Totally geodesic submanifolds in Riemannian symmetric spaces

Abstract

In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2.

Full text

arXiv:0810.4413v1 [math.DG] 24 Oct 2008
October 24, 2008 9:18 WSPC - Proceedings Trim Size: 9in x 6in tg
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Totally geodesic submanifolds in Riemannian symmetric spaces
Sebastian Klein
Universit¨at Mannheim, Lehrstuhl f¨ur Mathematik III,
Seminargeb¨aude A5, 68131 Mannheim, Germany
E-Mail: mail@sebastian-klein.de
In the first part of this expository article, the most important constructions and
classification results concerning totally geodesic submanifolds in Riemannian
symmetric spaces are summarized. In the second part, I describe the results
of my classification of the totally geodesic submanifolds in the Riemannian
symmetric spaces of rank 2.
To appear in the Proceedings volume for the conference VIII International
Conference on Differential Geometry, which took place in Santiago de Com-
postela in July 2008.
1. Totally geodesic submanifolds
A submanifold M′of a Riemannian manifold Mis called totally geodesic,
if every geodesic of M′is also a geodesic of M. In this article, we will
discuss totally geodesic submanifolds in Riemannian symmetric spaces; in
such spaces, a connected, complete submanifold is totally geodesic if and
only if it is a symmetric subspace.
There are several important construction principles for totally geodesic
submanifolds in Riemannian symmetric spaces M. First, we note that the
connected components of the fixed point set of any isometry fof Mare
totally geodesic submanifolds (this is in fact true in any Riemannian mani-
fold, see Ref. 11, Theorem II.5.1, p. 59). This construction principle is es-
pecially important in the case where fis involutive (i.e. f◦f= idM);
the totally geodesic submanifolds resulting in this case are called reflective
submanifolds; they have been studied extensively, for example by Leung
(see below).
Further constructions of totally geodesic submanifolds in Riemannian
symmetric spaces of compact type Mwere introduced by Chen and
Nagano:2,4 For p∈M, the connected components 6={p}of the fixed
point set of the geodesic reflection of Mat pare called polars or M+-

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submanifolds of M; note that they are in particular reflective submani-
folds of M. A pole of Mis a polar which is a singleton. It has been shown
by Chen/Nagano4that for every polar M+of Mand every q∈M+
there exists another reflective submanifold M−of Mwith q∈M−and
TqM−= (TqM+)⊥;M−is called a meridian or M−-submanifold of M.
For p1, p2∈M, a point q∈Mis called a center point between p1and
p2if there exists a geodesic joining p1with p2so that qis the middle
point on that geodesic. If p2is a pole of p1, then the set C(p1, p2) of cen-
ter points between p1and p2is called the centrosome of p1and p2; its
connected components are totally geodesic submanifolds of M(see Ref. 2,
Proposition 5.1).
Moreover, every symmetric space of compact type can be embedded in
its transvection group as a totally geodesic submanifold: Let M=G/K
be such a space, then there exists an involutive automorphism σof Gso
that Fix(σ)0⊂K⊂Fix(σ) . Because of this property, the Cartan map
f:G/K →G, gK 7→ σ(g)·g−1
is a well-defined covering map onto its image, which turns out to be a
totally geodesic submanifold of G. If Mis a “bottom space”, i.e. there
exists no non-trivial symmetric covering map with total space M, then we
have K= Fix(σ) , and therefore fis an embedding. In this setting fis
called the Cartan embedding of M.
It is a significant and interesting problem to determine all totally
geodesic submanifolds in a given symmetric space. Because totally geodesic
submanifolds are rigid (i.e. if M′
1, M ′
2are connected, complete totally
geodesic submanifolds of Mwith p∈M′
1∩M′
2and TpM′
1=TpM′
2,
then we already have M1=M2), they can be classified by determining
those linear subspaces U⊂TpMwhich occur as tangent spaces of totally
geodesic submanifolds of M.
The elementary answer to the latter problem is the following: There
exists a totally geodesic submanifold of Mwith a given tangent space
U⊂TpMif and only if Uis curvature invariant (i.e. we have R(u, v)w∈U
for all u, v, w ∈U, denoting by Rthe Riemannian curvature tensor of M).
Therefore the classification of totally geodesic submanifolds of Mre-
duces to the purely algebraic problem of the classification of curvature in-
variant subspaces of TpM. However, because of the algebraic complexity of
the curvature tensor, classifying the curvature invariant subspaces is by no
means an easy task, and therefore also the classification of totally geodesic
submanifolds remains a signification problem.

