in Sp(1)Sp(n).
Le Vbe a eal ec o subspace o he qua e nionic Euclidean space Hn. The K¨ahle
angle o a non-ze o ec o ∈Vwi h espec o a non-ze o J∈Jand Vis de ined o be he
angle be ween J and V. Equi alen ly, i is he alue ϕ∈[0, π/2] such ha hPJ , PJ i=
cos2(ϕ)h , i, whe e PJ:= πVJand we deno e by πV he o hogonal p ojec ion on o V.
The ollowing lemma was essen ially p o ed by Be nd and B ¨uck [2, Lemma 3]. We
s a e i in a somewha di e en o m ollowing [15, Theo em 3.1], whe e i was p o ed in
he mo e gene al con ex o subspaces o Cli o d modules.
16 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
only i ϕ36=π/2. Fu he mo e, we can easily check ha ¯
Pi¯
Pj=−¯
Pj¯
Pi, o i6=j. Indeed,
i , w ∈V, hen Lemma 2.2 yields
0 = h¯
Pi( +w),¯
Pj( +w)i=h¯
Pi , ¯
Pjwi+h¯
Pj , ¯
Piwi=−h¯
Pj¯
Pi , wi−h¯
Pi¯
Pj , wi.
Hence, Vhas a module s uc u e o e he Cli o d algeb a Cl(3) i ϕ36=π/2, o o e Cl(2)
i ϕ3=π/2. I is well known ha he e a e exac ly wo inequi alen i educible Cli o d
modules o e Cl(3), bo h o dimension 4 (we will deno e hem by V0and V1), whe eas
he e is exac ly one i educible Cl(2)-module up o equi alence, again o dimension 4 (we
will deno e i by V0). Mo eo e , Cli o d modules a e semisimple. This implies ha , i
ϕ36=π/2, we can decompose Vin o a di ec sum o i educible Cl(3)-modules as ollows
V= l0
MV0!⊕ l1
MV1!,
whe e l0+l1=l, whe eas i ϕ3=π/2 he Cl(2)-module Vcan be decomposed as
V=
l
MV0.
The abo e decomposi ions can be assumed o be o hogonal because he complex s uc u es
¯
Pia e o hogonal. This also implies ha wo di e en summands a e H-o hogonal: i
, w ∈Vbelong o wo di e en summands, hJk , wi=hPk , wi= 0 by he ¯
Pk-in a iance.
Finally, since ¯
Pilea es each ac o V ( ∈ {0,1}) in a ian , we deduce ha each V has
cons an qua e nionic K¨ahle angle (ϕ1, ϕ2, ϕ3). This leads us o s a e he ollowing:
Lemma 4.3. Le Vbe a eal subspace o Hno dimension 4l, wi h l∈N, and con-
s an qua e nionic K¨ahle angle (ϕ1, ϕ2, ϕ3). Assume ha he e exis s a canonical basis
{J1, J2, J3}o Jsuch ha he K¨ahle angle o any non-ze o ∈Vwi h espec o Jiand
Vis ϕi, o each i∈ {1,2,3}. Then, he e is an H-o hogonal decomposi ion
V=
l
M
=1
V ,
whe e each V has dimension 4and Φ(V ) = (ϕ1, ϕ2, ϕ3)as a subspace o Hn.
Con e sely, le Vbe a eal subspace o Hngi en by an H-o hogonal di ec sum V:=
Ll
=1 V , whe e each V has dimension 4, and Φ(V )=(ϕ1, ϕ2, ϕ3). Le {J1, J2, J3}be a
canonical s uc u e o Jsuch ha e e y non-ze o ec o in V has K¨ahle angle ϕiwi h e-
spec o Jiand V , o each i∈ {1,2,3}and each ∈ {1, . . . , l}. Then, Φ(V) = (ϕ1, ϕ2, ϕ3).
P oo . The i s asse ion has been p o ed abo e unde he assump ion ϕ26=π/2. I ϕ2=
π/2, he i s claim ollows om he classi ica ion o subspaces Vwi h Φ(V) = (ϕ, π/2, π/2),
ϕ∈[0, π/2] (c .
§
2.4 and [2, pp. 230-232]).
In o de o p o e he con e se, we i s no e ha πV(HV ) = V , o each ∈ {1, . . . , l}.
Indeed, o e e y ∈V and w∈Vs, 6=s,hπVJi , wi=hJi , wi= 0, whe e in he
las equali y we ha e used HV ⊥HVs. Hence, πVJ(V )⊂V , and since πV(V ) = V , we
deduce πV(HV ) = V .

HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 17
Now le =Pl
=1 ∈V, wi h ∈V o each ∈ {1, . . . , l}. Deno ing as usual
Pi=πVJi, o each i∈ {1,2,3}, we ha e
L (Ji, Jj) = hPi , Pj i=
l
X
,s=1hπVJi , πVJj si=
l
X
=1 hπVJi , πVJj i
=
l
X
=1 hPi , Pj i=
l
X
=1
cos2(ϕi)δijk k2= cos2(ϕi)δijk k2,
whe e in he hi d equali y we ha e used πV(HV ) = V and V ⊥Vs o all , s ∈ {1, . . . , l},
and in he i h one we ha e used ha he qua e nionic K¨ahle angle o wi h espec
o V and {J1, J2, J3}is (ϕ1, ϕ2, ϕ3). Since ∈Vis a bi a y, by Lemma 2.2 we conclude
ha Φ(V)=(ϕ1, ϕ2, ϕ3). 
