$QR$-submanifolds of maximal $QR$-dimension in quaternionic projective space

QR-SUBMANIFOLDS OF MAXIMAL QR-DIMENSION IN QUATERNIONIC PROJECTIVE SPACE

Hyang Sook Kim and Jin Suk Pak*

Abstract. The purpose of this paper is to study n-dimensional QR-submanifolds of maximal QR-dimension isometrically immersed in a quaternionic projective space and to give sucient conditions in order for such a submanifold to be a tube over a quaternionic invariant submanifold.

1. Introduction

Let M be a connected real n-dimensional submanifold of real codi- mension p of a quaternionic K¨ ahler manifold M with quaternionic K¨ ahler structure {F, G, H}. If there exists an r-dimensional normal distribution ν of the normal bundle T M ^⊥ such that

F ν x ⊂ ν x , Gν x ⊂ ν x , Hν x ⊂ ν x ,

F ν _x ^⊥ ⊂ T x M, Gν _x ^⊥ ⊂ T x M, Hν _x ^⊥ ⊂ T x M

at each point x in M , then M is called a QR-submanifold of r QR- dimension, where ν ^⊥ denotes the complementary orthogonal distribu- tion to ν in T M ^⊥ ([1, 10]). Real hypersurfaces, which are typical ex- amples of QR-submanifold with r = 0, have been investigated by many authors ([2, 9, 10, 11, 12, 14]) in connection with the shape operator and the induced almost contact 3-structure (for denition, see [7]).

In this paper we shall study QR-submanifolds of maximal QR-dimens- ion isometrically immersed in a quaternionic projective space QP ^(n+p)/4 and prove the following theorem which is an extension of theorem proved in [12, Theorem 10] to the case of QR-submanifolds with maximal QR- dimension :

2000 Mathematics Subject Classication: 53C40, 53C25.

Key words and phrases: quaternionic projective space, QR-submanifold, Ricci tensor, scalar curvature, mean curvature.

*This work was supported by ABRL Grant Proj. No. R14-2002- 003-01000-0 from

KOSEF.

Theorem 1.1. Let M be an n-dimensional QR-submanifold of max- imal QR-dimension in a quaternionic projective space QP ^(n+p)/4 . If the distinguished normal vector eld ξ is parallel with respect to the normal connection and the equalities appeared in (3.4) hold on M , then M is locally isometric to

π(S ⁴ⁿ

⁺³ (r 1 ) × S ⁴ⁿ

⁺³ (r 2 )) (r ² ₁ + r ² ₂ = 1)

for some integers n 1 , n 2 with 4n 1 + 4n 2 = n − 3, where π is the Hopf bration S ⁿ⁺⁴ (1) → QP ^(n+1)/4 .

Next, under the same assumptions as in Theorem 1.1, we bring into use an integral formula ([15]) which leads to an inequality among the Ricci curvature, the scalar curvature and the mean curvature of M . Us- ing this inequality, we provide the following theorem as quaternionic analogue to theorem given in [4, Theorem 4.2]. Theorem 1.2 is also a generalization of Lawson’s result ([11, Theorem 4]) for higher codimen- sion, but avoiding the condition of minimality:

Theorem 1.2. Let M be an n-dimensional compact QR-submanifold of (p − 1) QR-dimension in a quaternionic projective space QP ^(n+p)/4 , If the distinguished normal vector eld ξ is parallel with respect to the normal connection and the inequality

1

3 {Ric(U, U) + Ric(V, V ) + Ric(W, W )} + ρ − n ² ∥µ∥ ² ≥ (n ² + 8n − 1), then M is isometric to

π(S ⁴ⁿ

⁺³ (1/ √

2) × S ⁴ⁿ

⁺³ (1/ √ 2))

for some integers n 1 , n 2 with 4n 1 + 4n 2 = n − 3, where π is the Hopf bration S ⁿ⁺⁴ → QP ^(n+1)/4 .

2. Preliminaries

Let M be a real (n + p)-dimensional quaternionic K¨ ahler manifold.

Then, by denition, there is a 3-dimensional vector bundle V consisting with tensor elds 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,

F G = −GF = H, GH = −HG = F, HF = −F H = G.

(b) There is a Riemannian metric g which is hermite with respect to all of F , G and H.

(c) For the Riemannian connection ∇ with respect to g

(2.2)



 ∇F

∇H ∇G



 =



 0 r −q

−r 0 p

q −p 0







 F G H





where p, q and r are local 1-forms dened in U. Such a local basis {F, G, H} is called a canonical local basis of the bundle V in U ([5, 6]).

