On curvature pinching for totally real submanifolds of $HP^n (c)$

J. Korean Math. Soc. 34 (1997), No. 2, pp. 321-336

ON CURVATURE PINCHING FOR TOTALLY REAL SUBMANIFOLDS OF Hpn(c)

YOSHIO MATSUYAMA

ABSTRACT. LetSbe the Ricci curvature of an n-dimensional com- pact minimal totally real submanifoldM of a quaternion projective spaceHpn(c) of quaternion sectional curvature c. We proved that ifS

:5

3{~;2)c, then either S

== n4"l

^{c (i.e.} M istotally geodesic orS

==

3{~;2)c. All compact minimal totally real submanifolds of Hpn(c) satisfyin S

==

3(~;2)c are determined.

1.

Introduction

Let Hpn(c) be an n-dimensional quaternion projective space with constant quaternion sectional curvature c (> 0) and let M be an m-di- mensional totally real submanifold isometrically immersed in H pn (c).

Let h be the second fundamental form of M in H pn (c).

In [4] Funabashi showed: Let M be an n-dimensional totally real minimal submanifold isometrically immersed in Hpn(c). If

2

n+l

Ihl < 4(6n _ 2) c

if and only if M

is

totally geodesic and of constant curvature

~.

Recall the totally real imbeddings [4] and [11]:

Received August 30, 1995.

1991 Mathematics Subject Classification: 53C40, 53C42.

Key words and phrases: quaternion projective space, totally real, minimal, Ricci curvature.

322 Yoshio Matsuyama

and the first standard imbeddings of projective spaces:

'l/Jl : RP2( 1~)

^---*

^RP4( ~)

'l/J2 : CP2(~)

^---*

RP7(~)

'l/J3 " _. HP2(!) ₃

---*

RP13(!) ₄

'l/J4 : Cayp2(~)

^---*

RP25(~)

Moreover, Houh [6] proved: (A) Let M bean n-dimensional compact totally real minimal submanifold isometrically immersed in H pn(c).

If the sectional curvature

'Y

of M satisfies 'Y~. 4(2n _ n-2 1) c,

then either M is totally geodesic in Hpn(c) or n

=

2 and M is flat.

(B) Let M be an m-dimensional compact totally real minimal sub- manifold isometrically immersed in H pn (c)" If the sectional curvature

'Y

of M satisfies

m-1

'Y

~ ^{4(2m _ 1) c}

then either M is totally geodesic in Hpn(c) or m = ^{2, n}

~

4 and M is the Veronese surface in Hpn(c) with positive constant curvature t2"

Using the method of Chen and Ogiue [1], we can prove that: (A1) Let M be an n-dimensional compact totally real minimal submanifold isometrically immersed in H pn (c). If

I hl

²

< n(n + 1) - 4(2n-1)c,

then either M is totally geodesic in Hpn(2) or n

=

2 and M is flat.

(B1) Let M be an m-dimensional compact totally real minimal sub-

manifold isometrically immersed in Hpn(c). If

On curvature pinching for totally real submanifolds ofHP"'(c) 323

then either M is totally geodesic in H pn (c) or m

=

2, n 2:: 4 and M is the Veronese surface in H pn(c) with positive constant curvatrue

1~'

Recently, Coulton and Gauchman [3] proved the following: Let M be an m-dimensional compact totally real minimal submanifold iso- metrically immersed in H pn (c). Then

Ih(v,v)1

² :::;

12 1 c

for all unit tangent vector v

E

TxM if and only if one of the following conditions is satisfied: a) Ih(v,v)1

²

= ^{0 and M} is totally geodesic, b) Max {Ih(v, v)1

^2} = _112c

and M

is

either congruent to one of the imbeddings 'l/Ji = v

⁰

'l/Ji or to the immersion 'l/Js = 'l/Jl

^{0 1T',}

where

^{1T' :}

8 2U2 c) - Rp2 ( l2 c) is the covering map.

