On submersion and immersion submanifolds of a quaternionic ... - EMIS

Acta Univ. Sapientiae, Mathematica, 8, 1 (2016) 5–21 DOI: 10.1515/ausm-2016-0001

On submersion and immersion submanifolds of a quaternionic projective space Esmail Abedi

Zahra Nazari

Department of Mathematics, Azarbaijan shahid Madani University, Iran email: [email protected]

Department of Mathematics, Azarbaijan shahid Madani University, Iran email: [email protected]

Abstract. We study submanifolds of a quaternionic projective space, it is of great interest how to pull down some formulae deduced for submanifolds of a sphere to those for submanifolds of a quaternionic projective space.

1

Introduction

It is well known that an odd-dimensional sphere is a circle bundle over the quaternionic projective space. Consequently, many geometric properties of the quaternionic projective space are inherited from those of the sphere. Let M be a connected real n-dimensional submanifold of real codimension n+p p of a quaternionic K¨ ahler manifold M with quaternionic Kähler structure {F, G, H}. If there exists an r-dimensional normal distribution ν of the normal bundle TM⊥ such that Fνx ⊂ νx , Gνx ⊂ νx , Hνx ⊂ νx , ⊥ ⊂ T M, Hν⊥ ⊂ T M, Fν⊥ ⊂ T M, Gν x x x x x x 2010 Mathematics Subject Classification: 53C25, 53C40 Key words and phrases: quaternionic projective space, submersion and immersion submanifold

5

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E. Abedi, Z. Nazari

at each point x in M, then M is called a QR-submanifold of r QR-dimension, where ν⊥ denotes the complementary orthogonal distribution to ν in TM [2, 14, 16]. Equivalently, there exists distributions (Dx , D⊥ x ) of the tangent bundle TM, such that FDx ⊂ Dx , GDx ⊂ Dx , HDx ⊂ Dx , ⊥ , GD⊥ ⊂ T M⊥ , HD⊥ ⊂ T M⊥ , FD⊥ ⊂ T M x x x x x x where D⊥ x denotes the complementary orthogonal distribution to Dx in TM. Real hypersurfaces, which are typical examples of QR-submanifold with r = 0, have been investigated by many authors [3, 9, 14, 16, 18, 20] in connection with the shape operator and the induced almost contact 3-structure. Recently, Kwon and Pak have studied QR-submanifolds of (p − 1) QR-dimension ison+p metrically immersed in a quaternionic projective space QP 4 [14, 16]. Pak and Sohn studied n-dimensional QR-submanifold of (p−1) QR-dimension (n+p)

in a quaternionic projective space QP 4 [19]. Kim and Pak studied n-dimensional QR-submanifold of maximal QR-dimension isometrically immersed in a quaternionic projective space [13].

2

Preliminaries

Let M be a real (n + p)-dimensional quaternionic Kähler manifold. Then, by definition, there is a 3-dimensional vector bundle V consisting 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 F2 = −I,

G2 = −I,

FG = −GF = H,

H2 = −I,

GH = −HG = F,

(1) HF = −FH = 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      ∇F 0 r −q F  ∇G  = −r 0 p G (2) q −p 0 H ∇H where p, q and r are local 1-forms defined in U. Such a local basis {F, G, H} is called a canonical local basis of the bundle V in U (cf. [11, 12]).

On submersion and immersion submanifolds of a quaternionic

7

For canonical local basis {F, G, H} and {F 0 , G 0 , H 0 } of V in coordinate neighbor0 0 hoods of U and U , it follows that in U ∩ U  0   F F  G 0  = sxy  G  (x, y = 1, 2, 3) H H0 where sxy are local differentiable functions with (sxy ) ∈ SO(3) as a consequence of (1). As is well known [11, 12], every quaternionic Kähler 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 definition there is a unit normal vector field ξ such that ν⊥ x = span{ξ} at each point x in M. We set Fξ = −U,

Gξ = −V,

Hξ = −W.

(3)

Denoting by Dx the maximal quaternionic invariant subspace Tx M ∩ FTx M ∩ GTx M ∩ HTx M, ⊥ of Tx M, we have D⊥ x ⊃ Span{U, V, W}, where Dx means the complementary orthogonal subspace to Dx in Tx M. But, using (1), we can prove that D⊥ x = Span{U, V, W} [2, 16]. Thus we have

Tx M = Dx ⊕ Span{U, V, W},

∀x ∈ M,

which together with (1) and (3) imply FTx M, GTx M, HTx M ⊂ Tx M ⊕ Span{ξ}. Therefore, for any tangent vector field X and for a local orthonormal basis {ξα }α=1,...,p (ξ1 := ξ) of normal vectors to M, we have FX = ϕX + u(X)ξ,

GX = ψX + v(X)ξ,

Fξα = −Uα + P1 ξα , Hξα = −Wα + P3 ξα ,

HX = θX + ω(X)ξ,

(4)

Gξα = −Vα + P2 ξα , (α = 1, . . . , p).

