310 Submanifolds and Holonomy
Naitoh and Takeuchi proved in [242] that every n-dimensional totally real sym-
metric submanifold M of CP
n
is basically a product of the irreducible submanifolds
discussed above and a flat torus. A suitable product of n + 1 circles in S
2n+1
projects
via the Hopf map to a flat n-dimensional torus T
n
embedded in CP
n
as a totally real
symmetric submanifold. Naitoh and Takeuchi gave in [242] a unifying description
of all symmetric submanifolds in the Grassmann geometry G(RP
n
,CP
n
) using the
Shilov boundary of symmetric bounded domains of tube type.
11.3.6 Symmetric totally complex submanifolds of HP
n
The symmetric totally complex submanifolds of HP
n
of complex dimension n
were classified by Tsukada [326]. The reflective submanifold in the corresponding
Grassmann geometry is the totally geodesic CP
n
⊂ HP
n
. A basic tool for the classi-
fication is the twistor map CP
2n+1
→ HP
n
. Consider H
n+1
as a (right) vector space
and pick a unit quaternion, say i,whichturnsH
n+1
into a complex vector space
C
2n+2
. The twistor map CP
2n+1
→HP
n
maps a complex line in C
2n+2
to the quater-
nionic line in H
n+1
spanned by it. The fiber over each point is a complex projec-
tive line CP
1
⊂ CP
2n+1
. Alternatively, the set of all almost Hermitian structures in
the quaternionic K¨ahler structure of a quaternionic K¨ahler manifold
¯
M forms the so-
called twistor space Z of
¯
M, and the natural projection Z →
¯
M is the so-called twistor
map onto
¯
M. In the case of HP
n
the twistor space is just CP
2n+1
.
Now let M be a non-totally geodesic symmetric totally complex submanifold
of HP
n
belonging to the Grassmann geometry G(CP
n
,HP
n
).Thefirststepinthe
classificationistoshowthatM is a Hermitian symmetric space with respect to a
K¨ahler structure that is induced from the quaternionic K¨ahler structure of HP
n
.Then
one shows that M can be lifted to a K¨ahler immersion into the twistor space CP
2n+1
.
The main part of the proof is then to show, u sing representation theory of complex
semisimple Lie algebras, that this lift is one of the embeddings in CP
2n+1
as in Table
11.8.
TABLE 11.8: Symmetric totally complex submanifolds of HP
n
Mn= dim
C
M embedding Comments
Sp
3
/U
3
6 F
1
SU
6
/S(U
3
U
3
) 9 F
1
SO
12
/U
6
15 F
1
E
7
/E
6
U
1
27 F
1
CP
1
×SO
m
/SO
m−2
SO
2
m −1 F
1
⊗F
1
m ≥ 3
In the last case, the embedding is via the exterior tensor product of the first canon-
ical embedding of each factor; in the other cases, it is the first canonical embedding.
The four Hermitian symmetric spaces arising via the first canonical embedding are
precisely the irreducible, simply connected Riemannian symmetric spaces with root
system of type (C
3
) and for which the multiplicity of the shorter root is equal to o ne.
Note that, in the last case, the submanifold is isometric to CP
1
×CP
1
for m = 3and
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].
312 Submanifolds and Holonomy
We start with recalling the theory o f symmetric R-spaces from another viewpoint
(see Kobayashi and Nagano [177], Nagano [225], and Takeuchi [310] for details).
Let (
¯
g,
σ
) be a positive definite symmetric graded Lie algebra, that is,
¯
g is a real
semisimple Lie algebra with a gradation
¯
g =
¯
g
−1
⊕
¯
g
0
⊕
¯
g
1
so that
¯
g
−1
= {0} and the
adjoint action of
¯
g
0
on the vector space
¯
g
−1
is effective, and a Cartan involution
σ
satisfying
σ
(
¯
g
ν
)=
¯
g
−
ν
,
ν
∈{−1,0,1}. The positive definite symmetric graded Lie
algebras are completely classified (see [177], [310]).
By defining
τ
(X)=(−1)
ν
X for X ∈
¯
g
ν
we obtain an involutive automorphism
τ
of
¯
g that satisfies
στ
=
τσ
.Let
¯
g =
¯
k ⊕
¯
p be the Cartan d ecomposition induced
by
σ
.Thenwehave
τ
(
¯
k)=
¯
k and
τ
(
¯
p)=
¯
p.Let
¯
k = k
+
⊕k
−
and
¯
p = p
+
⊕p
−
be
the ±1-eigenspace decompositions of
¯
k and
¯
p with respect to
τ
. Obviously, we have
k
+
=
¯
k ∩
¯
g
0
, k
−
=
¯
k ∩(
¯
g
−1
⊕
¯
g
1
), p
+
=
¯
p ∩
¯
g
0
and p
−
=
¯
p ∩(
¯
g
−1
⊕
¯
g
1
).Since
¯
g is a
semisimple Lie algebra, there is a unique element Z ∈
¯
g
0
so that
¯
g
ν
= {X ∈
¯
g :ad(Z)X =
ν
X} ,
ν
∈{−1, 0, 1}.
