Codimension reduction for real submanifolds of quaternionic projective space

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CODIMENSION REDUCTION FOR REAL SUBMANIFOLDS OF QUATERNIONIC PROJECTIVE SPACE

JUNG-HWAN KWON AND JIN SUK PAK

ABSTRACT. In this paper we prove a reduction theorem of the codimension for real submanifold of quaternionic projective space as a quaternionic analogue corresponding to those in Cecil [4], Erbacher [5] and Okumura [9], and apply the theorem to quaternionic CR- submanifold of quaternionic projective space.

1. Introduction

In general, it is very hard to classify submanifolds immersed in a Riemannian manifold even though the ambient manifold is specified, and so the so-called codimension reduction problem is sometimes very important role in the theory of submanifolds.

The codimension reduction problem was investigated by Allendoer- fer [1] in the case that the ambient manifold is a Euclidean space and by Erbacher [5] in the case that the ambient manifold is a real space form.

On the other hand, as a complex analogue for submanifold of complex projective space, Cecil [4] proved a codimension reduction theorem for complex submanifold and Okumura [9] a theorem corresponding to those in [4] and [5] for real submanifold.

In this paper we prove a quaternionic analogue for real submanifold of quaternionic projective space which may correspond to those in [4], [5] and [9]. We mainly follow Okumura's method in his paper [9].

Received May 9, 1998.

1991 Mathematics Subject Classification: 53C40, 53C15.

Key words and phrases: quaternionic projective space, reduction theorem, quaternionically invariant subspace, quaternionic CR-submanifold, lift flat.

This research was supported by KOSEF, Project No. 981-0104-017-2 and (in part) by BSRI, 1998-015-D00030 and TGRC-KOSEF.

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2. Preliminaries

Let M be a real (n +

p

)-dimensional quaternionic Kahler manifold.

Then, by definition, there is a 3-dimensional vector bundle V con- sisting with tensor fields of type (1,1) over M satisfying the following conditions (a), (b) and (c):

(a) In any coordinate neighborhood U, there is a local basis {F, G, H} of V such that

(2.1) {F

²

= -I, G

²

= -I, H

²

= -I,

FG= -GF=H, GH= -HG=F, HF= -FH=G.

(b) There is a Riemannian metric 9 which satisfies the Hermitian property with respect to all of F, G and H.

(c) For the Levi-Civita connection V with respect to

9

(2.2) ( _VH ^~~) ⁼ (~r ~ ~q) (~)

_q _-p

₀ _H

where p, q and r are local I-forms defined in U. Such a local basis

{ F, G, H}

is called a

canonical local basis

of the bundle

V

in

U

(cf.

[6,7]).

For canonical local bases {F,G,H} and {'F,'G,'H} of V in coordi- nate neighborhoods U and 'u respectively, it follows from (2.1) that in

'unu

(x,

y

= 1, 2, 3)

with differentiable functions Sxy, where the matrix 8 = (sxy) is con- tained in 80(3). As is well known, every quaternionic Kahler manifold is orientable(ef. [6,7]).

Let

M

be an n-dimensional submanifold isometrically immersed in

M

and let

i

the isometric immersion. Then, for any tangent vector field X and normal vector field

~

to M, we have the following decompositions in tangential and normal components (In what follows we will delete i and its differential i* in our notation):

(2.4) FX = ^cjJX + u(X), GX = ^1jJX + veX), HX = ^OX + w(X),

(2.5)

F~

= -u{ +

PF~, G~

= -V{ +

PG~, H~

= -w{ +

PH~.

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Then </>, 'If; and 8 are skew-symmetric endomorphisms acting on the tangent bundle TM, and PF, Pc and PH define those on the normal bundle TM1... Also

U,

v and w are normal bundle valued I-forms on T M. It is easily verified that

(2.6) g(X,

U~J

= g(u(X),

~), g(X,~)

=

g(v(X),~),

g(X, Wd =

g(w(X),~)

for any X

E

T M,

_{~ E}

T M

1..,

where and in the sequel we denote the induced metric form that of M by the same letter

g.