(5)

Then it is easily seen that {ϕ, ψ, θ} and {P1 , P2 , P3 } are skew-symmetric endomorphisms acting on Tx M and Tx M⊥ , respectively. Also, from the hermitian properties g(FX, ξα ) = −g(X, Fξα ), g(HX, ξα ) = −g(X, Hξα ),

g(GX, ξα ) = −g(X, Gξα ), (α = 1, . . . , p).

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E. Abedi, Z. Nazari

It follows that g(X, Uα ) = u(X)δ1α , g(X, Vα ) = v(X)δ1α , g(X, Wα ) = w(X)δ1α , and hence g(X, U1 ) = u(X), g(X, V1 ) = v(X), g(X, W1 ) = w(X), Uα = 0, Vα = 0, Wα = 0, (α = 2, . . . p).

(6)

On the other hand, comparing (3) and (5) with α = 1, we have U1 = U, V1 = V, W1 = W, which together with (3) and (6) imply g(X, U) = u(X), u(U) = 1, Fξ = −U, Fξα=P1 ξα ,

g(X, V) = v(X), v(V) = 1, Gξ = −V, Gξα=P2 ξα

g(X, W) = w(X), w(W) = 1, Hξ = −W Hξα=P3 ξα , (α = 2, . . . , p).

Now, let ∇ be the Levi-Civita connection on M and ∇⊥ the normal connection induced from ∇ in the normal bundle TM⊥ of M. The Gauss and Weingarten formula are given by ∇X Y = ∇X Y + h(X, Y), ∇X ξα = −Aα X + ∇⊥ X ξα , (α = 1, . . . , p),

(7)

for any X, Y ∈ χ(M) and ξα ∈ Γ ∞ (T (M)⊥ ), (α = 1, . . . , p). h is the second fundamental form and Aα are shape operator corresponding to ξα . We have the following Gauss equation g(R(X, Y)Z, W) = g(R(X, Y)Z, W) −

p X

{g(Aa Y, Z)g(Aa X, W) − g(Aa X, Z)g(Aa Y, W)},

(8)

i=1

and Codazzi and Ricci equations g(R(X, Y)Z, ξa ) = (∇X Aa )Y − (∇Y Aa )X p X − {sba (X)g(Aa Y, Z) − sba (Y)g(Aa X, Z)}, b=1

g(R(X, Y)ξa , ξb ) = g([Ab , Aa ]X, Y) + g(R⊥ (X, Y)ξa , ξb ),

(9)

where R and R are the curvature tensors of M and M, respectively. sab are called the coefficients of the third fundamental form of M in M, such that satisfy sab = −sba .


3

9

The principal circle bundle S4n+3 (QPn , S3 )

Let Qn+1 be the (n + 1)-dimensional quaternionic space with natural quaternionic K¨ ahler structure ({F 0 , G 0 , H 0 }, h, i) and let S4n+3 be the unit sphere defined by S4n+3 = {(w1 , . . . , wn+1 ) ∈ Qn+1 |

n+1 X

wi (wi )∗ = 1}

i=1 1 1 1 1 n+1 n+1 n+1 n+1 = {(x1 , x2 , x3 , x4 , . . . , x1 , x2 , x3 , x4 ) n+1 X (xi1 )2 + (xi2 )2 + (xi3 )2 + (xi4 )2 = 1}. i=1

∈ R4n+4 |

such that w∗ = (x1 , −x2 , −x3 , −x4 ). The unit normal vector field ξ to S4n+3 is given by ξ=−

n+1 X

(xi1

i=1

∂ ∂ ∂ ∂ + xi2 i + xi3 i + xi4 i ). i ∂x1 ∂x2 ∂x3 ∂x4

From hF 0 ξ, ξi = hF 02 ξ, F 0 ξi = −hξ, F 0 ξi, hG 0 ξ, ξi = hG 02 ξ, G 0 ξi = −hξ, G 0 ξi, hH 0 ξ, ξi = hH 02 ξ, H 0 ξi = −hξ, H 0 ξi, it follows hF 0 ξ, ξi = 0, hG 0 ξ, ξi = 0, hH 0 ξ, ξi = 0, that is, F 0 ξ, G 0 ξ, H 0 ξ ∈ T (S4n+3 ). We put F 0 ξ = −ιU 0 ,

G 0 ξ = −ιV 0 ,

H 0 ξ = −ιW 0 ,

(10)

where ι denotes the immersion of S4n+3 into Qn+1 . From the Hermitian property of h, i, it is easily seen that U 0 , V 0 , W 0 are unit tangent vector field of S4n+3 . We put Hp (S4n+3 ) = {X 0 ∈ Tp (S4n+3 )|u 0 (X 0 ) = 0, v 0 (X 0 ) = 0, w 0 (X 0 ) = 0}. Then u 0 , v 0 , w 0 define a connection form of the principal bundle S4n+3 (QPn , S3 ) and we have Tp (S4n+3 ) = Hp (S4n+3 ) ⊕ span{Up0 , Vp0 , Wp0 }.