For canonical local bases {F, G, H} and { ^′ F, ^′ G, ^′ H } of V in coordi- nate neighborhoods U and ^′ U, it follows that in U ∩ ^′ U





′ F

′ G

′ H



 = (s xy )



 F G H



 (x, y = 1, 2, 3)

where s xy are local dierentiable functions with ( s xy ) ∈ SO(3) as a consequence of (2.1). As is well known ([5, 6]), every quaternionic K¨ ahler manifold is orientable.

Now let M be an n-dimensional QR-submanifold of maximal QR- dimension, that is, of (p − 1) QR-dimension isometrically immersed in

∨M. Then by denition there is a unit normal vector eld ξ such that ν _x ^⊥ = Span {ξ} at each point x in M. We set

(2.3) U = −F ξ, V = −Gξ, W = −Hξ.

Denoting by D x the maximal quaternionic invariant subspace T x M ∩ F T x M ∩ GT x M ∩ HT x M

of T x M , we have D ^⊥ x ⊃ Span{U, V, W }, where D ^⊥ x means the comple- mentary orthogonal subspace to D x in T x M . But, using (2.1), we can prove that D x ^⊥ = Span {U, V, W } ([1, 10]). Thus we have

T x M = D x ⊕ Span{U, V, W }, ^∀ x ∈ M, which together with (2.1) and (2.3) implies

F T x M, GT x M, HT x M ⊂ T x M ⊕ Span{ξ}.

Therefore, for any tangent vector eld X and for a local orthonormal basis {ξ α } α=1,...,p (ξ 1 := ξ) of normal vectors to M , we have

(2.4) F X = ϕX + u(X)ξ, GX = ψX + v(X)ξ, HX = θX + w(X)ξ,

(2.5) F ξ α = −U α + P 1 ξ α , Gξ α = −V α + P 2 ξ α , Hξ α = −W α + P 3 ξ α

(α = 1, . . . , p). Then it is easily seen that {ϕ, ψ, θ} and {P 1 , P 2 , P 3 } are skew-symmetric endomorphisms acting on T x M and T x M ^⊥ , respec- tively. Moreover, the hermitian property of {F, G, H} implies

g(X, ϕU α ) = −u(X)g(N 1 , P 1 ξ α ), g(X, ψV α ) = −v(X)g(N 1 , P 2 ξ α ), (2.6)

g(X, θW α ) = −w(X)g(N 1 , P 3 ξ α ), α = 1, . . . , p, g(U α , U β ) = δ αβ − g(P 1 ξ α , P 1 ξ β ),

g(V α , V β ) = δ αβ − g(P 2 ξ α , P 2 ξ β ), (2.7)

g(W α , W β ) = δ αβ − g(P 3 ξ α , P 3 ξ β ), α, β = 1, . . . , p.

Also, from the hermitian properties

g(F X, ξ α ) = − g(X, F ξ α ), g(GX, ξ α ) = −g(X, Gξ α ), g(HX, ξ α ) = −g(X, Hξ α ),

it follows that

g(X, U α ) = u(X)δ 1α , g(X, V α ) = v(X)δ 1α , g(X, W α ) = w(X)δ 1α

and hence

(2.8) g(U 1 , X) = u(X), g(V 1 , X) = v(X), g(W 1 , X) = w(X), U α = 0, V α = 0, W α = 0, α = 2, . . . , p.

On the other hand, comparing (2.3) and (2.5) with α = 1, we have U 1 = U , V 1 = V , W 1 = W , which together with (2.3) and (2.8) implies (2.9) g(U, X) = u(X), g(V, X) = v(X), g(W, X) = w(X),

u(U ) = 1, v(V ) = 1, w(W ) = 1.

Here and in the sequel we use the notations U , V , W instead of U 1 , V 1 , W 1 .

Next, applying F to the rst equation of (2.4) and using (2.5), (2.8), and (2.9), we have

ϕ ² X = −X + u(X)U, u(X)P 1 ξ = −u(ϕX)ξ.

Similarly we have

ϕ ² X = −X + u(X)U, ψ ² X = −X + v(X)V, (2.10)

θ ² X = −X + w(X)W,

u(X)P 1 ξ = − u(ϕX)ξ, v(X)P 2 ξ = −v(ψX)ξ, (2.11)

w(X)P 3 ξ = −w(θX)ξ,

from which, taking account of the skew-symmetry of P 1 , P 2 , and P 3 and using (2.6) with α = 1, we also have

(2.12)

u(ϕX) = 0, v(ψX) = 0, w(θX) = 0, ϕU = 0, ψV = 0, θW = 0,

P 1 ξ = 0, P 2 ξ = 0, P 3 ξ = 0.

