Volume 10 (2009), Issue 4, Article 119, 7 pp.
CONTACT CR-DOUBLY WARPED PRODUCT SUBMANIFOLDS IN KENMOTSU
SPACE FORMS
ANDREEA OLTEANU
FACULTY OF M ATHEMATICS AND C OMPUTER S CIENCE
U NIVERSITY OF B UCHAREST
S TR . ACADEMIEI 14
011014 B UCHAREST, ROMANIA
andreea_d_olteanu@yahoo.com
Received 26 February, 2009; accepted 06 November, 2009
Communicated by S.S. Dragomir
A BSTRACT. Recently, the author established general inequalities for CR-doubly warped products isometrically immersed in Sasakian space forms.
In the present paper, we obtain sharp estimates for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping functions (intrinsic invariants) for
contact CR-doubly warped products isometrically immersed in Kenmotsu space forms. The
equality case is considered. Some applications are derived.
Key words and phrases: Doubly warped product, contact CR-doubly warped product, invariant submanifold, anti-invariant
submanifold, Laplacian, mean curvature, Kenmotsu space form.
2000 Mathematics Subject Classification. Primary 53C40; Secondary 53C25.
1. I NTRODUCTION
In 1978, A. Bejancu introduced the notion of CR-submanifolds which is a generalization
of holomorphic and totally real submanifolds in an almost Hermitian manifold ([2]). Following this, many papers and books on the topic were published. The first main result on CRsubmanifolds was obtained by Chen [4]: any CR-submanifold of a Kaehler manifold is foliated
by totally real submanifolds. As non-trivial examples of CR-submanifolds, we can mention the
(real) hypersurfaces of Hermitian manifolds.
Recently, Chen [5] introduced the notion of a CR-warped product submanifold in a Kaehler
manifold and proved a number of interesting results on such submanifolds. In particular, he
established a sharp relationship between the warping function f of a warped product CRf and the squared norm of the second funsubmanifold M1 ×f M2 of a Kaehler manifold M
2
damental form ||h|| .
On the other hand, there are only a handful of papers about doubly warped product Riemannian manifolds which are the generalization of a warped product Riemannian manifold.
Recently, the author obtained a general inequality for CR-doubly warped products isometrically immersed in Sasakian space forms ([12]).
058-09
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In the present paper, we study contact CR-doubly warped product submanifolds in Kenmotsu
space forms.
We prove estimates of the squared norm of the second fundamental form in terms of the
warping function. Equality cases are investigated. Obstructions to the existence of contact
CR-doubly warped product submanifolds in Kenmotsu space forms are derived.
2. P RELIMINARIES
f, g) is said to be a Kenmotsu manifold if it
A (2m + 1)-dimensional Riemannian manifold (M
f, a vector field ξ and a 1-form η satisfying:
admits an endomorphism φ of its tangent bundle T M
φ2 = −Id + η ⊗ ξ,
(2.1)
η (ξ) = 1,
φξ = 0,
η ◦ φ = 0,
g (φX, φY ) = g (X, Y ) − η (X) η (Y ) , η (X) = g (X, ξ) ,
e X φ Y = −g (X, φY ) ξ − η (Y ) φX, ∇
e X ξ = X − η (X) ξ,
∇
f, where ∇
e denotes the Riemannian connection with respect to
for any vector fields X, Y on M
g.
f, i.e.,
We denote by ω the fundamental 2-form of M
f .
(2.2)
ω (X, Y ) = g (φX, Y ) ,
∀X, Y ∈ Γ T M
It was proved that the pairing (ω, η) defines a locally conformal cosymplectic structure, i.e.,
dω = 2ω ∧ η,
dη = 0.
f is called a φ-section if it is spanned by X and φX, where X is a
A plane section π in Tp M
unit tangent vector orthogonal to ξ. The sectional curvature of a φ-section is called a φ-sectional
curvature. A Kenmotsu manifold with constant φ-holomorphic sectional curvature c is said to
f (c).
be a Kenmotsu space form and is denoted by M
e of a Kenmotsu space form is given by [8]
The curvature tensor R
e (X, Y ) Z = c − 3 {g (Y, Z) X − g(X, Z)Y } + c + 1 {[η (X) Y − η (Y ) X] η (Z)
(2.3) R
4
4
+ [g (X, Z) η (Y ) − g (Y, Z) η (X)] ξ + ω (Y, Z) φX
− ω (X, Z) φY − 2ω (X, Y ) φZ}.
f be a Kenmotsu manifold and M an n-dimensional submanifold tangent to ξ. For any
Let M
vector field X tangent to M , we put
(2.4)
φX = P X + F X,
where P X (resp. F X) denotes the tangential (resp. normal) component of φX. Then P is an
endomorphism of the tangent bundle T M and F is a normal bundle valued 1-form on T M .
