International Journal of Mathematics and Mathematical Sciences

Volume 2013 (2013), Article ID 193697, 11 pages

http://dx.doi.org/10.1155/2013/193697

Research Article

-Submanifolds of   -Dimension in a Quaternionic Projective Space under Some Curvature Conditions

Hyang Sook Kim¹ and Jin Suk Pak²

¹Department of Applied Mathematics, Institute of Basic Science, Inje University, Gimhae 621-749, Republic of Korea

²Kyungpook National University, Daegu 702-701, Republic of Korea

Received 4 March 2013; Accepted 9 May 2013

Academic Editor: Luc Vrancken

Copyright © 2013 Hyang Sook Kim and Jin Suk Pak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to study n-dimensional -submanifolds of -dimension in a quaternionic projective space and especially to determine such submanifolds under some curvature conditions.

1. Introduction

Let be a connected real -dimensional submanifold of real codimension of a quaternionic Kähler manifold with quaternionic Kähler structure . If there exists an -dimensional normal distribution of the normal bundle such that

at each point in , then is called a QR-submanifold of -dimension, where denotes the complementary orthogonal distribution to in (cf. [1–3]). Real hypersurfaces, which are typical examples of -submanifold with , have been investigated by many authors (cf. [2–9]) in connection with the shape operator and the induced almost contact -structure (for definition, see [10–13]). In their paper [2, 3], Kwon and Pak had studied -submanifolds of -dimension isometrically immersed in a quaternionic projective space and proved the following theorem as a quaternionic analogy to theorems given in [14, 15], which are natural extensions of theorems proved in [6] to the case of -submanifolds with -dimension and also extensions of theorems in [16].

Theorem K-P. Let   be an -dimensional -submanifold of -dimension isometrically immersed in a quaternionic projective space , and let the normal vector field   be parallel with respect to the normal connection. If the shape operator   corresponding to   satisfies

then   is locally a product of   where    and    belong to some  -  and -dimensional spheres (  is the Hopf fibration  ).

On the other hand, when is a real hypersurface of , if is an Einstein space or a locally symmetric space, then has a parallel second fundamental form (cf. [4, 6, 7, 9]). Projecting the quantities on onto in , we can consider -submanifolds of -dimension with the conditions corresponding to or . In this paper, we will study such -submanifolds isometrically immersed in and obtain Theorem 3 and other results stated in the last Section 5 as quaternionic analogies to theorems given in [16, 17] and as the extensions of theorems given in [18] by using Theorem K-P.

2. Preliminaries

Let be a real -dimensional quaternionic Kähler manifold. Then, by definition, there is a -dimensional vector bundle consisting of tensor fields of type over satisfying the following conditions (a), (b), and (c).

(a) In any coordinate neighborhood , there is a local basis , , of such that

(b) There is a Riemannian metric which is Hermite with respect to all of , , and .(c) For the Riemannian connection with respect to ,

 where , , and are local -forms defined in . Such a local basis is called a canonical local basis of the bundle in (cf. [10, 19, 20]).

For canonical local bases and of in coordinate neighborhoods and , it follows that in

where are local differentiable functions with as a consequence of (3). As is well known (cf. [19]), every quaternionic Kähler manifold is orientable.

Now let be an -dimensional -submanifold of -dimension isometrically immersed in . Then by definition, there is a unit normal vector field such that at each point in . We set

Denoting by the maximal quaternionic invariant subspace of , we have , where means the complementary orthogonal subspace to in (cf. [1–3]). Thus, we have

which together with (3) and (6) implies

Therefore, for any tangent vector field and for a local orthonormal basis of normal vectors to , we have

. Then it is easily seen that and are skew-symmetric endomorphisms acting on and , respectively. Moreover, the Hermitian property of implies

Also, from the hermitian properties , , and , it follows that

and hence,

On the other hand, comparing (6) and (10) with , we have , , and , which together with (6) and (14) implies

In the sequel, we will use the notations , , and instead of , , and .

Next, applying to the first equation of (9) and using (10), (14), and (15), we have

Similarly, we have

from which, taking account of the skew symmetry of , , and and using (11), we also have

So (10) can be rewritten in the form

. Applying and to the first equation of (9) and using (3), (9), and (20), we have

and consequently,

Similarly, the other equations of (9) yield

From the first three equations of (20), we also have

Equations (14)–(17), (19), and (22)–(24) tell us that admits the so-called almost contact 3-structure and consequently for some integer (cf. [12]).

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

for , tangent to . Here denotes the second fundamental form and the shape operator corresponding to . They are related by . Furthermore, put

where is the skew-symmetric matrix of connection forms of .

Differentiating the first equation of (9) covariantly and using (4), (9), (10), (14) (25), and (26), we have

From the other equations of (9), we also have

Next, differentiating the first equation of (20) covariantly and comparing the tangential and normal parts, we have

From the other equations of (20), we have similarly

Finally the equation of Gauss is given as follows (cf. [21]):

for , and tangent to , where and denote the Riemannian curvature tensor of and , respectively.