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E. Abedi, Z. Nazari

We call Hp (S4n+3 ) and span{Up0 , Vp0 , Wp0 } the horizontal subspace and vertical subspace of Tp (S4n+3 ), respectively. By definition, the horizontal subspace Hp (S4n+3 ) is isomorphic to Tπ(p) (QPn ), where π is the natural projection from S4n+3 onto QPn . Therefore, for a vector field X on QPn , there exists unique horizontal vector field X 0 of S4n+3 such that π(X 0 ) = X. The vector field X 0 is called the horizontal lift of X and we denote it by X∗ . Proposition 1 As a subspace of Tp (Qn+1 ), Hp (S4n+3 ) is {F 0 , G 0 , H 0 }-invariant subspace. Proof. By definition (10) of the vertical vector field {U 0 , V 0 , W 0 }, for X 0 ∈ Hp (S4n+3 ), it follows hF 0 ιX 0 , ξi = −hιX 0 , F 0 ξi = hιX 0 , ιU 0 i = 0, hG 0 ιX 0 , ξi = −hιX 0 , G 0 ξi = hιX 0 , ιV 0 i = 0, hH 0 ιX 0 , ξi = −hιX 0 , H 0 ξi = hιX 0 , ιW 0 i = 0. This shows that F 0 ιX 0 , G 0 ιX 0 , H 0 ιX 0 ∈ Tp (S4n+3 ). In entirely the same way we compute hF 0 ιX 0 , ιU 0 i = −hιX 0 , F 0 ιU 0 i = hιX 0 , −ξi = 0, hG 0 ιX 0 , ιV 0 i = −hιX 0 , G 0 ιV 0 i = hιX 0 , −ξi = 0, hH 0 ιX 0 , ιW 0 i = −hιX 0 , H 0 ιW 0 i = hιX 0 , −ξi = 0. By use from relations F 0 V 0 = −H 0 ξ,

F 0 W 0 = G 0 ξ,

G 0 U 0 = H 0 ξ,

G 0 W 0 = F 0 ξ,

H 0 U 0 = −G 0 ξ,

H 0 V 0 = F 0 ξ,

we have hF 0 ιX 0 , ιV 0 i = −hιX 0 , F 0 ιV 0 i = hιX 0 , H 0 ξi = −hιX 0 , ιW 0 i = 0, hF 0 ιX 0 , ιW 0 i = −hιX 0 , F 0 ιW 0 i = hιX 0 , G 0 ξi = −hιX 0 , ιV 0 i = 0, hG 0 ιX 0 , ιU 0 i = −hιX 0 , G 0 ιU 0 i = hιX 0 , H 0 ξi = −hιX 0 , ιW 0 i = 0, hG 0 ιX 0 , ιW 0 i = −hιX 0 , G 0 ιW 0 i = hιX 0 , F 0 ξi = −hιX 0 , ιU 0 i = 0, hH 0 ιX 0 , ιV 0 i = −hιX 0 , H 0 ιV 0 i = hιX 0 , F 0 ξi = −hιX 0 , ιU 0 i = 0, hH 0 ιX 0 , ιU 0 i = −hιX 0 , H 0 ιU 0 i = hιX 0 , G 0 ξi = −hιX 0 , ιV 0 i = 0. and hence F 0 ιX 0 , G 0 ιX 0 , H 0 ιX 0 ∈ Hp (S4n+3 ), which completes the proof.


11

Therefore, the almost quaternionic structure {F, G, H} can be induced on Tπ(p) (QPn ) and we set (FX)∗ = F 0 ιX∗ ,

(GX)∗ = G 0 ιX∗ ,

(HX)∗ = H 0 ιX∗ .

(11)

Next, using the Gauss formula (7) for the vertical vector field {U 0 , V 0 , W 0 } and a horizontal vector fields X 0 of Tp (S4n+3 ), we compute ∇EX 0 U 0 = ∇X0 0 U 0 + g 0 (A 0 X 0 , U 0 )ξ = ∇X0 0 U 0 + hX 0 , U 0 iξ = ∇X0 0 U 0 ,

(12)

by similar computation for vector fields {V 0 , W 0 } we have ∇EX 0 V 0 = ∇X0 0 V 0 ,

(13)

