NAGURA, TOSHINOBU

Osaka City University Repository

Nagura, T.
Osaka J. Math.
11 (1974), 211-220
ON THE SECTIONAL CURVATURES OF R-SPACES
TOSHINOBU NAGURA
(Received September 8, 1973)
Introduction
Let g be a real semi-simple Lie algebra without compact factors, ϊ a
maximal compactly imbedded subalgebra of g, and g—ϊ+p the Cartan decom-
position of g relative to ϊ. We denote by B the Killing form of g. We regard
the subspace $ as a Euclidean space with the inner product < , > induced by the
restriction of B to p. Let Int (g) be the group of inner automorphisms of g
and, the Lie algebra of Int (g) being identified with g, K the connected Lie
subgroup of Int (g) corresponding to the Lie subalgebra ϊ of g. Then K
leaves the subspace p invariant and acts on the Euclidean space p as an isometry
group. Let S be the unit sphere of p and N an orbit of an element H0 in S.
Denoting by K* the stabilizer of H0 in K, the space N may be identified with
the quotient space K/K* and is called an R-space. We always assume that dim
N^2, and the space N is substantial, i.e. there exist no proper subspaces of p con-
taining N. The aim of this paper is to study the sectional curvatures of N with
respect to the ίΓ-invariant Riemannian metric < , > induced by the inner pro-
duct <( , > on p.
It is known (Takeuchi-Kobayashi [8]) that if the pair (K, K*) is a symmetric
pair, the metric < , > on N=K/K* coincides with the ^-invariant Riemannian
metric defined by a ^-invariant inner product on ϊ, and so the sectional
curvatures of N are always non-negative, and N has a positive sectional curvature
along each plane section if and only if the pair (K, K*) is of rank 1.
In this paper we shall show that in general cases the space N may have both
positive and negative sectional curvatures. Indeed, the curvatures are related
with the restricted root system r of g. Let Δ be a fundamental root system of
r. Then a subsystem Δ! of Δ corresponds to the space N (See section 3), and
we have:
(I) If the restricted root system r is irreducible and the cardinality |Δ—ΔJ
of Δ—Δ! is not less than 2, the space N has both positive and negative sectional
curvatures.
Furthermore we shall characterize the /^-spaces with strictly positive sectional
212 T. NAGURA
curvatures :
(II) The space N has a strictly positive sectional curvature along each plane
section if and only if N is the unit sphere S or the pair (K, K*) is a symmetric pair
of rank 1.
The class of Λ-spaces includes the spaces KIK*=SO(3)IQ, SU(3)/T,
Sp(3)/Sp(l)xSρ(l)xSp(l) and jF4/Spin (8), where Q (resp. T) denotes the sub-
group of all diagonal matrics in SO (3) (resp. in SU (3)). Although none of these
pairs is a symmetric pair of rank 1, Wallach [9] proved that each of these spaces
has a ^-invariant Riemannian metric with strictly positive sectional curvatures.
The charcterization (II) shows that the Riemannian metric of Wallach is not the
same as ours.
I wish to express my sincere gratitude to Professor M. Takeuchi for his
kind guidance and encouragements.
1. Preliminaries
1.1. The assumptions and the notation are the same as those in Introduc-
tion. Let α be a maximal abelian subspace in p. We shall identify α with the
dual space α* of α by means of the duality defined by the Killing form B of g.
For an element λeα, we define subspaces ϊ
λ
 and p
λ
 of g as follows:
ί ϊ
λ
 = {X(Ξ ϊ; ad(HγX = B(\, H)*X, for all
(
 ' ' I p
λ
 = μτ<Ξp; ad(H)2X = B(\, H)2X, for all
Then f_
λ
=f
λ
, p_
λ
=ψ
λ
 and p0=α. It is known (Satake [4]) that if we put
r is a root system in α. The root system r is called the restricted root system of
g. We denote by x+ the set of positive roots of ϊ with respect to a linear order in
the subspace α. Then we have the following orthogonal decompositions of ϊ
and p with respect to the Killing form B (cf. Helgason [2]).