The equation of Gauss is given by
e (X, Y, Z, W ) = R (X, Y, Z, W ) + g (h (X, W ) , h (Y, Z)) − g (h (X, Z) , h (Y, W ))
(2.5) R
for any vectors X, Y , Z, W tangent to M .
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 119, 7 pp.
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C ONTACT CR- DOUBLY WARPED P RODUCT S UBMANIFOLDS IN K ENMOTSU S PACE F ORMS
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Let p ∈ M and {e1 , ..., en , en+1 , ..., e2m+1 } be an orthonormal basis of the tangent space
f, such that e1 , ..., en are tangent to M at p. We denote by H the mean curvature vector, that
Tp M
is
n
1X
(2.6)
H (p) =
h (ei , ei ) .
n i=1
As is known, M is said to be minimal if H vanishes identically.
Also, we set
(2.7)
hrij = g (h (ei , ej ) , er ) ,
i, j ∈ {1, ..., n}, r ∈ {n + 1, ..., 2m + 1}
as the coefficients of the second fundamental form h with respect to {e1 , ..., en , en+1 , ..., e2m+1 },
and
n
X
2
g (h (ei , ej ) , h (ei , ej )) .
(2.8)
||h|| =
i,j=1
By analogy with submanifolds in a Kaehler manifold, different classes of submanifolds in a
Kenmotsu manifold were considered (see, for example, [13]).
A submanifold M tangent to ξ is called an invariant (resp. anti-invariant) submanifold if
φ (Tp M ) ⊂ Tp M , ∀ p ∈ M (resp. φ (Tp M ) ⊂ Tp⊥ M , ∀ p ∈ M ).
A submanifold M tangent to ξ is called a contact CR-submanifold ([13]) if there exists a pair
of orthogonal differentiable distributions D and D⊥ on M , such that:
(1) T M = D ⊕ D⊥ ⊕ {ξ}, where {ξ} is the 1-dimensional distribution spanned by ξ;
(2) D is invariant by φ, i. e., φ (Dp ) ⊂ Dp, ∀p ∈ M ;
(3) D⊥ is anti-invariant by φ, i. e., φ Dp⊥ ⊂ Dp⊥ , ∀p ∈ M .
In particular, if D⊥ = {0} (resp. D ={0}), M is an invariant (resp. anti-invariant) submanifold.
3. C ONTACT CR- DOUBLY WARPED P RODUCT S UBMANIFOLDS
Singly warped products or simply warped products were first defined by Bishop and O’Neill
in [3] in order to construct Riemannian manifolds with negative sectional curvature.
In general, doubly warped products can be considered as generalizations of singly warped
products.
Let (M1 , g1 ) and (M2 , g2 ) be two Riemannian manifolds and let f1 : M1 → (0, ∞) and
f2 : M2 → (0, ∞) be differentiable functions.
The doubly warped product M =f2 M1 ×f1 M2 is the product manifold M1 × M2 endowed
with the metric
(3.1)
g = f22 g1 + f12 g2 .
More precisely, if π1 : M1 × M2 → M1 and π2 : M1 × M2 → M2 are natural projections, the
metric g is defined by
(3.2)
g = (f2 ◦ π2 )2 π1∗ g1 + (f1 ◦ π1 )2 π2∗ g2 .
The functions f1 and f2 are called warping functions. If either f1 ≡ 1 or f2 ≡ 1, but not
both, then we obtain a warped product. If both f1 ≡ 1 and f2 ≡ 1, then we have a Riemannian
product manifold. If neither f1 nor f2 is constant, then we have a non-trivial doubly warped
product.
We recall that on a doubly warped product one has
(3.3)
∇X Z = Z (ln f2 ) X + X (ln f1 ) Z,
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 119, 7 pp.
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for any vector fields X tangent to M1 and Z tangent to M2 .
If X and Z are unit vector fields, it follows that the sectional curvature K (X ∧ Z) of the
plane section spanned by X and Z is given by
1
1
(3.4)
K (X ∧ Z) = { ∇1X X f1 − X 2 f1 } + { ∇2Z Z f2 − Z 2 f2 },
f1
f2
where ∇1 , ∇2 are the Riemannian connections of the Riemannian metrics g1 and g2 respectively.
By reference to [12], a doubly warped product submanifold M =f2 M1 ×f1 M2 of a Kenmotsu
f, with M1 a (2α + 1)-dimensional invariant submanifold tangent to ξ and M2 a βmanifold M
f is said to be a contact CR-doubly warped product
dimensional anti-invariant submanifold of M
submanifold.
