Similarly, for any vector elds X tangent to Mfand any vector eld V normal to Mf, we have
(∇Xφ)V =∇XφV −φ(∇XV)
=∇Xf V +∇XnV −φ(−AVX+∇e⊥XV)
=∇eXf V +eh(X, f V)−AnVX+∇eTXnV −φ(−AVX)−φ∇eTXV
=∇eXf V +eh(X, f V)−AnVX+∇eTXnV +f AVX+nAVX
−f∇eTXV −n∇eTXV
= (∇eXf)V +eh(X, f V)−AnVX+ (∇eXn)V +f AVX+nAnX.
That is,
exp(−σt)eg(φX, V)ξ+ω(φV)X−ω(V)φX −eg(X, φV)B = (∇eXf)V +eh(X, f V)
−AnVX+ (∇eXn)V +f AVX+nAnX. (3.46) Since φX =F X+N X, (3.46) becomes
exp(−σt)eg(φX, V)ξ+ω(φV)X−ω(V)(F X+N X)−eg(X, φV)(BT +BN)
= (∇eXf)V +eh(X, f V)−AnVX+ (∇eXn)V +f AVX+nAnX. (3.47) Lemma 3.2.4. Let Mfbe a contact CR-submanifold of an l.c. Kenmotsu M tangent to ξ. Then,
(∇eXf)V = exp(−σt)eg(φX, V)ξ+AnVX−f AVX+ω(φV)X−ω(V)F X
−eg(X, φV)BT, (3.48)
and (∇eXn)V =−nAnX−eh(X, f V)−ω(V)N X −eg(X, φV)BN, (3.49) for all vector eld X tangent toMf.
Integrability of distributions inCR-submanifolds 30
for any vector eldZ tangent to Mf. Since
(∇eXF)Y = exp(−σt){eg(φX, Y)ξ−η(Y)F X}+ω(φY)X−ω(Y)F X
−eg(X, φY)BT +eg(X, Y)F B+AN YX+feh(X, Y), (3.51) then, for any X, Y ∈Γ(De⊥) and Z ∈Γ(TM)f, we have that
0 =eg((∇eZF)X, Y) = exp(−σt){eg(φZ, X)η(Y)−η(X)eg(F Z, Y)}+ω(φX)eg(Z, Y)
−ω(X)eg(F Z, Y)−eg(Z, φX)eg(BT, Y) +eg(Z, X)eg(F B, Y) +eg(AN XZ, Y) +eg(feh(Z, X), Y). (3.52) since η(X) =η(Y) = 0. Equation (3.52) becomes
0 =eg((∇eZF)X, Y) =ω(φX)eg(Z, Y)−ω(X)eg(F Z, Y)−g(Z, φX)e eg(BT, Y)
+g(Z, Xe )eg(F B, Y) +eg(AN XZ, Y) +g(fe eh(Z, X), Y). (3.53) On the other hand, eg(F Z, Y) = 0,eg(Z, φX) = 0,eg(F B, Y) =−ω(φY) and
eg(feh(Z, X), Y) =eg(φeh(Z, X), Y)
=−eg(eh(Z, X), φY)
=−eg(eh(Z, X), N Y)
=−eg(AN YZ, X). (3.54)
Substituting (3.54) into (3.53) gives
0 =ω(φX)eg(Z, Y)−ω(φY)eg(Z, X) +eg(AN XZ, Y)−eg(AN YZ, X). (3.55) That is,
g(Ae N XY, Z)−eg(AN YX, Z) =−ω(φX)eg(Z, Y) +ω(φY)eg(Z, X). (3.56) Thus, we have the following Lemma.
Lemma 3.3.1. Let Mfbe a contact CR-submanifold of an l.c. Kenmotsu M tangent to ξ. Then,
AN XY −AN YX =ω(φY)X−ω(φX)Y (3.57) for all vector elds X and Y tangent to Mf.
Theorem 3.3.1. Let Mfbe a contact CR-submanifold of a(2n+ 1)-dimensional l.c.
Kenmotsu manifoldM tangent toξ. The distribution De⊥ is completely integrable and its maximal integral submanifold is a nite-dimensional anti-invariant submanifold of M normal to ξ.
Proof. LetX, Y ∈De⊥. Then we have that φ[X, Y] =F[X, Y] +N[X, Y]
=F∇eXY −F∇eYX+N[X, Y]
=∇eXF Y −(∇eXF)Y −∇eYF X+ (∇eYF)X+N[X, Y]
=−(∇eXF)Y + (∇eYF)X+N[X, Y]. (3.58) Substituting(3.51) into (3.58), gives
φ[X, Y] =ω(φX)Y −ω(φY)X+AN XY −AN YX+N[X, Y], (3.59) and by using(3.57), equation (3.59) simplies to
φ[X, Y] =N[X, Y]. (3.60) Therefore,[X, Y]∈Γ(De⊥).