(1.2) t = ϊ0+ΣΛ> P = α+Σ>
λe
r
+ \e
r
+
2. Second fundamental forms of /2-spaces
2.1. In Introduction we assume that the point H0^N is contained in the
unit sphere S. Moreover we may assume H0^S Γϊ α by virtue of the following
lemma (cf. Helgason [2]).
Lemma 1. For each element X^$y there exists an element k^K such that
kX is contained in the subspace α.
SECTIONAL CURVATURES OF Λ-SPACES 213
Let THo(N) be the tangent space of N at HQ. We can identify the tangent
space TffQ(N) with a subspace of p in a canonical manner, and we have
THo(N)=[t,H0].
Choose a linear order in the subspace α such that <λ, /ί0>^0 for each positive
root λ^t with respect to this order, and fix this order once for all. Put
rί=
Then the tangent space THQ(N) and the orthogonal complement TffQ(N) in p are
given by
(2.1)
Let
a:THo(N)xTHo(N)-»TH\(N)
be the second fundamental form at H0 of the submanifold N of p. In the same
way as in Takagi-Takahashi [6] we get the following:
Proposition 2. For -Y
λ
el>
λ
, F
λ
'eί>
λ
' (λ, λ'ex?, λ'^λ), ίfe second fun-
damental form a is given by
(2.2) (ii) ^-component of -_-L_[[H0,
0 otherwise .
3. Sectional curvatures of /^-spaces
3.1. Let R be the curvature tensor at H0 of the space N. By the equa-
tion of Gauss and the flatness of the space t> we have
(3.i) <#(*, y)y, xy = <α(*, ^ ), α(y, y)>-<α(^τ, y), α(^τ, y)>
for any X, Y^THo(N) (cf. Kobayashi-Nomizu [3]). For X, Y^THo(N) we
denote by K
χγ
 the sectional curvature along the plane section spanned by X
and y, X and y being assumed to be linearly independent.
Proposition 3. If there exist roots λ, λ^rj such that <λ, λ'><0, then the
space N has both positive and negative sectional curvatures.
214 T. NAGURA
Proof. Since λφλ', from (2.2) and (3.1) follows that
(3.2) KXκ,Yχ> = <α(*λ, XJ, α(Fλ', Γλ')>-<α(*λ, Fλ>), a(
for ^Γ
λ
et)
λ
, y
λ
'elv with <JΓ
λ
, J5Γ
λ
>=<y
λ
/, y
λ
/>=l. Since <λ,λ'><0, we
For each root μ,er we denote by Sμ the reflection of α with respect to μ.
Then λ//=5
λ
(λ/) is a root in r. Since <λ, λ'><0, we find that
<λ,λ>
and
- -<λ, λ'»0.
For Z
λ
»<Ξt>
λ
" with <Z
λ
",Z
λ
">=l, since <λ"— λ, #0»0, we have α(^Γλ, Zλ")
=0 by (2.2) (iii). So in the same way as above we have KXλtZλ">Q
3.2. In the following we assume that the root system r is irreducible. Let
Δ be the fundamental root system of t consisting of simple roots with respect to
the order chosen in section 2, and put
Let ^ be a Cartan subalgebra containing α. Let QC be the complexification of g,
§c the subspace of gc spanned by §, and ΐ)0 the real part of f)c. Let σ be the
conjugation of gc with respect to g, and choose a σ-order in the sense of Satake
[4] on ϊ)0, extending our order on α. In the root system of gc relative to ϊjc, let
Δ be the fundamental root system consisting of simple roots with respect to this
order, and denote the Satake diagram of Δ by the same symbol Δ. Then Δ
defines our fundamental root system Δ by the projection p of £)0 onto α: Δ
=p(Δ)— {0} . Let λ1=p~1(A1). It is known (Takeuchi [7]) that isomorphic pairs
(Δ, Δj) of Satake diagrams give rise to isomorphic pairs (Ky K*): We say that
the pair (Δ, Δ
x
) is isomorphic to the pair (Δr, Δ/) if there exists an isomorphism
φ of Δ onto Δx such that φ maps Δj_ onto Δ/, and the pair (K, K*) is said to
be isomorphic to the pair (K', ./£*') if there exists an isomorphism / of K onto
K' such that /maps K* onto K*'.
