R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI P ADOVA
C RISTIÁN U. S ÁNCHEZ
A NA L. C ALÍ
J OSÉ L. M ORESCHI
Homogeneous totally real submanifolds of complex projective space
Rendiconti del Seminario Matematico della Università di Padova, tome 101 (1999), p. 83-94
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of Complex Projective Space (*).
CRISTIÁN
U.SANCHEZ (**) -
ANA L.CALÍ (***) - JOSÉ
L.MORESCHI (***)
ABSTRACT - This paper is devoted to the
study
of afamily
oftotally
real submani- folds ofcomplex projective
space CP n . This is thefamily
of the so called R-spaces or real
flag
manifolds which have very natural immersions into certaincomplex projective
spaces. The main result determines thegeometric
proper- ties that characterize these immersions.1. - A natural immersion.
Every R-space (real flag manifold)
M n has a naturaltotally
real im-mersion into a
complex projective
space This immersion arisesnaturally
from the canonicalembedding
associated to theR-space.
Letus recall that an
R-space M n, by definition,
has a canonicalembedding
into defined as follows. Let g be a real
semisimple
Liealgebra
without
compact factors, f
a maximalcompactly
embeddedsubalgebra
ofg and g = f Q9 p the Cartan
decomposition
of g relative to ~. If we denoteby
B theKilling
form of gthen p
can be considered a Euclidean space with the innerproduct
definedby
the restriction of Bto p.
Let G ==
Int ( g )
be the group of innerautomorphisms
of g; whose Liealgebra
may be identified with g. Let K be the
analytic subgroup
of G corre-(*) This research
partially supported by
CONICET, SECYT-UNSL and SECYT-UNCOR ofArgentina.
(**) Indirizzo dell’A.: Fa. M.A.F., Universidad de C6rdoba, Medina Allende
y
Haya
de la Torre, Ciudad universitaria, 5000, C6rdoba,Argentina.
(***) Indirizzo
degli
AA.: Fac. Ci. Fi. Mat. y Nat., Universidad de San Luis, Chacabuco y Pedernera, 5700, San Luis,Argentina.
sponding to f;
then K iscompact
and acts on p as anisometry
group. TheR-space
M is the orbit of a non zero vector E E p i. e. M =AdG (K) E .
Aparticular
case of this situation is that of thecomplex flag
manifolds(see
for instance
[12])
where one takes acompact semisimple
Liealgebra
uand consider g =
uC
= uE9 iu
as a realsemisimple
Liealgebra.
Thenu E9 iu is a Cartan
decomposition
of g[5] [p. 185]
and U =Int (u) acting
on iu
by
theadjoint representation
has orbits which arecomplex flag
manifolds.
Let o :
M n ~ ( ~ , B )
be the canonicalembedding
of anR-space.
Wemay consider that it is isometric
by taking
on M n the induced metric. It is clear that theimage
of theembedding Q
lies on asphere.
This definesa new
embedding
where n + q + 1 is the dimension
of p.
We may take now the induced immersionwhich followed
by
the naturalembedding
Q:yields
animmersion
which
obviously
istotally
real.In the
present
paper we determine thegeometric properties that,
interms of the second fundamental
form,
characterize these immer- sions.One of the motivations for this work is a result mentioned in
[3]
(specifically
Theorem C in page354)
where the authorsstudy
minimaltotally
real submanifolds ofCP~ obtaining
a nice characterization of them in terms ofproperties
of the second fundamental form of the im- mersion under a restriction on the Ricci curvature.Another motivation is the known characterization of
R-spaces
as sub-manifolds of
R
withcanonically parallel
second fundamental form[8].
The paper is
organized
as follows. In Section 2 we find the relevantgeometric properties
of the immersion cp defined in(3).
Section 3 con- tains theproof
of the fact that theseproperties
determine thetotally
re-al immersion cp .