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This problem has been solved for the Riemannian symmetric spaces of
rank 1 by Wolf in Ref. 19, §3. Chen/Nagano claimed a classification for
the complex quadrics (which are symmetric spaces of rank 2) in Ref. 3, and
then for all symmetric spaces of rank 2 in Ref. 4, using their construction
of polars and meridians described above. However, it turns out that their
classifications are incorrect: For several spaces of rank 2, totally geodesic
submanifolds have been missed, and also some other details are faulty. In my
papers Refs. 7–10 I discuss these shortcomings and give a full classification
of the totally geodesic submanifolds in all irreducible symmetric spaces of
rank 2; Section 2 of the present exposition contains a summary of these
results. For symmetric spaces of rank ≥3 , the full classification problem
is still open.
However, there are several results concerning the classification of special
classes of totally geodesic submanifolds. Probably the most significant result
of this kind is the classification of reflective submanifolds in all Riemannian
symmetric spaces due to Leung; his results are found in final form in
Ref. 14, but also see Refs. 12,13. Another important problem of this kind
is the classification of the totally geodesic submanifolds M′of M=G/K
with maximal rank (i.e. rk(M′) = rk(M) ); this problem has been solved
for the symmetric spaces with rk(M) = rk(G) by Ikawa/Tasaki,6and
then for all irreducible symmetric spaces by Zhu/Liang.20
Further important classification results concern Hermitian symmetric
spaces M: In them, the complex totally geodesic submanifolds have been
classified by Ihara.5Moreover, the real forms of M(i.e. the totally real,
totally geodesic submanifolds M′of Mwith dimIR(M′) = dimC(M) ) are
all reflective; due to this fact Leung was able to derive a classification of
the real forms of all Hermitian symmetric spaces from his classification of
reflective submanifolds.15
Finally we mention a result by Wolf concerning totally geodesic sub-
manifolds M′of (real, complex or quaternionic) Grassmann manifolds
Gr(IKn) with the property that any two distinct elements of M′have zero
intersection, regarded as r-dimensional subspaces of IKn. Wolf showed18,19
that any such totally geodesic submanifold is isometric either to a sphere
or to a projective space over IR , C or IH; he was also able to describe em-
beddings for these submanifolds explicitly and to calculate their maximal
dimension in dependence of rand n.

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2. Maximal totally geodesic submanifolds in the
Riemannian symmetric spaces of rank 2
In the following, I list the isometry types corresponding to all the congru-
ence classes of totally geodesic submanifolds which are maximal (i.e. not
strictly contained in another connected totally geodesic submanifold) in
all Riemannian symmetric spaces of rank 2. In many cases I also briefly
describe totally geodesic embeddings corresponding to these submanifolds.
This is a summary of my work in Refs. 7–10, where it is proved that the
lists given here are complete, and where the totally geodesic embeddings
are described in more detail.
The invariant Riemannian metric of an irreducible Riemannian sym-
metric space is unique only up to a positive constant. In the sequel, we
use the following notations to describe the metric which is induced on the
totally geodesic submanifolds: For ℓ∈IN and r > 0 we denote by Sℓ
rthe
ℓ-dimensional sphere of radius r, and for κ>0 we denote by IRPℓ
κ,CPℓ
κ,
IHPℓ
κand OP2
κthe respective projective spaces, their metric being scaled
in such a way that the minimal sectional curvature is κ. ( IRPℓ
κis then of
constant sectional curvature κ, CPℓ
κis of constant holomorphic sectional
curvature 4κ, and we have the inclusions IRPℓ
κ⊂CPℓ
κ⊂IHPℓ
κof totally
geodesic submanifolds). For symmetric spaces of rank 2, we describe the ap-
propriate metric by stating the length aof the shortest restricted root of
the space as a subscript srr=a. For the three infinite families of Grassmann
manifolds G+
2(IRn) , G2(Cn) and G2(IHn) , we also use the notation srr=1∗
to denote the metric scaled in such a way that the shortest root occurring
for large nhas length 1 , disregarding the fact that this root might vanish
for certain small values of n.
The spaces in which the totally geodesic submanifolds are classified
below are always taken with srr=1∗(for the Grassmann manifolds) or srr=1
(for all others).
2.1. G+
2(IRn+2)
(a) G+
2(IRn+1 )srr=1∗
The linear isometric embedding IRn+1 →IRn+2 ,(x1, . . . , xn+1)7→
(x1,...,xn+1,0) induces a totally geodesic, isometric embedding
G+
2(IRn+1 )→G+
2(IRn+2 ) .
(b) Sn
r=1
Fix a unit vector v0∈IRn+2 , and let S:= {v∈IRn+2 | hv, v0i=
0,kvk= 1 }∼
=Sn
r=1 . Then the map S→G+
2(IRn+2 ), v 7→ IRv⊕IRv0is