For a set A we will denote by \A\ the cardinality of A.
Proposition 4. If the root system r is irreducible and \ Δ— Δ! | ^2, the space
N has both positive and negative sectional curvatures.
Proof. Assume that λ, μ,eΔ— Δ
x
 and λΦμ. Then there exist simple
SECTIONAL CURVATURES OF ^-SPACES 215
roots λ0, λj, •••, λseΔ such that the following conditions are satisfied:
λ0 = λ, \s =
(3 3) Γ < 0 if = ί) <λ,,λy> = /[= 0 other
where 0 ^  i <y ^  s
Put
λ" - SX^SA^-S^XO) and λ" -
Then by (3.3) we have mt >0 and λ"er£. Since
<λ", μ> — Wj-jO^-!, λ
s
> <0 ,
the proposition follows from Proposition 3.
An element H of α is called regular if <λ, #>ΦO for any λer. Then the
following Corollary is an immediate consequence of Proposition 4.
Corollary. If the root system ϊ is irreducible and the rank of r is not less
than 2, then the K-orbίt N through a regular element of α has both positive and ne-
gative sectional curvatures.
REMARK. If g is the direct sum of r-copies of 3/(2, R) (f^2), the space N
is the r-dimensional flat torus. According to the following result (A) and (β),
it follows that, except for the above case, the space N has always a plane section
along which the sectional curvature is strictly positive.
(A) Let M be an n-dίmensίonal manifold and G a compact connected transi-
tive Lie transformation group of M. If the universal covering manifold M of M
satisfies
, Z2) = {0} for any i > 0 ,
then G is the n-dimensional toral group , M is the n-dίmensional torus and G acts
on M as translations (M. Takeuchi).
(B) A simply connected complete Riemannian manifold with non-posi-
tive sectional curvatures is diffeomorphic with a Euclidean space (Theorem of
Cartan-Hadamard, cf. [3]).
3.3. Next we shall see under which conditions N has a strictly positive sec-
tional curvature along each plane section.
Assume that the space N has strictly positive sectional curvatures. Decom-
pose the root system r into the sum of irreducible components rco:
r = r
c l )U U rc
216 T. NAGURA
If we put (xc o)2=xc oΓΊX2 f°r 1 '^^ w, then the space N is substantial if and
only if (xα:>)2 Φ Φ for each /. Thus the root system x should be irreducible by
(3.2). If I Δ I =1, then Δ^φ and the space N coincides with the unit sphere S
itself. So we assume that | Δ | ^ 2 in the following. Then we have by Proposi-
tion 4 and (3.2):
(3.4) IΔ-ΔJ -1
(3.5) <λ, λ/»0 for all λ, λ'e xί
We remark that the properties (3.4) and (3.5) depend only on the root system
X. So we shall consider the following situation:
Let V be a real vector space and 8 a reduced irreducible root system in V.
Choosing a linear order in the space Vy let 8
+
 denote the set of positive roots and
Δ the fundamental root system of 8 consisting of simple roots. The Dynkin
diagram of Δ will be also denoted by the same symbol Δ. Let Δ
x
 be a subset
Δ of such that |Δ— ΔJ =1,
Δ = {λ,, X2, -, λ;} and Δ-Δ> =
We put
λ = Σ "*PW, i
We denote by λ0 the highest root in 8 and 5λ the reflection of V with respect to
the root λ e 8. We fix an inner product ( , ) on V invariant under the Weyl
group of 8. Since the root system 8 is irreducible, such an inner product is
unique up to a multiple of positive constant. The diagram obtained from the
Dynkin diagram Δ by adding — λ0 is called the extended Dynkin diagram of 8.
