R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI P ADOVA
M. J. F ERREIRA R. T RIBUZY
On the nullity index of isometric immersions of Kähler manifolds
Rendiconti del Seminario Matematico della Università di Padova, tome 105 (2001), p. 185-191
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of Kähler Manifolds.
M. J. FERREIRA - R.
TRIBUZY (**)
1. Introduction and statement of results.
Let M be a Kahler manifold with
complex
dimension m andcomplex
structure J. In the
present
work weanalyse
some obstructions to the existence of isometric immersions cp : M --~ N into certain Riemannian manifolds.Let
cp -1
TN be thepull-back
of thetangent
bundle of N. We will usethe
symbol V
torepresent
either the induced connection on orthe induced connection on
The covariant differential called the second fundamental
form,
may be understood as a smooth section where T 1 Nrepresents
the normal bundle.The
conjunction
of the second fundamental form with thecomplex
(*) Indirizzo dell’A.:
Faculty
of Sciences and CMAF of theUniversity
of Lisbon, Av. Prof. Gama Pinto 2, 1649-003 Lisboa,Portugal.
E-mail:
mjferr@lmc.fc.ul.pt
(**) Indirizzo dell’A.:
University
of Amazonas,Department
of Mathematics, ICE,Campus
Universitario, 69000 Manaus, AM, Brasil.E-mail:
tribuzy@buriti.com.br
Work
supported by
FCT, PRAXIS XXI,FEDER, project
PRA- XIS/2/2.1/MAT/125/94 and ICTP, Trieste.186
structure J
gives
rise to twooperators
where
X,
YEC(T(M)).
We can
get
a differentprospect
of theoperators working
in a com-plex framing.
Thecomplexification
of thetangent
bundle ofM,
denotedby decomposes
aswhere and are,
respectively,
theeigen-
bundles of the
complex
extension of J.The
decomposition
of a
according
totype
is such thatIt is a well-known fact that the
geometry
of the second fundamental form reflects thegeometry
of cp . From this viewpoint
one can ask up to which extent theoperators A
and C influence the behaviour of the map.The map cp is said to be
pluriminimal (or ( 1, 1 )-geodesic)
if = 0 .Equivalently,
rp ispluriminimal if,
for anyholomorphic
immersed curvec : c o cp : is
always
minimal.Trivial
examples
ofpluriminimal
immersions are the minimal immer- sions of a Riemannian surface. When N isflat, Dajczer
andRodriguez [D-R]
showed that thepluriminimal
immersions areexactly
the minimal ones,i.e.,
thevanishing
ofa ~ 1 ~ 1 ~
is theequivalent
of thevanishing
of themean curvature H = trace of a.
We define the index of relative
nullity
of a(resp.
of atIn the case
n>-2, Dajczer
andRodriguez
classifiedthe minimal immersions with v ~ 2 n - 4
everywhere.
In this
article,
when N is aconformally
flat manifold we obtain THEOREM 1. Let N be aconformally flat
Riemannianmanifold
with non-zero sectional curvatures.
If v(xo)
> 0 at somepoint
xoE M,
Mis a Riemann
surface.
Concerning
the relativenullity
index of a(1, 1),
we assume that N is aconformally
flat Riemannian manifold with dimension n, whose scalar curvature~(N)
never vanishes.Considering
the real numbers r ==
v)
=1,
X Es = inf and S = we can state:
XEM xeM
THEOREM 2. Let N be a
conformally flat
Riemannianmanifold
with
positive
scalarcurvature,
such thatat some
point
xo, M is a Riemannsurface.
THEOREM 3. Let N be a
conformaLly flat
Riemannianmanifold
with
negative
scalar curvature, such that at somepoint
xo, M is a Riemannsurface.
COROLLARY 1. Let N be a
conformally flat
Riemannianmanifold
whose Ricci curvature tensor
satisfies
oneof
thefollowing inequalities:
for
somepositive
real number d. Thenif,
at somepoint
xo,v(xo)
>0,
m=1.THEOREM 4. Let N be a Riemannian
manifold
whose sectionalcurvatures
satisfy
oneof
thefollowing conditions:
I for
some : 1188
2. Preliminaries.
A Riemannian manifold
(N, h)
is saidconformally
flat if there existsa smooth
function f :
N ~ R such that(N,
is flat. Riemannian ma-nifolds with constant sectional curvature are
particular examples
of con-formally
flat Riemannian manifolds. 2For each
y E N
we denote 2by Cy (N)
thesubspace
ofTy* N) (the 2-symmetric
forms onA T/ A0 consisting
of «curvature liketensors»;
that means, curvature tensors
satisfying
the first Bianchiidentity.