In the rest of this paper we assume that the distinguished normal vector field   is parallel with respect to the normal connection . Then it follows from (27) that , and consequently, (30)-(31) imply

On the other hand, since the curvature tensor of is of the form

for , and tangent to , (32) reduces to

3. Fibrations and Immersions

From now on -dimensional -submanifolds of -dimension isometrically immersed in only will be considered. Moreover, we will use the assumption and the notations as in Section 2.

Let be the hypersphere of radius (>0) in the quaternionic space of quaternionic dimension , which is identified with the Euclidean -space . The unit sphere will be briefly denoted by . Let be the natural projection of onto defined by the Hopf fibration . As is well known (cf. [10, 11, 20]), admits a Sasakian -structure whereby are mutually orthogonal unit Killing vector fields. Thus it follows that

where denotes the Riemannian connection with respect to the canonical metric on (cf. [6, 9–13]). Moreover, each fibre of in is a maximal integral submanifold of the distribution spanned by , and . Thus the base space admits the induced quaternionic Kähler structure of constant -sectional curvature (cf. [10, 11]). We have especially a fibration which is compatible with the Hopf fibration . More precisely speaking, is a fibration with totally geodesic fibers such that the following diagram is commutative:

xy(37)

where and are isometric immersions.

Now, let , and be the unit vector fields tangent to the fibers of such that , , and . (In what follows, we will again delete the and in our notation.) Furthermore, we denote by the horizontal lift of a vector field tangent to . Then, the horizontal lifts () of the normal vectors to form an orthonormal basis of normal vectors to in . Let and be the corresponding shape operators and normal connection forms, respectively. Then, as shown in [3, 9, 22], the fundamental equations for the submersion are given by

where denotes the Riemannian metric of induced from in and the Levi-Civita connection with respect to . The same equations are valid for the submersion by replacing , , and (resp., , , and ) with , , and (resp., , , and ), respectively. We denote by the normal connection of induced from . Since the diagram is commutative, implies

because of (10), (26), and (38), from which, comparing the tangential part, we have

Next, calculating and using (10), (26), and (40), we have

which yields

and similarly

Hence, (43) and (46) with imply

4. Co-Gauss Equations for the Submersion

In this section, we derive the co-Gauss and co-Codazzi equations of the submersion for later use.

Differentiating (38) with covariantly along and using (24), (38), and (39), we have

Similarly (38) with and (38) with give

respectively. On the other hand, it follows from (19), (24), (38), and (39) that

By means of (48) and (51), we have

where denotes the curvature tensor of with respect to the connection . Using (30), (31), (33), and (35), we can easily see that

By the same method, we can easily verify that (49), (50), (52), and (53) yield

Differentiating (38) with covariantly along and using (24), we have

Similarly, (38) with and (38) with , respectively, give

Differentiating (38) also covariantly in the direction of and using (24), we have

Similarly, differentiating (38) covariantly in the direction of and , respectively, we have

On the other hand, (38) and (39) with imply

Similarly, from (39) with and (39) with , respectively, we find that

Using (28)–(31), it follows from (57), (60), and (63) that

from which, taking account of (35) and using (24) and (33), we obtain

Similarly, by using (58), (59), (61), (62), (64), and (65), we can easily obtain

5. Main Results

It is well known [3] that if is locally symmetric then which implies identities (2) in Theorem K-P. In this point of view, we consider the following assumptions in (69) which are weaker conditions than the locally symmetry of .

In order to obtain our main results, let be -dimensional -submanifolds of -dimension in with the assumptions

We notice here that the above curvature conditions in (69) are different from those in [18] due to Pak and Sohn.

We first consider the assumption

Differentiating (55) covariantly in the direction of and using (19), (36), and (40), and the assumption , we have

from which, taking the vertical component of , and , respectively, and using (22)–(24) and (55) itself, we can get

Putting in (72) and using (19) and (24), we have

and consequently,

Putting and in (73) and using (15) and (24) yield

Also, putting and in (74) and using (15) and (24), we have

Summing up, we have

Thus we get the following lemma.

Lemma 1. Let be an -dimensional -submanifold of -dimension in a quaternionic projective space , and let the normal vector field be parallel with respect to the normal connection. If the equalities in (69) are established, then

Next, we assume the additional condition

Differentiating (67) covariantly in the direction of and using (36), (40) and the assumption , we have

from which, taking the vertical component of , and , respectively, and using (22)–(24) and (67) itself, we can find

Taking the skew-symmetric part of (83) with respect to and , we have

Replacing with in (86) and using (19) and (22)–(24), we have

Now we consider the following orthonormal basis:

which will be called -basis, where . Taking the trace of the above equation with respect to the -basis and using (76), we can easily see ; that is,

Replacing also with in (86) and using (19) and (22)–(24), we have

Substituting (89) and (90) into (75), we have and hence

On the other hand, replacing with in (84) and using (19), (22), (23), and (90), we obtain

from which, taking the trace with respect to the -basis and using (15) and (24), we find ; that is,

which together with (84) and (90) implies

Replacing with in the above equation and using (17) and (93), we can easily see that

for any vector tangent to .

Summing up, we have the following lemma.

