TOTALLY UMBILICAL PSEUDO-SLANT SUBMANIFOLDS OF NEARLY KAEHLER MANIFOLDS MERAJ ALI KHAN In the present talk, we will discuss totally umbilical pseudo-slant submanifolds of nearly Kaehler manifolds. The classification theorem of totally umbilical pseudo-slant submanifold of a nearly Kaehler manifold has been proved.

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1. Introduction The geometry of slant immersions was initiated by B.Y. Chen [3] as a natural generalization of both holomorphic and totally real immersions. Many authors have studied slant immersions in almost Hermitian manifolds. A. Lotta [8] introduced the notion of slant immersions in contact geometry. On the line of semi-slant submanifolds A. Carriazo [1]defined and studied bislant submanifolds of almost Hermitian and alomst contact metric manifolds and then he gave the notion of pseudo-slant submanifold [2]. Later on, V.A. Khan and M.A. Khan [6] studied pseudo-slant submanifolds of a Sasakian manifold. The purpose of the present paper is to study totally umbilical pseudo-slant submanifolds of nearly Kaehler manifolds. In section 2, we review and collect some necessary results. In section 3, we define pseudo-slant submanifolds in the setting of nearly Kaehler manifolds and work out integrability conditions of involved distributions in this setting and in last section 4, we obtained a classification theorem for totally ¯. umbilical pseudo-slant submanifold M of a nearly Kaehler manifold M

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2. Preliminaries In this section, we give some terminology and notions used throughout this paper. We recall some necessary facts and formulas from the theory of almost Hermitian manifolds and their submanifolds. ¯ be an almost Hermitian manifold with almost complex structure Let M ¯ denote the tangent bundle of J and a Riemannian metric g. Further let T M ¯ and ∇, ¯ the covariant differential operator on M ¯ with respect to g. If the M almost complex structure J satisfies ¯ X J)Y + (∇ ¯ Y J)X = 0 (∇

(2.1)

¯ , then the manifold M ¯ is called nearly Kaehler manifold. for any X, Y ∈ T M Obviously, every Kaehler manifold is nearly Kaehler manifold. ¯ , the Gauss For an arbitrary submanifold M of a Riemannian manifold M and Weingarten formulae are respectively given by ¯ X Y = ∇X Y + h(X, Y ) ∇

(2.2)

¯ X V = −AV X + ∇⊥ ∇ XV

(2.3)

and, for all X, Y ∈ T M , where ∇ is the induced Riemannian connection on M , ¯ , h is the second fundamental form of M , ∇⊥ V is a vector field normal to M is the normal connection in the normal bundle, T ⊥ M and AV is the shape operator of the second fundamental form. Moreover, g(AV X, Y ) = g(h(X, Y ), V )

(2.4)

¯ as well as the metric induced where g denotes the Riemannian metric on M on M . The mean curvature vector H of M is given by n

1X h(ei , ei ) H= n α=1

(2.5)

where n is the dimension of M and {e1 , e2 , · · · , en } is a local orthonormal frame of vector fields on M . A distribution D on a submanifold M of an almost Hermitian manifold ¯ is said to be slant distribution if for each U ∈ Dx . The angle θ between JU M and Dx is constant i.e., independent of x ∈ M and U ∈ Dx . A submanifold 3

¯ is said to be slant submanifold if the tangent bundle T M of M is M of M slant. ¯ is a submanifold which admits two A pseudo-slant submanifold M of M orthogonal complementary distributions D1 and D2 such that D1 is slant and D2 is totally real i.e., JD2 ⊆ T ⊥ M and D2 is slant with slant angle θ 6= π/2. Moreover, for a slant distribution Dθ , we have T 2 X = cos2 θX

(2.6)

g(T X, T Y ) = cos2 θg(X, Y )

(2.7)

g(N X, N Y ) = sin2 θg(X, Y )

(2.8)

Using (2.6), we can calculate

for all X, Y ∈ Dθ . For any X ∈ T M and V ∈ T ⊥ M , the transformation JX and JV are decomposed into tangential and normal parts respectively as JX = T X + N X

(2.9)

JV = tV + f V

(2.10)

Let T M denote the tangent bundle on M . If µ is the invariant subspace of the normal bundle T ⊥ M . Then, in the case of pseudo-slant submanifold the normal bundle T ⊥ M can be decomposed as follows T ⊥ M = µ ⊕ N D1 ⊕ N D2 .

(2.11)

Now, denoting by PX Y and QX Y the tangential and normal parts of ¯ (∇X J)Y and making use of equations (2.9), (2.10) and Gauss and Weingarten formulae, the following equations may easily be obtained PX Y = (∇X T )Y − AF Y X − th(X, Y )

(2.12)

¯ X N )Y + h(X, P Y ) − f h(X, Y ) QX Y = (∇

(2.13) ¯ X J)V Similarly, for V ∈ T ⊥ M , denoting tangential and normal parts of (∇ by PX V and QX V respectively, we find PX V = (∇X t)V + T AV X − Af V X

(2.14)

QX V = (∇X f )V + h(tV, X) + N AV X

(2.15)

where the covariant derivative of T and N are defined by (∇X T )Y = ∇X T Y − T ∇X Y 4

(2.16)

¯ X N )V = ∇⊥ N Y − N ∇X Y (∇ X

(2.17)

for all X, Y ∈ T M and V ∈ T ⊥ M . ¯ is said to be A submanifold M of an almost Hermitian manifold M totally umbilical submanifold if the second fundamental form satisfies h(X, Y ) = g(X, Y )H.