∇EX 0 W 0 = ∇X0 0 W 0 ,

where ∇E denotes the Euclidean connection of E4n+4 , ∇ 0 denotes the connection of S4n+3 and A 0 denotes the shape operator with respect to ξ. Now, using relations (10), (12) and (13) and the Weingarten formula (7), we conclude ∇X0 0 U 0 = −∇EX 0 F 0 ξ 0 = −(∇EX 0 F 0 )ξ − F 0 ∇EX 0 ξ = −(r(X 0 )G 0 ξ − q(X 0 )H 0 ξ) − F 0 ∇EX 0 ξ = r(X 0 )V 0 − q(X 0 )W 0 + F 0 (ιA 0 X 0 )

(14)

= r(X 0 )V 0 − q(X 0 )W 0 + F 0 ιX 0 by similar computation for vector fields {V 0 , W 0 } we have ∇X0 0 V 0 = −∇EX 0 G 0 ξ 0 = −r(X 0 )U 0 + p(X 0 )W 0 + G 0 ιX 0 , ∇X0 0 W 0 = −∇EX 0 H 0 ξ 0 = q(X 0 )U 0 − p(X 0 )V 0 + H 0 ιX 0 .

(15)

Consequently, according to notation (11), relations (14) and (15) can be written as ∇X0 ∗ U 0 = r(X∗ )V 0 − q(X∗ )W 0 + (FX)∗ , ∇X0 ∗ V 0 = −r(X∗ )U 0 + p(X∗ )W 0 + (GX)∗ , ∇X0 ∗ W 0

∗

0

∗

0

(16)

∗

= q(X )U − p(X )V + (HX) .

We note that, since by definition, the Lie derivative of a horizontal lift of a vector field with respect to a vertical vector field is zero, it follows 0 = LU 0 X∗ = [U 0 , X∗ ] = ∇U0 0 X∗ − ∇X0 ∗ U 0 , 0 = LV 0 X∗ = [V 0 , X∗ ] = ∇V0 0 X∗ − ∇X0 ∗ V 0 0 ∗ 0 0 0 = LW 0 X∗ = [W 0 , X∗ ] = ∇W 0 X − ∇X ∗ W

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E. Abedi, Z. Nazari

and using (16), we conclude ∇U0 0 X∗ = r(X∗ )V 0 − q(X∗ )W 0 + (FX)∗ , ∇V0 0 X∗ = −r(X∗ )U 0 + p(X∗ )W 0 + (GX)∗ , 0 ∗ ∇W 0X

∗

0

∗

0

(17)

∗

= q(X )U − p(X )V + (HX) .

We define a Riemannian metric g and a connection ∇ in QPn respectively by g(X, Y) = g 0 (X∗ , Y ∗ ), ∇X Y =

π(∇X0 ∗ Y ∗ ).

(18) (19)

Then (∇X0 Y)∗ is the horizontal part of ∇X0 ∗ Y ∗ and therefore ∇X0 ∗ Y ∗ = (∇X0 Y)∗ + g 0 (∇X0 ∗ Y ∗ , U 0 )U 0 + g 0 (∇X0 ∗ Y ∗ , V 0 )V 0 + g 0 (∇X0 ∗ Y ∗ , W 0 )W 0 .

(20)

Using relations (16) and (18), we compute g 0 (∇X0 ∗ Y ∗ , U 0 ) = −g 0 (Y ∗ , ∇X0 ∗ U 0 ) = −g 0 (Y ∗ , r(X∗ )V 0 − q(X∗ )W 0 + (FX)∗ ) = −g(Y, FX), g 0 (∇X0 ∗ Y ∗ , V 0 ) = −g(Y, GX), g 0 (∇X0 ∗ Y ∗ , V 0 ) = −g(Y, HX), and, using (20), we conclude ∇X0 ∗ Y ∗ = (∇X Y)∗ − g(Y, FX)U 0 − g(Y, GX)V 0 − g(Y, HX)W 0 .

(21)

Proposition 2 ∇ is the Levi-Civita connection for g. Proof. Let T be the torsion tensor field of ∇. Then we have T (X, Y) = ∇X Y − ∇Y X − [X, Y] = π(∇X0 ∗ Y ∗ ) − π(∇Y0 ∗ X∗ ) − [πX∗ , πY ∗ ] = π(∇X0 ∗ Y ∗ − ∇Y0 ∗ X∗ − [X∗ , Y ∗ ]) = π(T 0 (X∗ , Y ∗ )) = 0, hence ∇ is torsion free. We now show that ∇ is a metric connection. (∇X g)(Y, Z) = X(g(Y, Z)) − g(∇X Y, Z) − g(Y, ∇X Z) = X∗ (g 0 (Y ∗ , Z∗ )) − g 0 ((∇X Y)∗ , Z∗ ) − g 0 (Y ∗ , (∇X Z)∗ ).