Applying F to the first equation of (2.4) and using (2.1), (2.4) and (2.5), we have

</>2X = ^-X +

^Uu(X),

u(</>X) = ^-PFu(X).

Similarly we have (2.7)

</>2X

=

-X +

^Uu(X),

^'If;2X

⁼

-X +

^{Vv(X) ,}

⁸ ² ^X

⁼

-X +

^Ww(X),

(2.8)

u(</>X)

=

-PFu(X), v('If;X)

=

-Pcv(X), w(8X)

=

-PHW(X).

Next, applying G and H to the first

~quation

of (2.4), respectively and using (2.1), (2.4) and (2.5), we have

8X + w(X) = -'lj;(</>X) - v(</>X) +

^{Vu(X) -}

^Pcu(X),

1j;X + veX) = 8(</>X) + w(</>X) -

Wu(x)

+ PHU(X), and consequently

(2.9) 'lj;</>X

=

-8X +

^Vu(x),

v(</>X) + ^Pcu(X)

⁼

^-w(X),

(2.10) 8</>X =

^'lj;X

+

^Wu(x),

w(</>X) + ^PHu(X) = ^veX).

Similarly we have from the other equations of (2.4) (2.11)

(2.12) (2.13) (2.14)

</>'lj;X = 8X + Uv(X), 8'lj;X = -</>X +

^Wv(x),

</>8X = -'lj;X +

^Uw(x),

'lj;8X

=

</>X +

^Vw(x),

u('l/JX) + ^PFV(X)

=

w(X), w('l/JX) + PHV(X) = -u(X),

u(8X) + ^PFw(X)

⁼

^-veX),

v(8X) + Pew (X) = u(X).

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By the quite similar method as above, we have from (2.5) that (2.15) PF2~ = -~ + u(U~), PG2~ = -~ + v(~),

PH2~ = -~ + _w(W~),

(2.16)

cf>(U~) = -UPF~' 1/J(~) = -VPG~'

8(Wd =

-WPH~'

(2.17)

_W~

=

-VPF~ -1/JU~, PGPF~

=

-PH~

+

v(U~),

(2.18)

V~

=

WPF~

+

8U~, PHPF~

=

PG~

+

w(U~),

(2.19)

W~

=

UPG~

+

cj>~, PFPG~

=

PH~

+ u(ve), (2.20)

_U~ = -WPG~--O~, PHPG~= -PF~

+

w(~),

(2.21)

V~

=

-UPH~ - cj>W~, PFPH~

=

-PG~

+

u(W~),

(2.22)

U~ = _VPH~

+

1/JW~, PGPH~

=

PF~

+

v(W~).

We denote by V" the Levi-Civita connection of M and by V".L the normal connection induced from V" to T M.L. Then they are related by the Gauss and Weingarten equations (In what follows we will again delete i and its differential i* in our notation):

(2.23) (2.24)

V"XY = V"xY + heX, Y),

- .L

V"

x~= _-A~X

+ V"

x~,

where h is the second fundamental form and

_A~

the shape operator with respect to the normal vector field

~.

Differentiating the first equation of (2.4) covariantly and using (2.2), (2.4), (2.5), (2.23) and (2.24), we have

(V"ycj»X = r(Y)1/JX - q(Y)8X - Uh(y,X) + Au(x)Y, (2.25) *

(V"yu)X = r(Y)v(X) - q(Y)w(X) - hey, cj>X) + PFh(Y, X),

* *

where (V"yu)X is defined by (V"yu)X = V"¥u(X) - u(V"y X).