5. Low dimensional subspaces wi h cons an qua e nionic K¨
ahle angle
As a consequence o P oposi ion 3.2, we only ha e o s udy subspaces o dimensions 3
and mul iples o 4. The la e can be educed o s udying subspaces o dimension 4 by
i ue o Co olla y 4.2 and Lemma 4.3. We will de o e his sec ion o he classi ica ion o
(no necessa ily p o ohomogeneous) eal subspaces o dimensions k∈ {3,4}wi h cons an
qua e nionic K¨ahle angle. The main ool ha we will use in his sec ion is he isospec-
ali y o he K¨ahle angle map Ω in oduced in (1). We s a wi h a lemma ha p o ides
an app op ia e basis o he subspace.
Lemma 5.1. Le Vbe a eal subspace o Hno dimension k∈ {3,4}wi h Φ(V) =
(ϕ1, ϕ2, ϕ3). Le e0∈Vbe a uni ec o . Then, he e exis s a canonical basis {J1, J2, J3}
o Jand ec o s ei∈HnHe0,i∈ {1, . . . , k −1}, such ha
(2) cos(ϕi)Jie0+ sin(ϕi)Jiei, i ∈ {0, . . . , k −1},
cons i u e an R-o hono mal basis o V, whe e we pu J0:= Id and ϕ0= 0.
Mo eo e , o each i∈ {0, . . . , k −1}wi h ϕi6=π/2, we ha e
¯
Pie0= cos(ϕi)Jie0+ sin(ϕi)Jiei,
whe e ¯
Pi=Pi/cos(ϕi) = πVJi/cos(ϕi).
Finally, i ϕi= 0 we ake ei= 0, whe eas i ϕi>0, hen eiis a uni ec o .
P oo . Le e0∈Vbe a uni ec o . By Lemma 2.2, he e is a canonical basis {J1, J2, J3}
o Jsuch ha e0has K¨ahle angle ϕiwi h espec o Ji o i∈ {1,2,3}, and hPie0, Pje0i=
cos2(ϕi)δij. In pa icula , h¯
Pie0,¯
Pje0i= 0 o any i,j∈ {0, . . . , k −1},i6=j, wi h ϕi,
ϕj6=π/2.
Fix i∈ {1,2,3}. I ϕi= 0, hen we ake ei= 0. Le us assume i s ha ϕi∈(0, π/2).
By ega ding Hnas a complex ec o space C2nwi h espec o he complex s uc u e Ji,
[2, Lemma 2] yields he exis ence o a uni ec o ei∈HnspanR{e0, Jie0}sa is ying
¯
Pie0= cos(ϕi)Jie0+ sin(ϕi)Jiei.
18 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
We ha e o see ha ei∈HnHe0. Obse e ha HnspanR{e0, Jie0}coincides wi h he
o hogonal sum (HnHe0)⊕spanR{Ji+1e0, Ji+2e0}, whe e indices a e aken modulo 3. Le
a, b ∈R. Then
hei, aJi+1e0+bJi+2e0i=−1
sin(ϕi)hJi¯
Pie0+ cos(ϕi)e0, aJi+1e0+bJi+2e0i
=1
sin(ϕi) cos(ϕi)(ahPie0, Pi+2e0i−bhPie0, Pi+1e0i)=0,
whe e in he las equali y we ha e used Lemma 2.2. The e o e, ei∈HnHe0.
Now i ϕ2=π/2, subspaces Vwi h Φ(V) = (ϕ, π/2, π/2), ϕ∈[0, π/2], a e classi ied
(see
§
2.4) and hey can be spanned by a basis as in he s a emen (see [2, p. 232] and no e
ha he {ei}in he s a emen do no ha e o be H-o hono mal).
Thus, we inally ha e o deal wi h he case k= 4, ϕ26=π/2, and ϕ3=π/2. Then, by
he p e ious a gumen , he e exis s a uni ec o ∈Hnsuch ha {e0,¯
P1e0,¯
P2e0, }is an
R-o hono mal basis o V, whe e ¯
Pie0= cos(ϕi)Jie0+ sin(ϕi)Jiei,i∈ {1,2}. Recalling he
de ini ion o he K¨ahle angle map (1), we ha e
(Ω(e0)) =
3
X
i=1 hPie0, Pie0i=
3
X
i=1 hPie0, e0i2+hPie0,¯
P1e0i2+hPie0,¯
P2e0i2+hPie0, i2
= cos2(ϕ1) + cos2(ϕ2) +
3
X
i=1 hJie0, i2,
whe e we ha e used Lemma 2.2 and ¯
Pie0=Pie0/cos(ϕi). Since Φ(V) = (ϕ1, ϕ2, π/2), he
eigen alues o Ω(e0) a e cos2(ϕ1), cos2(ϕ2) and 0, and hence we deduce ha ∈HnHe0.
Thus, aking e3=−J3 yields he esul . 