13

Since Z∗ is horizontal vector field, from relation (20), it follows that g 0 ((∇X Y)∗ , Z∗ ) = g 0 (∇X0 ∗ Y ∗ , Z∗ ) − g 0 (∇X0 ∗ Y ∗ , U 0 )g 0 (U 0 , Z∗ ) − g 0 (∇X0 ∗ Y ∗ , V 0 )g 0 (V 0 , Z∗ ) − g 0 (∇X0 ∗ Y ∗ , U 0 )g 0 (U 0 , Z∗ ) = g 0 (∇X0 ∗ Y ∗ , Z∗ ) and we compute (∇X g)(Y, Z) = X∗ (g 0 (Y ∗ , Z∗ )) − g 0 (∇X0 ∗ Y ∗ , Z∗ ) − g 0 (∇X0 ∗ Z∗ , Y ∗ ) = (∇X0 ∗ g 0 )(Y ∗ , Z∗ ), where we have used the fact that ∇ 0 is the Levi-Civita connection for g 0 . Thus ∇ is the Levi-Civita connection for g and the proof is complete. Now, by (21), it follows [X∗ , Y ∗ ] = [X, Y]∗ + g 0 ([X∗ , Y ∗ ], U 0 )U 0 + g 0 ([X∗ , Y ∗ ], V 0 )V 0 + g 0 ([X∗ , Y ∗ ], W 0 )W 0 = [X, Y]∗ + g 0 (∇X0 ∗ Y ∗ − ∇Y0 ∗ X∗ , U 0 )U 0 + g 0 (∇X0 ∗ Y ∗ − ∇Y0 ∗ X∗ , V 0 )V 0 + g 0 (∇X0 ∗ Y ∗ − ∇Y0 ∗ X∗ , W 0 )W 0 = [X, Y]∗ + g 0 ((∇X0 Y)∗ − g(Y, FX)U 0 − g(Y, GX)V 0 − g(Y, HX)W 0 , U 0 )U 0 − g 0 ((∇Y X)∗ − g(FY, X)U 0 − g(GY, X)V 0 − g(HY, X)W 0 , U 0 )U 0 + g 0 ((∇X Y)∗

(22)

− g(Y, FX)U 0 − g(Y, GX)V 0 − g(Y, HX)W 0 , V 0 )V 0 − g 0 ((∇Y X)∗ − g(FY, X)U 0 − g(GY, X)V 0 − g(HY, X)W 0 , V 0 )V 0 + g 0 ((∇X Y)∗ − g(Y, FX)U 0 − g(Y, GX)V 0 − g(Y, HX)W 0 , W 0 )W 0 − g 0 ((∇Y X)∗ − g(FY, X)U 0 − g(GY, X)V 0 − g(HY, X)W 0 , W 0 )W 0 = [X, Y]∗ − 2g(Y, FX)U 0 − 2g(Y, GX)V 0 − 2g(Y, HX)W 0 . Consequently, using (16), (17), (21) and (22), the curvature tensor R of QPn is calculated as follows: R(X, Y)Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y] Z 0 ∗ = π{∇X0 ∗ (∇Y Z)∗ ) − ∇Y0 ∗ (∇X Z)∗ ) − ∇[X,Y] ∗ Z )}

= π{∇X0 ∗ (∇Y0 ∗ Z∗ + g(Z, FY)U 0 + g(Z, GY)V 0 + g(Z, HY)W 0 ) − ∇Y0 ∗ (∇X0 ∗ Z∗ + g(Z, FX)U 0 + g(Z, GX)V 0 + g(Z, HX)W 0 ) 0 ∗ − ∇[X ∗ ,Y ∗ ]+2g(Y,FX)U 0 +2g(Y,GX)V 0 +2g(Y,HX)W 0 Z )}

14

E. Abedi, Z. Nazari = π{∇X0 ∗ ∇Y0 ∗ Z∗ + g(Z, FY)∇X0 ∗ U 0 + g(Z, GY)∇X0 ∗ V 0 + g(Z, HY)∇X0 ∗ W 0 − ∇Y0 ∗ ∇X0 ∗ Z∗ − g(Z, FX)∇Y0 ∗ U 0 0 ∗ − g(Z, GX)∇Y0 ∗ V 0 − g(Z, HX)∇X0 ∗ W 0 − ∇[X ∗ ,Y ∗ ] Z 0 ∗ − 2g(Y, FX)∇U0 0 Z∗ − 2g(Y, GX)∇V0 0 Z∗ − 2g(Y, HX)∇W 0Z }

