# A Classification of a Totally Umbilical Slant

Authors: Siraj Uddin; Cenap Ozel; Viqar Azam Khan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study slant submanifolds of a cosymplectic manifold.
It is shown that a totally umbilical slant submanifoldM of a cosymplectic
manifold M̅ is either an anti-invariant submanifold or a 1−dimensional
submanifold. We show that every totally umbilical proper slant submanifold
of a cosymplectic manifold is totally geodesic.


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## Body

## 1. Introduction

The study of slant submanifolds in complex spaces was initiated by Chen as a natural generalization of both holomorphic and totally real submanifolds [1, 2]. Since then, many research papers have appeared concerning the existence of these submanifolds as well as on the geometry of the existent slant submanifolds in different known spaces (cf. [3, 4]). The slant submanifolds of an almost contact metric manifold were defined and studied by Lotta [4]. Later on, these submanifolds were studied by Cabrerizo et al. in the setting of Sasakian manifolds [3].Recently, Şahin proved that a totally umbilical proper slant submanifold of a Kaehler manifold is totally geodesic [5]. Our aim in the present paper is to investigate slant submanifolds in contact manifolds. Thus, we study slant submanifolds of a cosymplectic manifold. We have shown that a totally umbilical slant submanifold M of a cosymplectic manifold M̅ is either an anti-invariant submanifold or the dimM=1 or the mean curvature vector H∈Γ(μ), and then we have obtained an interesting result for a totally umbilical proper slant submanifold of a cosymplectic manifold.


## 2. Preliminaries

LetM̅ be a (2n+1)-dimensional manifold with (1,1) tensor field ϕ satisfying [6]:(2.1)ϕ2=-I+η⊗ξ,
where I is the identity transformation, ξ a vector field, and η a 1-form on M̅ satisfying ϕξ=η∘ϕ=0 and η(ξ)=1. Then M̅ is said to have an almost contact structure. There always exists a Riemannian metric g on M̅ such that(2.2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),
for all vector fields X,Y, on M̅. From (2.2), it is easy to observe that(2.3)g(ϕX,Y)+g(X,ϕY)=0.The fundamental 2-formΦ is defined as: Φ(X,Y)=g(X,ϕY). If [ϕ,ϕ]+dη⊗ξ=0, then the almost contact structure is said to be normal, where [ϕ,ϕ](X,Y)=ϕ2[X,Y]+[ϕX,ϕY]-ϕ[ϕX,Y]-ϕ[X,ϕY]. If Φ=dη, the almost contact structure is a contact structure. A normal almost contact structure such that Φ is closed and dη=0 is called cosymplectic structure. It is well known [7] that the cosymplectic structure is characterized by(2.4)(∇̅Xϕ)Y=0,(∇̅Xη)Y=0,
for all vector fields X, Y, on M̅, where ∇̅ is the Levi-Civita connection of g. From the formula ∇̅Xϕ=0, it follows that ∇̅Xξ=0.LetM be submanifold of an almost contact metric manifold M̅ with induced metric g and let ∇ and ∇⊥ be the induced connections on the tangent bundle TM and the normal bundle T⊥M of M, respectively. Denote by ℱ(M) the algebra of smooth functions on M and by Γ(TM) the ℱ(M)-module of smooth sections of a vector bundle TM over M, then Gauss and Weingarten formulae are given by
(2.5)∇̅XY=∇XY+h(X,Y),(2.6)∇̅XN=-ANX+∇X⊥N,
for each X,Y∈Γ(TM) and N∈Γ(T⊥M), where h and AN are the second fundamental form and the shape operator (corresponding to the normal vector field N), respectively for the immersion of M into M̅. They are related as(2.7)g(h(X,Y),N)=g(ANX,Y),
where g denotes the Riemannian metric on M̅ as well as the one induced on M [8].For anyX∈Γ(TM), we write(2.8)ϕX=PX+FX,
where PX is the tangential component and FX is the normal component of ϕX. Similarly for any N∈Γ(T⊥M), we write(2.9)ϕN=tN+fN,
where tN is the tangential component and tN is the normal component of ϕN. If we denote the orthogonal complementary distribution of F(TM) in T⊥M by μ, then we have the direct sum(2.10)T⊥M=F(TM)⊕μ.
We can see that μ is an invariant subbundle with respect to ϕ. Furthermore, the covariant derivatives of the tensor fields P and F are defined as
(2.11)(∇̅XP)Y=∇XPY-P∇XY,(∇̅XF)Y=∇X⊥FY-F∇XY,
for any X,Y∈Γ(TM).A submanifoldM is said to be invariant if F is identically zero, that is, ϕX∈Γ(TM) for any X∈Γ(TM). On the other hand, M is said to be anti-invariant if P is identically zero, that is, ϕX∈Γ(T⊥M), for any X∈Γ(TM).A submanifoldM of an almost contact metric manifold M̅ is called totally umbilical if(2.12)h(X,Y)=g(X,Y)H,
for any X,Y∈Γ(TM). The mean curvature vector H is given by
(2.13)H=∑i=1mh(ei,ei),
where m is the dimension of M and {e1,e2,…,em} is the local orthonormal frame on M. A submanifold M is said to be totally geodesic if h(X,Y)=0 for each X,Y∈Γ(TM) and is minimal if H=0 on M.


