Kawamoto, S-I.

Portal de Revistas Científicas Complutenses

REVISTA MATEMATICA de la
Universidad Complutense de M,d,id
Valunien 7, udinera 1: 1994
Codimension Reduction for Real Submanifolds
of a Complex l-lyperbolic Space
SEIN-ICE! KAWAMOTO
ABSTRACT. Wc study real submanifalds af a complex hyperbolic space
and prove a cbdimension reductian thearem.
O. INTRODUCTION.
Recently Oknmura ([3]) defined itolaniorpitic first normal space for
real submanifolds of a Kaeitler manifaid and praved a cadimettsiasi re-
ductiasi titeorem far real submasiifalds of a complex projective space.
Namely, ite sitawed following:
Theorem. La M be a conneeted u-dimensional real submanifold
of a real (u + p)-dirnensionul complez projective space ~p(n+v)/2 aud
Jet No(x) be the orthogonul complement of first normal space itt 2’4(M).
We prd Ho(x) = JNo(x)flNo(x) aud Jet H(x) be a J-invuriunt subspuce
of Ho(x) where J is complez structure of CP(n+v)/2. If ihe orthogonul
eornplement H
2(x) of H(z) itt 2’J(M) is invariant under purallel trata-
lution with respecí to ihe normal connection and {f ej is tite constunt
1991 Mathem.tics Subject Clasaification: 53H25, 53H30.
Editorial Complutense. Madrid, 1994.
http://dx.doi.org/10.5209/rev_REMA.1994.v7.n1.17784
120 Sbin-ichi K,w,mato
di,nestsion of H2(x), then tites-e exists u real (u + q)-dimensional tota¿ly
geodesie complez projective subspace c£(n+Q)/2 in CP(tt+P)/
2 such that
M c ~p(n+q)/2
Tite purpose of titis paper is to prove titat tite similar result to tite
aboye theorem Ls still itold in asubrnanifold of complex ityperbolic space.
Tite autiter wonid like to express bis titanlcs to Prafessors M. Oku-
niura and M. Kimura for titeir valuable suggestiosis.
1. CODIMENSION REDUCTION POR SUBMANIFOLDS
OF ANTI-DE SITTER SPACE.
Let R~+l be a real vector space of (n+ 1) dlmension with a psendo-
Riemannian metric 4 of signature (u — 1, 2) given by
n
4(x,y) = —xoyo—xíyí+>jx~y
1 (1.1)
where x ~ (x ,xí,...,x,3, y ~ (Yo,Y1,...,Yn) E Rn+í. Let H’ =
{x E R¿+l 1 g(x,x) = —1}. Titen tite itypersurface Hf is a Lorentzian
manifold witit tite isiduced Loresitzian metric 4 of constasit sectional
curvature —1. We cail it n.dirnensional anti-De Sitter space.
Let ffr+P be asi (u + p)-dimettsiosial anti-De Sitter space and let
i: M —~ Hr+P be asi isametric imifiersion of a connected ii-dimensional
Loretttzian manifaid witit tite Larentzian rnetric g into Hr+P. Titesi tite
tangent bundle T(M) is identified witb a subitundie af T(Jfl+P) and
tite normal busidle 2’
1(M) is a subbusidle of 2’(JIr~~) cosisisting of alí
elernent itt T(Rt+~) whicb are ortitagona] to 2’(M) witb respect to 4.
We denote by ‘~ and sy tite Levi-Civita cannection of M and R?+P
respectLvely and D tite induced normal cosinection fram y to 2”(M).
Titen titey are related by tite followisig Ganss and Weingartesi formulae:
VíxiY = iSZxY+h(X,Y) (1.2)
= —iA¿X + Dx¿ (1.3)
Cadimension Reductian fo, Rnl Subm,nifald, of, Complez Hyperbalic Space 121
witere E E T’(M), h(X,Y) Ls tite secosid fundamesital forní asid A~ Ls
a symmetric linear transforníation of T(M) witicit Ls called tite sitape
operator witit respect to E• Titey satisfy
4(h(X,Y),E) = g(A~X,Y). (1.4)
Next let No(z) = {E E T2-(M) ¡ = 0}. Tite first siormal space
Ní(x) is defined to be tite oTtitogonal complernent of No(x) in 2’4}M).