= π{R 0 (X∗ , Y ∗ )Z∗ + g(Z, FY)(r(X∗ )V 0 − q(X∗ )W 0 + (FX)∗ ) + g(Z, GY)(−r(X∗ )U 0 + p(X∗ )W 0 + (GX)∗ ) + g(Z, HY)(q(X∗ )U 0 − p(X∗ )V 0 + (HX)∗ ) + g(Z, FX)(r(Y ∗ )V 0 − q(Y ∗ )W 0 + (FY)∗ ) + g(Z, GX)(−r(Y ∗ )U 0 + p(Y ∗ )W 0 + (GY)∗ ) + g(Z, HX)(q(Y ∗ )U 0 − p(Y ∗ )V 0 + (HY)∗ ) + 2g(Y, FX)(r(Z∗ )V 0 − q(Z∗ )W 0 + (FZ)∗ ) + 2g(Y, GX)(−r(Z∗ )U 0 + p(Z∗ )W 0 + (GZ)∗ ) + 2g(Y, HX)q(Z∗ )U 0 − p(Z∗ )V 0 + (HZ)∗ }. Since the curvature tensor R 0 of S4n+3 satisfies R 0 (X∗ , Y ∗ )Z∗ = g 0 (Y ∗ , Z∗ )X∗ − g 0 (X∗ , Z∗ )Y ∗ = g(Y, Z)X∗ − g(X, Z)Y ∗ ,

(23)

we conclude that the curvature tensor of QPn is given by R(X, Y)Z = g(Y, Z)X − g(X, Z)Y + g(FY, Z)FX − g(FX, Z)FY − 2g(FX, Y)FZ + g(GY, Z)GX − g(GX, Z)GY − 2g(GX, Y)GZ

(24)

+ g(HY, Z)HX − g(HX, Z)HY − 2g(HX, Y)HZ.

4

Submanifolds of quaternionic manifolds n+p

Let M be an n-dimensional submanifold of QP 4 and π−1 (M) be the circle bundle over M which is compatible with the Hopf map π : Sn+p+3 −→ QP

n+p 4

.

Then π−1 (M) is a submanifold of Sn+p+3 . The compatibility with the Hopf map is expressed by πoι 0 = ιoπ where ι 0 and ι are the immersions of M into n+p QP 4 and π−1 (M) into Sn+p+3 , respectively. n+p Let ξa , a = 1, . . . , p be orthonormal normal local fields to M in QP 4 and


15

ξ∗a be the horizontal lifts of ξa . Then ξ∗a are mutually orthonormal normal local fields to π−1 (M) in Sn+p+3 . At each point y ∈ π−1 (M) we compute gS (ι 0 X∗ , ξ∗a ) = gS ((ιX)∗ , ξ∗a ) = g(ιX, ξa ) = 0, gS (ι 0 U, ξ∗a ) = gS (U 0 , ξ∗a ) = 0, gS (ι 0 V, ξ∗a ) = gS (V 0 , ξ∗a ) = 0, gS (ι 0 W, ξ∗a ) = gS (W 0 , ξ∗a ) = 0, gS (ξ∗a ξ∗b ) = g(ξa , ξb ) = δab , n+p

where gS and g denote the Riemannian metric on Sn+p+3 and QP 4 , respectively. Here U 0 = ι 0 U, V 0 = ι 0 V, W 0 = ι 0 W are unit tangent vector field of Sn+p+3 defined by relation (10). Now, let ∇S , ∇ 0 , ∇ and ∇ be the Riemannian connections of Sn+p+3 , π−1 (M), n+p QP 4 and M, respectively. By means of the Gauss formula (7) and relations (4) and (21), we compute ∇SX∗ ι 0 Y ∗ = ∇SX∗ (ιY)∗ = (∇X ιY)∗ − g(FιX, ιY)ι 0 U − g(GιX, ιY)ι 0 V − g(HιX, ιY)ι 0 W = (ι∇X Y + h(X, Y))∗ − g(ιϕX, ιY)U 0 − g(ιψX, ιY)V 0 − g(ιθX, ιY)W 0

(25)

= ι 0 (∇X Y)∗ + (h(X, Y))∗ − g(ϕX, Y)U 0 − g(ψX, Y)V 0 − g(θX, Y)W 0 where g is the metric on M. On the other hand, we also have ∇SX∗ ι 0 Y ∗ = ι 0 ∇X0 ∗ Y ∗ + h 0 (X∗ , Y ∗ ) = ι 0 ((∇X Y)∗ − g(ϕX, Y)U − g(ψX, Y)V − g(θX, Y)W) + h 0 (X∗ , Y ∗ )

(26)

where h and h 0 denote the second fundamental form of M and π−1 (M), respectively. Comparing the vertical part and horizontal part of relations (25) and (26), we conclude (h(X, Y))∗ = h 0 (X∗ , Y ∗ ),

(27)

that is, p X a=1

p p X X g 0 (Aa0 X∗ , Y ∗ )ξ∗a = ( g(Aa X, Y)ξa )∗ = g(Aa X, Y)ξ∗a , a=1

a=1

16

E. Abedi, Z. Nazari

where Aa and Aa0 are the shape operators with respect to normal vector fields ξa and ξ∗a of M and π−1 (M), respectively. Consequently, we have g 0 (Aa0 X∗ , Y ∗ ) = g(Aa X, Y), (a = 1, ..., p). Next, using the weingarten formula, we calculate ∇SX∗ ξ∗a as follows ∇SX∗ ξ∗a = −ι 0 Aa0 X∗ + ∇X0⊥∗ ξ∗a = −ι 0 Aa0 X∗ +

p X

0 sab (X∗ )ξ∗b .