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a totally geodesic, isometric embedding.
(c) (Sℓ
r=1 ×Sℓ′
r=1)/ZZ2, where ℓ+ℓ′=n
The map
Sℓ
r=1 ×Sℓ′
r=1 →G+
2(IRn+2 )
((x0,...,xℓ),(y0, . . . , yℓ′)) 7→ IR (x0, . . . , xℓ,0,...,0) ⊕IR (0, . . . , 0, y0, . . . , yℓ′)
is a totally geodesic, isometric immersion, and a two-fold covering map
onto its image in G+
2(IRn+2 ) .
(d) For n≥4 even: CPn/2
κ=1/2
Let us fix a complex structure Jon IRn+2 . Then the complex-1-
dimensional linear subspaces of (IRn+2 , J ) are in particular real-2-
dimensional oriented linear subspaces of IRn+2 . Therefore the com-
plex projective space IP ∼
=CPn/2over (IRn+2, J ) is contained in
G+
2(IRn+2 ) ; it turns out to be a totally geodesic submanifold.
(e) For n= 2 : CP1
κ=1/2×IRP1
κ=1/2
The image of the Segr´e embedding CP1×CP1→CP3(see for exam-
ple Ref. 17, p. 55f.) is a 2-dimensional complex quadric in CP3; such a
quadric is isometric to G+
2(IR4) . Thereby we see that G+
2(IR4) is iso-
metric to CP1
κ=1/2×CP1
κ=1/2. Let Cbe the trace of a (closed) geodesic
in CP1
κ=1/2; then Cis isometric to IRP1
κ=1/2, and CP1
κ=1/2×Cis a
totally geodesic submanifold of CP1
κ=1/2×CP1
κ=1/2∼
=G+
2(IR4) .
(f) For n= 3 : S2
r=√5
To describe this totally geodesic submanifold, as well as similar to-
tally geodesic submanifolds occurring in G2(C6) and G2(IH7) (see
Sections 2.2(g) and 2.3(f) below), we note that there is exactly one
irreducible, 14-dimensional, quaternionic representation of Sp(3) (see
Ref. 1, Chapter VI, Section (5.3), p. 269ff.). It can be constructed as fol-
lows: The vector representation of Sp(3) on C6induces a representation
of Sp(3) on V3C6. This 20-dimensional representation decomposes into
two irreducible components: One, 6-dimensional, is equivalent to the vec-
tor representation of Sp(3) ; the other, acting on a 14-dimensional linear
space V, is the irreducible representation we are interested in.
It turns out that the restriction of the representation of Sp(3) on Vto
an SO(3) embedded in Sp(3) in the canonical way, is a real representa-
tion, and that in any real form VIR of V, two linear independent vectors
are left invariant. By splitting off the subspace of V′spanned by these
vectors, we get a real-5-dimensional representation V′
IR of SO(3) , which
turns out to be irreducible (and equivalent to the Cartan representation
SO(3) ×End+(IR3)0→End+(IR3)0,(B, X )7→ BXB−1). It turns out