We state the following estimate of the squared norm of the second fundamental form for
contact CR-doubly warped products in Kenmotsu manifolds.
f (c) be a (2m+1)-dimensional Kenmotsu manifold and M =f2 M1 ×f1 M2
Theorem 3.1. Let M
an n-dimensional contact CR-doubly warped product submanifold, such that M1 is a (2α + 1)dimensional invariant submanifold tangent to ξ and M2 is a β-dimensional anti-invariant subf (c). Then:
manifold of M
(i) The squared norm of the second fundamental form of M satisfies
(3.5)
||h||2 ≥ 2β||∇ (ln f1 ) ||2 − 1],
where ∇ (ln f1 ) is the gradient of ln f1 .
(ii) If the equality sign of (3.5) holds identically, then M1 is a totally geodesic submanifold
f. Moreover, M is a minimal submanifold
and M2 is a totally umbilical submanifold of M
f.
of M
Proof. Let M =f2 M1 ×f1 M2 be a doubly warped product submanifold of a Kenmotsu manf, such that M1 is an invariant submanifold tangent to ξ and M2 is an anti-invariant
ifold M
f.
submanifold of M
For any unit vector fields X tangent to M1 and Z, W tangent to M2 respectively, we have:
e Z φX, φZ
(3.6)
g (h (φX, Z) , φZ) = g ∇
e
e
= g φ∇Z X, φZ = g ∇Z X, Z = g (∇Z X, Z) = X ln f1 ,
g (h (φX, Z) , φW ) = (X ln f1 ) g (Z, W ) .
f is a Kenmotsu manifold, it is easily seen
On the other hand, since the ambient manifold M
that
(3.7)
h (ξ, Z) = 0.
Obviously, (3.3) implies ξ ln f1 = 1. Therefore, by (3.6) and (3.7), the inequality (3.5) is
immediately obtained.
Denote by h00 the second fundamental form of M2 in M . Then, we get
g (h00 (Z, W ) , X) = g (∇Z W, X) = − (X ln f1 ) g (Z, W, )
or equivalently
(3.8)
h” (Z, W ) = −g (Z, W ) ∇ (ln f1 ) .
If the equality sign of (3.5) identically holds, then we obtain
(3.9)
h (D,D) = 0, h D⊥ ,D⊥ = 0, h D,D⊥ ⊂ φD⊥ .
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 119, 7 pp.
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C ONTACT CR- DOUBLY WARPED P RODUCT S UBMANIFOLDS IN K ENMOTSU S PACE F ORMS
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The first condition (3.9) implies that M1 is totally geodesic in M . On the other hand, one has
e X φY, φZ = g (∇X Y, Z) = 0.
g (h (X, φY ) , φZ) = g ∇
f.
Thus M1 is totally geodesic in M
The second condition in (3.9) and (3.8) imply that M2 is a totally umbilical submanifold in
f
M.
f.
Moreover, by (3.9), it follows that M is a minimal submanifold of M
In particular, if the ambient space is a Kenmotsu space form, one has the following.
f (c) be a (2m + 1)-dimensional Kenmotsu space form of constant φCorollary 3.2. Let M
sectional curvature c and M =f2 M1 ×f1 M2 an n-dimensional non-trivial contact CR-doubly
warped product submanifold, satisfying
||h||2 = 2β ||∇ (ln f1 ) ||2 − 1 .
Then, we have
f (c). Hence M1 is a Kenmotsu space
(a) M1 is a totally geodesic invariant submanifold of M
form of constant φ-sectional curvature c.
f (c). Hence M2 is a real space
(b) M2 is a totally umbilical anti-invariant submanifold of M
c−3
form of sectional curvature ε > 4 .
Proof. Statement (a) follows from Theorem 3.1.
f (c). The Gauss equation
Also, we know that M2 is a totally umbilical submanifold of M
implies that M2 is a real space form of sectional curvature ε ≥ c−3
.
4
Moreover, by (3.3), we see that ε = c−3
if
and
only
if
the
warping
function f1 is constant. 4
4. A NOTHER I NEQUALITY
In the present section, we will improve the inequality (3.5) for contact CR-doubly warped
product submanifolds in Kenmotsu space forms. Equality case is characterized.
f (c) be a (2m + 1)-dimensional Kenmotsu space form of constant φTheorem 4.1. Let M
sectional curvature c and M =f2 M1 ×f1 M2 an n-dimensional contact CR-doubly warped
product submanifold, such that M1 is a (2α + 1)-dimensional invariant submanifold tangent to
f (c). Then:
ξ and M2 is a β-dimensional anti-invariant submanifold of M
(i) The squared norm of the second fundamental form of M satisfies
(4.1)
||h||2 ≥ 2β ||∇ (ln f1 ) ||2 − ∆1 (ln f1 ) − 1 + αβ (c + 1) ,
where ∆1 denotes the Laplace operator on M1 .