(28)

b=1

where ∇ 0⊥ is normal connection π−1 (M) in Sn+p+3 . On the other hand, from relation (21), it follows ∇SX∗ ξ∗a = (∇X ξa )∗ − g(FιX, ξa )ι 0 U − g(GιX, ξa )ι 0 V − g(HιX, ξa )ι 0 W ∗ = (−ιAa X + ∇⊥ X ξa ) −

p X {ub (X)g(ξa , ξb )U 0 b=1

+ vb (X)g(ξa , ξb )V 0 + wb (X)g(ξa , ξb )W 0 } = −ι 0 (Aa X)∗ +

p X (sab (X∗ )ξb )∗ − ua (X)U 0 b=1 a

− v (X)V − w (X)W 0 , a

0

(29)

where ∇⊥ is normal connection M in QP We have put Fιx = ιϕX +

n+p 4

p X

Gιx = ιψX +

.

ua (X)ξa ,

a=1 p X

va (X)ξa ,

a=1

Hιx = ιθX +

p X

wa (X)ξa .

(30)

a=1

Comparing relations (28) and (29), we obtain Aa0 X∗ = (Aa X)∗ + ua (X)U 0 + va (X)V 0 + wa (X)W 0 = (Aa X)∗ + g(Ua , X)U 0 + g(Va , X)V + g(Wa , X)W 0 , ∗ ∇X0⊥∗ ξ∗a = (∇⊥ X ξa ) ,

(31)


17

0 (X∗ ) = s (X)∗ , where U , V , W are defined by that is, sab a a a ab

Fξa = −Ua + Gξa = −Va +

p X

P1ab ξb ,

b=1 p X

P2ab ξb ,

b=1 p X

Hξa = −Wa +

(32)

P3ab ξb .

b=1

Pp

where, b=1 Piab ξb = Pi ξa , (i = 1, 2, 3). Now, we consider ∇SU ξ∗a and using relations (17) and (32) imply ∇SU ξ∗a = (Fξa )∗ = −ιUa∗ + P1 ξ∗a , ∇SV ξ∗a = (Gξa )∗ = −ιVa∗ + P2 ξ∗a , ∇SW ξ∗a = (Hξa )∗ = −ιWa∗ + P3 ξ∗a .

(33)

On the other hand, from the Weingarten formula, it follows ∇SU ξ∗a

= −ι

0

Aa0 U

+

∇U0⊥ ξ∗a

= −ι

0

Aa0 U

+

p X

0 sab (U)ξ∗b ,

b=1

∇SV ξ∗a = −ι 0 Aa0 V + ∇V0⊥ ξ∗a = −ι 0 Aa0 V +

p X

0 sab (V)ξ∗b ,

b=1 p X

0⊥ ∗ ξa = −ι 0 Aa0 W + ∇SW ξ∗a = −ι 0 Aa0 W + ∇W

0 sab (W)ξ∗b .

(34)

b=1

Consequently, using (33) and (34), we obtain Aa0 U = U∗a , Aa0 V = Va∗ , Aa0 W = Wa∗ , 0 0 0 sab (U) = P1 , sab (V) = P2 , sab (W) = P3 ,

(35)

∇U0⊥ ξ∗a ∇V0⊥ ξ∗a 0⊥ ∗ ∇W ξa

(36)

∗

= (FX) + ιUa∗ , = (GX)∗ + ιVa∗ , = (HX)∗ + ιWa∗ .

The first relations of (31) and (35), we get g 0 (Aa0 Ab0 X∗ , Y ∗ ) = g(Aa Ab X, Y) + ub (X)ua (Y) + vb (X)va (Y) + wb (X)wa (Y),

(37)

18

E. Abedi, Z. Nazari

and especially, g 0 (Aa02 X∗ , Y ∗ ) = g(A2a X, Y) + ua (X)ua (Y) + va (X)va (Y) + wa (X)wa (Y),

(38)