Similarly, from the other equations of (2.4), we have (V"y1/J)X = p(Y)OX - r(Y)</>X - Vh(y,X) + Av(x)Y, (2.26)

*

(V"yv)X = p(Y)w(X) - r(Y)u(X) - h(Y,1/JX) + PGh(Y,X),

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(2.28)

(2.30)

('VyO)X

=

q(Y)4>X - p(Y)'lj;X - Wh(y,X) + Aw(x)Y, (2.27) *

('Vyw)X

=

q(Y)u(X) - p(Y)v(X) - h(Y,OX) + PHh(Y,X),

* *

where ('Vyv)X = 'V¥v(X) - v('VyX) and ('Vyw)X = 'V¥w(X) - w('VyX).

Next, differentiating the first equation of (2.5) covariantlyand mak- ing use of (2.2), (2.4), (2.5), (2.23) and (2.24), we have

'VyU~

=

r(Y)V~

-

q(Y)W~

+

4>A~Y

-

ApF~Y

+

UIV~~'

('V¥PF)€

=

r(Y)PG€ - q(Y)PH€ - u(A~Y) + h(Y,U~), where ('V¥PF)€ is defined by ('V¥PF)€

=

'V¥(PF€) - PF('V¥€).

Similarly, from the other equations of (2.5), we have

'VyV~= _-r(Y)U~

+

p(Y)W~

+

'lj;A~Y

-

_ApG~Y

+

Vv~~,

(2.29)

('V¥PG)€

=

-r(Y)PF€ + p(Y)PH€ - v(A~Y) + hey, Vd,

'VyW~= _q(Y)U~

-

p(Y)V~

+

OA~Y

-

ApH~Y

+

Wv~~,

('V¥PH)€ = q(Y)PF€ - p(Y)PG€ - w(A~Y) + hey, W~), where ('V¥PG)€ = 'V¥(PG€) - PG ('Vj;€) and ('V¥PH)€ = 'V¥(PH€)- PH('V¥€).

3. Quaternionically holomorphic first normal space

Let No(x)

:=

{€

E

TxMJ... I

A~ =

O}. The first normal space N1(x) is defined to be the orthogonal complement to No(x) in TxMJ.... We put

Ho(x)

^:=

No(x) n FNo(x) n GNo(x) n HNo(x).

Then Ho(x) is the maximal quaternionically invariant(or briefly Q- invariant) subspace of No(x), that is,

FHo(x) c Ho(x), GHo(x) c Ho(x), HHo(x) c Ho(x).

Since F, G and H are isomorphisms, it is clear that

FHo(x)

=

Ho(x), GHo(x)

=

Ho(x), HHo(x)

=

Ho(x).

Taking account of (2.5), we can easily verify

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LEMMA

3.1. For any

_~E

Ho(x), we have

A~ =

0 and

U~ = ~ =

We

=

o.

DEFINITION.

The quaternionically holomorphic (or Q-holomorphic) first normal space H

1

(x) is the orthogonal complement of Ho(x) in TxM.l.

By definition, it is clear that Nl(X)

C

H

1

(x) in TxM.l. Moreover we have

LEMMA

3.2. If M is a Q-invariant submanifold of a quaternionic Kii.hler manifold, then H

1

(x) = N

₁

(x). .

Proof Since H

₁

(x) and N

₁

(x) are the orthogonal complements of Ho(x) and No(x), respectively, we have only to show that Ho(x) = No(x). Since TxM.l is Q-invariant, it follows from (2.2), (2.23) and (2.24) that

V x(F~) =r(X)G~ - _q(X)H~ + _F(Vi:~ - AeX)

= -ApeX + Vi:(F~)

and consequently

Ap~X

=

FA~X.

Similarly we have AGeX = ^GAeX

and AHeX = HAeX. Thus, if

_~ E

No(x), then

Ape

=

0 and

~ E

FNo(x), AGe

=

0 and

~ E

GNo(x), A He = ^{0 and}

~ E

HNo(x).

This shows that

~ E

No(x) implies

~ E

Ho(x), which completes the

prooL 0

LEMMA

3.3. Let H(x) be a Q-invariant subspace of Ho(x) and

H

₂

(x) its orthogonal complement in TxM.l. Then TxM

$

H

₂

(x)

is

a Q-invariant subspace ofTxM.

