Submanifolds of Symmetric Spaces 305
Proof Each reflective submanifold M obviously has parallel second fundamental
form and, by definition, each of its normal spaces
ν
p
M is a curvature-invariant sub-
space of T
p
¯
M. According to Proposition 11.3.1, M is a symmetric submanifold.
Conversely, assume that M is a totally geodesic symmetric submanifold. Again,
Proposition 11.3.1 tells us that each normal space
ν
p
M of M is a cu rvature-invariant
subspace of T
p
¯
M. Thus, both T
p
M and
ν
p
M are Lie triple systems for all p ∈ M,
which means that M is a reflective submanifold o f
¯
M.
11.3.3 Grassmann geometries
The obvious question now is:
Are there any non-totally geodesic symmetric submanifolds in a given Riemann-
ian symmetric space?
We will discuss this question in the framework of Grassmann geometries.
Let
¯
M be a Riemannian manifold. The isometry group I(
¯
M) of
¯
M acts in a natural
way on the Grassmann bundle G
m
(T
¯
M) of m-planes in the tangent bundle T
¯
M.Anm-
dimensional connected submanifold M of
¯
M is said to belong to the (m-dimensional)
Grassmann geometry of
¯
M if all its tangent spaces lie in the same orbit of the action
of I(
¯
M) on G
m
(T
¯
M). For example, any homogeneous submanifold of
¯
M belongs
to some Grassmann geometry of
¯
M.IfM belongs to some Grassmann geometry of
¯
M,theGrassmann geometry associated to M is the set G(M,
¯
M) of all connected
submanifolds of
¯
M whose tangent spaces lie in the same orbit as those of M.For
example, the Grassmann g eometry G(S
m
,S
n
) associated with an m-sphere in S
n
is the
geometry of all m-dimensional submanifolds of S
n
. Also, the Grassmann geometry
G(CP
m
,CP
n
) associated with an m-dimensional complex projective subspace in CP
n
is the geometry of all m-dimensional complex submanifolds in CP
n
.
Now assume that M is an m-dimensional symmetric submanifold of
¯
M.Letp
1
and p
2
be two different points in M. Connecting p
1
and p
2
by a geodesic in M,
the symmetry
σ
p
0
at the midpoint p
0
on
γ
between p
1
and p
2
is an isometry of
¯
M
leaving M invariant and interchanging p
1
and p
2
. This shows in particular that M is
a homogeneous submanifold and hence belongs to a Grassmann geometry of
¯
M.
From now on we assume that
¯
M is a Riemannian symmetric space. As we have
seen above, each tangent space and each normal space of a symmetric submanifold
of
¯
M is a Lie triple system. This implies:
Proposition 11.3.3 Each symmetric submanifold of a Riemannian symmetric space
¯
M belongs to the Grassmann geometry G(M,
¯
M) associated with a suitable reflective
submanifold M of
¯
M.
This proposition m otivates the investigation of the Grassmann geometries asso-
ciated with reflective submanifolds of Riemannian symmetric spaces in more detail.
For simply connected Riemannian symmetric spaces of compact type this was done
by Naitoh in a series of papers [237–241]. His proof also works for the Riemannian
symmetric spaces of noncompact type.
306 Submanifolds and Holonomy
Theorem 11.3.4 (Naitoh) All Grassmann geometries associated with reflective sub-
manifolds o f simply connected irreducible Riemannian symmetric spaces have only
totally geodesic submanifolds with the following exceptions:
1. G(S
m
,S
n
) and G(RH
m
,RH
n
) (1 ≤ m ≤ n −1), that is, the geometry of m-
dimensional submanifolds of S
n
resp. RH
n
.
2. G(CP
m
,CP
n
) and G(CH
m
,CH
n
) (1 ≤ m ≤ n −1), that is, the geometry of
m-dimensional complex submanifolds of CP
n
resp. CH
n
.
3. G(RP
n
,CP
n
) and G(RH
n
,CH
n
), that is, the geometry of n-dimensional to-
tally real submanifolds of CP
n
resp. CH
n
.
4. G(CP
n
,HP
n
) and G(CH
n
,HH
n
), that is, the geometry of n-dimensional to-
tally complex submanifolds of HP
n
resp. HH
n
.
5. G(M,
¯
M), where the rank of
¯
M is greater than one and the isotropy represen-
tation of
¯
M has a symmetric orbit M , that is, the geometries associated with
irreducible symmetric R-spaces and their noncompact dual geometries.
Naitoh also obtained a decomposition theorem for the reducible case, see [239].
It remains to classify the symmetric submanifolds in these five Grassmann ge-
ometries. This has been carried out by various authors whose results we will now
describe (see also [243] for a survey about symmetric submanifolds of symmetric
spaces of rank one). We already discussed the symmetric submanifolds of spheres in
Section 2.8.
11.3.4 Symmetric complex submanifolds of CP
n
In this part we describe the classification of symmetric complex submanifolds
in complex projective spaces. All these s ubmanifolds arise from so-called canonical
embeddings of certain Hermitian symmetric spaces.