Rema k 5.2.Whene e ϕ1>0, he o hogonali y o (2) yields hJ3e1, e2i= 0, and i k= 4,
also hJ1e2, e3i=hJ2e3, e1i= 0.
5.1. Subspaces o dimension 3.
In his subsec ion we classi y 3-dimensional eal subspaces o Hnwi h cons an qua e -
nionic K¨ahle angle.
P oposi ion 5.3. Le V⊂Hnbe a eal subspace o dimension 3. Then, Vhas cons an
qua e nionic K¨ahle angle i and only i Φ(V)=(ϕ, ϕ, π/2),ϕ∈[0, π/2], and o any uni
e0∈V, he e is a canonical basis {J1, J2, J3}o Jsuch ha
(3) {e0,cos(ϕ)J1e0+ sin(ϕ)J1e1,cos(ϕ)J2e0+ sin(ϕ)J2e2}
is an o hono mal basis o V, whe e, i ϕ6= 0,e1, e2a e uni ec o s sa is ying e1, e2∈
HnHe0,e2∈Hn(Im H)e1, and ei he he1, e2i= cos(ϕ)/(cos(ϕ)−1) wi h ϕ∈[π/3, π/2],
o he1, e2i= cos(ϕ)/(cos(ϕ) + 1) wi h ϕ∈(0, π/2].
P oo . By P oposi ion 3.2, we ha e ha ϕ1=ϕ2=ϕ∈[0, π/2] and ϕ3=π/2. Le us
assume ha Vis spanned by he basis desc ibed in Lemma 5.1 wi h k= 3. I ϕ= 0 o
HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 19
ϕ=π/2 he claim ollows om he classi ica ion o subspaces wi h cons an qua e nionic
K¨ahle angle (0,0, π/2) o (π/2, π/2, π/2); see
§
2.4.
Thus, le us assume ϕ∈(0, π/2). Then, o each l∈ {1,2}and unde s anding he
subsc ip l+ 1 ∈ {1,2}modulo 2,
Ω( ¯
Ple0)ij =hPi¯
Ple0, Pj¯
Ple0i=hJi¯
Ple0, e0ihJj¯
Ple0, e0i+
2
X
=1hJi¯
Ple0,¯
P e0ihJj¯
Ple0,¯
P e0i,
=h¯
Ple0, Pie0ih¯
Ple0, Pje0i+hJi¯
Ple0,¯
Pl+1e0ihJj¯
Ple0,¯
Pl+1e0i,
whe e in he second equali y we ha e calcula ed he o hogonal p ojec ion o ec o s on o
Vby using he o hono mal basis {e0,¯
P1e0,¯
P2e0}o V. Hence, o l∈ {1,2}, using
Lemma 5.1 we ha e
(4)
Ω( ¯
Ple0)ll = cos2(ϕ) + hel, Jl+1el+1i2sin4(ϕ),
Ω( ¯
Ple0)l+1,l =he1, J1e2ihe1, J2e2isin4(ϕ),
Ω( ¯
Ple0)l+1,l+1 =hel+1, Jleli2sin4(ϕ),
Ω( ¯
Ple0)13 =he2, J2e1isin2(ϕ)(cos2(ϕ) + he1, e2isin2(ϕ)),
Ω( ¯
Ple0)23 =−he1, J1e2isin2(ϕ)(cos2(ϕ) + he1, e2isin2(ϕ)),
Ω( ¯
Ple0)33 = (cos2(ϕ) + he1, e2isin2(ϕ))2.
Now, since Ω( ¯
Ple0) is symme ic wi h eigen alues cos2(ϕ) (o mul iplici y 2) and 0, by he
min-max heo em, one ob ains
0≤Ω( ¯
Ple0)ll ≤cos2(ϕ), l ∈ {1,2}.
This implies he1, J2e2i=he2, J1e1i= 0, which oge he wi h Rema k 5.2 yields e2∈
Hn(Im H)e1. Taking again in o accoun he spec um o Ω( ¯
P1e0), we ha e he ollowing
ela ion o i s ace,
2 cos(ϕ)2= (Ω( ¯
P1e0)) = (cos2(ϕ) + he1, e2isin2(ϕ))2+ cos2(ϕ).
F om his and he ac ha e1and e2a e uni ec o s, we deduce ha ei he he1, e2i=
cos(ϕ)/(cos(ϕ)−1) whe e ϕ∈[π/3, π/2) o he1, e2i= cos(ϕ)/(1 + cos(ϕ)) whe e ϕ∈
(0, π/2). This p o es he necessi y o he s a emen .
Fo he con e se we ake an a bi a y uni ec o ∈Vwhich we w i e as
=x0e0+x1cos(ϕ)J1e0+ sin(ϕ)J1e1+x2cos(ϕ)J2e0+ sin(ϕ)J2e2.
Then, i ε∈ {±1}is such ha he1, e2i= cos(ϕ)/(1 + εcos(ϕ)), we ha e
Ω( ) = cos2(ϕ)
x2
0+x2
1x1x2−εx0x2
x1x2x2
0+x2
2εx0x1
−εx0x2εx0x1x2
1+x2
2
.
Since is a uni ec o , x2
0+x2
1+x2
2= 1, and i is now easy o see ha Ω( ) has a double
eigen alue cos2(ϕ), and a simple eigen alue 0. 