Lemma 2. Let be as in Lemma 1, and let the normal vector field be parallel with respect to the normal connection. If the equalities in (69) and (5.2) are established, then

Finally, we will prove our main theorem.

Theorem 3. Let be an -dimensional -submanifold of -dimension in a quaternionic projective space , and let the normal vector field be parallel with respect to the normal connection. If the equalities in (69) and (5.2) are established, then is locally a product of where and belong to some - and -dimensional spheres ( is the Hopf fibration ).

Proof. By means of (96), it follows easily from (83) that

By the quite same method, we can obtain

Combining with those equalities and Theorem K-P, we complete the proof.

Corollary 4. Let be an -dimensional -submanifold of -dimension in a quaternionic projective space , and let the normal vector field be parallel with respect to the normal connection. If the following equalities:

are established, then is locally a product of where and belong to some - and -dimensional spheres.

Acknowledgment

This work was supported by the 2013 Inje University research grant.

References

A. Bejancu, Geometry of CR-Submanifolds, D. Reidel Publishing Company, Dordrecht, The Netherlands, 1986.

J.-H. Kwon and J. S. Pak, “Scalar curvature of QR-submanifolds immersed in a quaternionic projective space,” Saitama Mathematical Journal, vol. 17, pp. 89–116, 1999.

J.-H. Kwon and J. S. Pak, “QR-submanifolds of QR-dimension in a quaternionic projective space ,” Acta Mathematica Hungarica, vol. 86, no. 1-2, pp. 89–116, 2000.

H. B. Lawson, Jr., “Rigidity theorems in rank- symmetric spaces,” Journal of Differential Geometry, vol. 4, pp. 349–357, 1970.

A. Martínez and J. D. Pérez, “Real hypersurfaces in quaternionic projective space,” Annali di Matematica Pura ed Applicata, vol. 145, pp. 355–384, 1986.

J. S. Pak, “Real hypersurfaces in quaternionic Kaehlerian manifolds with constant -sectional curvature,” Kōdai Mathematical Seminar Reports, vol. 29, no. 1-2, pp. 22–61, 1977.

J. D. Pérez and F. G. Santos, “On pseudo-Einstein real hypersurfaces of the quaternionic projective space,” Kyungpook Mathematical Journal, vol. 25, no. 1, pp. 15–28, 1985.

J. D. Pérez and F. G. Santos, “On real hypersurfaces with harmonic curvature of a quaternionic projective space,” Journal of Geometry, vol. 40, no. 1-2, pp. 165–169, 1991.

Y. Shibuya, “Real submanifolds in a quaternionic projective space,” Kodai Mathematical Journal, vol. 1, no. 3, pp. 421–439, 1978.

S. Ishihara, “Quaternion Kählerian manifolds and fibred Riemannian spaces with Sasakian -structure,” Kōdai Mathematical Seminar Reports, vol. 25, pp. 321–329, 1973.

S. Ishihara and M. Konishi, “Fibred riemannian space with sasakian 3-structure,” in Differential Geometry in Honor of K. Yano, pp. 179–194, Kinokuniya, Tokyo, Japan, 1972.

Y.-Y. Kuo, “On almost contact -structure,” The Tohoku Mathematical Journal, vol. 22, pp. 325–332, 1970.

Y. Y. Kuo and S.-I. Tachibana, “On the distribution appeared in contact -structure,” Taiwanese Journal of Mathematics, vol. 2, pp. 17–24, 1970.

Y.-W. Choe and M. Okumura, “Scalar curvature of a certain CR-submanifold of complex projective space,” Archiv der Mathematik, vol. 68, no. 4, pp. 340–346, 1997.

M. Okumura and L. Vanhecke, “-dimensional real submanifolds with -dimensional maximal holomorphic tangent subspace in complex projective spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 43, no. 2, pp. 233–249, 1994.

J. S. Pak and W.-H. Sohn, “Some curvature conditions of -dimensional CR-submanifolds of CR-dimension in a complex projective space,” Bulletin of the Korean Mathematical Society, vol. 38, no. 3, pp. 575–586, 2001.

W.-H. Sohn, “Some curvature conditions of n-dimensional CR-submanifolds of (n−1) CR-dimension in a complex projective space,” Communications of the Korean Mathematical Society, vol. 25, pp. 15–28, 2000.

J. S. Pak and W.-H. Sohn, “Some curvature conditions of -dimensional QR-submanifolds of QR-dimension in a quaternionic projective space ,” Bulletin of the Korean Mathematical Society, vol. 40, no. 4, pp. 613–631, 2003.

S. Ishihara, “Quaternion Kählerian manifolds,” Journal of Differential Geometry, vol. 9, pp. 483–500, 1974.

K. Yano and S. Ishihara, “Fibred spaces with invariant Riemannian metric,” Kōdai Mathematical Seminar Reports, vol. 19, pp. 317–360, 1967.

B. Y. Chen, Geometry of Submanifolds, Marcel Dekker, New York, NY, USA, 1973.

B. O'Neill, “The fundamental equations of a submersion,” The Michigan Mathematical Journal, vol. 13, pp. 459–469, 1966.