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Proof We first note that

TxM = TxM EI1 H

2

(x) EI1 H(x).

Since FH(x) = H(x), for any {

^E

H(x) there exists."

^E

H(x) such that P,., = {. Now let Z

E

TxM EI1 H2(X). Then for any €

E

H(x), (FZ,{)

=

(Z,.,,)

=

O. This means that FZ

^E

TxM EI1H

2

(x). By quite similar method, we can verify that TxM EI1 H

₂

(x) is Q-invariant. This

completes the proof. 0

Now we recall that an (n+p+3)-dimensional sphere

sn+p+3

of radius 1 in a Euclidean (n+p+4)-space is a principal S3-bundle over QP~.

Then the Hopf-fibration 7f :

sn+p+3 ^{- t}

QP~ defines a Riemannian submersion. We construct the S3-bundle over the submanifold M in such a way that the diagram

M

ⁱ QP~

is commutative (i, i being the isometric immersions). We denote by X* the horizontal lift of X

E

T M and by e that of the normal vector field {

E

TM.L. We put

N~(X')= {{' E

T

X^I7r-

1 _(M).L I A{l

=

O}, x'

^E ^7r-¹

(M),

where A{l denotes the shape operator with respect to the normal vector field e ^to

^7r-¹

(M). Then, as shown ill [8], we have

(3.1)

_N~(X')

= {C I Ae = 0, Uf. =

~

= Wf. = O}.

Applying the reduction theorem due to Erbacher [5], we prove

THEOREM

3.4. Let M be an n-dimensional real submanifold of a real (n + p)-dimensional quaternionic projective space QP~ ^and ^let H(x) a Q-invariant subspace of Ho(x). If the orthogonal complement H

2

(x) of H (x) in T

_x

M.L is invariant under parallel translation with respect to the normal connection and q is the constant dimension of H2, then there exists a real (n+q)-dimensional totally geodesic quaternionic

. . .!!tl .!!tl

prOjectIve subspace QP

⁴

such that M c QP

⁴ ^•

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Proof

Let €

^E

H(x). Then €

^E

Ho(x), which and Lemma 3.1 give A€

=

0 and U€

=

v€

=

W€

=

0 and consequently

A~. =

0 because of (3.1). This means that for a point x' with 7r(x')

=

x

H(x)*

=

{C I €

^E

H(x)} c NMx').

Hence the orthogonal complement H2(x)* ⁼ {€* I €

^E

H2(x)} of H(x) * in T

x

'(7r-

¹

(M).L is a subspace of T

x

'(7r-

¹

(M)).L such that Hfex') c

H

2

(x)

*.

Since H

2

(x) is invariant under parallel translation with respect to the normal connection, so does H (x). This shows that for &llY

€

^E

H(x),

\7~€^E

H(x), which and

\7'f.€*

⁼

(\7~€)*

^E

H(x),*

\7~C = -(G€)*

^E

H(x),*

\7~C = -(F€)*

^E

H(x),*

\7~C

⁼

-(H€)*

^E

H(x)*

imply that H (x)

* is

invariant under parallel translation with respect to the normal connection \7'1. of 7r-

¹

(M). From the reduction theorem ([5], p. 339), we know that there exists a totally geodesic submanifold

sn+q+3

such that 7r-

¹

(M)

C sn+q+3.

Let U(x') be a neighborhood of a point x' with 7r(x')

=

x. Then the tangent space T

y,(sn+q+3)

of the totally geodesic submanifold at

y' E

U(x')

is

where

y =

7r(Y'). Since

sn+q+3 is

totally geodesic in

sn+p+3,

the maximal integral submanifold

S3

of the distribution y'

^I--+-

T

y

'(7r-

¹

(y»

is a 3-dimensional great sphere of

sn+p+3.