So (2.5) can be rewritten of the form

(2.13)

F ξ = −U, Gξ = −V, Hξ = −W, F ξ α = P 1 ξ α =

∑ p β=2

P _1αβ ξ β , Gξ α = P 2 ξ α =

∑ p β=2

P _2αβ ξ β ,

HN α = P 3 N α =

∑ p β=2

P _3αβ ξ β (α = 2, . . . , p).

Applying G and H to the rst equation of (2.4) and using (2.1), (2.4), and (2.13), we have

θX + w(X)ξ = −ψ(ϕX) − v(ϕX)ξ + u(X)V, ψX + v(X)ξ = θ(ϕX) + w(ϕX)ξ − u(X)W, and consequently

(2.14) ψ(ϕX) = −θX + u(X)V, v(ϕX) = −w(X),

θ(ϕX) = ψX + u(X)W, w(ϕX) = v(X).

Similarly the other equations of (2.4) yield

(2.15) ϕ(ψX) = θX + v(X)U, u(ψX) = w(X), θ(ψX) = −ϕX + v(X)W, w(ψX) = −u(X), (2.16) ϕ(θX) = −ψX + w(X)U, u(θX) = −v(X),

ψ(θX) = ϕX + w(X)V, v(θX) = u(X).

From the rst three equations of (2.13), we can also easily obtain

(2.17)

ψU = −W, v(U) = 0, θU = V, w(U) = 0, ϕV = W, u(V ) = 0, θV = −U, w(V ) = 0, ϕW = −V, u(W ) = 0, ψW = U, v(W ) = 0.

The equations (2.8)–(2.10), (2.12), and (2.14)–(2.17) tell us that M admits the so-called almost contact 3-structure (for denition, see [7]) and consequently n = 4m + 3 for some integer m.

Now let ∇ be the Levi-Civita connection on M and let ∇ ^⊥ the normal connection induced from ∇ in the normal bundle T M ^⊥ of M . Then Gauss and Weingarten formulae are given by

∇ X Y = ∇ X Y + h(X, Y ), (2.18)

∇ X ξ α = −A α X + ∇ ^⊥ X ξ α , α = 1, . . . , p (2.19)

for vector elds X, Y tangent to M . Here h denotes the second funda- mental form and A α the shape operator corresponding to ξ α . They are related by

h(X, Y ) =

∑ p α=1

g(A α X, Y )ξ α

Furthermore, we put

(2.20) ∇ ^⊥ X ξ α =

∑ p β=1

s αβ (X)ξ β ,

where (s αβ ) is the skew-symmetric matrix of connection forms of ∇ ^⊥ . Dierentiating the rst equation of (2.4) covariantly and using (2.2), (2.4), (2.18), and (2.19), we have

(2.21) ( ∇ Y ϕ)X = r(Y )ψX − q(Y )θX + u(X)A 1 Y − g(A 1 Y, X)U,

( ∇ Y u)X = r(Y )v(X) − q(Y )w(X) + g(ϕA 1 Y, X).

From the other equations of (2.4) we also have

(2.22) ( ∇ Y ψ)X = −r(Y )ϕX + p(Y )θX + v(X)A 1 Y − g(A 1 Y, X)V, (∇ Y v)X = −r(Y )u(X) + p(Y )w(X) + g(ψA 1 Y, X),

(2.23) ( ∇ Y θ)X = q(Y )ϕX − p(Y )ψX + w(X)A 1 Y − g(A 1 Y, X)W, ( ∇ Y w)X = q(Y )u(X) − p(Y )v(X) + g(θA 1 Y, X).

Next, dierentiating the rst equation of (2.13) covariantly and using (2.2), (2.13), (2.18), and (2.19), we have

(2.24)

∇ Y U = r(Y )V − q(Y )W + ϕA 1 Y, g(A α U, Y ) = −

∑ p β=2

s 1β (Y )P _1βα , α = 2, . . . , p.

From the other equations of (2.13), we have similarly

(2.25)

∇ Y V = −r(Y )U + p(Y )W + ψA 1 Y, g(A α V, Y ) = −

∑ p β=2

s 1β (Y )P _2βα , α = 2, . . . , p,

(2.26)

∇ Y W = q(Y )U − p(Y )V + θA 1 Y, g(A α W, Y ) = −

∑ p β=2

s 1β (Y )P 3βα , α = 2, . . . , p.

