TASAKI, HIROYUKI

Tasaki, Hiroyuki

Osaka University Knowledge Archive

The cut locus and the diastasis of a Hermitian symmetric space of compact type

Osaka City University Repository

Tasaki, H.
Osaka J. Math.
22 (1985), 863-870
THE CUT LOCUS AND THE DI AST AS IS OF A
HERMITIAN SYMMETRIC SPACE OF COMPACT TYPE
HIROYUKI TASAKI
(Received October 9, 1984)
1. Introduction
For a complete Riemannian manifold M and a point p in Λf, we denote
by CP(M) the cut locus of M with respect to p. As a property of the cut locus
of a simply connected compact symmetric space My it is known in [2] that the
cut locus CP(M) coincides with the first conjugate locus of M with respect to
p. Sakai [5] proved that in general the cut locus of a compact symmetric space
is determined by that of its maximal totally geodesic flat submanifold (see Sec-
tion 4 for details). Using this, Sakai [6] and Takeuchi [8], [9] gave stratifi-
cations of the cut loci of compact symmetric spaces.
Calabi [1] introduced the notion "diastasis" to study Kahler imbeddings.
The diastasis of a Kahler manifold M is a real analytic function defined on its
domain of real analyticity in MxM containing the diagonal set and behaves
like as the square of the geodesic distance in the small (see Section 2 for the
definition). The most characteristic property of the diastasis proved by Calabi
will be that the diastasis of a Kahler submanifold N of a Kahler manifold M
coincides with the restriction of the diastasis of M to N. Making use of these
properties, Calabi obtained various fundamental results of Kahler imbeddings.
In particular, he proved the rigidity of a Kahler submanifold of a space of con-
stant holomorphic sectional curvature.
It seems to be interesting to study relations between the geodesic distance
and the diastasis in the large. In this note we shall show a relation between
the cut locus and the diastasis of a Hermitian symmetric space of compact
type. More precisely, the main result of this note is the following:
Theorem. Let M be a Hermitian symmetric space of compact type and
D be the diastasis of M. Then, for each point p in My the cut locus CP(M) is
equal to the set of points q at which D(p, q) cannot be defined.
In other words, M—CP(M) is the domain of real analyticity of the real
analytic function q\-+D(p, q). This result gives a relation between the cut
locus of a Hermitian symmetric space of compact type and that of its symmetric
864 H. TASAKI
Kahler sub manifold (Corollary 8). In particular, the cut locus of a symmetric
Kahler submanifold of a complex projective space is a hyperplane section of
the submanifold.
2. The diastasis
In this section we shall give the definition of the diastasis of a Kahler mani-
fold due to Calabi [1] and state some basic properties of the diastasis. At the
end of this section we shall show that, for P=P\C)X ••• xP\C) with the
product metric of Hermitian symmetric metrics on P\C) and a point p in P,
the cut locus CP(P) is the set of points q at which the diastasis D(p, q) cannot
be defined.
Let M be a ^-dimensional complex manifold with an analytic Kahler metric
and M be its conjugate manifold. For each point p in Λf, the point corre-
sponding to p in M is denoted by p. For each complex coordinate system
(s1, #2, •••, **) inΛf, put
Then (s1*, ••-, #**) is a complex coordinate system in M and (z1, •••, #*, s1*, •••,
#**) is a complex coordinate system in the product complex manifold MxM.
Imbedding M into MxM as the diagonal set {(p, p) p e M} , we can uniquely
extend a real analytic functional element in M to a complex analytic functional
element ΊnMxM.
Let
df = g&(x,S)iU*d*r (1)
be the Kahler metric of Λf, then there exists a real analytic function Φ(#, z)
such that
We can extend Φ(#, z) to a complex analytic function defined on an open sub-
set of MxM. For each points p and q in the open set on which the complex
coordinate system (#*, ••-, zk) is defined, we define the functional element
D(p, q) = φ(<p), ,
l j
These functional elements generate a real analytic function, which is called