Moreover, using the methods of Gauchman [5] and Xia [12], we can prove that: (A2) Let M be an n-dimensional compact totally real minimal submanifold isometrically immersed in H pn (c). If n is odd and

2

n+ 1

Ih(v,v)1 :::; 12n-S c,

then M is totally geodesic. (B2) Let M be an m-dimensional compact totally real minimal submanifold isometrically immersed in Hpn(c). If m is odd and

Ih(v,v)1

² :::;

12::-S

c,

the M is totally geodesic. (A3) Let M be an n-dimensional compact totally real minimal submanifold isometrically immersed in Hpn(c). If

I ^{hl2 <} _- ^{n+1 c} _{6 '}

then either M is totally geodesic or n = 2 and M is flat. (B3) Let M be an m-dimensional compact totally real minimal submanifold iso- metrically immersed in H pn (c). If

Ih12:::; ;c,

then either M is totally geodesic in H pn (c) or m

=

2, n 2:: 4 and M is the Veronese surface in H pn (c) with positive constant curvature

^lc2'

The purpose of this paper is to prove the following:

324

THEOREM

1. Let M be an n-dimensionaJ compact totally real min- imal submanifold isometrically immersed in Hpn(c). Then the Ricd curvature 8 of M satisfies

8>3(n-2)c

- 16

if and only if the following condition is satisfied: M is totally geodesic and M is an imbedded submanifold congruent to one of the standard immbedding

i :

Rpn

(~) ~

H pn (c) or to the standard immersion

i 01r : 8n(~) ~

Hpn(c), where

1r : 8n(~) ~

Rpn(c) or n

= 2

and M

is Bat.

THEOREM

2. Let M be an m-dimensional compact totally real minimal submanifold isometrically immersed in Hpn(c). Then 8 of M satisfies

8> (m -1 _ 3(m + 2) )c

- 4 8(2m+5)

if and only if one of the following conditions is satisfied:

A) 8 = ^{milc and} M is totally geodesic,

B) 8

=

(mi

^{l -}

~~:~l»c ^{and M is an} imbedded submanifold con- gruent to one of the imbeddings 'l/Ji

=

v

0

'l/Ji or to the immersion

'l/J5

=

'l/Jl

^{0 1r,}

where

1r :

8 2( l2 c)

~

Rp2 ( l2 c) is the covering map.

2. Preliminaries

Let M be a differentiable manifold of dimension 4n, and assume that there is a 3-dimensional vector bundle V, [7], consisting of tensors of type (1,1) over M satisfying the following condition: in any coordinate neighborhood U of M there is a local base {I, J,K} of V called a

canonical local base of V such that (2.1) ¹² = J2 = K 2 = -Id,

IJ=-JI=K;JK

=

-KJ=I;KI= -IK= J,

where Id denotes the identity tensor field of type (1,1). If M is a

manifold and V is a bundle over M satisfying the above condition then

(M, V) is called an almost quatemion manifold. If 9 is a Riemannian

On curvature pinching for totally real submanifolds ofHpn(c) 325

metric for (M, V) such that g(c.pX, Y) + ^{g(X, c.pY)}

⁼

0, holds for any cross section c.p of V, with X, Y

E

T M, then (M, V, g) is called an

almost quaternion metric manifold.

Assume that the Riemannian connection V' of (M, V,9) satisfies the following condition: if c.p is a local cross section of the bundle V, then V'xc.p is also a local cross section of V, where X is an arbitrary vector field. In this case M

=

(M, V, g) is called a Kaehler quaternion manifold.

Let x

^E

M and X

E

TxM. Consider the 4-dimensional subspace Q(X) in TxM defined by

Q(X) = SpanR{X,IX, JX,KX}

We call this the Q-section generated by X. If for all x

^E

M, X

E

TxM and Y, Z

E

Q(X) the sectional curvature cr(Y, Z)

=

c (a constant), then we say that M is a Kaehler quaternion manifold of constant Q-sectional curvature c. In addition, such a manifold is called a quaternion space form.