20 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
Rema k 5.4.We will deno e by Vϕ
+and Vϕ
−any eal subspace o Hncons uc ed as in
P oposi ion 5.3, depending on whe he he1, e2i= cos(ϕ)/(cos(ϕ) + 1) o ϕ∈(0, π/2], o
he1, e2i= cos(ϕ)/(cos(ϕ)−1) o ϕ∈[π/3, π/2], espec i ely. No e ha he subspaces Vϕ
±
can be cons uc ed as subspaces o any Hnwi h n≥3. One can easily check ha he only
one ha i s in o an H2is Vπ/3
−(bu i canno i in o H).
P oposi ion 5.5. Le Vbe a subspace o Hnwi h cons an qua e nionic K¨ahle angle and
dimension 3. Then Vis p o ohomogeneous.
P oo . We know om P oposi ion 5.3 ha Φ(V)=(ϕ, ϕ, π/2). We can assume ha
ϕ∈(0, π/2) since, o he wise, Vis known o be p o ohomogeneous (see
§
2.4).
Le e0∈Vbe an a bi a y uni ec o . By Lemma 2.2 he e is a canonical basis
{J1, J2, J3}o Jsuch ha e0has K¨ahle angle ϕwi h espec o J1and J2, and K¨ahle
angle π/2 wi h espec o J3. In iew o Lemma 5.1 and P oposi ion 5.3, le us conside
he uni ec o s ei∈HnHe0,i∈ {1,2}, gi en by
(5) ei:= −(Ji¯
Pie0+ cos(ϕ)e0)/sin(ϕ), i ∈ {1,2},
whe e ¯
Pi:= πVJi/cos(ϕ). On he one hand, by (5) we ha e
(6) he1, e2i=1
sin2(ϕ)hJ1¯
P1e0+ cos(ϕ)e0, J2¯
P2e0+ cos(ϕ)e0i
=1
sin2(ϕ)hJ1¯
P1e0, J2¯
P2e0i−cos2(ϕ).
On he o he hand, again by P oposi ion 5.3, he1, e2ican ake wo possible alues. We will
i s see ha , gi en V,he1, e2iis independen o e0.
Le S2deno e he uni sphe e o V. We de ine Θ: S2→Rby Θ(e0) = he1, e2i. We claim
ha Θ is well de ined. Le {J0
1, J0
2, J3}be ano he canonical basis o Jsuch ha e0has
K¨ahle angle ϕwi h espec o J0
i,i∈ {1,2}, and le e0
i:= −(J0
i¯
P0
ie0+ cos(ϕ)e0)/sin(ϕ)
whe e ¯
P0
i:= πVJ0
i/cos(ϕ) o i∈ {1,2}. Then, he e is θ∈[0,2π) such ha J0
i=
cos(θ)Ji+ (−1)i+1 sin(θ)Ji+1 o i∈ {1,2}and subsc ip s modulo 2. Thus,
(7) J0
i¯
P0
i= (cos(θ)Ji+ (−1)i+1 sin(θ)Ji+1)(cos(θ)¯
Pi+ (−1)i+1 sin(θ)¯
Pi+1)
= cos2(θ)Ji¯
Pi+ sin2(θ)Ji+1 ¯
Pi+1 + (−1)i+1 cos(θ) sin(θ)(J1¯
P2+J2¯
P1).
Consequen ly, using Equa ion (6) wice, and hen (7), we ge , a e some calcula ions,
he1, e2i−he0
1, e0
2i=1
sin2(ϕ)hJ1¯
P1e0, J2¯
P2e0i−hJ0
1¯
P0
1e0, J0
2¯
P0
2e0i= 0,
which implies ha Θ is well-de ined.
Now no e ha he assignmen e0∈S27→ span{J1, J2} ∈ G2(J), whe e G2(J) is he
G assmannian o 2-planes o J∼
=R3, is con inuous due o he con inuous dependence o
he quad a ic o m J∈J7→ L (J, J) = hPJ , PJ i ∈ Ron . Hence, he map Θ is also
con inuous. Bu , as men ioned jus a e (6), Θ(S2) has a mos wo elemen s. The e o e,
Θ is cons an on S2.
HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 21
Finally, we p o e ha Vis p o ohomogeneous. Le e0,e0
0be a bi a y uni ec o s in V.
Le {J1, J2, J3},{J0
1, J0
2, J0
3}be canonical bases o J, and e1, e2, e0
1, e0
2be uni ec o s in V
such ha bo h (3), and (3) wi h e0
iins ead o eiand J0
iins ead o Ji, a e o hono mal
bases o V. Bo h se s o ec o s {e0, e1, e2}and {e0
0, e0
1, e0
2}span a o ally eal subspace
o Hn, and since Θ is cons an , hei, eji=he0
i, e0
ji o all i, j ∈ {0,1,2}. I hen ollows
ha he e exis s an elemen T∈Sp(1)Sp(n) such ha Tei=e0
i o each i∈ {0,1,2}, and
TJj=J0
jT o each j∈ {1,2,3}. Thus, by (5) we ge T¯
Pie0=¯
P0
ie0
0 o i∈ {0,1,2}, whe e
¯
P0=¯
P0
0= Id. The e o e, Tis an elemen o Sp(1)Sp(n) such ha TV =Vand Te0=e0
0.