Hence the Hopf-fibration

sn+q+3 ^---7

QP~ by

^S3

is compatible with the Hopf-fibration "if :

+ +3

~ !!.±!l •

sn p ^---7

QP

⁴

and the tangent space of QP

⁴

at x

¹⁸

TxM

Ef)

H

2

(x). Moreover, by Lemma 3.3, QP~

is

Q-invariant in QP~.

This completes the proof. 0

For a Q-invariant submanifold, by Lemma 3.2 we see that Ho(x) =

No(x) at any x in M. Thus we have

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COROLLARY

3.5. Let M be a real n-dimensional Q-invariant sub- manifold of QP~. Assume that a Q-invariant subspace of the first normal space N

₁

(x) has constant dimension q and is invariant under parallel translation with respect to the normal connection. Then there exists a totally geodesic real (n+q)-dimensional quaternionic projective

~ ~

space QP

⁴

such that M c QP

⁴ ^•

4. Quaternionic C R-submanifolds

In this section, let M be an n-dimensional real submanifold of a quaternionic Kahler manifold,

If

there is a Q-invariant distribution

1) :

x

~

1)x c TxM such that its complementary orthogonal distribution

1)J... :

x

~1);' in

T M is anti-quaternionic, that is,

then M

is

called a quaternionic CR-submanifold ([2,3]). In particular, if dim1)x

=

0 for any x in M, the quaternionic CR-submanifold

is

called an anti-quaternionic submanifold ([3,10]).

Let M be a quaternionic CR-submanifold of a quaternionic Kahler manifold M. Then, by definition, the tangent space

Tx~\i

at x

in

M

is

decomposed as

(4.1)

where NxM is the orthogonal complement of F1);' EB CV;' EEl H1);'

in

TxMl..

LEMMA

4.1. NxM is a Q-invariant, tha.t is,

Proof Let X

E

TxM EB F1);' EB C1);' EEl

H1)~

and

~ E

NxM. Since

X is decomposed as

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for some

Xl E

'Ox and X

_{2 ,}

YI, Y2, Y3

E

V;', it is clear that (X,F€)

=

-(FX,€)

=

-(FX

I

,€) - ( ^FX2,€) + ^(YI,€) + (Y2,e) + ^(Y

3

,€)

=0.

Similarly we have (X, G€)

=

(X, H€)

=

0, and consequently

This completes the proof. 0

LEMMA

4.2. Assume that NM is invariant under parallel transla- tion with respect to the normal connection. Then, for any €

E

N M and'fJETM.l..,

Proof. By means of Lemma 4.1 and our assumption, it follows that for any €

^E

^NM

F€ = Pp€, G€ = Pa €, H€ = PH€, Vi€,

FVi€ = ^ppVi€, GVi€ = ^PaVi€, ^HVi€ = ^PHVi€

are all contained in N M. Differentiating the first three equations of those covariantly, we have

(4.2)

Also we have (4.3)

- .1.

Vx (F€) = r(X)Pa€ - q(X)PH€ - 4>Af.X - u(Af.X) + ^PpVx€,

Vx(G€)

=

p(X)PH€ - r(X)Pp€ - 'l/;Af.X - v(Af.X) + PaVi€,

V x(H€)

=

q(X)Pp€ - p(X)Pa€ - OAf.X - w(Af.X) + PHvif

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We notice that U,

=

V,

=

W,

=

° ^{for any (}

^E

NM. Consequently (2.6) implies

u(X), v(X), w(X)

E

F'D.L

$

G'D.L

$

H'D.L

for any X

in

TM. Comparing the normal parts of (4.2) and (4.3), we have

u(A~X)=

0,

v(A~X)=

0,

w(A~X) =

o.

Thus for any

TJ E

T M.L

and consequently

A~UT/ = A~

VT/

= A~

WT/

=

o. This completes the

prooL 0

THEOREM

4.3. Let M be an n-dimensional anti-quaternionic sub- manifold of QP.!!{£. If N M

is

invariant under parallel translation with respect to the normal connection, then there exists a real 4n- dimensional totally geodesic quaternionic projective space Qpn of Q p~ such that M

^is

an anti-quaternionic submanifold of Q pn.