The curvature operator R of a quaternionic space form M

=

(M, V,9) has the form:

(2.2)

R(X, Y)Z =~{g(Y,Z)X -g(X,Z)Y +g(IY,Z)IX -g(IX,Z)IY +2g(X,IY)IZ +g(JY,Z)JX - g(JX, Z)JY + 2g(X, JY)JZ + g(KY, Z)KX -g(KX,Z)KY + 2g(X, KY)KZ} ,

where c is the Q-sectional curvature. It is well known that the quater- nion projective space Hpn(c) is a compact 4n-dimensional quaternion space form.

Let (M, V, g) be a Kaehler quaternion manifold and let M be a lliemannian submanifold isometrically immersed in M. We say that M is a totally real submanifold of M, [4], if

for any x

E

M, and any ()

E

V

x ,

where V

x

is the fibre of V over x.

Recall that h is the second fundamental form.

326

(2.3)

LEMMA

1. Assume that M

is

a totally real submanifold of a Kaehler quaternion manifold. Then

g(h(X, Y),IZ),g(h(X, Y),JZ) and g(h(X, Y),KZ) are symmetric with respect to X, Y and Z E TxM, x EM.

Let M be a compact Riemannian manifold, U M its unit tangent bundle, and UM

x

the fibre of U M over a point x of M. We denote by dX,dv and dvx denote the canonical measures on M, UM and UMx respectively.

For any continuous function f : UM

~

R, we have [ fdv = [ ( [ fdvx)dx.

JUM JM JUM",

If T

is

a k-covariant tensor on M and VT is its covariant derivative, then we have:

[ {f (VT) (ei' ei, v,·" ,v)}dv = 0,

JUM ^i=l

where

ell . . .

,em

is

an orthonormal basis of TxM, x EM.

Now, we suppose that M is an m-dimensional compact Riemannian manifold isometrically immersed in a Riemannian manifold M. To simplify notation, we henceforth write

g( , )

=< , >. We denote by

< , > the metric of M as well as that induced on M. Let h be the

second fundamental form of the immersion.

Let X, Y, Z and W denote the tangent vector fields on M. Then if Vh and V

²

h denote the first and second covariant derivatives of h, respectively, one has that Vh is symmetric and V

²

h satisfies the following relation:

(V

²

h) (X, Y, z, W) =(V

²

h)(Y; X, Z, W) + Ri-(X, Y)h(Z, W) -h(R(X, Y)Z, W) - h(Z, R(X, Y)W), where RJ.. and R are the curvature operators of the normal and tangent bundles over M, respectively.

If S is the Ricci curvature of M and M is minimally immersed in M, from Gauss equation we have :

m m

(2.4) S(v,w)

⁼

LR(v,ei,ei'w) - L ^< Ah(v,ei)ei,W >,

i=l i=l

On curvature pinching for totally real submanifolds ofHpn(c) 327

where R is the curvature operator of M.

Now let v E UM

_{x ,}

x EM. If

e2, ... ,em

are orthonormal vectors in UM

x

orthogonal to v, then we can consider

{e2,··· ,em}

as an orthonormal basis of Tv(UMx ). We remark that {v

= eI,e2,··· ,em}

is an orthonormal basis of TxM. If we denote the Laplacian of U M x

^~

sm-l

by A, then Ai

= e2e2i

+..

^·+ememi,

where I is a differentiable function on U M

x .

Define a function !I on UMx, x

^E

M, by h(v) =IA

h

(v,v)vI

²

m

= L ^< h(v,v),h(v,ei) >2.

i=l

Using the minimality of M we can prove that (Ah)(v)

= - 6(m

+ 4)h(v)

(2.5)

m

+ 8 L ^<

Ah(v,v)V, Ah(v,ei)ei

>

i=l m

+ 8 L ^<

Ah(v,v)ei, Ah(v,ei)V

>

i=l m

+ 8 L ^<

Ah(v,ei)V, Ah(v,ei)V

>

i=l m

+

²

L ^<

Ah(v,v)ei, Ah(v,v)ei

> .

i=l

Similarly, define 12,13,

^i4,

is, 16, 17, Is, f9 and ho by

m

h(v) = L ^<

Ah(v,ei)V, Ah(v,ei)V

>,

i=l m

fg(v)