Finally if the ambient manifold ∨M is of constant Q-sectional curva- ture c, the equations of Gauss and Codazzi are given by

R(X, Y )Z = c

4 {g(Y, Z)X − g(X, Z)Y (2.27)

+ g(ϕY, Z)ϕX − g(ϕX, Z)ϕY − 2g(ϕX, Y )ϕZ + g(ψY, Z)ψX − g(ψX, Z)ψY − 2g(ψX, Y )ψZ + g(θY, Z)θX − g(θX, Z)θY − 2g(θX, Y )θZ}

+ ∑

α

g(A α Y, Z)A α X − ∑

α

g(A α X, Z)A α Y,

g(( ∇ X A 1 )Y − (∇ Y A 1 )X, Z) (2.28)

= c

4 {g(ϕY, Z)u(X) − g(ϕX, Z)u(Y ) − 2g(ϕX, Y )u(Z) + g(ψY, Z)v(X) − g(ψX, Z)v(Y ) − 2g(ψX, Y )v(Z) + g(θY, Z)w(X) − g(θX, Z)w(Y ) − 2g(θX, Y )w(Z)}

+ ∑

β

{g(A β X, Z)s β1 (Y ) − g(A β Y, Z)s β1 (X) },

respectively. Moreover, (2.9), (2.10), and (2.27) yield (2.29)

Ric(X, Y )

= c

4 {(n + 8)g(X, Y ) − 3(u(X)u(Y ) + v(X)v(Y ) + w(X)w(Y ))}

+ ∑

α

{(trA α )g(A α X, Y ) − g(A ² α X, Y ) },

(2.30) ρ = c

4 (n + 9)(n − 1) + n ² ∥µ∥ ² − ∑

α

trA α 2 ,

where Ric and ρ denote the Ricci tensor and the scalar curvature, re- spectively, and

(2.31) µ = 1

n

∑

α

(trA α )ξ α

is the mean curvature vector ([3]).

3. Some properties of the shape operator A 1

In this section, let M be an n-dimensional QR-submanifold of maxi- mal QR dimension in a quaternionic space form M ^(n+p)/4 (c) with con- stant Q-sectional curvature c. In what follows, we assume that the distin- guished normal vector field ξ 1 := ξ is parallel with respect to the normal connection ∇ ^⊥ , that is,

∇ ^⊥ X ξ = 0

for any vector eld X tangent to M . Then it follows from (2.20) that s 1β = 0 and consequently (2.24)–(2.26) imply

(3.1) A α U = 0, A α V = 0, A α W = 0, α = 2, . . . , p.

Moreover, since s 1β = 0, (2.20) yields

(3.2) ∇ ^⊥ X ξ α =

∑ p β=2

s αβ (X)ξ β , α = 2, . . . , p.

From now on we denote by A the shape operator A 1 corresponding to the normal vector ξ = ξ 1 and prepare a lemma for later use.

Lemma 3.1. Let M be an n-dimensional QR-submanifolds of max- imal QR-dimension in a quaternionic space form M ^(n+p)/4 (c). If the distinguished normal vector eld ξ is parallel with respect to the normal connection and if the equalities

(3.4) h(X, ϕY ) = h(ϕX, Y ), h(X, ψY ) = h(ψX, Y ), h(X, θY ) = h(θX, Y ),

hold for any vector elds X, Y tangent to M , then

(3.5) Aϕ = ϕA, Aψ = ψA, Aθ = θA,

and A α = 0 for α = 2, · · · p.

Proof. Sine n = 4m + 3 and p = 4t + 1 for some integers m and t, and since the subspace ν is quaternionic invariant (see also (2.13)), we can take a local orthonormal basis {ξ, ξ a , ξ a

, ξ a

, ξ a

} a=1,...,t of normal vectors M such that

ξ 1

:= F ξ 1 , . . . , ξ t

:= F ξ t , ξ 1

:= Gξ 1 , . . . , ξ t

:= Gξ t , ξ 1

:= Hξ 1 , . . . , ξ t

:= Hξ t .

Then we can express the second fundamental form h as

h(X, Y ) = g(AX, Y )ξ +

∑ t a=1

{g(A a X, Y )ξ a + g(A a

X, Y )ξ a

+ g(A a

X, Y )ξ a

+ g(A a

X, Y )ξ a

}.