2. - Geometric
properties
of the immersion cp.We describe now the fundamental
properties
of the immersion cp.THEOREM 1. The second
fundamental form
aof
the isometric im- mersion cp has thefollowing properties:
i)
a iscanonically parallel,
ii) If
J denotes thecomplex
structure on cpn + q at any EE
cp(M)
thenPROOF. It is
known, [8],
that the canonicalembedding (2
of the mani- fold M ninto = l~ n + q +
1 describedabove,
hascanonically parallel
sec-ond fundamental form in the sense of
[9]
withrespect
to any one of thepossible
canonical connections that can be defined on M n .Recall that a linear connection V’ on M n is said to be canonical if it satisfies the
following properties
ii)
V~D =0 ,
where V is the Riemannian connection of the induced metric(.,.)
on M n andWith the canonical connection V’ on M n we can define as in
[9]
the canonical covariant derivativeof
the secondfundamental form
a 0 of theembedding e by
the formulaWe say that a o is
canonically parallel
ifLet a 1 be the second fundamental form of the
embedding Q
1 definedin
(1)
and let a 2 be that of(2).
It is easy to see that = 0 im-plies 1 )
= 0 which in turnimmediately yields (Vx a 2)
= 0 . Since Q isa
totally geodesic embedding
we haveproved
the firstpart
of theproposition.
Let us prove
(ii).
We do this in twosteps.
Let us assume first that M "is a
complex flag manifold
and let Q: g be one of its canonical em-beddings.
There exists on M n anintegrable
almostcomplex
structureJ1
(see
for instance[1])
which commutes with theshape operators A~
of theimbedding
o(see [10]
and referencestherein)
i.e. for each X ETp (M), and p e M "
Then the second fundamental form a 0 satisfies
and since
where h is a real valued
symmetric
bilinear formand ~
is alocally
de-fined normal vector
field,
weclearly
haveIn turn a 2 and a
satisfy
the sameequalities.
On the otherhand,
since theimmersion cp
istotally real,
its second fundamental form a has tosatisfy
the
following identity
where J is thecomplex
structure in thecomplex projective
spaceCP’ (see
for instance[7,
p.431])
Then we have
because
a(X, J1 X)
=0,
VX eTp (M)
and therefore theidentity (ii)
holdsin this case.
Let us consider now the case in which M n is a real
flag
manifold(i.e.
an
R-space)
which is not acomplex flag manifold.
We have then the data associated to theR-space
M n and its canonicalimbedding namely
areal
semisimple
Liealgebra
g withoutcompact factors, f
a maximal com-pactly
imbeddedsubalgebra
of g and g = fQ3 p
the Cartandecomposition
of g relative to f. Let B denote the
Killing
form of g;then p
can be con-sidered a Euclidean space with the inner
product
definedby
the restric- tion of B to ~. Let G =Int(G)
be the group of innerautomorphisms
of g.We may
identify
g with the Liealgebra
of G. Let K be theanalytic
sub-group of G
corresponding
tof;
K iscompact
and acts on p as a group of isometries. TheR-space
M n is the orbit of a non zero vector i.e.M=Ad(K)E.
Let ac p be a maximal abelian
subspace;
we may assume E E a[5,
p.247].
Let us extend a, to a Cartansubalgebra b
=t Q3 a .
Considerthe
complexification
Qc of g and let cv be thecorresponding conjugation.
Since g = ~
Q9 p
is a Cartandecomposition
there exists acompact
realform gu of Qc such that
Let
Gc
be thecomplex, simply connected, semisimple
Lie group asso- ciated to thecomplex
Liealgebra
Qc and letG1
andGu
be theanalytic subgroups
ofGc corresponding
to thesubalgebras
g and gurespectively.
They
are closed inG, by [5,
p.152, 4, (ii)].
LetK1
be theanalytic
sub-group of
Gc corresponding to t , clearly K1
cG1
nGu
and Ad:G1
- G is ananalytic homomorphism
onto G such thatAdG, (K1 )
= K.The manifold c gu is
again
ourR-space
M because therepresentations of K1 in p and ip
areequivalent. By [5][p. 180], if X,
Y E e p thenand so, if we take on
ip
the Euclidean metric inducedby - Bu
then M nand are isometric.