=

L ^<

Ah(v,ei)V, Ah(v,v)ei

>,

i=l m

14(V) =

L ^<

Ah(ej,ei)ej, Ah(v,v)ei

>,

i,j=l

328 Yoshio Matsuyama

m

f5(V) = L ^< Ah(v,v)v,Ah(v,ei)ei >,

i=l m

f6(V) = L ^< Ah(ej,ei)ej, Ah(v,ei)V >,

i,j=l

m

h(v) = L ^< Ah(ei,V)ei, Ah(v,ej)ej >,

i,j=l

f8(V)

=

L ^< Ah(v,v)ei, Ah(v,v)ei >,

i=l

m

ho(v) = L ^< Ah(v,ei)ei,V >,

i=l

respectively. Then we know that

(2.6)

(~h)(v) = -

4(m + 2)h(v) + 4f6(V)

m

+

⁴

L ^< Ah(ej,ei)V, Ah(v,ei)ej >

i,j=l

m

+

2

L ^< Ah(ej,e.)v, Ah(ej,ei)V >

i,j=l

m

+ 2 L ^< Ah(v,ei)ej, Ah(v,ei)ej >,

i,j=l

(2.7)

_(~h)(v)

= - 4(m + ^2)h(v) + ^2!4(v)

m

+ 4 L ^< Ah(ej,ei)V, Ah(ej,v)ei >

i,j=l

m

+ 4 L ^< Ah(v,ei)ej, Ah(ej,v)ei >,

i,j=l

(2.8)

(2.9)

On curvature pinching for totally real submanifolds ofHpn(c) 329

(D.-fs)(v)

= -

4(m + 2)fs(v) + 4f6(V) +4h(v) + 2f4(V),

171

(2.10) (D.-f6)(V) = -2mf6(v) +

2

L ^<

Ah(ej,ei)ej, Ah(ek,ei)ek

>,

i,j,k=l

171

(2.11) (D.-h)(v) = ^-2mh(v) + 2 L ^<

Ah(ej,ei)ej, Ah(ek,ei)ek

>,

i,j,k=l

(2.12) (D.-fs)(v)

= -

4(m + 2)fs(v)

171

+ 8 L ^<

Ah(ej,v)ei, Ah(ej,v)ei

>,

i,j=l

171

(2.13) (D.-fg)(v) = -4(m + 2)fg(v) + 8 L ^<

Ah(v,ei)ei,

v> .

i=l

(2.14) (D.-flO)(V)

=

-2mflO(v) + 21h1

²•

Then we have the following (See [8] and [9]):

LEMMA

2. Let M be an m-dimensional compact minimal subman- ifold isometrically immersed in M. Then for all x EM, we have

(2.15)

where

{ei}~l

is an orthonormal basis of the tangent space TxM to M at x.

The following is the well known Chern-Do Carmo-Kobayashi in-

equality (Lemma 1 in [2]):

LEMMA

3. Under the same assumption of Lemma 2 we have

(2.16)

Since

(m + 4) 1

_UM:z;

^h(v)dv

^{x -}

⁴¹

_UM:z;

^fs(v)dv

^x

1 ^-4 1

⁴

- 3

fs(v)dv

x ~ ⁽ 2)

Ihl dv

x

UM:z;

m m+

UM:z;

1

m m

2" I:(V'2!9)(ei,ei'v) ⁼ I: ^{< (V'}

²

h)(ei,ei'v, v), h(v, v) >

i=l i=l

+ I: ^< (V'h)(ei, v, v), (V'h)(ei' v, v) >,

i=l we have

LEMMA

4. Let M be an m-dimensional totally real minimal sub- manifold isometrically immersed in H pn (c). Then for v

E

UM x we have

(2.17)

1

^{m m}

2" I:(V'

²

fg)(ei,ei'v)