Let g be a complex simple Lie algebra, h a Cartan subalgebra of g and Δ the
corresponding set of roots. We choose a Weyl canonical basis {H
α
,X
α
},
α
∈Δ,ofg
and define a compact real form g
u
of g by
g
u
=
∑
α
∈Δ
R(iH
α
) ⊕
∑
α
∈Δ
R(X
α
+ X
−
α
) ⊕
∑
α
∈Δ
Ri(X
α
−X
−
α
).
Let
α
1
,...,
α
l
∈ Δ be a set of simple roots for Δ. For each j ∈{1 ,...,l} we put
Δ
j
=
α
=
l
∑
ν
=1
n
ν
α
ν
∈Δ : n
1
,...,n
l
< 0
/
,
and define a complex subalgebra l
j
of g by
l
j
= h ⊕
∑
α
∈ΔΔ
j
CX
α
Submanifolds of Symmetric Spaces 307
and a subalgebra h
u, j
of g
u
by
h
u, j
= g
u
∩l
j
.
Let G be the simply connected complex Lie grou p with Lie algebra g and L
j
be
the connected complex Lie subgroup of G with Lie alg ebra l
j
.ThenG/L
j
is a sim-
ply connected compact homogeneous complex manifold. Let G
u
and H
u, j
be the
connected real Lie subgroups of G with Lie algebras g
u
and h
u, j
, respectively. The
inclusion G
u
→ G induces a homeomorphism from M
j
= G
u
/H
u, j
onto G/L
j
and
turns M
j
into a C-space, that is, a simply connected compact complex homogeneous
space, on which G
u
acts transitively by holomorphic transformations. Note that the
second Betti number b
2
(M
j
) of M
j
is equal to one. Conversely, as was shown by
Wang [337], every irreducible C-space M with b
2
(M)=1 arises in this m anner.
We now describe a family of holomorphic embeddings of M
j
into complex pro-
jective spaces. Let p be a positive integer and
ρ
: G → gl(C
n(p)+1
) be the irreducible
representation of G with highest weight pΛ
j
,whereΛ
j
is the fundamental weight
corresponding to the simple root
α
j
. Denote by V ⊂ C
n(p)+1
the one-dimensional
eigenspace of
ρ
corresponding to pΛ
j
. Then the map
G → CP
n(p)
, g →
ρ
(g)V
induces a full holomorphic embedding of M
j
= G
u
/H
u, j
= G/L
j
into CP
n(p)
,which
is called the p-th canonical embedding of M
j
into a complex projective space. The
submanifold M
j
of CP
n(p)
is the unique compact orbit of the action of the complex
Lie group G on CP
n(p)
. The dimension n(p) can be calculated explicitly by means of
Weyl’s dimension formula. The induced metric on M
j
⊂ CP
n(p)
is K¨ahler-Einstein.
Note that M
j
is Hermitian symmetric if and only if n
j
= −1 for all roots
α
∈Δ
j
,and
every Hermitian symmetric space arises in this m anner.
It follows from Proposition 11.3.1 that a complete complex submanifold M of
CP
n
is symmetric if and only if its second fundamental form is parallel. The com-
plex submanifolds of CP
n
with parallel second fundamental form were classified by
Nakagawa and Takagi [244].
Theorem 11.3.5 (Nakagawa-Takagi) A co nnected complete complex su bmanifold
MofCP
n
is a symmetric submanifold if and only if it is either a totally geodesic
complex projective subspace or congruent to one of the models in Table 11.5.
In Table 11.5, F
1
resp. F
2
denotes the first resp. second canonical embedding,
and F
1
⊗F
1
is the embedding that is induced by the exterior tensor product of the two
representations associated to the first canonical embedding F
1
.
The first canonical embedding of SO
m+2
/SO
m
SO
2
is congruent to the standard
embedding of the complex quadric Q
m
= {[z] ∈ CP
m+1
: z
2
0
+ ...+ z
2
m
= 0} into
CP
m+1
.
The first canonical embedding of SU
m+2
/S(U
m
U
2
) is congruent to the Pl¨ucker
embedding of the complex 2-plane Grassmannian G
2
(C
m+2
) of complex 2-
dimensional linear subspaces of C
m+2
into CP
n
= P(Λ
2
C
m+2
) given by
V = Cv
1
⊕Cv
2
→ C(v
1
∧v
2
).
308 Submanifolds and Holonomy
TABLE 11.5: Symmetric complex submanifolds of CP
n
Mn Embedding Remarks
SO
m+2
/SO
m
SO
2
m + 1 F
1
m ≥ 1
SU
m+2
/S(U
m
U
2
)
1
2
(m + 1)(m + 2) −1 F
1
m ≥ 3
SO
10
/U
5
15 F
1
E
6
/Spin
10
U
1
26 F
1
CP
m
1
2
(m + 1)(m + 2) −1 F
2
m ≥ 2
CP
a
×CP
b
ab + a + bF
1
⊗F
1
1 ≤ a ≤ b
The first canonical embedding of SO
10
/U
5
into CP
15
is induced by the positive
half-spin representation of Spin
10
(C) on C
16
.