Theorem 3.3.2. Let Mfbe a contact CR-submanifold of a(2n+ 1)-dimensional l.c.
Kenmotsu manifold M tangent to ξ. Then the distribution De ⊕ {ξ} is integrable if and only if
eh(X, F Y)−eh(Y, F X) = 2g(Y, φX)BN, (3.61) for any X, Y ∈Γ(De⊕ {ξ}).
Proof. LetX, Y ∈Γ(De ⊕ {ξ}). Then, N X =N Y = 0 and φ[X, Y] =F[X, Y] +N[X, Y]
=F[X, Y] +N(∇eXY)−N(∇eY)X
=F[X, Y] +∇eXN Y −(∇eXN)Y −∇eYN X + (∇eYN)X
=F[X, Y] + (∇eYN)X−(∇eXN)Y. (3.62) Substituting(3.45) into (3.62), gives
φ[X, Y] =F[X, Y] +neh(Y, X)−eh(Y, F X)−ω(X)N Y −eg(Y, φX)BN +eg(Y, X)N B−neh(X, Y) +eh(X, F Y) +ω(Y)N X+eg(X, φY)BN
−eg(X, Y)N B, (3.63)
which becomes
φ[X, Y] =F[X, Y]−eh(Y, F X)−2eg(Y, φX)BN
+eh(X, F Y). (3.64)
Thus, we see that[X, Y]∈Γ(De⊕ {ξ})ieh(X, F Y)−eh(Y, F X) = 2eg(Y, φX)BN.
Integrability of distributions inCR-submanifolds 32
Proposition 3.3.1. Let Mfbe a contact CR-submanifold of a (2n+ 1)-dimensional l.c. Kenmotsu manifold M tangent to ξ. Then the distribution De ⊕ {ξ} is integrable if and only if
eh(X, F Y)−eh(Y, F X) = 2eg(Y, φX)BN, (3.65) for any X, Y ∈Γ(De⊕ {ξ}).
Proof. LetX, Y ∈Γ(D)e . Then,N X =N Y = 0 and φ[X, Y] =F[X, Y] +N[X, Y]
=F[X, Y] +N(∇eXY)−N(∇eY)X
=F[X, Y] +∇eXN Y −(∇eXN)Y −∇eYN X + (∇eYN)X
=F[X, Y] + (∇eYN)X−(∇eXN)Y. (3.66) Substituting(3.45) into (3.66), gives
φ[X, Y] =F[X, Y] +neh(Y, X)−eh(Y, F X)−ω(X)N Y −eg(Y, φX)BN +eg(Y, X)N B−neh(X, Y) +eh(X, F Y) +ω(Y)N X+eg(X, φY)BN
−eg(X, Y)N B, (3.67)
which becomes
φ[X, Y] =F[X, Y]−eh(Y, F X)−2eg(Y, φX)BN +eh(X, F Y), (3.68) Thus, we see that[X, Y]∈Γ(De ⊕ {ξ}) if and only if
eh(X, F Y)−eh(Y, F X) = 2eg(Y, φX)BN, and that completes the proof.
The relation (3.65) in the Theorem 3.3.2 is also equivalent to the integrability of the distribution De.
Assume that De is integrable. Then, the relation (3.65) can be rewritten as, eh(F X, F Y) = eh(X, Y)−2eg(X, Y)BN, (3.69) for any X, Y ∈Γ(D)e .
Let M0 be a leaf of De. Then M0 is a maximal integral submanifold immersed in M. For any X, Y ∈Γ(T M0),
∇XY =∇0XY +h0(X, Y), (3.70) where
h0 =eh+σ+{−ω(ξ)g+g◦h}ξ, (3.71) is the second fundamental form of M0, immersed as a submanifold in M with σ a vector eld in De⊥, and ∇0 the Levi-Civita connection on M0.
Theorem 3.3.3. Let Mfbe a contact CR-submanifold of a(2n+ 1)-dimensional l.c.
Kenmotsu manifold M tangent to ξ. Assume that the distribution De is integrable.