Since e0, e0
0a e a bi a y, his p o es ha Vis p o ohomogeneous. 
Finally we show ha he wo ypes o subspaces Vϕ
+and Vϕ
−in oduced in Rema k 5.4
a e indeed inequi alen o ϕ6=π/2. Recall ha Vϕ
+is de ined o all ϕ∈(0, π/2], bu Vϕ
−
only o ϕ∈[π/3, π/2].
P oposi ion 5.6. Le ϕ∈[π/3, π/2]. Then he e exis s T∈Sp(1)Sp(n)such ha TV ϕ
+=
Vϕ
−i and only i ϕ=π/2.
P oo . I ϕ=π/2, hen Vπ/2
+and Vπ/2
−a e o ally eal, he e o e equi alen . Le us assume
ha ϕ6=π/2 and ha he e is T∈Sp(1)Sp(n) such ha TV ϕ
+=Vϕ
−. By applying an
elemen o Sp(1)Sp(n) i necessa y, we can assume ha he e is a uni ec o e0∈Vϕ
+∩Vϕ
−
and ha e0has qua e nionic K¨ahle angle (ϕ, ϕ, π/2) wi h espec o bo h Vϕ
+and Vϕ
−and
a common canonical basis {J1, J2, J3}o J. Then, by Lemma 5.1 and P oposi ion 5.3, Vϕ
±is
spanned by he basis {e0,¯
P±
1e0,¯
P±
2e0}, whe e ¯
P±
i:= πVϕ
±Ji/cos(ϕ), i∈ {1,2}. Mo eo e ,
¯
P±
ie0= cos(ϕ)Jie0+ sin(ϕ)Jie±
i,wi h he±
1, e±
2i=cos(ϕ)
cos(ϕ)±1,
and e±
i∈HnHe0,i∈ {1,2}.
By P oposi ion 5.5, we can assume ha Te0=e0. Le J0=TJ3T−1∈J. Since
J3e0∈HnVϕ
+, we ha e J0e0=J0Te0=TJ3e0∈HnVϕ
−. This implies TJ3=εJ3T,
whe e ε∈ {−1,1}, because ±J3a e he only complex s uc u es in Jwhich send e0 o
HnVϕ
−. The e o e, he e exis s θ∈[0,2π) such ha
(8) TJi=εi(cos(θ)Ji+ (−1)i+1 sin(θ)Ji+1)T, i ∈ {1,2},and TJ3=εJ3T.
Using (8) and Te0=e0, we ha e
(9) T¯
P+
1e0= cos(ϕ)TJ1e0+ sin(ϕ)TJ1e+
1
=εcos(ϕ)(cos(θ)J1e0+ sin(θ)J2e0) + εsin(ϕ)(cos(θ)J1Te+
1+ sin(θ)J2Te+
1).
By P oposi ion 5.5, Vϕ
±is p o ohomogeneous, and no e ha SO(3) is he only connec ed
subg oup o Sp(1)Sp(n)⊂SO(4n) ha ac s ansi i ely and e ec i ely on he uni sphe e
o Vϕ
±. Thus, we can assume ha T¯
P+
1e0=ε¯
P−
1e0, jus by composing Twi h some
elemen in he iso opy o he ac ion o SO(3) on Vϕ
−a e0. Bu inse ing (9) and ¯
P−
1e0=
cos(ϕ)J1e0+sin(ϕ)J1e−
1in o he equali y T¯
P+
1e0=ε¯
P−
1e0, and analyzing he He0and Hn
He0componen s (no e ha e±
1∈HnHe0,Te0=e0, and Tp ese es H-o hono mali y)
we ge θ= 0 and Te+
1=e−
1. Mo eo e , by (8) we ge TJi=εiJiT,i∈ {1,2,3}.

22 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
Since Te0=e0,T¯
P+
1e0=ε¯
P−
1e0and TV ϕ
+=Vϕ
−, we mus ha e T¯
P+
2e0=±¯
P−
2e0.
Then, inse ing ¯
P±
2e0= cos(ϕ)J2e0+sin(ϕ)J2e±
2in he las equali y, and using TJ2=J2T,
we deduce ha Te+
2=e−
2. Bu his join ly wi h Te+
1=e−
1yields a con adic ion wi h
he ac ha Tis an o hogonal ans o ma ion o Hn, because he+
1, e+
2i 6=he−
1, e−
2i o all
ϕ6=π/2. 
5.2. Subspaces o dimension ou .
The aim o his subsec ion is o classi y 4-dimensional eal subspaces o Hnwi h cons an
qua e nionic K¨ahle angle.
We s a by es ic ing ou a en ion o subspaces wi h ϕ1= 0.
P oposi ion 5.7. Le V⊂Hnbe a eal subspace o dimension 4wi h cons an qua e nionic
K¨ahle angle (0, ϕ2, ϕ3). Then, ϕ2=ϕ3∈[0, π/2].