The table of extended Dynkin diagrams is seen, for instance, in Borel-de Sie-
benthal [1]. We shall consider the condition
(3.6) (λ, λ')>0 for all λ, λ'eΞ8ί
and prove the following:
Proposition 5. The pair (Δ, Δj) satisfies the condition (3.6) if and only if the
Dynkin diagram Δ is of type At and λj. is one of the terminals of Δ, i.e. the vertex
\! is connected with only one vertex in Δ.
Proof. Let Δ be of type At and Xx one of the terminals of Δ. We may
assume that there exist an (/+l)-dimensional vector space W^V with an inner
product ( , ) and an orthonormal basis e19 •••, el+l of W such that the following
conditions are satisfied:
( i ) The restriction to Fof the inner product ( , ) on PFis invariant under
the Weyl group of 8.
SECTIONAL CURVATURES OF /^-SPACES 217
(ϋ) «={*,-*,; ι^ί,y^
(iii) Δ={έ?t— ei+l\ l^i^l}, \l=el—e2
Then we have
8+ = fa-e^ 2^i^
Hence the pair (Δ, ΔJ satisfies the condition (3.6).
It remains to prove that if the pair (Δ, Δj) satisfies (3.6), then the Dynkin
diagram Δ is of type At and Xx is one of the terminals of Δ.
Lemma 6. If the pair (Δ, Δj) satisfies (3.6), the inner product (X0, λ^
Proof. Since the highest root λ0 is contained in ££> we have (X0, λ^
Lemma 7. If the pair (Δ, Δ
α
) satisfies (3.6), λ
x
 is one of the terminals of Δ.
Proof. Suppose that λj is not a terminal. Then there exist two simple
roots X, μ^Δ such that (λ, X^ΦO and (μ,, XJΦO. Since the Dynkin diagram
Δ contains no cycles, we have (λ, μ)=0. It is known that (λ, λj)<0 and (μ, λ
α
)
<0. Then it follows that \+^ is a root (cf. Serre [5]). Since (λ+λ^ μ)<0,
it follows also that λ+λt+μ is a root. On the other hand λj+λ+μ belongs
to §J by definition, so that (λ^fλ+μ, Xj)>0. Thus we see that \+μ
^(λ+λi+μ)— \! is a root.
Now let
λ— />μ> •"> λ— μ, λ, λ+μ, •••, \+qμ
be a μ-series of roots such that neither λ— (p-\-l)μ nor \+(q+l)μ is a root.
Then we have (cf. Serre [5]).
q — p = — 2(λ>//>)
(μ>μ)
Since (λ, μ)=0 and p=Q, we get <7=0, which contradicts to X+μ,G7.
Lemma 8. Suppose that the pair (Δ, ΔJ satisfies the condition (3.6). If the
Dynkin diagram Δ is one of the classical types, then Δ is of type At and \λ is one of
the terminals of Δ.
Proof. It suffices to show that the Dynkin diagram Δ is of type At. Suppose
that Δ is of type J97(/Ξ>3) or of type Z)/(/>4). Then, looking at their extended
Dynkin diagrams, we see from Lemma 6 and Lemma 7 that there exist no pairs
(Δ, Δi) satisfying the condition (3.6).
Suppose that the pair (Δ, Δ^ satisfies (3.6) and the Dynkin diagram Δ is of
type C/(/^2). Then we may assume that there exists an orthonormal basis
(eι> •••> et} of V such that the following conditions are satisfied:
218 T. NAGURA
(i) »={±ei±ej
(ii) Δ= fa— *2, e2—ezj — , */_!—*„ 2*,}
By Lemma 6 and Lemma 7 we should have \l=el— e2. So we have
For el—e2> e^e^^z we have (e1—e2, ^-)- 2^)=0, which is a contradiction.
Lemma 9. If the Dynkin diagram Δ is one of the exceptional types, there
exist no pairs (Δ, Δ
x
) satisfying the condition (3.6).