Hence the assumption (3.4) implies

(3.6)

Aϕ = ϕA, Aψ = ψA, Aθ = θA,

A a

ϕ = ϕA a

, A a

ψ = ψA a

, A a

θ = θA a

, A a

ϕ = ϕA a

, A a

ψ = ψA a

, A a

θ = θA a

A a

ϕ = ϕA a

, A a

ψ = ψA a

, A a

θ = θA a

. On the other hand,

ξ a

= F ξ a , ξ a

= Gξ a , ξ a

= Hξ a (a = 1, . . . , t) give, respectively,

(3.7)

∇ X F ξ a = −A a

X + ∇ ^⊥ X ξ a

,

∇ X Gξ a = −A a

X + ∇ ^⊥ X ξ a

,

∇ X Hξ a = −A a

X + ∇ ^⊥ X ξ a

, a = 1, . . . , t.

Hence, using (2.2), (2.13), (2.19), and (3.2), it follows from the rst equation of (3.7) that

− A a

X + ∇ ^⊥ X ξ a

= r(X)G(ξ a ) − q(X)H(ξ a ) + F (−A a X +

∑ t b=1

{s ab

(X)ξ b

+ s ab

(X)ξ b

+ s ab

(X)ξ b

}), from which, taking the tangential part, we can easily obtain

(3.8) ϕA a = A a

, a = 1, . . . , t.

Similarly, from the other equations of (3.7), we have (3.9) ψA a = A a

, θA a = A a

, a = 1, . . . , t.

Therefore, for any vectors X, Y tangent to M , it is clear from (3.8) that g(A a

ϕX, Y ) = −g(A a ϕX, ϕY )

and consequently

g(A a

ϕX, Y ) = g(A a

ϕY, X) = −g(ϕA a

X, Y ), that is,

A a

ϕ = −ϕA a

,

which and (3.6) imply ϕA a

= 0. Thus we have from (3.8) that ϕ ² A a =

0, which together with (2.10) and (3.1) yields A a = 0. Hence (3.9) yields

A a = 0, A a

= 0, A a

= 0, A a

= 0. 

4. Codimension reduction and proof of Theorem 1.1

In this section we assume that the ambient manifold is a quaternionic projective space Q of constant Q-sectional curvature 4. Let

N 0 (x) = {η ∈ T x M ^⊥ : A η = 0 }

and let H 0 (x) be the maximal quaternionic invariant subspace of N 0 (x), that is,

H 0 (x) = N 0 (x) ∩ F N 0 (x) ∩ GN 0 (x) ∩ HN 0 (x).

Then Kwon and the rst author of this paper [8] have proved the following theorem :

Lemma 4.1. Let M be an n-dimensional real submanifold of an (n + p)-dimensional quaternionic projective space QP ^(n+p)/4 . If the orthogo- nal complement H 1 (x) of H 0 (x) in T M ^⊥ is invariant under the parallel translation with respect to the normal connection and q is the constant dimension of H 1 (x), then there exists a real (n + q)-dimensional to- tally geodesic quaternionic projective space QP ^(n+p)/4 such that M ⊂ QP ^(n+p)/4 .

In our cases, N 0 (x) = Span {ξ 2 (x), . . . , ξ p (x) }. In fact, as a conse- quence of Lemma 3.1, A α = 0 for α = 2, . . . , p. Hence

Span {ξ 2 (x), . . . , ξ p (x) } ⊂ N 0 (x).

On the other hand, for any η in N 0 (x), we can put η = ∑ p

α=1 λ ^α ξ α . But

A η =

∑ p α=1

λ ^α A α = λ ¹ A 1 = 0

since A α = 0 for α = 2, . . . , p. Hence λ ¹ = 0 and consequently η ∈ Span{ξ 2 (x), . . . , ξ p (x)}.

Hence we have

N 0 (x) = H 0 (x) = Span {ξ 2 (x), . . . , ξ p (x) }.

Thus H 1 (x) = Span {ξ(x)} and so our assumption yields that H 1 (x)

is invariant under parallel translation with respect to the normal con-

nection. Therefore we can apply Lemma 4.1 and obtain the following

theorem.

Theorem 4.2. Let M be an n-dimensional QR-submanifold of max- imal QR-dimension in a quaternionic projective space QP ^(n+p)/4 . If the distinguished normal vector eld ξ is parallel with respect to the normal connection and the equalities appeared in (3.4) hold on M , then there exists a real (n + 1)-dimensional totally geodesic quaternionic projective space QP ^(n+1)/4 such that M ⊂ QP ^(n+1)/4 .