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that the corresponding action of SO(3) on G+
2(V′
IR)∼
=G+
2(IR5) has
exactly one totally geodesic orbit; this orbit is isometric to S2
r=√5.
2.2. G2(Cn+2)
(a) G2(Cn+1)srr=1∗
The linear isometric embedding Cn+1 →Cn+2,(z1, . . . , zn+1)7→
(z1,...,zn+1 ,0) induces a totally geodesic, isometric embedding
G2(Cn+1)→G2(Cn+2 ) .
(b) G2(IRn+2)srr =1∗
The map G2(IRn+2)→G2(Cn+2 ),Λ7→ Λ⊕iΛ is a totally geodesic,
isometric embedding.
(c) CPn
κ=1
Fix a unit vector v0∈Cn+2 . Then the map CP((Cv0)⊥)→
G2(Cn+2),Cv7→ Cv⊕Cv0is a totally geodesic, isometric embedding.
(d) CPℓ
κ=1 ×CPℓ′
κ=1 with ℓ+ℓ′=n
Let Cn+2 =W⊖W′be a splitting of Cn+2 into complex-linear sub-
spaces of dimension ℓ+ 1 resp. ℓ′+ 1 . Then CP(W)×CP(W′)→
G2(Cn+2),(Cv, Cv′)7→ Cv⊕Cv′is a totally geodesic, isometric em-
bedding.
(e) For neven: IHPn/2
κ=1/2
Let us fix a quaternionic structure τon Cn+2 (i.e. τ: Cn+2 →Cn+2
is anti-linear with τ2=−id ). Then the quaternionic-1-dimensional
linear subspaces of (Cn+2, τ ) are in particular complex-2-dimensional
linear subspaces of Cn+2 . Therefore the quaternionic projective space
IP ∼
=IHPn/2over (Cn+2, τ ) is contained in G2(Cn+2 ) ; it turns out
that IP is a totally geodesic submanifold of G2(Cn+2 ) .
(f) For n= 2 : G+
2(IR5)srr=√2and (S3
r=1/√2×S1
r=1/√2)/ZZ2
Note that G2(C4)srr=1∗is isometric to G+
2(IR6)srr=√2. This isometry
can be exhibited via the Pl¨ucker map G2(Cm)→CP(V2Cm),Cu⊕
Cv7→ C(u∧v) , which is an isometric embedding for any m; for m= 4
its image in CP(V2C4)∼
=CP5turns out to be a 4-dimensional com-
plex quadric; such a quadric is isomorphic to G+
2(IR6) . Thus G2(C4)
is isometric to G+
2(IR6) , hence its maximal totally geodesic submani-
folds are those given in Section 2.1 for n= 4 , namely: G+
2(IR5)srr=√2,
(S3
r=1/√2×S1
r=1/√2)/ZZ2, (S2
r=1/√2×S2
r=1/√2)/ZZ2,S4
r=1/√2,CP2
κ=1 .
The first two of these submanifolds are those which are listed under
this point; the remaining submanifolds have already been listed above

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(note that (S2×S2)/ZZ2and S4are isometric to G2(IR4) and IHP1,
respectively).
(g) For n= 4 : CP2
κ=1/5
Let us consider the 14-dimensional quaternionic, irreducible represen-
tation Vof Sp(3) described in Section 2.1(f). The restriction of that
representation to a SU(3) canonically embedded in Sp(3) leaves a to-
tally complex 6-dimensional linear subspace VCof Vinvariant; the
resulting 6-dimensional representation VCof SU(3) is irreducible. It
turns out that the induced action of SU(3) on G2(VC)∼
=G2(C6) has
exactly one totally geodesic orbit; this orbit is isometric to CP2
κ=1/5.
2.3. G2(IHn+2 )
(a) G2(IHn+1 )srr=1∗
The linear isometric embedding IHn+1 →IHn+2 ,(q1, . . . , qn+1)7→
(q1,...,qn+1 ,0) induces a totally geodesic, isometric embedding
G2(IHn+1 )→G2(IHn+2 ) .
(b) G2(Cn+2)srr=1∗
We fix two orthogonal imaginary unit quaternions iand j, and let
C = IR ⊕IRi. Then the map G2(Cn+2 )→G2(IHn+2 ),Λ7→ Λ⊕Λjis
a totally geodesic, isometric embedding.
(c) IHPn
κ=1
Fix a unit vector v0∈IHn+2 . Then the map IHP((v0IH)⊥)→
G2(IHn+2 ), vIH 7→ vIH ⊕v0IH is a totally geodesic, isometric embed-
ding.
(d) IHPℓ
κ=1 ×IHPℓ′
κ=1 with ℓ+ℓ′=n
Let IHn+2 =W⊖W′be a splitting of IHn+2 into quaternionic-linear
subspaces of dimension ℓ+ 1 resp. ℓ′+ 1 . Then IHP(W)×IHP(W′)→
G2(IHn+2 ),(vIH, v′IH) 7→ vIH ⊕v′IH is a totally geodesic, isometric
embedding.
(e) For n= 2 : Sp(2)srr=√2and (S5
r=1/√2×S1
r=1/√2)/ZZ2
Let U∈G2(IH4) be given, then U⊥is the only pole corresponding to
Uin G2(IH4) . The centrosome between this pair of poles is a totally
geodesic submanifold of G2(IH4) which is isometric to Sp(2) . This
Sp(2) is also a reflective submanifold of G2(IH4) , the complementary
reflective submanifold is isometric to (S5
r=1/√2×S1
r=1/√2)/ZZ2.
(f) For n= 5 : IHP2
κ=1/5
We again consider the irreducible, quaternionic 14-dimensional repre-
sentation Vof Sp(3) introduced in Section 2.1(f); we now view Vas