=

I: 1(V'h)(ei,v,v)1

²

+ : clh(v,v)1

²

i=l i=l

m

+2 I: ^<

Ah(v,v)ei, Ah(ei'v)V

>

i=l

m

-2 I: ^<

Ah(v,ei)ei, Ah(v,v)V

>

i=l

- I:

m

^<

Ah(v,v)ei' Ah(v,v)ei

>

i=l

m

C " " ( ( ) 2 2

+4

^LJ

< h v,v ,Iei > + < h(v,V),Jei >

i=l

+ < h(v, v), Kei >2)

On curvature pinching for totally real submanifolds ofHpn(c) 331

3. Proof of theorem 1 Since by (2.4) it holds

n-1 ~

S(v,w)

=

- 4 -c - L...J < Ah(v,ei)ei,W >,

i=l

we have only to prove Theorem 1 under the assumption of

n

n+2

(3.1) L ^< Ah(v,ei)ei, v>:::; --We.

i=l

Let v

E

U M

_{x ,}

x

E

M. Since M is totally real, the following equations hold:

(3.2)

(3.3)

L

n

^< Ah(ej,ei)V, Ah(ej,v)ei >

i,j=l n

=

L ^< Ah(v,ei)ej, Ah(v,ej)ei >

i,j=l

L

n

^< Ah(ej,ei)V, Ah(ej,ei)V >

i,j=l

n

=

L ^< Ah(v,ei)ej,Ah(v,ei)ej > .

i,j=l

In terms of (2.5), (2.6), (2.7), (2.8), (2.9), (2.10), (2.11), (2.12), (2.17), (3.2) and (3.3) we obtain

(3.4)

1

n

1 1

2" _~(V'2 fg)(ei' ei, v) - 6(~fd(v)

^-

³⁽ⁿ + 2) (~h)(v)

1 1 1

+ 6(n + 2) (~h)(v) + 3n(n + 2) (df4) (v) + 6(n + 2) (~f5)(V)

1 1 1

3n(n + 2) (df6) (v) + 3n(n + 2) (~h)(v) + 6(n + 2) (~fg)(v)

~

²

ⁿ⁺¹

= _~

1(V'h)(ei, v, v)1 + ^-4-cfg(v)

+ ⁽ⁿ + 4)!I(v) - 4f5(V) - 2fg(v).

332

(3.5)

(3.6)

From (2.15) we get

02: kM", t.1(Vh)(ei,V,v)12dVz

1 ^{n+l 2n}

+ (-4- cfg (v) - - - 2 fs (v) - 2fs(v»dvz .

UM",

n+

Define a self-adjoint operator L : TzM - TzM by Lv

= _L:~=l

Ah(v,ei)ei.

If

el, . . .

,en is an orthonormal basis of TzM,

X

EM, such that Lei

=

Uiei

from (3.1) we have

fs(v)

=

< Lv, Ah(v,v)V >

:$ 1 6cfg (v) n+2 Since

n

f8(V)

=

Ih(v, v)1 2

L ^< AI:t:::~1 ^ei, AI:t:::~1

^ei

^>,

i=l

we have

(3.7) f8(V) :$ 16cf9(v) n+2

Combining (3.5) and (3.6) with (3.7), we obtain 02: kM", t.1(Vh)(ei, v, v)1

²

dvz

I ^{n+l n n+2}

+. ^(-4- C- SC - -s-c)fg(v)dVz .

UM",

Thus we see that M is a submanifold of H pn (c) with parallel second fundamental form.

4. Proof of theorem 2

As in the proof of Theorem 1 by (2.4) since it holds

m-I ~

S(v,w)

=

- 4 -c - L...J < Ah(v,ei)ei'W >,

i=l

On curvature pinching for totally real submanifolds ofHpn (c) 333

we have only to prove Theorem 2 under the assumption of

(4.1)

Let v

E

U M

x ,

x EM. Since M is totally real, in this case the following equations also hold:

(4.2)

(4.3)