the diastasis of M and denoted by D.
The following proposition is due to Calabi [1].
Proposition 1. Let M be a Kahler submanifold of a Kahler manifold N
CUT LOCUS AND THE DlASTASIS 865
with an analytic Kάhler metric. Then the diastasίs of M is the restriction of the
dίastasis of N to M.
The following lemma follows from (1), (2), and (3).
Lemma 2. Let M19 •••, Mn be Kάhler manifolds with analytic Kάhler
metrics ahd Di be the diastasίs of M{ for i=l, ••-, n, then the diastasis of MλX •••
χM
n
 is equal to D
λ
+ ••• +D
n
.
At the end of this section we consider the diastasis of Pl(C)X ~xP\C).
Proposition 3. For P = P1(C)X ••• XP1(C) with the product metric of
Hermίtian symmetric metrics on Pl(C) and a point p in P, the cut locus CP(P) is
equal to the set of points q at which the diastasis D(p, q) cannot be defined.
Proof. We denote by [2°, zs1] the homogeneous coordinate of P\C). With-
out loss of generality it may be assumed that the homogeneous coordinate of
p is [1, 0]. Let
O is an open subset containing p and ^r1/^0 is a complex coordinate on O. The
diastasis D of P\C) is given by
for some α>0. Let p' be the point whose homogeneous coordinate is [0, 1],
then the set of points q at which D(p, q) cannot be defined is {p'}> which is
equal to the cut locus CP(P\C)}. This proves Proposition 3 for P=P\C).
Proposition 3 follows from Lemma 2 and the fact that, for complete Rieman-
nian manifolds Mly •••, Mn and points pi in M, (\^i^n)y
-• xM
n
U
3. Certain submanifolds of a Hermitian symmetric space of com-
pact type
Let M be a Hermitian symmetric space of compact type. In this section
we shall construct a maximal totally geodesic flat submanifold A of M and
a totally geodesic Kahler submanifold P of M which includes A. For details
about the results without proofs, see Helgason [3].
Let (it, θ) be the orthogonal symmetric Lie algebra associated with M.
We have the canonical direct sum decomposition of it :
866 H. TASAKI
tt =
where
I = {^Γeu; Θ(X) = X} and t> =
Take a maximal Abelian sub algebra § of t. We denote the complexifica-
tions of it, I, t>, and ϊj by g, ϊ, f>, and § respectively, ξ is a Cartan subalgebra
of g. Let Δ be the set of nonzero roots of g with respect to §. For each root
a in Δ, put
g* = {Zeg; [H, X] = a(H)X for each #eξ} .
Since ξίcϊ, for each a in Δ, grtcϊ or g*cp. A root a is called compact
[resp. noncompact], if gΛCΪ [resp. β*cf>]. By the root space decomposition
of g, we obtain
ϊ = %+ Σ g*, ρ= Σ gβ.
fl> : compact β : non-compact
We introduce a lexicographic order in the dual of the real vector space
\/— 1 ^ . Note that each root is real valued on \J — I f).
Let Q be the set of positive noncompact roots in Δ and r be the rank of
M. Then there is a strongly orthogonal root system {γj, •••, y
r
} in Q, that is,
γ iyyφΔ for l^iyj^r. We can choose nonzero vectors XΛ€ΐQΛ for each
roots a in Δ such that
(4)
where fία, is the dual vector of a with respect to the Killing form of g. Since
% ±OOΦΔ,
\X±Ί, X±,:\ = [H*,,, X±yj] = 0, if f φι . (6)
By this property
αp = Σ ΛV=T (Λ ^ +^ -,,) (7)
is a maximal Abelian subspace in £.
Let C7 be a simply connected Lie group with Lie algebra it and K be the
analytic subgroup of U with Lie algebra ϊ. Since M is simply connected,
M= U/K. The action of U on M is isometric and holomorphic.
Put
A = exp(dp)0 ,
where o is the origin of M=UIK. The submanifold ^4 is a maximal totally
CUT LOCUS AND THE DlASTASIS 867
geodesic flat submanifold of M, because dp is a maximal Abelian subspace
in p.
We define a linear map φ, : 3u(2) -> u by
_ i J ^  Xti—X-ii >
By (4) and (5) φ, is a well-defined injective Lie algebra homomorphism of
3u(2) to u. Since
[φ,<§u(2)), φ/3u(2))] = {0}, l^iΦ j^r ,
by (6), we can define an injective Lie algebra homomorphism φ from the r-fold
direct sum §tt(2)r of 3u(2) into u by
..-, X,) = ± φ,(X,) for jr|
φ also denotes the homomorphism from the r-fold direct product SU(2)r
of SU(2) into C7 induced by the Lie algebra homomorphism φ. Then φ
induces an equivariant holomorphic imbedding