The first canonical embedding of E
6
/Spin
10
U
1
into CP
26
is induced by the 27-
dimensional fundamental representation o f E
6
(C) on C
27
.
The second canonical embedding of CP
m
is also known as the second Veronese
embedding and is explicitly given by
[z
0
: ... : z
m
] → [z
2
0
: ... : z
2
m
:
√
2z
0
z
1
: ... :
√
2z
i
z
j
: ... :
√
2z
m−1
z
m
](i < j).
Finally, the embedding of CP
a
×CP
b
is also known as the Segre embedding and
is explicitly given by
([z
0
: ...: z
a
],[w
0
: ...: w
b
]) → [z
0
w
0
: ...: z
ν
w
μ
: ... : z
a
w
b
]
(all possible products of the coordinates).
11.3.5 Symmetric totally real submanifolds of CP
n
The classification of n -dimensional totally real symmetric submanifolds in CP
n
was established by Naitoh [233] (for the irreducible case) and by Naitoh and
Takeuchi [242] (for the general case). The reflective submanifold in the correspond-
ing Grassmann geometry is the totally geodesic real projective subspace RP
n
⊂CP
n
.
The crucial observation for the classification is that an n-dimensional totally real
symmetric submanifold M of CP
n
is symmetric if and only if its inverse image
˜
M =
π
−1
(M) under the Hopf map
π
: S
2n+1
→ CP
n
is a symmetric submanifold
of S
2n+1
. According to Theorem 2.8.14 this indicates a close relation with symmetric
R-spaces. In the irreducible case, the relevant symmetric R-spaces
˜
M are U
m
/SO
m
,
U
m
, U
2m
/Sp
m
and E
6
U
1
/F
4
. Among all the standard embeddings of irreducible sym-
metric R-spaces they are characterized by the property that the dimension of the am-
bient Euclidean space is exactly twice the dimension of the symmetric R-space. So, if
n + 1 denotes the dimension of the symmetric R-space
˜
M, its image lies in the sphere
S
2n+1
⊂ R
2n+2
= C
n+1
. It turns out that
˜
M is invariant under the canonical S
1
-action
on S
2n+1
and hence projects via the Hopf map to an n-dimensional submanifold M of
CP
n
. Each of these submanifolds M is a totally real symmetric submanifold of CP
n
.
We list all of them in Table 11.6.
Submanifolds of Symmetric Spaces 309
TABLE 11.6: Symmetric totally real subman ifolds of CP
n
Mn= dim M Remarks
SU
m
/SO
m
1
2
(m −1)(m + 2) m ≥ 3
SU
m
(m −1)(m + 1) m ≥ 3
SU
2m
/Sp
m
(m −1)(2m + 1) m ≥ 3
E
6
/F
4
26
Note that these are precisely the irreducible, simply connected, Riemannian sym-
metric spaces of compact type whose root system is of type (A
n
) with n ≥ 2.
These embeddings can be described explicitly in a different way. Consider the
natural action of SL
m
(C) on J
m
(R)⊗C, the complexification of the real Jordan alge-
bra J
m
(R) of all symmetric m ×m-matrices with real coefficients, given by
(A,X) → AXA
T
for A ∈ SL
m
(C) and X ∈ J
m
(R) ⊗ C. The complex dimension of J
m
(R) ⊗ C is
m(m + 1)/2 and hence this action induces an action of SL
m
(C) on CP
n
with n =
m(m + 1)/2 −1 =(m −1 )(m + 2)/2. This action has exactly m orbits, which are
parametrized by the rank of the matrices. The subgroup of SL
m
(C) preserving com-
plex conjugation on CP
n
is SL
m
(R).Nowfix a maximal compact subgroup SO
m
of
SL
m
(R). The restriction to SO
m
(C) of the action of SL
m
(C) on J
m
(R) ⊗C splits off
a one-dimensional trivial factor corresponding to the trace. This means that SO
m
(C),
and hence SO
m
, fixes the point o in CP
n
given by complex scalars of the identity
matrix in J
m
(R) ⊗C. The maximal compact subgroup SO
m
of SL
m
(R) determines a
maximal compact subgroup SU
m
of SL
m
(C). The orbit of the action of SU
m
through
o gives an embedding of SU
m
/SO
m
in CP
n
as a totally real symmetric submanifold
of real dimension n. The other three embeddings can be constructed in a similar fash-
ion by using the real Jordan algebras J
m
(C), J
m
(H) and J
3
(O). The corresponding
subgroups are listed in Table 11.7.
TABLE 11.7: Some subgroups of some complex Lie groups
RC H O
SL
m
(C) SL
m
(C) ×SL
m
(C) SL
2m
(C) E
6
(C)
SL
m
(R) SL
m
(C) SU
∗
2m
E
−26
6
SO
m
SU
m
Sp
m
F
4
SU
m
SU
m
×SU
m
SU
2m
E
6
Theorem 11.3.6 (Naitoh) A complete irreducible n-dimensional totally real subma-
nifold M of CP
n
is a symmetric submanifold if and only if it is a totally geodesic real
projective subspace RP
n
⊂CP
n
, or if it is congruent to one of the embeddings listed
in Table 11.6.
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