Proof of Theorem 1.1. From now on we shall give the proof of the theorem stated in Section 1. By means of Theorem 4.2 the submanifold M can be regarded as a real hypersurface of QP ^(n+1)/4 which is totally geodesic in QP ^(n+p)/4 . Tentatively we denote QP ^(n+1)/4 by M ^′ and by i 1 the immersion of M into M ^′ and by i 2 the totally geodesic immersion of M ^′ into QP ^(n+p)/4 . Then it is clear from (2.18) that

(4.1) ∇ ^′ i

X i 1 Y = i 1 ∇ X Y + h ^′ (X, Y ) = i 1 ∇ X Y + g(A ^′ X, Y )ξ ^′ , where ∇ ^′ is the induced connection on M ^′ from that of QP ^(n+p)/4 , h ^′ the second fundamental form of M in M ^′ and A ^′ the corresponding shape operator to a unit normal vector eld ξ ^′ to M in M ^′ . Since i = i 2 ◦ i 1

and M ^′ is totally geodesic in QP ^(n+p)/4 , we can easily see that (2.18) and (4.1) imply

(4.2) ξ = i 2 ξ ^′ , A 1 = A ^′ .

Since M ^′ is a quaternionic invariant submanifold of QP ^(n+p)/4 , for any vector eld X tangent to M

(4.3) F i 2 X = i 2 F ^′ X, Gi 2 X = i 2 G ^′ X, Hi 2 X = i 2 H ^′ X

are valid, where {F ^′ , G ^′ , H ^′ } is the induced quaternionic K¨ahler struc- ture on M ^′ . Thus it follows from (4.2), (4.3), and the rst equation of (2.4) that

F iX = F i 2 ◦ i 1 X = i 2 F ^′ i 1 X = i 2 (i 1 ϕ ^′ X + u ^′ (X)ξ ^′ ) = iϕ ^′ X + u ^′ (X)ξ for any vector eld X tangent to M . Comparing this equation with the rst equation of (2.4), we have ϕ = ϕ ^′ and u = u ^′ . Similarly we have

ϕ = ϕ ^′ , ψ = ψ ^′ , θ = θ ^′ , u = u ^′ , v = v ^′ , w = w ^′ , which and Lemma 3.1 imply

A ^′ ϕ ^′ = ϕ ^′ A ^′ , A ^′ ψ ^′ = ψ ^′ A ^′ , A ^′ θ ^′ = θ ^′ A ^′ .

Now applying the theorem proved in [12, Theorem 10, pp. 57] by the rst author of this paper, we may conclude that M is locally isometric to

π(S ⁴ⁿ

⁺³ (r 1 ) × S ⁴ⁿ

⁺³ (r 2 )) (r ² ₁ + r ² ₂ = 1)

for some n 1 , n 2 with 4n 1 + 4n 2 = n − 3, where π is the Hopf bration

S ⁿ⁺⁴ → QP ^(n+1)/4 . 

5. An integral formula for the model space π(S ⁴ⁿ

⁺³ (1/ √ 2) × S ⁴ⁿ

⁺³ (1/ √

2))

Let M be an n-dimensional QR-submanifold of maximal QR di- mension in a quaternionic space form ∨M ^(n+p)/4 (c) with constant Q- sectional curvature c. We put

T := ∇ U U + ∇ V V + ∇ W W − (divU)U − (divV )V − (divW )W and take an orthonormal basis {U, V, W, e a , e a

, e a

, e a

} a=1,...,m of tan- gent vectors to M such that

e a

:= ϕe a , e a

:= ψe a , e a

= θe a .

Then it follows from (2.8), (2.9), (2.12), and (2.24)–(2.26) that T = ϕAU + ψAV + θAW,

(5.1)

g(T, U ) = g(T, V ) = g(T, W ) = 0.

(5.2)

Here and in the sequel we also denote by A the shape operator A 1

corresponding to the distinguished normal vector ξ = ξ 1 . We note that T is a global vector eld dened on M . For later use we compute

divT =

∑ n i=1

g(e i , ∇ e

T ).

Dierentiating (5.1) covariantly and using (2.21)–(2.26), we have

∇ X T = {u(AU) + v(AV ) + w(AW )}AX

− g(A ² U, X)U − g(A ² V, X)V − g(A ² W, X)W + ϕAϕAX + ψAψAX + θAθAX

+ ϕ( ∇ X A)U + ψ( ∇ X A)V + θ( ∇ X A)W,

from which, taking account of (2.8)–(2.10), (2.12), and (2.14)–(2.17), divT

= {u(AU) + v(AV ) + w(AW )}trA − u(A ² U ) − v(A ² V ) − w(A ² W ) + tr(ϕAϕA) + tr(ψAψA) + tr(θAθA) − g((∇ V A)W − (∇ W A)V, U )