Theorem 1.1. Let i : M —. H?+P be us aboye. Leí N2(z) be
o subspace of TJ,-(M) such thai Ní(x) c N2(x). If N2(x) is invarianí
under parallel tranelution with respect to the normal connection and if ej
is Me constaní dimension of N2(x), then ihere exisis a totufly geodesic
anti-De Sities- subspuce ~ of 11fl+p snch tital i(M) c ~n+q
Praof. We consider H~+P as a itypersurface of ft~+P+í• Let x E M
asid let ¿ = 1(x) be tite positiott vector. Titen «x) is normal to H2+P
and 4(E(x),E(x)) = —1 witere 4 Is tite metric of ~ Let 7 be
tite Levi-Civita cosisiectiosi on R2+P+í witit respect to ~ asid <P be asi
irnníersiosi froní fffl+P to R+P+í. Titen
= soX (1.5)
= <PVXYY(X,Y)¿ (1.6)
witere X,Y E 2’dHr~~). For x e M let P(x) = 2’~(M) + N2(x).
For any z E M titere exist ortitosiormal norma] vector fields 6, . . . ,
defined in a sieigitboritood U of x sucit titat:
(a) For any yE U, &(v)~... ,~(y) spasi N2(y), and
,$,(y) span N(y) witere N(x) is tite ortitogonal complemnent
of N2(x) in 2”(M).
(b)~’íx¿a = OisiUfora=q+landXtangenttoM.
(e) {F(y) 1 y e U} ½invariant under parailel transíation with
respect to tite consiection si along any csirve in U (see [1]). Titesi
= S7¿XEO for X tangent to M. Let D’ be tite nor-
mal cosinection itt tite siormal bundie T’(M) of M isi R~+~~+í. Titen
122 Shín.ichi K.wamata
N2(x) + span{~(x)} is invariant under parallel transíation witit respect
to D’. Furtiter,
W(x) 71(M) + N2(x) + span{¿(x)} (1.7)
is invariant under parallel trasislation witit respect to ‘7. Next we sitan
show titat titere exists a totally geodesLc subníanifold ~1fl+q ofHr+P such
that i(M) C ~ Define functions fa on U by f~, = 4(i(x),wEa) for
a =q + 1.
9,(iX) fa 4(V~~(Ix) i(X),<P&,) + 4(i(z),<P~a),7~,(IX>SO~a) = O
Thus fq+i,. .. ,f~, are constant. Put
fa = C0( constasit) (a =q + 1). (1.8)
And pnt i(z) = (za,. ..,z,.+,) asid ~ = (a,,... ,e;~o. Titen (1.6)
can be wrLtten
( n+ p
—E~+1xo — E~+ix + E ~+ix1 =
n+p (1.9)
— E~xs + E ¿,x~ = Ci,.i2
Since ~ . . ,.% are linearly isidependent, U lies ir< tite intersectiosi of
p— q ityperplanes asid tite dimesision of tite ityperplane Ls n+ ej + 1. As tite
normal vectors of tite intersectiosi W’ are ‘fq+i,.. . ,4,, titey spasi N(x).
Since W’ is afflne space, W’ is tite ortitogosial compleníent of N(x) in
2’r(R~+~~+í). On tite otiter itand, tite ortitogona] coníplement of N(x) itt
T~(R~+P+í) is 2’~(M)+N2(x)+spasi{«x)} (= W(x)). Titerefore W’ =
W. Wemay assnmethat titepoint (1,0,...,O)isin U. W(x) cotttains¿,
and if~ = (1,O,...,O), titen W(x)passestitrougittiteorigin ~
lEus W(x) = ~n+q+í• Moreover since M is Lorentzian snbmanifold
and ~ is tite position vector, tite signature of tite induced metric of
Codimension Reductian fon Real Subm,nifalds of, Complez Hyperbalic Sp,ce 123
ftfl+q+i Ls (u + ej — 1,2). Titen W’ = ~‘+~+‘• Titus fffl+P <-.~
2Ls totally geodesic H~+P, titat LS,
i(U) a — o R~~<’ (1.10)
Hence Titeorem 1.1. is true locally. itt entirely tite same way as in [1],
we can get tite global result. Titis completes tite proof.
2. REAL SUBMANIFOLDS OF A KAEHLER MANIFOLD
AND HOLOMORPHIC FIRST NORMAL SPACE.
Let M be a real (n+p)-dirnesisional ¡Caehíer manifoid witit Kaeitler
structure (J, <,>), titat Ls, J Ls tite ettdomorphism of tite tangesit bundle
2’(Ñ) satisfying fi — idesitity asid <,> tite RLemanniasi metric of M
satisfyisig tite Hermitian conditiosi < jí,JV > = < Ñ,Y > for asiy
X,V E T(At.
Let M be a cosinected n-dimesisional snbmanifold and let i be tite
isoníetric imrnersiosi. For any X E 2’(M) tite transform JiX is writtesi
as a sum of its tangential parts iFX and tite norma] parts u(X) in tite
followisig way:
JiX = iEX + u(X) (2.1)
Titesi E Ls an endmorphism on tite tangent bundie T(M) asid u is a
normal valued 1-form on tite tangesit busidie. Itt tite same way, for any
¿ E T’(M), tite transform J¿ Ls written
= —iU¿ + P¿, (2.2)
witere P defisies asi endomorpitisní osi tite normal bundie 2”(M). It Ls
easily verified titat
g(X, 1J~) = < u(X),¿ >, (2.3)
witere y Ls tite Riemansilan metric whicit Ls induced froní tite Rieníannian
metric <,>.