P oo . Fi s o all, i ϕ2= 0, hen ϕ3= 0 by a combina ion o [2, P oposi ion 9] and
he ac ha subspaces wi h Φ(V) = (0,0, π/2) ha e dimension 3 (see
§
2.4). Hence,
le us assume ha ϕ26= 0. Lemma 5.1 yields a basis {e0, J1e0, 2, 3}o V, whe e i=
cos(ϕi)Jie0+sin(ϕi)Jiei, o ce ain uni ei∈HnHe0,i∈ {2,3}. The e o e, a compu a ion
as in Equa ions (4), o each i∈ {2,3}, gi es
Ω( i)11 =cos(ϕ2) cos(ϕ3) + he2, e3isin(ϕ2) sin(ϕ3)2,
Ω( i)22 = cos(ϕi)2+hJ3e3, e2i2sin2(ϕ2) sin2(ϕ3),
Ω( i)33 = cos(ϕi)2+hJ2e2, e3i2sin2(ϕ2) sin2(ϕ3).
Hence, by he isospec ali y o Ω,
0 = (Ω( 2)) − (Ω( 3)) = 2 cos2(ϕ2)−2 cos2(ϕ3),
om whe e we conclude ϕ2=ϕ3.
In iew o P oposi ion 5.7, all eal subspaces Vo Hnwi h Φ(V) = (0, ϕ2, ϕ3) ac ually
sa is y Φ(V) = (0, ϕ, ϕ). No e ha such subspaces ha e been classi ied (see
§
2.4).
Thus, in he ollowing esul s we will analyze he case ϕ1>0. We conside he basis o
Vgi en in Lemma 5.1.
Lemma 5.8. Le V⊂Hnbe a eal subspace o dimension 4such ha Φ(V)=(ϕ1, ϕ2, ϕ3)
wi h ϕ1>0. Fo each i∈ {1,3}wi h ϕi6=π/2, we ha e hei, Jjeji= 0 o all j∈ {1,2,3}.
P oo . Acco ding o Lemma 5.1, e0has K¨ahle angle ϕiwi h espec o Vand Ji∈J o
each i∈ {1,2,3}. Le us ega d Hnas a complex Euclidean space C2nwhose complex
s uc u e is Ji, o i∈ {1,2,3}. By [16, Theo em 2.7] he e is a non-emp y ini e subse
Ψi⊂[0, π/2] such ha V=Lϕ∈ΨiVi
ϕis a C-o hono mal decomposi ion o Vand Vi
ϕ⊂
C2nis a eal subspace wi h cons an K¨ahle angle ϕ∈Ψi. I ollows ha any non-ze o
∈Vi
ϕhas K¨ahle angle ϕwi h espec o Vand Ji, and he minimum ( esp. maximum) o
Ψicoincides wi h he minimum ( esp. maximum) K¨ahle angle o a non-ze o ec o ∈V
wi h espec o Vand Ji.
HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 23
We claim ha ϕ1∈Ψ1. On he one hand, i he e exis ed ϕ∈Ψ1such ha ϕ<ϕ1, hen
he e would be ec o s in Vwhose K¨ahle angle wi h espec o Vand J1∈Jis ϕ<ϕ1,
hus con adic ing he minimali y o ϕ1by Lemma 2.2. On he o he hand, i ϕ > ϕ1 o
all ϕ∈Ψ1, hen we would ge a con adic ion wi h he ac ha e0has K¨ahle angle ϕ1
wi h espec o J1. Analogously, we ge ha ϕ3∈Ψ3.
Now assume ϕ16=π/2. By [16, p. 1190–1191] and he discussion abo e, we ha e a
decomposi ion V=Vϕ1⊕Vψ1in o eal subspaces o cons an K¨ahle angle wi h espec o
he complex s uc u e J1, whe e ψ1∈Ψ1( he possibili y ψ1=ϕ1is allowed). We also ha e
ha ¯
P1:= πVϕ1J1/cos(ϕ1) = πVJ1/cos(ϕ1)|Vϕ1de ines a complex s uc u e on Vϕ1. As e0∈
Vϕ1, we ge Vϕ1= spanR{e0,¯
P1e0}. Mo eo e , CVϕ1⊥CVψ1, so Vψ1= spanR{¯
P2e0,¯
P3e0},
and o j∈ {2,3}, using Lemma 5.1,
0 = h¯
P1e0, J1¯
Pje0i= sin(ϕ1) sin(ϕj)he1, Jjeji.
Since ϕ1>0, we ge he1, Jjeji= 0. A simila a gumen wo ks o ϕ3, i ϕ36=π/2. 
Be o e add essing he classi ica ion, we s a e a lemma ha e ines [15, Lemma 5.1].
Lemma 5.9. Assume 0< ϕ1≤ϕ2≤ϕ3≤π/2, and le ε∈ {−1,1}. Then, he e exis s a
subse {e1, e2, e3}o uni ec o s o R3wi h inne p oduc s
hei, ei+1i=εcos(ϕi+2)−cos(ϕi) cos(ϕi+1)
sin(ϕi) sin(ϕi+1) o each i∈ {1,2,3}
i and only i cos(ϕ1) + cos(ϕ2)−εcos(ϕ3)≤1.
Fu he mo e, he subspace spanR{e1, e2, e3}has dimension 2i and only i cos(ϕ1) +
cos(ϕ2) + εcos(ϕ3) = 1, and dimension 3o he wise.