Then the integral manifolds of the distributionDe are l.c. Kähler manifolds with mean curvature vector eld given by
H0 = 1 rank(D)e {
rank(D)e
X
i=1
eh(ei, ei) + trace|M0σ} −4BN −2ω(ξ)ξ}. (3.72) Proof. Assume that the distributionDe is integrable. Let M0 be an integral manifold of De. The tensor elds φt and gt induce an almost complex structure Jt = J and a Hermitian metricgt0 onM0. Then, for anyX ∈Γ(T M0), we haveφ02tX =−X+ηt0Xξt =
−X = Jt2X = J2X and dΦt = 0, so M0 is an l.c. Kähler. Using (3.71), the second fundamental form of M0 is explicitly given by
eh0(X, Y) =eh(X, Y) +σ(X, Y) +{−ω(ξ)eg(X, Y) +eg(hX, Y)}ξ. (3.73) Fixing a local orthonormal frame{e1,· · · , en, φe1,· · · , φen}inT M0 and applying the properties on h, one has,
H0 = 1 rank(D)e {
rank(D)e
X
i=1
eh0(ei, ei) +
rank(D)e
X
i=1
eh0(φei, φei)}
= 1
rank(D)e {trace|M0eh+ trace|M0σ+
rank(D)e
X
i=1
(−ω(ξ)eg(ei, ei) +eg(hei, ei))ξ
+
rankD)
X
i=1
(−ω(ξ)eg(φei, φei) +eg(hφei, φei))ξ}
= 1
rank(D)e {trace|M0eh+ trace|M0σ−2ω(ξ)
rank(D)e
X
i=1
eg(ei, ei)ξ
= 1
rank(D)e {trace|M0eh+ trace|M0σ−2rank(D)ω(ξ)ξ}.e (3.74) Now, using the identity (3.16) and (3.69), the trace ofeh on M0 is given,
trace|M0eh=
rank(D)e
X
i=1
eh(ei, ei) +
rank(D)e
X
i=1
eh(φei, φei)
=
rank(D)e
X
i=1
eh(ei, ei)−4rank(D)Be N. (3.75) Putting the relation (3.75) into (3.74) completes the proof.
Integrability of distributions inCR-submanifolds 34
Corollary 3.3.1. Let Mf be a contact CR-submanifold of a (2n + 1)-dimensional l.c. Kenmotsu manifoldM tangent to ξ. Assume that the distribution De is integrable with a non-vanishing normal component of the vector eld B ( or a non-vanishing function ω(ξ)), then the integral manifolds of the distribution De cannot be minimal.
Conclusion and Perspectives
We introduced a new concept of almost contact structures, namely, l.c. almost Ken- motsu structures, supported by an example. The latter was characterized by the existence of a closed 1-form ω such that
dη =ω∧η and dΦ = 2{exp(−σt)η+ω} ∧Φ,
with ω = dσt. We proved that a locally conformal almost Kenmotsu manifold with the smooth1-form, ω, that is proportional to the contact structure, η, i.e. ω=αη, is aβ-Kenmotsu manifold with β =α+ exp(−σt).
In addition, the integrability of distributions was studied and the results have shown that the contact distribution D= kerη admits foliations whose leaves are l.c.
almost Kählerian with mean curvature vector eldH0 =−ω(ξ)ξ. We also proved that there exist classes of almost contact structures that admit foliations whose leaves are Kählerian and umbilical.
We investigated CR-submanifolds of l.c. almost Kenmotsu manifold and paid attention of distributions De, De⊥ and De ⊕ {ξ}. As a result, we concluded that the distribution Db⊥ is completely integrable and its maximal integral submanifold is a nite-dimensional anti-invariant submanifold ofM normal to ξ. Furthermore, De and De ⊕ {ξ} are integrable if and only if
eh(X, F Y)−eh(Y, F X) = 2eg(Y, φX)BN.
The latter can then be extended to the foliation on the ambient manifolds M. That is, if the metric on an l.c. (almost) Kenmotsu manifold is bundle-like for the foliation F, as one of the perspectives, we would like to know whether the leaves of F are (almost) Kähler or admit another geometric structure.
However, studies indicated that we are only at the beginning of the rst study of locally conformal (almost) Kenmotsu manifolds. A lot can still be done in an l.c.
(almost) Kenmotsu manifold. One may also look at which properties can be preserved in an l.c. (almost) Kenmotsu manifold with reference to other types of manifolds.
Our study revealed that l.c. almost Kenmotsu manifolds are not necessary Ken- motsu and that the vector eldξis not a Killing vector eld for l.c. almost Kenmotsu
Integrability of distributions inCR-submanifolds 36
manifolds. Among other orientations, we shall study the topology of l..c almost Ken- motsu geometry. It is known that there are more than four thousand almost contact structures, so we are planing to pay attention to some of them and see how we can ex- tract their even-dimensional structure in order to see the light of proving the Golberg Conjecture which says that: Any Einstein symplectic structure is Kähler.
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