We define tite itolomorphic first norma] space. We pnt IJo(x) —
JNo(x) O N0(x). Titen Ho(x) Ls tite maximal J-invariasit subspace of
124 Shin-ichi K,wamata
No(x). Since J is isomorpitism, we see titat JHo(x) = Ho(x). Making
use of (2.2), we can easily prove tite following
Proposition 2.1. ([3]) For uny E E fIo(x), me huye AE = O and
lI~ = O.
Definition ([3]) The holomorpitie fis-st normal space Hi(x) is tite
ortitogonal complement of flo(x) itt Tr’(M).
Propositian 2.2. ([3]) IfM isa complex submanifold of u Haehíer
manzfo¿d, titen Hs(x) = Ni(x).
Proposition 2.3. ([3]) Let 11(x) be u J-invariant subspace of
Ho(x) and leí H2(x) be tite orthogona¿ complement of H(x) itt 2’t(M).
Titen 2’r(M) + H2(x) is u J-inyariant subspace of 2’~(KI).
3. CODIMENSION REDUCTION FOIl. SUBMANIFOLDS
OF COMPLEX HYPERBOLIC SPACE.
In titis section, we consider tite case titat tite ambient manifoid
M is a coníplex ityperbolic space C11(n+P)/
2 witit tite Bergmasin met-
nc of cosistant itolomorphic sectiosial curvature —4. Given a real si-
diiuensional submanifold M of CH(n+P)/2, osie casi construct a Lorent-
zian submanifold M’ witit tiníe like totally geodesic fibres and projection
ir’ : M’ — M sucit that tite diagram ([2])
5
M —~ CH(n+P)/2
5
is cominutative (i being tite isometric immersiosi). Let V’ be tite unit
vector fleid tangent to tite libre of M’. Titen iV’ is tite unit vector
fleld tangesit to tite libre of H
1fl+P+í. We desiote by g’ and 7’ tite
Lorentzian níetric and tite Levi.Civita connection of M’ respectively.
Also we desiote by E asid X tite fusidamental tensor of tite subrnersion
Codímension Reduction far Real Subrn,nifolds of, Complez Hyperbolic Space 125
ir’ asid tite horizosital Iift for X E 1(M) respectively. Itt tite sanie way,
~ is tite itorizosital hift of tite siormal fleid ~ E 2”(M). Tite fusidamesital
equatiosis for tite subrnersion ir’ are given as following ([4]):
‘7’x * y5 = (VxYY + g’((FX~,Y~)V’, (3.1)
* V’ = = (EX), (3.2)
witere y LS tite Levi-Civita connection of M. Tite similar equations are
valid for the submersion ir ~ CH<’4”)/~ witen we replace E
and y’ witit J asid iV’ respectively. Let 9, ~, A’ and D’ be respectively
tite Lorentzian metric of
111n+p+í, tite Levi-Civita connectiosi for 4, tite
sitape operator and tite normal connectiosi of M’, and let A and D be
tite sitape operator asid tite normal consiection of M respectively. Titesi
([3]) we itave
* = (A~ * X)’ — g(U¿,X)V’, (3.3)
* = (DrEY, (3.4)
= U¿, (3.5)
= (~¿)* (3.6)
Lxi fact, froní tite commutativity of tite diagrain, (2.3) and (3.1) imply
S7<íx> * = (~íXEY + g((JiX)tC)~V’ (3.7)
= —(iA¿X)
5 + (Dxfl5+ < JiX,¿ 9 iV’
= —~(A¿X)+ <s4X),¿ > iV’ + (DXU5
= ~~4{(A~X)*— g((4, X)V’} + (DXfl*
Qn tite otiter itasid, by tite Weisigartesi formula, we get
* = —iA~ * X + D~ * (3.8)
Comparing (3.7) asid (3.8), we itave (3.3) axid (3.4).
126 Sbin-áchi K,w,moto
Lemma 3.1. ([3]) Fos- a poirtt x’ such thai ir(x’) = x, me itave
N¿(x’) = {Q~ A~ = 0, U¿ = 0}.