It can be easily seen that Z ∈
¯
p and hence Z ∈ p
+
.
We denote by B the Killing form of
¯
g. The restriction of B to
¯
p ×
¯
p is a positive
definite inner product on
¯
p and will be denoted by ·, ·. This inner product is invariant
under the adjoint action of
¯
k on
¯
p and under the involution
τ
|
¯
p
. In particular, p
+
and
p
−
are perpendicular to each other. Let
¯
G be the simply connected Lie group with
Lie algebra
¯
g and
¯
K be the connected Lie subgroup of
¯
G corresponding to
¯
k,and
define the homogeneous space
¯
M =
¯
G/
¯
K.Let
π
:
¯
G →
¯
M be the natural projection
and put o =
π
(e),wheree ∈
¯
G is the identity. The restriction to
¯
p of the differential
d
e
π
:
¯
g → T
o
¯
M of
π
at e yields a linear isomorphism
¯
p → T
o
¯
M. In the following
we will always identify
¯
p and T
o
¯
M via this isomorphism. From the Ad(
¯
K)-invariant
inner product ·,· on
¯
p
∼
=
T
o
¯
M we get a
¯
G-invariant Riemannian metric on
¯
M.Then
¯
M =
¯
G/
¯
K is the Riemannian symmetric space of noncompact type that is associated
with (
¯
g,
σ
,·,·).
We put
K
+
= {k ∈
¯
K :Ad(k)Z = Z}.
Then K
+
is a closed Lie subgroup with Lie algebra k
+
. The homogeneous space M
=
¯
K/K
+
is diffeomorphic to the orbits Ad(
¯
K) ·Z ⊂
¯
p and
¯
K ·
π
(Exp(Z)) ⊂
¯
M,where
Exp :
¯
g →
¯
G denotes the Lie exponential map from
¯
g into
¯
G. We equip M
with the
induced Riemannian metric from
¯
M.ThenM
is a compact Riemannian symmetric
space associated with the orthogonal symmetric Lie algebra (
¯
k,
τ
|
¯
k
),where
τ
|
¯
k
is the
restriction of
τ
to
¯
k. The symmetric spaces M
arising in this manner are precisely the
symmetric R-spaces. If
¯
g is simple, then M
is an irreducible symmetric R-space.
The subspace p
−
is a Lie triple system in
¯
p = T
o
¯
M and [p
−
,p
−
] ⊂ k
+
. Thus
there exists a complete totally geodesic submanifold M of
¯
M with o and T
o
M = p
−
.
Since M is the image of p
−
under the exponential map of
¯
M at o, we see that M
is simply connected. We define a subalgebra g of
¯
g by g = k
+
⊕p
−
and denote by
G the connected Lie subgroup of
¯
G with Lie algebra g. Then, by construction, M
is the G-orbit containing o. We denote by K
+
the isotropy group at o of the action
of G on
¯
M. The Lie algebra of K
+
is k
+
.SinceM = G/K
+
is simply connected,
Submanifolds of Symmetric Spaces 313
K
+
is connected. The restriction
τ
|
g
of
τ
to g is an involutive automorphism of g
and (g,
τ
|
g
) is an orthogonal symmetric Lie algebra dual to (
¯
k,
τ
|
¯
k
). Moreover , M is
a Riemannian symmetric space of noncompact type associated with (g,
τ
|
g
).Since
both p
−
and p
+
are Lie triple systems, M is a reflective submanifold of
¯
M.The
corresponding Grassmann g eometry G(M ,
¯
M) is a g eometry according to Theorem
11.3.4 (5).
We will construct a one-parameter family of symmetric submanifolds in
¯
M con-
sisting of submanifolds belonging to the Grassmann geometry that contains the to-
tally geodesic submanifold M and the symmetric R-space M
. For each c ∈ R we
define a subspace p
c
of p
−
⊕k
−
=
¯
g
−1
⊕
¯
g
1
by
p
c
= {X + cad(Z)X : X ∈ p
−
}.