− g((∇ W A)U − (∇ U A)W, V ) − g((∇ U A)V − (∇ V A)U, W )

−

∑ m a=1

{g((∇ e

A)e a

− (∇ e

A)e a + (∇ e

A)e a

− (∇ e

A)e a

, U ) + g(( ∇ e

A)e a

− (∇ e

A)e a + ( ∇ e

A)e a

− (∇ e

A)e a

, V ) + g(( ∇ e

A)e a

− (∇ e

A)e i + ( ∇ e

A)e a

− (∇ e

A)e a

, W ) }, or equivalently

divT = {u(AU) + v(AV ) + w(AW )}trA (5.3)

− u(A ² U ) − v(A ² V ) − w(A ² W ) + 3(n − 3)

4 c

+ tr(ϕAϕA) + tr(ψAψA) + tr(θAθA)

because of (2.28) with s β1 = 0. On the other hand, using (2.8)–(2.10) and (2.12), we can easily obtain that

∥ϕA − Aϕ∥ ² + ∥ψA − Aψ∥ ² + ∥θA − Aθ∥ ²

= 6 trA ² − 2{u(A ² U ) + v(A ² V ) + w(A ² W ) } + 2 {tr(ϕAϕA) + tr(ψAψA) + tr(θAθA)}, which together with (5.3) yields

(5.4)

divT = 1

2 {∥ϕA − Aϕ∥ ² + ∥ψA − Aψ∥ ² + ∥θA − Aθ∥ ² } + 3(n − 3)

4 c − 3trA ² + trA {u(AU) + v(AV ) + w(AW )}.

On the other hand, it follows from (2.29)–(2.31) and (3.1) that Ric(U, U ) = c

4 (n + 5) + (trA)u(AU ) − u(A ² U ), Ric(V, V ) = c

4 (n + 5) + (trA)v(AV ) − v(A ² V ), Ric(W, W ) = c

4 (n + 5) + (trA)w(AW ) − w(A ² W ), trA ² = −ρ + c

4 (n + 9)(n − 1) + n ² ∥µ∥ ² −

∑ p α=2

trA α 2 ,

from which and (5.4), we have

(5.4) ^′

divT = 1

2 {∥ϕA − Aϕ∥ ² + ∥ψA − Aψ∥ ² + ∥θA − Aθ∥ ² } + Ric(U, U ) + Ric(V, V ) + Ric(W, W )

+ 3 {ρ − c

4 (n ² + 8n − 1) − n ² ∥µ∥ ² } + 3

∑ p α=2

trA ² _α + ∥AU∥ ² + ∥AV ∥ ² + ∥AW ∥ ² .

Thus we have

Lemma 5.1. Let M be an n-dimansional compact QR-submanifold of (p − 1) QR-dimension in a quaternionic projective space QP ^(n+p)/4 . If the distinguished normal vector eld ξ is parallel with respect to the normal connection and the inequality

1

3 {Ric(U, U) + Ric(V, V ) + Ric(W, W )} + ρ − n ² ∥µ∥ ² ≥ n ² + 8n − 1, holds on M , then

Aϕ = ϕA Aψ = ψA, Aθ = θA, A α = 0, α = 2, . . . , p and AU = AV = AW = 0.

Proof. Applying Stokes’s theorem to (5.4) ^′ , we can easily obtain the

conclusions. 

Combining Lemma 4.1 and Lemma 5.1, we have

Theorem 5.2. Let M be an n-dimensional compact QR-submanifold of (p − 1) QR-dimension in a quaternionic projective space QP ^(n+p)/4 . If the distinguished normal vector eld ξ is parallel with respect to the normal connection and the inequality

1

3 {Ric(U, U) + Ric(V, V ) + Ric(W, W )} + ρ − n ² ∥µ∥ ² ≥ n ² + 8n − 1,

holds on M , then there exists an (n + 1)-dimensional totally geodesic

quaternionic projective space QP ^(n+1)/4 such that M ⊂ QP ^(n+1)/4 .

Proof. By means of Lemma 5.1

A α = 0, α = 2, . . . , p.