(2.18)

The submanifold M is totally geodesic if h(X, Y ) = 0, for all X, Y ∈ T M and minimal if H = 0. 3. Integrability conditions of distributions In this section we shall discuss the integrability of involved distributions. Proposition 3.1. Let M be a pseudo-slant submanifold of a nearly Kaehler manifold then the anti-invariant distribution D⊥ is integrable iff AJY X − AJX Y + 2PX Y ∈ D⊥ for all X, Y ∈ D⊥ . Proof. For any X, Y ∈ D⊥ and Z ∈ Dθ , then by (2.9), we have ¯ Y JX − (∇ ¯ Y J)X − ∇ ¯ X JY + (∇ ¯ X J)Y, Z) g([X, Y ], T Z) = g(∇ Using (2.2) and (2.3) the above equation reduced to g([X, Y ], T Z) = g(AJY X − AJX Y + 2PX Y, Z).

(3.1)

Thus the assertion follows from (3.1). Proposition 3.2. Let M be a pseudo-slant submanifold of a nearly Kaehler ¯ , then the distribution Dθ is integrable iff manifold M ⊥ h(Z, T W ) − h(W, T Z) + ∇⊥ Z N W − ∇W N Z − 2QZ W ∈ N Dθ .

Proof. For any X ∈ D⊥ and Z, W ∈ Dθ , then from equations (2.9), (2.2), ¯ Z J)W we have (2.3) and the normal components of (∇ g(N [Z, W ], N X) = g(h(Z, T W ) − h(W, T Z) ⊥ + ∇⊥ Z N W − ∇W N Z − 2QZ W, N X).

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(3.2)

The result follows from equation (3.2) and the fact that N Dθ and N D⊥ are orthogonal. 4. Totally umbilical pseudo-slant submanifolds Through out this section we shall conisder M as a totally umbilical ¯ . We have the folpseudo-slant submanifold of a nearly Kaehler manifold M lowing prepatory results: Theorem 4.1 [5]. Let M be a totally umbilical pseudo-slant submanifold of a nearly Kaehler manifold, then following are equivalent (a) There is a nearly Kaehler character (T, g) on the slant distribution Dθ . (b) H ∈ µ. Theorem 4.2 [5]. Let M be a totally umbilical pseudo-slant submanifold with nearly Kaehler structure (T, g) on slant distribution of a nearly Kaehler ¯ , such that f is parallel then M is totally geodesic. manifold M We consider M as a pseudo-slant submanifold of a nearly Kaehler mani¯ and Dθ and D⊥ are integrable distributions corresponding to the slant fold M ¯ , respectively. Now, for any U ∈ T M , we and totally real submanifolds of M ¯ U J)U = 0, using this fact, we get have (∇ ¯ Z J)Z = 0 (∇

(4.1)

for any Z ∈ D⊥ . Therefore the tangential and normal parts of the above equation are PZ Z = 0 and QZ Z = 0, respectively. From (2.12) and the tangential componet of (4.1), we obtain PZ Z = 0 = (∇Z T )Z − AF Z Z − th(Z, Z). That is, (∇Z T )Z = AF Z Z + th(Z, Z).

(4.2)

As M is totally umbilical and Z ∈ D⊥ , T Z = 0, using this fact and equation (2.16), the above equation takes the form T ∇Z Z = −g(H, F Z)Z − |Z|2 tH.

(4.3)

Taking the product in equation (4.3) with W ∈ D⊥ , we obtain g(H, F Z)g(Z, W ) − |Z|2 g(tH, W ) = 0. Thus the equation (4.4) has a solution if one of the following holds: 6

(4.4)

(a) dimD⊥ = 1 (b) H ∈ µ Now, we are in the position to prove our main result. Theorem 4.3. Let M be a totally umbilical pseudo-slant submanifold of a ¯ . Then atleast one of the following statement is nearly Kaehler manifold M true (i) M is totally real, (ii) The slant distribution Dθ has a nearly Kaehler character (T, g), (iii) M is totally geodesic when f is parallel, (iv) dimD⊥ = 1. Proof. If Dθ = {0}, then by definition M is totally real which is case (i). If Dθ 6= {0} and H ∈ µ, then by Theorem (4.1) (T, g) has nearly Kaehler character on Dθ , this is case (ii). Moreover, if H ∈ µ and f is parallel then by Theorem (4.2) M is totally geodesic, which proves case (iii). Finally, if H 6∈ µ, then equation (4.4) has a solution if dimD⊥ = 1 which is case (iv). Thus the theorem porves completely.

References [1] J.L. Cabrerizo, A. Carriazo, L.M. Fernandez and M. Fernandez, Semislant submanifolds of a Sasakian manifold, Geom. Dedicata 78 (1999), 183-199. [2] A. Carriazo, New Developments in Slant submanifolds theory, Narosa Publishing House, New Delhi, India, 2002. [3] B.Y. Chen, Slant immersions, Bull. Austral. Math. Soc. 41 (1990), 135-147. [4] K.A. Khan, V.A. Khan and S.I. Husain, Totally umbilical CR-submanifolds of a nearly Kaehler manifold, Geom. Dedicata 50 (1994), 47-51. [5] M.A. Khan, Siraj Uddin and Khushwant Singh, Slant submanifold of a nearly Kaehler manifold, (Submitted). [6] V.A. Khan and M.A. Khan, Pseudo-Slant Submanifolds of a Sasakian Manifold, Indian J. pure appl. Math. 38 (2007), 31-42. [7] V.A. Khan and M.A. Khan, Semi-slant submanifolds of a nearly Kaehler manifold, Turk. J. Math. 31 (2007), 1-13. 7

[8] A. Lotta, Slant submanifolds in contact geometry, Bull. Math. Soc. Roumanie 39 (1996), 183-198. Authors’ addresses: M. A. Khan School of Mathematics and Computer Applications Thapar University, Patiala-147 004, India E-mail: [email protected]

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