In particular, p
1
=
¯
g
1
and p
−1
=
¯
g
−1
are abelian subalgebras of
¯
g.Theng
c
= k
+
⊕p
c
is a
τ
-invariant subalgebra of
¯
g and (g
c
,
τ
|
g
c
) is an orthogonal symmetric Lie algebra.
We denote by G
c
the connected Lie subgroup of
¯
G with Lie algebra g
c
and by M
c
the
orbit of G
c
in
¯
M through o.
Proposition 11.3.8 For each c ∈ R, the orbit M
c
is a symmetric submanifold of
¯
M
belonging to the Grassmann geometry G(M,
¯
M). The submanifolds M
c
and M
−c
are
congruent via the geodesic symmetry s
o
of
¯
M at o. The submanifolds M
c
, 0 ≤ c < 1,
form a family of noncompact symmetric submanifolds that are homothetic to the re-
flective submanifold M. The submanifolds M
c
, 1 < c < ∞, form a family of compact
symmetric submanifolds that are homothetic to the symmetric R-space M
. The sub-
manifold M
1
is a flat symmetric space that is isometric to a Euclidean space. The
second fundamental form
α
c
of M
c
is given by
α
c
(X,Y )=c[ad(Z)X,Y ] ∈ p
+
=
ν
o
M
c
, X,Y ∈ p
−
= T
o
M
c
.
In particular, all submanifolds M
c
, 0 ≤ c < ∞, are pairwise noncongruent.
It was proved in [33] that every non-totally geodesic symmetric submanifold of
an irreducible Riemannian symmetric space of noncompact type and rank ≥2arises
in this way. The crucial point for the proof is a generalization of the Fundamental
Theorem of Submanifold Geometry in space forms to certain Grassmannian geome-
tries.
11.4 Submanif olds with parallel second fundamental form
11.4.1 ... in real space forms
With our achievements so far the classification of submanifolds with parallel sec-
ond fundamental form in spaces of constant curvature becomes very simple. When
314 Submanifolds and Holonomy
¯
M has constant curvature, every subspace of any tangent space of
¯
M is curvature-
invariant. From Proposition 11.3.1 we therefore get the following.
Corollary 11.4.1 A complete submanifold of S
n
, R
n
or RH
n
has parallel second
fundamental form if and only if it is a symmetric submanifold.
11.4.2 ... in complex space forms
When the ambient space has nonconstant curvature, one cannot expect that com-
plete submanifolds with parallel second fundamental form are symmetric subman-
ifolds. This can be seen most easily in complex p rojective space CP
n
. A totally
geodesic real projective space RP
k
, k ∈{1,...,n −1}, is complete and obviously has
parallel second fundamental form. But, at each point, the normal space is isomorphic
to the subspace R
k
⊕C
n−k
⊂ C
n
, and this cannot be the tangent space of a totally
geodesic submanifold of CP
n
. Hence, the normal spaces are not curvature-invariant,
and it follows that RP
k
is not a symmetric submanifold of CP
n
.
The classification of submanifolds with parallel second fundamental form in
complex projective space CP
n
and complex hyperbolic space CH
n
was obtained by
Naitoh in [234, 235].
Theorem 11.4.2 (Naitoh) Let M be a complete submanifold of CP
n
or CH
n
,n≥2,
with parallel second fundamental form and assume that M is not totally geodesic.
Then M is
(i) a complex submanifold, or
(ii) a submanifold that is contained in a totally geodesic RP
k
⊂CP
n
resp. RH
k
⊂
CH
n
for some k ∈{1 ,...,n},or
(iii) a k-dimensional totally real submanifold that is contained in a totally geodesic
CP
k
⊂ CP
n
resp. CH
k
⊂ CH
n
for some k ∈{1,...,n}.
The normal spaces of complex submanifolds in CP
n
and CH
n
are always
curvature-invariant. Thus, the classification of complex submanifolds with parallel
second fundamental form reduces to the one of symmetric complex submanifolds,
which we discussed above (see Theorem 11.3.5) in the case o f complex projective
space. In the case of complex hyperbolic space, Kon [ 187] proved:
Theorem 11.4.3 (Kon) Let M be a complex submanifold of complex hyperbolic
space CH
n
. If M has parallel second fundamental form, then M is totally geodesic.
In case (ii), M also has parallel second fundamental form when considered as a
submanifold in RP
k
resp. RH
k
. So this case reduces to the corresponding problem in
real space forms, which we briefly discussed above.
In the last case (iii), M has parallel second fundamental form when considered as
a submanifold in CP
k
resp. CH
k
.SinceM is totally real and has half the dimension
of these smaller ambient spaces, this case reduces to the study of half-dimensional
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