Proof. Since M is anti-quaternioriic, the tangential parts of (4.3) vanish identically. Comparing the tangential parts of (4.2) and (4.3), we have

ApF~ =

0,

ApG~ =

0,

ApH~ =

0 for all

~

in N M. But, in an anti-quaternionic submanifold, Pp, PG and PH are all isomorphisms on N M and consequently

A~=

° ^{for any}

~

in NM. Thus by means of Lemma 4.1 NM c HoM.

Conversely, let

~ E

Ho(x). Then for any X, YI, Y2, Ys

^E

TxM, we have

(~,X

+ FY

1

+ GY2 + HY

3 ) =

0 since Ho(x) is Q-invariant. Thus

_~

belongs to the orthogonal comple-

ment of TxM$ FTxMEB GTxMEBHTxM, that is,

~ E

NxM. Hence

NM

=

HoM and consequently FTxM

$

GTxM EB HTxM is the Q-

holomorphic first normal space. Applying Theorem 3.4, we conclude

that there is a real4n-dimansional totally geodesic quaternionic projec-

tive space Qpn of QPE:P such that M is anti-quaternionic in Qpn.o

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In [8J and [11J it was already proved that the normal connection of

7r-1

(M) in

sn+p+3

is fiat if and only if the following conditions are satisfied on 11,1:

(a) R.L(X, y)e

= -

2g(</>X, Y)PFe - 2g('lj;X, Y)PGe

(4.4) - 2g(OX, Y)PHe,

(b) The structure induced on the normal bundle

is

parallel (for the definition, see [11]).

In this sense the normal connection of M in QP ~

^is

^{said to be} ^lift fiat if the conditions (a) and (b) are valid.

LEMMA

4.4. Let M be a quaternionic CR-submanifold of QP~

with lift Bat normal connection. Then

A~

A1)

= _A1)A~

for e

^E

^Nx'M

and "I

E

TxM.L.

Proof Since the normal connection is lift flat, the equation of Ricci and (4.4) (a) implies

.0

=h(A~X,

Y) -

h(A~Y,X)

+ g(Y,

U~)u(X)

- g(X,

U~)u(Y)

+ g(Y,

V~)v(X)

- g(X,

V~)v(Y)

+ g(Y; Wdw(X) - g(X,

W~)w(Y).

In particular, for e

^E

^NxM,

^U~ ⁼ ^V~ ⁼ ^W~ ⁼

0, and consequently (4.5)

Hence, if e

^E

^{NxM and} ^"I

^E

TxM.L, we have

g«A1)A~

-

_A~A1))X,

Y)

=

_g(A1)A~X,

Y) -

_g(A~A1)X,

Y)

=

g(h(A~X,

Y), "I) -

g(h(A~Y,X),

"I)

=

0 because of (4.5). This completes the proof. o

THEOREM

4.5. Assume that the normal connection of a quater- nionic CR-submanifold M ofQP~ is lift Bat and that NM

is

invari- ant under parallel translation with respect to the normal connection.

Then there is a totally geodesic quaternionic projective space QP~

such that M

is

a quaternionic C R-submanifold of the quaternionic

projective space.

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Proof. By means of Theorem 4.3, it suffices to show that N M =

HoM. We choose orthonormal normal vector fields 6, ... , e

^p ⁱⁿ

^such

a way that

(q must be a multiple of three) and denote by A

_o

the shape operator for eo' Since N M is not only invariant under parallel translation with respect to the normal connection, but also Q-invariant, it follows that

p

(4.6) V~eo

⁼

L ^sOA(X)6, ^a

⁼ ^q

⁺ ^{1, ...}

^,p,

A=q+l

(4.7) { ^Feo

⁼

^PFeo

⁼ E~=q+l(PF)OA6,

Ceo. = ^PGeo =

E~=q+l(PG)o.A6,

a =

q

+ 1, ...