## 3. Slant Submanifolds

Throughout the section, we assume thatM is a slant submanifold of a cosymplectic manifold M̅. We always consider such submanifold tangent to the structure vector field ξ. For each nonzero vector X tangent to M at x, we denote by 0≤θ(X)≤π/2, the angle between ϕX and TxM, known as the Wirtinger angle of X. If the Wirtinger angle θ(X) is constant, that is, independent of the choice of x∈M and X∈TxM-{ξ}, then M is said to be a slant submanifold [4]. In this case the constant angle θ is called slant angle of the slant submanifold. Obviously if θ=0,M is invariant and if θ=π/2, M is an anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant submanifold. If M is a slant submanifold of an almost contact metric manifold, then the tangent bundle TM is decomposed as(3.1)TM=D⊕〈ξ〉,
where 〈ξ〉 denotes the distribution spanned by the structure vector field ξ and 𝒟 is the complementary distribution of 〈ξ〉 in TM, known as the slant distribution.We recall the following result for a slant submanifold.Theorem 3.1 (see [3]).
LetM be a submanifold of an almost contact metric manifold M̅,  such that ξ∈TM.  Then, M is slant if and only if there exists a constant λ∈[0,1] such that
(3.2)P2=λ(-I+η⊗ξ).
Furthermore, if θ is slant angle, then λ=cos2θ.The following relations are straightforward consequence of (3.2):(3.3)g(PX,PY)=cos⁡2θ[g(X,Y)-η(X)η(Y)],(3.4)g(FX,FY)=sin⁡2θ[g(X,Y)-η(X)η(Y)],
for any X, Y tangent to M.Now, we prove the following.Theorem 3.2.
LetM be a totally umbilical slant submanifold of a cosymplectic manifold M̅.  Then at least one of the following statements is ture:(i)
M is an anti-invariant submanifold;(ii)
M is a 1-dimensional submanifold;(iii)
IfM is a proper slant submanifold, then H∈Γ(μ),
whereH is the mean curvature vector of the submanifold M.Proof.
LetM be a totally umbilical slant submanifold of a cosymplectic manifold M̅, then for any X,Y∈Γ(TM), we have
(3.5)h(PX,PY)=g(PX,PY)H.
From (2.5) and (3.3), we deduce that
(3.6)∇̅PXPX-∇PXPX=cos⁡2θ{g(X,X)-η(X)η(X)}H.
Using (2.8) and the fact that M̅ is cosymplectic we obtain that
(3.7)ϕ∇̅PXX-∇̅PXFX-∇PXPX=cos⁡2θ{‖X‖2-η2(X)}H.
Then from (2.5) and (2.6), we get
(3.8)ϕ∇PXX+ϕh(X,PX)+AFXPX-∇PX⊥FX-∇PXPX=cos⁡2θ{‖X‖2-η2(X)}H.
Thus by (2.8) and (2.12), we obtain
(3.9)P∇PXX+F∇PXX+g(PX,X)ϕH+AFXPX-∇PX⊥FX-∇PXPX=cos⁡2θ{‖X‖2-η2(X)}H.
Equating the normal components, we get
(3.10)F∇PXX-∇PX⊥FX=cos⁡2θ{‖X‖2-η2(X)}H.
On the other hand, from (3.4), we have
(3.11)g(FX,FX)=sin⁡2θ{g(X,X)-η(X)η(X)},
for any X∈Γ(TM). Taking the covariant derivative of the above equation with respect to PX, we obtain
(3.12)2g(∇̅PXFX,FX)=2sin⁡2θg(∇̅PXX,X)-2sin⁡2θη(X)g(∇̅PXX,ξ)-2sin⁡2θη(X)g(X,∇̅PXξ).