Theorem 3.2. Let i M .-. CH(tL+P)/2 be att isometrie immer-
szoit of a connected n-dirnenszonal real submanifold into u real (n + p)-
dimensionaC complex hyperbolic space CH~~~~>/2 asid let H(x) be aznvariant subspuce of Ho(x). 4 ihe ortitogonul complemer¿t H
2(x) of
H(x) itt 2’4-(M) i.s invarianí under parallel trattslation mith respecí to
tite nos-muí connection asid if the dimension ej of H2(x) is corwtant, ihen
tites-e exisis a real (n+q)-dimensional totully geodesic complez hyperbolic
subspuce Cff(»+~>/
2 itt CH<’~~>/2 such that i(M) c ~ff(fl+q)/2
Proof. We construct tite prisicipal circie hundie M’ over M witit
time like totally geodesic libre ~1• We sitail sitow titat H
2(x) is invari-
ant usider parallel transíation witit respect to tite normal consiection.
Assume ¿ E H(x). Titen ¿ E Ho(x) and by Propositiosi 2.1., we itave
= O asid 14 = 0. (3.9)
From Lemma 3.1., titis yields
= 0. (3.10)
Titis sitows titat, for a point x’ ssicit that ir(x’) = x, H(x) =
ff1 ¿ E H(x)} is a subspace of N¿(x’). Hence, tite ortitogonal
pleznent H2(x) = ¿ c ¡12(x)} of H(x) isi 2’~4(M’) is a subspace
of T4(M’) such that N«x’) c ¡12(x)t Since H2(x) is invariant uxider
parailel traxislation witit repect to tbe normal connection, so is 11(x),
that is, for ~ E 11(x), Dx¿ E 11(x), hence, from (3.4) asid (3.5), we
itave D~ * = (Dx4j E ff(x)* and Dj,I¿ = (P¿)~ E H(x)s.
Since H(x)* ½isivariant under transíation witit respect to tite normal
connection of M’, so is ff2(x)~. llere from Titeorem 1.1., titere exists a
totally geodesic submanifold 11n+q+l sucit titat i(M’) cE Hn+q+l Let
U(x’) be a sieigitboritood of z’ which satisfles ir(x’) = x. For y’ E U(x’)
with y = ir’(y’), we get
= T~.(M’) + H2(y)*
= (2’~(M) +
112(y)) + span{V’} (3.11)
Codimension Reduction lar Real Subm,nifalds of, Camplez Hyperbolic 5p.ce 127
Tite integral curveS’ of iv is time llke totally geodesie fibre isi »~+~+‘.
Since H~+~+í Ls totally geodesic , tite isitegral curve 51 j~
11,,+q+í¡8~• Titesi tite Hopf fffl+P+la geodesie of H~+~+l. We denote by ~ff(n+q)/
2 tite quotient space1 fibration gn+q+1 ~ff(fl+q>/2 by tite
geodesic 5’ Ls conípatible witb tite Hopf fibration ir :
CH(~+P)/2 axid sittce H~+~+í LS totaily geodesic in
111n+p+l, ~~(»±q)/
2
is totally geodesic itt C11~~~~>’2. Hence tite diagrarn
1 -.-.-.~ Hr+P+í
.1’ .1,
is commutative. Sisice tite tasigesit space of tite
011(n+q)/2 at a, Ls
71(M) + 112(x), by Proposition 2.3., ~J~(n+q)/2 is J-isivariant subspace
of ~ Titis completes tite proof.
For a complex submanifold M, from Proposition 2.2. asid Titeorern
3.2., we have
Corollary. Leí M be un nj2-dimettsional complez submanifold of
CH(n+P)/a. Suppose a J-invariant subspace of tite fis-st normal space
N,(x) itas constaní dimen.sion ej and N,(x) is parallel with respecí lo
tite normul connection. Titen tites-e exists a totully geodesic (si + q)-
dimensional complez ityperbolic snbspuce ~ff(”+q)/
2 such thut M c
~jj(n+q)/2
References.
[1] Erbaciter, J., Reduction of tite codimertajon of an isometrie immer-
sion, J. Differential Geornetry. 5 (1971), 333-MO.
[2] Moxitiel, 5., Real hypersurfaces of a complex ityperboric space, J.
Matit. Soc. Japan. 37 (1985), 515-535.
[3]Okumura, M., Codimensior¿ reduction problemfor real submanifold of
complez projective space, to appear itt Colloqula Matitematica Societatis
Janos Bolyai. 56 (1991).
128 Shin-icbi K,nmato
[4] O’Neil, E. Tite fundamental equutions of u submes-sion, Micitigan
Matit. J. 13 (1966), 459-469.
[>epartmiiental Matheni,tics,
FacuIty al Science,
S,it,ni, Un¡versity, Urawa,
SaIL,ína 338, J,p,n.
Recibido: 15 de octubre de 1992


Codimension Reduction for Real Submanifolds of a Complex Hyperbolic Space

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Codimension Reduction for Real Submanifolds of a Complex Hyperbolic Space

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