p: SU(2)rIS(U(l)x U(l)Y -> M
xS(U(l)xU(l)Y^φ(x)o for
Note that the r-fold direct product P\C)r of P\C) is canonically identified
with SU(2)r/S(U(ί)x U(l)γ.
We denote by P the image of the imbedding p. By the definition of φ
and (7), αp c φ(3u(2)r), so A cP. Since
φ(§u(2)r) = I n φ(§u(2)r)+ί> n Φ(3u(2)'),
P is a totally geodesic submanifold of M. The induced metric on P\C)r is
the product metric of Hermitian symmetric metrics on P\C), because the im-
bedding p is equivariant.
By a theorem of Cartan to the effect that M is given by
M=(J kA,
we obtain
M= U kP.
The following proposition summarizes this section.
868 H. TASAKI
Proposition 4. a) A is a maximal totally geodesic flat submanifold of M
through o.
b) P^P\CY is a totally geodesic Kάhler submanifold of M which includes A
and its metric is the product metric of Hermίtian symmetric metrics on P\C).
c) M = U kP .k<=&
REMARK. The imbedding p was used by Takagi and Takeuchi [6] in
order to determine the degree of symmetric Ka'hler submanifolds of a complex
projective space.
4. The cut locus and the diastasis of a Hermitian symmetric space
of compact type
In this section we shall prove the following main theorem stated in Intro-
duction.
Theorem 5. Let M be a Hermitian symmetric space of compact type and
D be the diastasis of M. Then, for each point p in M, the cut locus Cp(M) is equal
to the set of points q at which D(p, q) cannot be defined.
We retain the notations in Section 3.
Lemma 6.
C0(M) nA = C0(A) and C0(M) - U kC0(A) .k(ΞK
This lemma is due to Sakai [5].
Lemma 7.
C0(M) = U kC0(P) .
k^K
Proof. Put
U, = φ(SU(2)r) and Kl=UlΓ\K.
Then (C/!, K^ is a Riemannian symmetric pair and P=UJK1.
Since P is a totally geodesic submanifold of M, A is also a maximal totally
geodesic flat submanifold of P. Applying Lemma 6 to P= U^K ^ and A, we
have
C.(F) = U kβ.(A)
*ι<=jq
= U kι(AΓ(C.(M))
hence from c) of Proposition 4
CUT LOCUS AND THE DlASTASIS 869
= C0(M).
Now we shall prove Theorem 5. Without loss of generality we may as-
sume that p is the origin o of M= U/K. Since P is a Kahler submanifold of
M, the restriction of D to P is the diastasis of P by Proposition 1. The action
of K on Λf is isometric and holomorphic, hence
\ D(p, q) cannot be defined)
= U k{q<=P\ D(p, q) cannot be defined}
= ΪV.(P)
= C.(M) .
This completes the proof of Theorem 5.
Corollary 8. Let M1 and M2 be Hermitian symmetric spaces of compact
type. If MΊ is a Kάhler submanifold of M2, then
for each point p in M
λ
.
REMARK. In case of M2=P'
n(C), Theorem 4.3 in Nakagawa and Takagi
[4] implies that the imbedding of M1 into Pn(C) is equivariant. So we can de-
scribe the behavior of a geodesic of M1 in P
n(C) and directly show the assertion
of Corollary 8 in this case.
References
[1] E. Calabi: Isometric imbedding of complex manifolds, Ann. of Math. 58 (1953),
1-23.
[2] R. Crittenden: Minimum and conjugate points in symmetric spaces, Canad. J.
Math. 14 (1962), 320-328.
[3] S. Helgason: Differential geometry, Lie groups, and symmetric spaces, Aca-
demic Press, New York, 1978.
[4] H. Nakagawa and R. Takagi: On locally symmetric Kaehler submanifolds in a
complex protective space, J. Math. Soc. Japan 28 (1976), 638-667.
[5] T. Sakai: On cut loci of compact symmetric spaces, Hokkaido Math. J. 6 (1977),
136-161.
[6] T. Sakai: On the structure of cut loci in compact Riemannian symmetric spaces,
Math. Ann. 235 (1978), 129-148.
[7] R. Takagi and M. Takeuchi: Degree of symmetric Kahlerian submanifolds of a
complex projective space, Osaka J. Math. 14 (1977), 501-518.
870 H. TASAKI
[8] M. Takeuchi: On conjugate loci and cut loci of compact symmetric spaces I, Tsuku-
ba J. Math. 2 (1978), 35-68.
[9] M. Takeuchi: On conjugate loci and cut loci of compact symmetric spaces II, Tsuku-
ba J. Math. 3 (1979), 1-29.
Department of Mathematics
Tokyo Gakugei University
Koganei, Tokyo 184
Japan


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The cut locus and the diastasis of a Hermitian symmetric space of compact type

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