Using the property of metric connection ∇̅, the last two terms of the right-hand side are cancelling each other, thus we have
(3.13)g(∇̅PXFX,FX)=sin⁡2θg(∇̅PXX,X).
Then by (2.5) and (2.6), we derive
(3.14)g(∇PX⊥FX,FX)=sin⁡2θg(∇PXX,X).
Now, taking the inner product in (3.10) with FX, for any X∈Γ(TM), then
(3.15)g(F∇PXX,FX)-g(∇PX⊥FX,FX)=cos⁡2θ{‖X‖2-η2(X)}g(H,FX).
Then from (3.4) and (3.14), we obtain
(3.16)-sin⁡2θη(X)η(∇PXX)=cos⁡2θ{‖X‖2-η2(X)}g(H,FX),
or
(3.17)-sin⁡2θη(X)g(∇PXX,ξ)=cos⁡2θ{‖X‖2-η2(X)}g(H,FX).
Using (2.5), we derive
(3.18)-sin⁡2θη(X)g(∇̅PXX,ξ)=cos⁡2θ{‖X‖2-η2(X)}g(H,FX).
Since ∇̅ is the metric connection, then the above equation can be written as
(3.19)sin⁡2θη(X)g(X,∇̅PXξ)=cos⁡2θ{‖X‖2-η2(X)}g(H,FX).
As M̅ is cosymplectic thus using the fact that ∇̅PXξ=0, the left hand side of the above equation vanishes identically, then
(3.20)cos⁡2θ{‖X‖2-η2(X)}g(H,FX)=0.
Thus from (3.20), it follows that either θ=π/2 or X=ξ or H∈Γ(μ), where μ is the invariant normal subbundle orthogonal to FTM. This completes the proof.Theorem 3.3.
Every totally umbilical proper slant submanifoldM of a cosymplectic manifold M̅ is totally geodesic,  provided ∇X⊥H∈Γ(μ), for any X∈TM.Proof.
AsM̅ is cosymplectic, then we have
(3.21)∇̅UϕV=ϕ∇̅UV,
for any U,V∈Γ(TM̅). Using this fact and formulae (2.5) and (2.8) we obtain that
(3.22)∇̅XPY+∇̅XFY=P∇XY+F∇XY+ϕh(X,Y),
for any X,Y∈Γ(TM). Then from (2.5), (2.6) and (2.12), we get
(3.23)∇XPY+h(X,PY)-AFYX+∇X⊥FY=P∇XY+F∇XY+g(X,Y)ϕH.
Taking the inner product in (3.23) with ϕH and using the fact that H∈Γ(μ) (by Theorem 3.2), we obtain
(3.24)g(h(X,PY),ϕH)+g(∇X⊥FY,ϕH)=g(X,Y)g(ϕH,ϕH).
Then from (2.2) and (2.12), we derive
(3.25)g(X,PY)g(H,ϕH)+g(∇X⊥FY,ϕH)=g(X,Y)g(H,H).
That is,
(3.26)g(∇X⊥FY,ϕH)=g(X,Y)‖H‖2.
Now, we consider
(3.27)∇̅XϕH=ϕ∇̅XH,
for any X∈Γ(TM). From (2.6), we obtain
(3.28)-AϕHX+∇X⊥ϕH=ϕ(-AHX+∇X⊥H).
Thus, on using (2.8), (2.9), we get
(3.29)-AϕHX+∇X⊥ϕH=-PAHX-FAHX+t∇X⊥H+f∇X⊥H.
Taking the inner product with FY, for any Y∈Γ(TM), then
(3.30)g(∇X⊥ϕH,FY)=-g(FAHX,FY)+g(f∇X⊥H,FY).
Since f∇X⊥H∈Γ(μ), then by (3.4) the above equation takes the form
(3.31)g(∇X⊥ϕH,FY)=-sin⁡2θ{g(AHX,Y)-η(AHX)η(Y)}.
Using (2.6), (2.7), and (2.12), we get
(3.32)g(∇̅XϕH,FY)=-sin⁡2θ{g(X,Y)-η(X)η(Y)}‖H‖2.
The above equation can be written as
(3.33)g(∇̅XFY,ϕH)=sin⁡2θ{g(X,Y)-η(X)η(Y)}‖H‖2.
Again using the fact that H∈Γ(μ), then by (2.6), we obtain
(3.34)g(∇X⊥FY,ϕH)=sin⁡2θ{g(X,Y)-η(X)η(Y)}‖H‖2.
From (3.26) and (3.34), we derive
(3.35){cos⁡2θg(X,Y)+sin⁡2θη(X)η(Y)}‖H‖2=0.
Thus, (3.35) implies either H=0 or θ=tan-1(-g(X,Y)/η(X)η(Y)), which is not possible, because the slant angle θ∈(0,π/2). Hence, M is totally geodesic in M̅.



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*Source: 716967-2012-02-29.xml*