,p,

Heo. = PHeo = E~=q+l(PH)OA6,

from which we have

(4.8) {

FV~eo. = L:tJL=q+l So.A(X)(PFhJLeJL ,

Cv~eo. = L:tJL=q+l so>.(XHPG».JLeJL , a = q + 1, ... ,po HV-;eo. = L:~'JL=q+l so>.(X)(PH)>.JLeJL ,

On the other hand, comparing the tangential parts of (4.2) and (4.3), and using (4.7), we obtain

(4.9) { <jJA>.X

=

E~=q+l (PF)>'JLAJLX,

'l/;A>.X

=

E~=q+l(PG)>'JLAJLX, A _~ q+

l.

OA>.X

=

L:~=q+l(PH)>'/-LA/-LX,

Substituting A>.X for X

in

(4.9) and summing over A =

^q

+ ^{1, ...}

^,p,

we have

p p

<jJ L ^A>.

²

^X

⁼

L (PF)>'/-LA>.A/-LX

=

0 >.=q+l >',JL=q+l

(14)

because (PF)

_>.~ is

skew-symmetric with respect to

oX

and J.L, but

_A.xA~

=

A~A.x

by Lemma 4.4. Thus we have

p p P

</J2 L ^A.x2X = - L ^A.x2X ⁺ L

^Uu(A,,2X)

⁼ o.

.x=q+l .x=q+l

a=q+l

However, as already shown

in

the proof of Lemma 4.2, u(A

_a²

X) = 0 and consequently

_L:~=q+l

A.x

²

X = 0, that is, A>. = 0,

oX~ ^q

+ 1. Thus Nr;M

^is

a Q-invariant subspace of NoM. Since HoM

^is

maximal, it follows that NxM

C

HoM. Let e

^E

HoM and 11

E

F'Dl.. Ef)G'DJ.. Ef)H'Dl.. . Then there exist Y

1 ,

Y2, 1'3

^E

'D.l.

C

T M such that

11 = FYi + GY2 + HY3.

Therefore it

is

clear that

since HoM is Q-invariant. This means that e

E

N M and consequently

NM = HoM. This completes the proof. 0

REMARK. As

already shown

in

the proof of Lemma 4.4, it suffices to assume only the condition (4.4) (a) instead of the condition "lift flatness" in order to prove Theorem 4.5.

References

[1J

C. B. Allendoerfer, Rigidity for spaces of class greater than one, Amer. Jour.

Math. Soc. 61 (1939), 633-644.

[2J

M. Barros, B. Y. Chen and F. Urbano, Quaternion CR-submanifolds of a quaternion manifolds, Kodai Math.J. 4 (1981), 399-418.

[31 A. Bejancu, Geometry of CR-submanifolds, D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Toyko, 1886.

[4J T. E. Cecil, Geometric applications of critical point theory to submanifolds of complex projective space, Nagoya Math J. 55 (1974), 5-3l.

[5J

J. Erbacher, Reduction of the codimension of an isometric immersion,J. Diff.

Geom. 5 (1971), 333-340.

[6] S. Ishihara, Quaternion Kaehlerian manifolds, J. Diff. Geom. 9 (1974), 483- 500.

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[7] S. Ishihara and M. Konishi, Differential geometry of jibred spaces, vol. 8, Pub- lication of the study group of geometry, Toyko, 1973.

[8] J.-H. Kwon and J. S. Pak, QR-submanifolds of(p - 1) QR-dimension in a quaternionic projective space QP(n+^p)/4,preprint.

[9] M. Okumura, Codimension reduction problem for real submanifold of com- plex projective space, Collo. Math. Soc. Janos Bolyai 56, Diff. Geom., Eger (Hungary) (1989), 573-585.

[10] J. S. Pak, Anti-quaternionic submanifolds of a quaternionic projective space, Kyungpook Math. J. 18 (1981), 91-115.

[11] Y. Shibuya, Real submanifolds in a quaternionic projective space, Kodai Math.

Codimension reduction for real submanifolds of quaternionic projective space