Let
M,
= this isobviously
acomplex flag manifold,
contains M
and,
if we consider onM,
the Riemannian metric inducedby
the inner
product
on gu definedby - Bu,
it is clear that M n is isometri-cally
embedded inM~ .
Our construction of the immersion not
changed
ifwe use
io :
instead of g :M~2013>p.
Now we can associate to cp an immersioncp’:
defined as the bottom line in the nextdiagram
where both
spheres
have radius . In fact M = nMe and cp
=(cp elM).
Wekeep
the notation above(a
1 is the second fundamental form of Q 1, or a 2 that of ~0 2 ,etc.)
wejust
add a «c» toindicate the
corresponding
second fundamental forms forMp.
Let y de- note the second fundamental form of M inM~ .
Then,
forX ,
we haveand hence
Then,
for and thecomplex
structure J inCP(S( gu»,
wehave that
In view of the fact that
(ii)
holds forcomplex flag
manifolds we have~ a ~ (X , X), JX)
= 0 and sinceM,
istotally
real inCP(S(gu»,
it is clearthat
(y(X, X), JX)
= 0. Therefore we obtainand the
proof
of the theorem iscomplete.
3. - The characterization.
In this section we prove that the conditions of Theorem 1
essentially
determine the immersion cp. In fact we have
THEOREM 2. Let M n be a
simply connected, compact,
Riemannianmanifold
with a canonical connectionand f :
an isometricimmersion which
satisfies
thefollowing
conditions.i) f
istotally
real.ii)
The secondfundamental form
aof
the immersionf
is canoni-cally parallel (V’a
=0).
iii) If
J is thecomplex
structure on CP n + q thenThen Mn is a
real flag manifold (R-space)
and there is a vector E e p such that andf
isof
theform
PROOF. The
proof
will be divided into several lemmas.Let us take a
point
p E M ~ which we shallkeep fixed for
the restof
theproof.
We shall use thefollowing
notation.is the
first
normaL space at p andthe
first osculating
space at p. Also will denote theorthogonal complement
of in Since M istotally
real in cpn+q wehave
J(Tp(M»
c .As it is well
known,
the way to constructsymmetric subspaces (i.e.
totally geodesic submanifolds)
of asymmetric
space, such asis to consider a Lie
triple system
in thetangent
space at apoint
p E .
The
following
lemma is amodification,
to our conditions(Vc
a = 0 in- stead of Va =0),
of[6,
p.101, 13].
LEMMA 3. Under the conditions
of
Theorem2, ifR
denotes the cur-vature tensor in CP n +
q ( c )
thenPROOF. Since
f
istotally real, (A)
follows from[2,
p.260, 3.1].
To prove(B)
we takeX , H E Np (M) and; E Np (M) 1.. Clearly Ai
--- 0 onTp (M).
Let us prove that We may assume that H = for some
U,
V E thenbecause
V.k a
= 0 . This shows that(X , Y) H
may be written in terms of elementsbelonging
to and thenR 1 (X ,
Now
and therefore
R(X, Y)
H EOp (M).
Now for Z E we have
and
by part (A)
the first term is zero soR(X, Y) H E Np(M)
andpart (B)
isproved.
LEMMA 4. Under the conditions
of
Theorem2,
at the chosenpoint
p E Mn, Jp(Tp(M)) C Np(M)+.
For a
totally
realimmersion,
the second fundamental form satisfies thefollowing identity [7,
p.431].
For everyX, Y,
Z ETp(M),
Then the three-linear function
Y), JpZ)
issymmetric
onTp(M). By
condition(iii)
of Theorem 2 it vanishes on thediagonal
ofTp (M)
xTp (M)
xTp (M)
and hence it isidentically
zero. Thisclearly
means that
as was to be proven. -
LEMMA 5. Under the
hypothesis of
Theorem 2PROOF.
According
to Lemma 4 we haveThe
proof
of thepresent
lemma isformally anaLogous
to thatof [7,
p.433].
Theproof
is omitted.LEMMA 6. Under the
hypothesis of
Theorem 2PROOF. Let
X, Y,
Z be vector fieldstangent
toM~
and H a sectionof
N(M).