P oo . A subse {e1, e2, e3}o he Euclidean space R3sa is ies he inne p oduc ela ions
in he s a emen i and only i he associa ed G am ma ix G= (hei, eji)1≤i,j≤3is posi i e
semi-de ini e. This happens p ecisely when all p incipal mino s o Ga e non-nega i e; in
his p oo , by Gij we deno e he ma ix o o de 2 esul ing om dele ing he i- h ow
and he j- h column o G. Le xi:= cos(ϕi) o each i∈ {1,2,3}. Hence, Gis posi i e
semi-de ini e i and only i de Gii ≥0 o all i∈ {1,2,3}and de G≥0. We compu e
de (G) = (ε+x1−x2−x3)(−ε+x1+x2−x3)(−ε+x1−x2+x3)(ε+x1+x2+x3)
(1 −x2
1)(1 −x2
2)(1 −x2
3).
Taking in o accoun ha 1 > x1≥x2≥x3≥0, one can check ha de G≥0 i and only
i −1+x1+x2−εx3≤0. Simila ly, de (Gii) = (1−x2
1−x2
2−x2
3+2εx1x2x3)/Qj6=i(1−x2
j),
i∈ {1,2,3}. Hence, de (Gii)≥0 i and only i
(10) 1 −x2
1−x2
2−x2
3+ 2εx1x2x3≥0.
Now, i 1 > x1≥x2≥x3≥0, one can show ha (10) holds p o ided ha −1 + x1+x2−
εx3≤0. This comple es he p oo o he i s claim o he lemma.
Assume ha we a e in he si ua ion o he i s asse ion o he s a emen . Then
{e1, e2, e3}spans a 3-dimensional subspace i and only i Gis posi i e de ini e, which in his
si ua ion amoun s o de G > 0. This happens p ecisely when x1+x2−εx3<1. Hence,
24 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
he p oo o he lemma will be comple e i we show ha spanR{e1, e2, e3}canno ha e
dimension 1. Assume his is he case. Then he ank o Gis 1. Hence x1= 1 −x2+εx3,
and he mino de (G33) anishes, i.e.
0 = de (G33) = −2(1 + x3)
(1 + x2)(−2 + x2−x3).
The e o e, x3=−ε, which yields a con adic ion. This inishes he p oo . 
We a e now in posi ion o comple e he desc ip ion o 4-dimensional eal subspaces o
Hnwi h cons an qua e nionic K¨ahle angle.
P oposi ion 5.10. Le V⊂Hnbe a eal subspace o dimension 4and e0∈Va uni
ec o . Then Vhas cons an qua e nionic K¨ahle angle Φ(V)=(ϕ1, ϕ2, ϕ3), wi h ϕ1>0,
i and only i he e is a canonical basis {J1, J2, J3}o J,ε∈ {−1,1}, and uni ec o s
e1, e2, e3∈HnHe0wi h ei∈Hn(Im H)ej,i,j∈ {1,2,3}, such ha
(i) 0 < ϕ1≤ϕ2≤ϕ3≤π/2,
(ii) cos(ϕ1) + cos(ϕ2)−εcos(ϕ3)≤1,
(iii) o all i∈ {1,2,3}and indices modulo 3,
hei, ei+1i=εcos(ϕi+2)−cos(ϕi) cos(ϕi+1)
sin(ϕi) sin(ϕi+1),
(i ) {cos(ϕi)Jie0+ sin(ϕi)Jiei:i= 0,1,2,3}is an o hono mal basis o V, whe e o
simplici y we pu ϕ0:= 0 and J0:= Id.
Mo eo e , i Vis as abo e, he K¨ahle angle o any non-ze o ∈Vwi h espec o Ji
and Vis ϕi, o each i∈ {1,2,3}.
P oo . In o de o p o e he necessi y, le us assume ha Vis spanned by he basis de-
sc ibed in Lemma 5.1 wi h k= 4. No ice ha ϕ2=π/2 implies Φ(V)=(ϕ, π/2, π/2);
such subspaces a e classi ied (see
§
2.4), and [2, p. 232], oge he wi h some s aigh o wa d
calcula ions, show ha hey can be spanned by a basis as abo e. Thus, we can suppose
ϕ1, ϕ2∈(0, π/2).
Le us i s assume ϕ36=π/2. A long bu elemen a y calcula ion, simila o he one used
o ob ain Equa ions (4), using he isospec ali y o Ω, Rema k 5.2 and Lemma 5.8, yields
3
Y
j=1
cos2(ϕj) = de (Ω( ¯
Pie0)) = cos2(ϕi)
3
Y
j=1
j6=i
(cos(ϕi) cos(ϕj) + hei, ejisin(ϕi) sin(ϕj))2,
o each i∈ {1,2,3}. This implies
(11) cos2(ϕi+2) = (cos (ϕi) cos(ϕi+1) + hei, ei+1isin (ϕi) sin (ϕi+1)) 2, i ∈ {1,2,3}.