Theaction of the
orthogonal
group0(n) (n
= dimM)
onCy (N) gives
rise tothe
following decomposition
into irreduciblesubspaces:
where
= RId A Ty Nand ERy (N)
is formedby
the «traceless Ricci»tensors,
that is to say, those tensors 0 whose Ricci contractionc(O) ( c( 8 )( a , b ) = trace 8( a , . , b , . ) )
vanishes. Theorthogonal complement Wy (N)
of Q9 inCy (N)
is called the space ofWeyl
tensors.The
Weyl
tensor of a Riemannian manifold is theWeyl part
of its curva-ture tensor.
The
Weyl
curvature tensor W is the main invariant under conformalchanges
of the metric. Thevanishing
of W characterisescompletely
theconformally
flat Riemannian manifolds.It is then
easily
seen that the Riemannian curvature tensorR N
of aconformally
flat Riemannian manifold(N, h)
with Ricci curvatureRicN
and normalised scalar curvature
s(N)
isgiven by
where @ represents
the Kulkarni-Nomizuproduct
of thesymmetric
2-tensors,
defined in thefollowing
way:if z, k E
(D2 Ty*
N.Let d be a
positive
real number. The Riemannian manifold(N, h)
issaid to be
positively (resp. negatively) d-pinched
at apoint YEN
if thereexists a
positive
real number r such thatfor any 2-dimensional
subspace
a ofTy N.
N is said to bepositively (resp.
negatively) d-pinched
if it ispositively (resp. negatively) d-pinched
at ea-ch
point ?/eA7B
LEMMA 1.
[B]
Let N be a Rieman,nianmanifold
whose sectional curvaturessatisfy
oneof
thefollowing inequalities:
Then
if X, Y, Z,
W is a local orthonormalframe field,
thefollowing inequality
holds:Let M be a Kahler manifold with
complex
structure J.Denoting
re-spectively by
a’and yr" theprojections
of thecomplexified tangent
bun-dle
TG M
into itsholomorphic
andanti-holomorphic parts,
andTO, 1 M,
we use thefollowing
notation:3. Proof of the statements.
PROOF OF THEOREM. 1. Choose a
point
xo E M such andconsider
X , Y E C( TM)
such thatX(xo ) eL1xo’ (X, Y) = (X, JY)
= 0 and|X| = |Y| = 0.
It is clear from Gauss
equation
that for all W EC(TM)
From
(1)
we obtain190
(2) (RN (X, Y) JX, JF)
=0,
so that(3) ~R N (X , Y~ = o ,
which cannothappen. Thus,
dim M =1.PROOF OF THEOREMS
2,
3 AND 4 As above we consider xo E M with andX , Y E C( TM)
such that .Using
thecomplex
multilinear extension of Gaussequation,
we can writeIn the case of theorems 2 and 3 we now follow section 3 of
[F-R-T]
anduse
equation (1)
toget
Under the
assumptions
of theorem 2 we havea contradiction.
The conditions of theorem 3
imply
thatwhich cannot
happen,
hence dim M = 1.In the case of theorem
4, following
section 2 of[F-R-T],
according
to Lemma 1.REFERENCES
[B] M. BERGER, Les Variétés
Riemanniennes 1/4
-pincées, Ann. Scuola Nor-mal
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KählerSubmanifolds,
Du-ke Math. J., 53 (1986), pp. 211-220.
[E-L] J. EELLS - L. LEMAIRE, Another Report on Harmonic
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Bull. Lon-don Math. Soc. No. 86, 20 (1988), pp. 324-385.
[F-T] M. FERREIRA - R. TRIBUZY, On the
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Kähler
Manifolds,
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Stable HarmonicMaps,
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Complex Analicity of
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Lecture Notes in Mathematics, 1468 (1991), pp. 371-407.Manoscritto pervenuto in redazione il 24 marzo 2000.