Since issymmetric
we haveH) Y =
0 .Then
We may assume, as
above,
that H =a( U, ~. Then,
since V~ a =0,
we haveand therefore
We may
proceed
then as we did with H and henceWe conclude that
which
yields (E).
Now we prove
(F). Let X,
Y be vector fields on M n andH, W,
sec- tions ofN(M).
We have(VyR)(X, H)
W = 0 and thenAs in the
proof
of(E)
we may show thatV~
H andV~
Wbelong
toN(M)
and thereforeWe may write
R(X, H)
W = U+ ~
with U ET(M)
and thereforewhich
clearly belongs
toOp (M).
It follows that
R (a(X , Y),
this proves(F)
and com-pletes
theproof
of the lemma.The contents of Lemmas
3,
5 and 6clearly yield
thefollowing
LEMMA 7. Under the
hypothesis of
Theorem2,
thefirst osculating
space
Op (M)
is a Lietriple system
inThen there exists a
totally geodesic
submanifold thatTp(L) = Op (M).
The submanifold L iscomplete.
Since c > 0 we have two
possibilities
for L: eitherBy
Lemma 4 the manifold L can not be oftype cpn+s(c).
We have to show now that
f (M)
c L and to that end we needTHEOREM 8. Let be an isometric immersion
of
acompact
connected Riemannian
manifold
M into a Riemannianmanifold M1
and assume that:
i)
M admits a canonical connection.ii)
The secondfundamental form of
the immersionf
is canonical-ly parallel
i. e. Vc a = 0 .Then
for
eachpoint
p E M and eachunitary
vector X ETp (M), if
y is theVC-geodesic
on M ndefined by X ,
thenc(t) = f ( y ( t ) )
isa W-curve in
Ml , [4,
p.57], of osculating
1 suchthat
for j
=1,
... , r thej-th
elementof
the Frenetframe
can beurritten as
V ( t )
= + wherePj
andQj
are thetangent
andnormal
components respectively
andsatisfy
REMARK. This is a
generalization
of apart
of Theorem 13 of[11]
which is proven in that article for The extension of theproof given there,
to thegeneral
case, isstraightforward.
Now the fact that VC a == 0
implies
thatQi(0), ... , Qr ( 0 )
ENp (M)
andtherefore
Vi(0),
... , ETp (L).
Letb(t)
be the Frenet curve on L ofosculating
rank r with initial conditionsc(0), Vi(0),...,
and theconstant curvatures ... ,
kr
of the W-curvec(t).
The curveb(t)
is a Frenet curve in the ambient space because L istotally geodesic
incpn+q(c). But,
sincec( t )
andb(t) satisfy
the sameequations
and have the same initial conditions in
cpn+q(c)
we havec( t )
=b(t),
Vt .Since in a manifold with a linear connection every
pair
ofpoints
can bejoined by
a brokengeodesic,
we haveproved
We observe that for each q E M n we have
Tq (L)
=Oq (M)
because due to Lemma 9 we have and since the dimension ofOq (M)
isconstant
along
M n(recall
that V’a =0)
and at the chosenwe have
Tp(L)
=Op (M)
we have thatthey
coincideeverywhere
in M n .We have
Since Mn is
simply
connected we have alifting
and since M n has
canonically parallel
second fundamental form incpn+q(c)
andRpn+s(c/4)
istotally geodesic
incpn+q(c)
then M n hascanonically parallel
second fundamental form inRpn+s(c/4)
and hencethe immersion Il has the same
property.
It is easy to show that in this situation theimmersion u
considered as an immersion inR n + s +
1 hascanonically parallel
second fundamental form. In view of[8,
p.195]
theproof
of Theorem 2 is nowcomplete.
REMARK. If the manifold Mn is not
simply
connected we need an extracondition to
get
the existence of thelifting.
This condition isobviously _ ~ 1 ~ in .7l1 (RP n + S).
To have this we mayask,
instead of thesimple
connectedness ofM’,
that the immersionf
be such that:For every
totally geodesic submanifold
N c cpn+q such that ccNccpn+q
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