Using (11), we can also calcula e o i∈ {1,3}
3
X
j=1
cos2(ϕj) = (Ω( ¯
Pie0)) =
3
X
j=1
cos2(ϕj) + he2, Jieii2sin2(ϕi) sin2(ϕ2),
HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 25
which implies he2, Jieii= 0, i∈ {1,3}. This, along wi h Rema k 5.2 and Lemma 5.8, shows
ha ei∈Hn(Im H)ej,i, j ∈ {1,2,3}. Fu he mo e, (11) gi es ise o he wo possible
exp essions o hei, ei+1iin he s a emen (co esponding o ε= 1 o ε=−1). No e ha
such exp essions a e incompa ible o a ixed V, ha is, i o some i∈ {1,2,3}we ha e
hei, ei+1i=cos(ϕi+2)−cos(ϕi) cos(ϕi+1)
sin(ϕi) sin(ϕi+1),hei+1, ei+2i=−cos(ϕi) + cos(ϕi+1) cos(ϕi+2)
sin(ϕi+1) sin(ϕi+2),
hen one can check ha de Ω((e0+¯
Pi+1e0)/√2)= 0, which gi es a con adic ion wi h
he assump ion ϕ36=π/2. Finally, he inequali y in i em (ii) o he s a emen ollows om
Lemma 5.9.
Now assume ha ϕ3=π/2. Le {e0,¯
P1e0,¯
P2e0, J3e3}be he o hono mal basis p o-
ided by Lemma 5.1. A simila compu a ion as in (4), using he isospec ali y o Ω and
Lemma 5.8, yields
2
X
i=1
cos2(ϕi) = (Ω( ¯
P1e0)) + (Ω( ¯
P2e0)) − (Ω(J3e3))
= cos2(ϕ1) + cos2(ϕ2) + 2 sin2(ϕ1) sin2(ϕ2)hJ1e1, e2i2
+ 2 (cos(ϕ1) cos(ϕ2) + sin(ϕ1) sin(ϕ2)he1, e2i)2.
Then,
(12) he2, J1e1i= 0 and he1, e2i=−co (ϕ1) co (ϕ2).
Also, using (12), i i∈ {1,2}we ge
0 = de Ω1
√2e0+1
√2
¯
Pie0=1
4he3, Jieii2cos2(ϕ1) cos2(ϕ2) sin2(ϕi).
Thus,
(13) he3, Jieii= 0, i ∈ {1,2}.
Taking in o accoun (12) and (13), we can calcula e
2
X
i=1
cos2(ϕi) = (Ω( ¯
P1e0)) = cos2(ϕ1) + he1, e3i2sin2(ϕ1),
2
X
i=1
cos2(ϕi) = (Ω( ¯
P2e0)) = cos2(ϕ2) + sin2(ϕ2)(he2, e3i2+he2, J3e3i2),
whence
(14) he1, e3i=εcos(ϕ2)/sin(ϕ1) and cos2(ϕ1) = sin2(ϕ2)(he2, e3i2+he2, J3e3i2),
o some ε∈ {−1,1}. Using hese ela ions we compu e
2
X
i=1
cos2(ϕi) = Ω1
√2
¯
P1e0+1
√2
¯
P2e0=
2
X
i=1
cos2(ϕi) + ε1
2he2, J3e3isin(2ϕ2),
32 J. C. D´
IAZ-RAMOS, M. DOM´
INGUEZ-V´
AZQUEZ, AND A. RODR´
IGUEZ-V´
AZQUEZ
Fu he mo e, when n < k ≤2n, we ha e Φ(V) = (0, π/2, π/2), whe eas when k≤n
we ha e Φ(V) = (ϕ, π/2, π/2), o some ϕ∈[0, π/2].
(4) Case k= 3. By P oposi ion 3.2, Φ(V) = (ϕ, ϕ, π/2) o some ϕ∈[0, π/2]. I
n≥3, P oposi ions 5.3 and 5.6 gua an ee ha , o each iple (ϕ, ϕ, π/2) in R−
3( esp.
in R+
3 R−
3) we ha e exac ly wo ( esp. one) inequi alen subspaces wi h Φ(V) =
(ϕ, ϕ, π/2). By Rema k 5.4, i n= 2, we ha e Φ(V)∈ {(0,0, π/2),(π/3, π/3, π/2)},
whe eas i n= 1, Φ(V) = (0,0, π/2). 
P oo o Theo em B. This ollows om combining Theo em A wi h he heo y o cohomo-
genei y one ac ions on symme ic spaces o non-compac ype and ank one (c .
§
2.2). We
jus ha e o no e ha he ac ion p oducing he sol able olia ion ( esp. he ac ion wi h a
o ally geodesic singula o bi HH`,`∈ {1, . . . , n}) can be eco e ed by he me hod ha
yields he ac ions wi h a non- o ally geodesic singula o bi by aking Vas a 1-dimensional
subspace o gα∼
=Hn( esp. by aking Vas a qua e nionic subspace Hn−`+1 in gα∼
=Hn). 
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Depa men o Ma hema ics, Uni e si y o San iago de Compos ela, Spain.
Email add ess:[email p o ec ed]
Depa men o Ma hema ics, Uni e si y o San iago de Compos ela, Spain.
Email add ess:[email p o ec ed]
Depa men o Ma hema ics, Uni e si y o San iago de Compos ela, Spain.
Email add ess:[email p o ec ed]