As shown in the proof of Theorem 4.2, applying Lemma 4.1, we may conclude that there exists an (n+1)-dimensional totally geodesic quater- nionic projective space QP ^(n+1)/4 such that M ⊂ QP ^(n+1)/4 .  Proof of Theorem 1.2. From now on we shall give the proof of Theo- rem 1.2 stated in Section 1. By means of Theorem 5.2 the submanifold M can be regarded as a real hypersurface of QP ^(n+1)/4 which is totally geodesic in QP ^(n+p)/4 . By the same method as in the proof of Theorem 1.1, we can easily see that

A ^′ ϕ ^′ = ϕ ^′ A ^′ , A ^′ ψ ^′ = ψ ^′ A ^′ , A ^′ θ ^′ = θ ^′ A ^′ ,

where A ^′ is the shape operator to a unit normal vector eld ξ ^′ to M in QP ^(n+1)/4 and {ϕ ^′ , ψ ^′ , θ ^′ } denote the almost contact 3-structure induced on M from the quaternionic K¨ ahler structure on QP ^(n+1)/4 . Now, ap- plying the theorem proved in [12, Theorem 10, pp. 57] by the rst author of this paper, we may conclude that M is isometric to

π(S ⁴ⁿ

⁺³ (r 1 ) × S ⁴ⁿ

⁺³ (r 2 )) (r ² ₁ + r ² ₂ = 1)

for some n 1 , n 2 with 4n 1 + 4n 2 = n − 3, that is, M is a tube over a totally geodesic submanifold QP ⁿ

. Moreover, A ^′ U = A ^′ V = A ^′ W = 0 implies that the radius of the tube is ^π ₄ (for details, see [2]). Thus M is isometric to

π(S ⁴ⁿ

⁺³ (1/ √

2) × S ⁴ⁿ

⁺³ (1/ √

2)) (4n 1 + 4n 2 = n − 3).  Remark. We consider special generalized Cliord tori in

S ⁿ⁺⁴ := {(x 1 , . . . , x n+5 ) ∈ R ⁽ⁿ⁺⁵⁾ |

n+5 ∑

i=1

x ² _i = 1 } dened by

S ⁴ⁿ

⁺³ (1/ √

2) × S ⁴ⁿ

⁺³ (1/ √ 2)

= {(x 1 , . . . , x n+5 ) ∈ S ⁿ⁺⁴ |

4n ∑

+4 i=1

x ² _i = 1 2 ,

n+5 ∑

i=4n

+5

x ² _i = 1

2 },

where 4n 1 + 4n 2 = n − 3 and n = 4s + 3 for some integer s. Then, since S ⁴ⁿ

⁺³ (1/ √

2) × S ⁴ⁿ

⁺³ (1/ √

2) is a real hypersurface of S ⁿ⁺⁴ , its shape operator ˜ A is of the form

A = diag(1, ˜ −1)

for suitable orthonormal basis. The multiplicities of 1 and −1 are 4n 1 +3 and 4n 2 + 3, respectively ([13]). By choosing the spheres so that they lie in quaternionic subspace, we have brations

S ³ → S ⁴ⁿ

⁺³ (1/ √

2) × S ⁴ⁿ

⁺³ (1/ √

2) → M n ^Q

,n

compatible with the Hopf bration π : S ⁿ⁺⁴ → QP ^(n+1)/4 , where we put

M _n ^Q

_,n

= π(S ⁴ⁿ

⁺³ (1/ √

2) × S ⁴ⁿ

⁺³ (1/ √ 2)).

In this case we can easily see that the geodesic distance from QP ⁿ

to M _n ^Q

_,n

is ^π ₄ and that its principal curvatures are 1, −1, and 0 with multiplicities n − 3 − 4n 1 , 4n 1 , 3, respectively (for details, see [2]). Fur- thermore, let ξ be a unit normal vector eld of M _n ^Q

_,n

and let {F, G, H}

be the canonical quaternionic K¨ ahler structure of QP ^(n+1)/4 . Then U = −F ξ, V = −Gξ, W = −Hξ

are principal vectors corresponding to the principal curvature 0, that is AU = 0, AV = 0, AW = 0,

where A denote the shape operator of M _n ^Q

_,n

in QP ^(n+1)/4 . Applying (2.29)–(2.31) to the real hypersurface M _n ^Q

_,n

, we obtain

trA = n − 3 − 8n 1 , trA ² = n − 3,

Ric(U, U ) + Ric(V, V ) + Ric(W, W ) = 3(n + 5), ρ = n ² + 7n − 6 + (n − 3 − 8n 1 ) ² .

Hence, for M _n ^Q

_,n

, we have 1

3 {Ric(U, U) + Ric(V, V ) + Ric(W, W )} + ρ − n ² ∥µ∥ ² = n ² + 8n − 1.

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Hyang Sook Kim

Department of Computational Mathematics School of Computer Aided Science

Inje University

Kimhae 621-749, Korea E-mail : [email protected] Jin Suk Pak

Department of Mathematics Kyungpook National University Daegu 702-701, Korea

E-mail : [email protected]