Proof. Suppose that the pair (Δ, Δ
x
) satisfies the condition (3.6) and the
Dynkin diagram Δ is of type E6. By Lemma 6 and the extended Dynkin diagram
we should have
Λ2 λ>3 X4 X5 Λ6
O-
-λ00
We put
6
§' = {λe§; λ — Σ W j λ. , w2 — m6 = 0} ,
i=l
Δ7 = {λ^ X3, X4, X5} and Δ/ = {λ3, X4, X5} -
Then §x is a root system of type D4 with the fundamental root system Δ7. The
subset (§')? of §7 corresponding to the pair (Δr, Δ/) is given by (8/)?=^2 Π^.
It follows from Lemma 8 that there exist two roots X, μ,e(§x)2 such that (X, μ)
^0. This is a contradiction.
In the cases of type E7 and E8 we can prove the lemma in the same way, mak-
ing use of Lemma 8 for Δ of type D5 and D7 respectively.
In the case of type F4 we can prove in the same way, making use of Lemma
8 for Δ of type J53.
In the case of type G2 the lemma is easily proved.
By Lemma 8 and Lemma 9 we have the complete proof of Proposition 5.
Now we give the table of pairs (Δ, Δt) of Satake diagrams (up to an isomor-
phism) with the following properties.
1) The Satake diagram Δ has no compact factors.
2) The pair (Δ, Δ t) of fundamental root systems obtained from (Δ, Δt)
SECTIONAL CURVATURES OF ^-SPACES 219
by the projection has the properties:
(i) The Dynkin diagram Δ is of type Al with /^>2.
(ii) The set Δ—Δ
x
 consists of one of the terminals of Δ.
Here the vertexes in Δ—Δ
x
 are represented by ©. pl(F) denotes the /-
dimensional projectίve space over F where F denotes R, C, H (the algebra of
real quaternions), or A" (the algebra of real Cayley numbers).
Λ
AI
All
EIV
(ΔΛ)
® f~\ τ τ . ΛΛ(^J {_J
• I I I (^2)
® ΛΛ T 7 T τ τ τ t S~\(^_J {J
I
® C~\ , t τ t , t τ τ τ τ .. ,τ. -. . /"^LJ LJ
7
o
- 'ί'/"*^
2/+1
foi A A A n
1
N
SU(l+l)
/S(U(l)xU(l))=P'(C)
SO(l+ί)
/S(0(l)χO(l))=P'(R)
Sp(l+l)
/Sp(l)xSp(l)=P'(H)
F4/Spin (9)=P2(K)
Each of the pairs (K, K*) appeared in this table is a symmetric pair of rank
1 and the space N has strictly positive sectional curvatures as we have noted in
Introduction. Thus we have proved the following:
Proposition 10. The space N has strictly positive sectional curvatures if and
only if N is the unit sphere S or the pair (K, K*) is a symmetric pair of rank 1.
OSAKA UNIVERSITY
Bibliography
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de Lie dos, Comment. Math. Helv. 23 (1949-50), 200-221.
[2] S. Helgason: Differential Geometry and Symmetric Spaces, Academic press,
New York, 1962.
220 T. NAGURA
[3] S. Kobayashi & K. Nomizu: Foundations of Differential Geometry II, Interscience,
New York, 1969.
[4] I. Satake: On representations and compactifications of symmetric Riemannian spaces,
Ann. of Math. 71 (1960), 77-110.
[5] J. P. Serre: Algebres de Lie Semi-simples Complexes, W. A. Benjamin, inc.
New York, 1966.
[6] R. Takagi & T. Takahashi: On the principal curvatures of homogeneous hyper surf aces
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[7] M. Takeuchi: Cell decompositions and Morse equalities on certain symmetric spaces,
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Geometry 2 (1968), 203-215.
[9] N. R. Wallach: Compact homogeneous Riemannian manifolds with strictly positive
curvature, Ann. of Math. 96 (1972), 277-295.


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