L

'In

^< Ah(ej,e.)v, Ah(ej,v)ei >

i,j=l

'In

=

L ^< Ah(v,ei)ej, Ah(v,ej)ei >,

i,j=l

L

'In

^< Ah(ej,ei)V, Ah(ej,ei)V >

i,j=l

'In

=

L ^< Ah(v,ei)ej, Ah(v,e.)ej >, i,j=l

(4.4)

1

^'In

1 1

2" ~(V2f9)(ei' ^{ei, v) -} 6 (.6.!I) (v) - 3(m + ²⁾ (.6.h) (v)

1 1 1

+ 6(m + 2) (.6.f3) (v) + 3m(m + 2) (.6.f4) (v) + 6(m + 2) (.6.fs) (v)

1 1 1

3m(m + 2) (.6.f6) (v) + ^3m(m + 2) (.6.h) (v) + ^6(m + 2) (.6.fs)(v)

'In

= L !(Vh)(ei, v, v)1

²

+ : Cf9(V) + ^(m + 4) !I(v) - 4f5(V) - 2f8(V)

i=l

'In

C ' " ( ) 2 ) 2 ( 2)

+"4 L..,;« h V,V ,Iei> + < h(v,v ,Jei > + < h v,V),Kei > .

i=l

(4.6)

Integrating (4.4) and multiplying it by

_~,

we have

3{ _~

2

3m{

0=2" JT' ^LJ I(Vh)(ei,v,v)/ dv

^x

+ 8

^C

JL ^fg(v)dv

^x

UMx i=l UMx

+~(m + 4) { h(v)dv

x -

6 { f5(V)dv

_{x -}

3 ( fs(v)dv

_x

Ju~ Ju~ Ju~

+ ~C 1 f« ^h(v,v),Iei

^>2

^{+ <} h(v,V),Jei >2

UMx i=l

+ < h(v,V),Kei >2)dv

x .

Using (2.16), we get (4.5)

3{ _~

²

3m{

02::2"

JUM

x

~ I(Vh)(ei, v, v)1 dv

^x

+ BC

JUM x

fg(v)dv

x

+-2 1

(m+4) { h(v)dv

x

-2 { f5(V)dv

_{x -}

(4 ) { Ihl

⁴

dv

_x

J UMx JUM",

m m +

² JUMx

+ ~ 1 ^{f« h(v,v),Iei}

^>2

^{+ <} h(v,v), Jei'

^>2

UM", i=l

+ < h(v,V),Kei >2)dv

x .

From (2.15) we know that

~(m+4) ^{{ h(v)dv}

^x

^{-2 { f5(V)dv}

^x

JUMx JUM",

2:: -m

2 ( f5(V)dv

_{x .}

m+

JUMx

By means of (4.1) we obtain (4.7)

(4.8)

3(m+2)

2

f5(V) ::; 8(2m + 5) clh(v, v)1 ,

Ihl2 < 3m(m + 2) c.

- 8(2m+5)

On curvature pinching for totally real submanifolds ofHpn(c) 335

Combining (4.5), (4.6) and (4.7) with (4.8), we have

Noting (2.13) and (2.14), we get

1 Ihl dv

² x =

^m(m+2)

2

1 Ih(v,v)1 dv

² x .

UMx UMx

Hence,

This proves Theorem 2.

References

[1) B.-Y. Chen andK. Ogiue, On totally real submanifolds, Trans.Amer. Math.

Soc. 193 (1974), 257-266.

[2) S. -So Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of sphere with second fundamental form of a constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, III 1968), Springer, New York (1970), 59-75.

336 Yoshio Matsuyama

[3] P. Coulton and H. Gauchman, Submanifolds of quaternion projective space with bounded second fundamental form, Kodai Math. J. 12 (1989),296-307.

[4] S. Funabashi, Totallyrealsubmanifolds of a quaternionic Kaehlerian manifold, Kodai Math. J. 29 (1978), 261-270.

[5] H. Gauchman, Pinching theorems for totally real minimal submanifolds of

CP"(c),

Tohoku Math. J. 41 (1989), 249-257.

[6] C. -So Houh, Some totallyreal submanifolds in a quaternion projective space, Rocky Mountain J. Math. 10 (1980), 509-515.

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