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a quaternionic-7-dimensional linear space. The representation of Sp(3)
on Vinduces an action of Sp(3) on the quaternionic 2-Grassmannian
G2(V)∼
=G2(IH7) ; again it turns out that this action has exactly one
totally geodesic orbit, which is isometric to IHP2
κ=1/5.
(g) For n= 4 : S3
r=2√5
According to the present list, two of the maximal totally geodesic sub-
manifolds of the 2-Grassmannian G2(IH7) are isometric to G2(IH6) and
IHP2
κ=1/5, respectively. The intersection of these two totally geodesic
submanifolds is a totally geodesic submanifold of G2(IH6) , which turns
out to be isometric to S3
r=2√5.
2.4. SU(3)/SO(3)
(a) (S2
r=1 ×S1
r=√3)/ZZ2
(b) IRP2
κ=1/4
2.5. SU(6)/Sp(3)
(a) IHP2
κ=1/4
(b) CP3
κ=1/4
(c) SU(3)srr=1
The map SU(3) →SU(6)/Sp(3), B 7→ B0
0B−1·Sp(3) is a totally
geodesic embedding of this type.
(d) (S5
r=1 ×S1
r=√3)/ZZ2
2.6. SO(10)/U(5)
In the descriptions of the embeddings for this symmetric space, we consider
both U(5) and SO(10) as acting on C5∼
=IR10 ; in the latter case, this
action is only IR-linear.
(a) CP4
κ=1
(b) G2(C5)srr=1
(c) CP3
κ=1 ×CP1
κ=1
Let G:= SO(6) ×SO(4) be canonically embedded in SO(10) in such
a way that its intersection with U(5) is maximal. Then G/(G∩U(5))
is a totally geodesic submanifold of SO(10)/U(5) which is isometric to
(SO(6)/U(3)) ×(SO(4)/U(2)) ∼
=CP3×CP1.

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(d) G+
2(IR8)srr=√2
Let G:= SO(8) be canonically embedded in SO(10) in such a way
that its intersection with U(5) is maximal. Then G/(G∩U(5)) is
a totally geodesic submanifold of SO(10)/U(5) which is isometric to
SO(8)/U(4) ∼
=G+
2(IR8) .
(e) SO(5)srr=1
The map SO(5) →SO(10)/U(5), B 7→ B0
0B−1·U(5) is a totally
geodesic embedding of this type.
2.7. E6/(U(1) ·Spin(10))
(a) OP2
κ=1/2
(b) CP5
κ=1 ×CP1
κ=1
(c) G+
2(IR10 )srr=√2
(d) G2(C6)srr=1
(e) (G2(IH4)/ZZ2)srr=1
(f) SO(10)/U(5)srr=1
2.8. E6/F4
(a) OP2
κ=1/4
(b) IHP3
κ=1/4
(c) ((SU(6)/Sp(3))/ZZ3)srr=1
(d) (S9
r=1 ×S1
r=√3)/ZZ4
2.9. G2/SO(4)
(a) SU(3)/SO(3)srr=√3
(b) (S2
r=1 ×S2
r=1/√3)/ZZ2
(c) CP2
κ=3/4
(d) S2
r=2
3√21
2.10. SU(3)
(a) SU(3)/SO(3)srr=1
The Cartan embedding f: SU(3)/SO(3) →SU(3) is a totally geodesic
embedding of this type.
(b) (S3
r=1 ×S1
r=√3)/ZZ2

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(c) CP2
κ=1/4
The Cartan embedding f: SU(3)/S(U(2) ×U(1)) →SU(3) is a totally
geodesic embedding of this type.
(d) IRP3
κ=1/4
2.11. Sp(2)
(a) G+
2(IR5)srr=1
The Cartan embedding f: Spin(5)/(Spin(2) ×Spin(3)) →Spin(5) ∼
=
Sp(2) is a totally geodesic embedding of this type.
(b) Sp(1) ×Sp(1)
The canonically embedded Sp(1) ×Sp(1) ⊂Sp(2) is a totally geodesic
submanifold of this type.
(c) IHP1
κ=1/2
The Cartan embedding f: Sp(2)/(Sp(1) ×Sp(1)) →Sp(2) is a totally
geodesic embedding of this type.
(d) S3
r=√5
2.12. G2
(a) G2/SO(4)srr=1
The Cartan embedding f:G2/SO(4) →G2is a totally geodesic em-
bedding of this type.
(b) (S3
r=1 ×S3
r=1/√3)/ZZ2
(c) SU(3)srr=√3
Regard G2as the automorphism group of the division algebra of the
octonions O and fix an imaginary unit octonion i. Then the subgroup
{g∈G2|g(i) = i}is isomorphic to SU(3) and a totally geodesic
submanifold of this type.
(d) S3
r=2
3√21
Acknowledgments
This work was supported by a fellowship within the Postdoc-Programme
of the German Academic Exchange Service (DAAD).
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