For x ∈ M, let {e1 , . . . , en }be an orthonormal basis of Tx M and y be a point of π−1 (M) such that π(y) = x. We take an orthonormal basis {e∗1 , . . . , e∗n , U, V, W} of Ty (π−1 (M)). Then, using the first relations (35) and (39), we compute p X

trace Aa02

=

a=1

p X n X

g 0 (Aa02 e∗i , e∗i ) + g 0 (Aa02 U, U)

a=1

i=1 + g (Aa02 V, V) p X n X 0

=

+ g 0 (Aa02 W, W)}

g(Aa02 ei , ei ) + ua (ei )ua (ei )

a=1 a

i=1 a

a

(39)

a

+ v (ei )v (ei ) + w (ei )w (ei ) + g 0 (Aa0 U, Aa0 U) + g 0 (Aa0 V, Aa0 V) + g 0 (Aa0 W, Aa0 W)} =

p X

{trace A2a + 2g(Ua , Ua ) + 2g(Va , Va ) + 2g(Wa , Wa )},

a=1

we conclude Proposition 3 Under the above assumptions, the following inequality p X a=1

trace Aa02 ≥

p X

trace A2a

a=1

is always valid. The equality holds, if and only if M is a {F, G, H}- invariant submanifold. Corollary 1 [8] M is a totally geodesic submanifold if and only if relation Aξ = 0 holds for any normal vector field ξ of M. Particularly, M is totally geodesic if and only if A1 = . . . = Ap = 0 for an orthonormal frame field ξ1 , . . . , ξp of T ⊥ (M) Proposition 4 Under the condition stated above, if π−1 (M) is a totally geodesic submanifold of Sn+p+3 , then M is a totally geodesic {F, G, H}- invariant submanifold.


19

Proof. Since π−1 (M) is a totally geodesic submanifold of Sn+p+3 , using Corollary (26), it follows Aa0 = 0. The first Relation (31) then implies Aa = 0 and Ua = Va = Wa = 0, which, using relation (32), completes the proof. Further, for the normal curvature of M in QP second relation (9), we obtain

n+p 4

, using relation (24) and the

g(R⊥ (X, Y)ξa , ξb ) = g([Aa , Ab ]X, Y) + ub (X)ua (Y) − ua (X)ub (Y) + vb (X)va (Y) − va (X)vb (Y) + wb (X)wa (Y) − wa (X)wb (Y)

(40)

− 2g(ϕX, Y)P1 − 2g(ψX, Y)P2 − 2g(θX, Y)P3 Therefore, if M be a totally geodesic submanifold of {F, G, H}- invariant submanifold, we conclude g(R⊥ (X, Y)ξa , ξb ) = −2g(ϕX, Y)P1 − 2g(ψX, Y)P2 − 2g(θX, Y)P3

(41)

In this case the normal space Tx⊥ (M) is also {F, G, H}- invariant and P1 , P2 , P3 never vanish. We have thus proved Proposition 5 The normal curvature of a totally geodesic submanifold of {F, G, H}- invariant submanifold of a quaternionic projective space never vanishes. This proposition show that the normal connection of the quaternionic projective space which is immersed standardly in a higher dimensional quaternionic projective space not flat. Finally, we give a relation between the normal curvature R⊥ and R 0⊥ of M n+p and π−1 (M), respectively, where M is a n-dimensional submanifold of QP 4 and π−1 (M) is the circle bundle over M which is compatible with the Hopf map π. Using relation (37), we obtain g 0 ([Aa0 , Ab0 ]X∗ , Y ∗ ) = g([Aa , Ab ]X, Y) + ub (X)ua (Y) − ua (X)ub (Y) + vb (X)va (Y) − va (X)vb (Y) + wb (X)wa (Y) − wa (X)wb (Y), and therefore, from the second relation (9), it follows −gS (R 0S (ι 0 X∗ , ι 0 Y ∗ )ξ∗a , ξ∗b ) + gS (R 0⊥ (X∗ , Y ∗ )ξ∗a , ξ∗b ) = −g(R(ιX, ιY)ξa , ξb ) + g(R⊥ (X, Y)ξa , ξb ) + ub (X)ua (Y) − ua (X)ub (Y) + vb (X)va (Y) − va (X)vb (Y) + wb (X)wa (Y) − wa (X)wb (Y).

20

E. Abedi, Z. Nazari

Using the expression (23) and (24) , for curvature of Sn+p+3 and QP respectively and relations (30) and (32) imply

n+p 4

(C),

gS (R⊥ (X∗ , Y ∗ )ξ∗a , ξ∗b ) = g(R⊥ (X, Y)ξa , ξb ) + 2g(ϕX, Y)P1 + 2g(ψX, Y)P2 + 2g(θX, Y)P3

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[12] S. Ishihara, Differential geometry of fibred spaces, Publication of the study group of geometry, vol. 8, Tokyo, 1973. [13] H. S. Kim, J. S. Pak, QR-submanifolds of maximal QR-dimension in quaternionic projective space, J. Korean Math. Soc, 42 (2005), 655–672. [14] J. -H. Kwon, J. S. Pak, Scalar curvature of QR-submanifolds immersed in a quaternionic projective space, Saitama Math. J., 17 (1999), 89–116. [15] K. Kenmotsu, A class of almost Contact Riemannian manifolds, Tohoku Math. J., 24 (1972), 93–103. [16] J. H. Kwon, J. S. Pak, QR-submanifolds of (p − 1) QR-dimension in a quaternionic projective space QP 89–116.

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Received: October 26, 2014