Submanifolds of Symmetric Spaces 311
isometric to CP
1
×CP
1
×CP
1
for m = 4. The embedding of G
3
(C
6
)=SU
6
/S(U
3
U
3
)
into CP
19
is the Pl¨ucker embedding. The image of each of these embeddings under
the Hopf map CP
2n+1
→HP
n
is indeed an n-dimensional symmetric totally complex
submanifold of HP
n
. Tsukada proved:
Theorem 11.3.7 (Tsukada) A connected complete totally complex submanifold M
of HP
n
with dim
C
M = n is a symmetric submanifold if and only if it is either a totally
geodesic subspace CP
n
⊂HP
n
or congruent to one of the embeddings listed in Table
11.8.
11.3.7 Symmetric submanifolds associated with irreducible symmetric
R-spaces
The pairs (M,
¯
M) mentioned in part 5 of Theorem 11.3.4 are, for irreducible
¯
M,
precisely the pairs (K ·X,G) and (K ·X,G/K) listed in Tables A.6 and A.7. The
embedding of M in
¯
M can be described as follows. Write
¯
M = G/K with (G,K) a
Riemannian symmetric pair and put o = eK ∈
¯
M.Letg = k ⊕p be the corresponding
Cartan decomposition of g. Then there exists an element Z ∈ p so that the eigenval-
ues of ad(Z) are +1,0,−1. The element Z de termines a closed geodesic
γ
in
¯
M.The
antipodal point q to o on
γ
is a pole of o,thatis,afixed point of the action o f K on
¯
M.Thereflective submanifold M is the centrosome of o and q, that is, the orbit of
K through the midpoint on
γ
between o and q (it does not matter which of the two
possible midpoints is selected). The orbits of K through the other points on
γ
and dis-
tinct from o and q are non-totally geodesic symmetric submanifolds of
¯
M belonging
to the Grassmann geometry G(M,
¯
M). In this way, we get a one-parameter family of
non-congruent symmetric submanifolds of
¯
M, and every symmetric submanifold in
G(M,
¯
M) arises in this way up to congruence (Naitoh [236]). In particular, any non-
totally geodesic symmetric submanifold of
¯
M arises as an orbit of the action of the
isotropy group of
¯
M = G/K. It is worthwhile to mention that, among th e reflective
submanifolds in
¯
M, the symmetric R-spaces are precisely those for which the totally
geodesic submanifolds tangent to the normal spaces of M are locally reducible with
a one-dimensional flat factor.
11.3.8 Symmetric submanifolds of symmetric spaces of noncompact
type
In this part we describe the classification o f symmetric submanifolds of Riemann-
ian symmetric spaces of noncompact type. For the real hyperbolic space RH
n
,this
was already done in Section 2.8. It was shown by Kon [187] respectively Tsukada
[326] that every symmetric submanifold in G(CH
m
,CH
n
) resp. G(CH
n
,HH
n
) is to-
tally geodesic. The classification of symmetric submanifolds in G(RH
n
,CH
n
) was
obtained b y Naitoh [235]. Here we want to describe the classification of symmet-
ric submanifolds in the remaining Grassmann geometry G(M,
¯
M) listed in Theo-
rem 11.3.4 (5). This classification was obtained by Berndt, Eschenburg, Naitoh, and
Tsukada [33].