(ii) The equality sign of (4.1) holds identically if and only if we have:
f (c). Hence M1 is a Kenmotsu
(a) M1 is a totally geodesic invariant submanifold of M
space form of constant φ-sectional curvature c.
f (c). Hence M2 is a real
(b) M2 is a totally umbilical anti-invariant submanifold of M
c−3
space form of sectional curvature ε ≥ 4 .
Proof. Let M =f2 M1 ×f1 M2 be a contact CR-doubly warped product submanifold of a (2m +
f (c), such that M1 is an invariant submanifold tangent
1)-dimensional Kenmotsu space form M
f (c).
to ξ and M2 is an anti-invariant submanifold of M
We denote by ν be the normal subbundle orthogonal to φ (T M2 ). Obviously, we have
T ⊥ M = φ (T M2 ) ⊕ ν,
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 119, 7 pp.
φν = ν.
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For any vector fields X tangent to M1 and orthogonal to ξ and Z tangent to M2 , equation
(2.3) gives
e (X, φX, Z, φZ) = c + 1 g(X, X)g(Z, Z).
R
2
On the other hand, by the Codazzi equation, we have
e (X, φX, Z, φZ) = −g ∇⊥ h (φX, Z) − h (∇X φX, Z) − h (φX, ∇X Z) , φZ
(4.2) R
X
+ g ∇⊥
φX h (X, Z) − h (∇φX X, Z) − h (X, ∇φX Z) , φZ .
By using the equation (3.3) and structure equations of a Kenmotsu manifold, we get
g ∇⊥
X h (φX, Z) , φZ
= Xg (h (φX, Z) , φZ) − g h (φX, Z) , ∇⊥
X φZ
e
= Xg (∇Z X, Z) − g h (φX, Z) , φ∇X Z
= X ((X ln f1 ) g (Z, Z)) − (X ln f1 ) g (h (φX, Z) , φZ) − g (h (φX, Z) , φhν (X, Z))
= X 2 ln f1 g (Z, Z) + (X ln f1 )2 g (Z, Z) − ||hν (X, Z) ||2 ,
where we denote by hν (X, Z) the ν-component of h (X, Z).
Also, by (3.6) and (3.3), we obtain respectively
g (h (∇X φX, Z) , φZ) = ((∇X X) ln f1 ) g (Z, Z) ,
g (h (φX, ∇X Z) , φZ) = (X ln f1 )g (h (φX, Z) , φZ) = (X ln f1 )2 g (Z, Z) .
Substituting the above relations in (4.2), we find
e (X, φX, Z, φZ) = 2||hν (X, Z) ||2 − X 2 ln f1 g (Z, Z) + ((∇X X) ln f1 ) g (Z, Z)
(4.3) R
− (φX)2 ln f1 g (Z, Z) + ((∇φX φX) ln f1 ) g (Z, Z) .
Then the equation (4.3) becomes
c+1
2
g(X, X) + X 2 ln f1 − ((∇X X) ln f1 )
(4.4) 2||hν (X, Z) || =
2
i
+ (φX)2 ln f1 − ((∇φX φX) ln f1 ) g(Z, Z).
Let
{X0 = ξ, X1 , ..., Xα , Xα+1 = φX1 , ..., X2α = φXα , Z1 , ..., Zβ }
be a local orthonormal frame on M such that X0 , ..., X2α are tangent to M1 and Z1 , ..., Zβ are
tangent to M2 .
Therefore
(4.5)
2
β
2α X
X
||hν (Xj , Zt ) ||2 = αβ (c + 1) − 2β∆1 (ln f1 ).
j=1 t=1
Combining (3.5) and (4.5), we obtain the inequality (4.1).
The equality case can be solved similarly to Corollary 3.2.
f (c) be a Kenmotsu space form with c < −1. Then there do not exist
Corollary 4.2. Let M
f (c) such that ln f1 is a
contact CR-doubly warped product submanifolds f2 M1 ×f1 M2 in M
harmonic function on M1 .
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 119, 7 pp.
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C ONTACT CR- DOUBLY WARPED P RODUCT S UBMANIFOLDS IN K ENMOTSU S PACE F ORMS
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Proof. Assume that there exists a contact CR-doubly warped product submanifold f2 M1 ×f1 M2
f (c) such that ln f1 is a harmonic function on M1 .Then (4.5) implies
in a Kenmotsu space form M
c ≥ −1.
f (c) be a Kenmotsu space form with c ≤ −1. Then there do not exist
Corollary 4.3. Let M
f (c) such that ln f1 is a
contact CR-doubly warped product submanifolds f2 M1 ×f1 M2 in M
non-negative eigenfunction of the Laplacian on M1 corresponding to an eigenvalue λ > 0.
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