We prove that the contact distributions admit foliations whose leaves are locally conformal, almost Kählerian, with a mean curvature vector proportional to the characteristic vector field. We also prove that integral manifolds immersed in the CR submanifolds of the locally conformal Kenmotsu are locally conformal Kählerian and which cannot be minimal under some conditions. The class of locally conformal Kähler manifolds is one of sixteen classes of nearly Hermitian manifolds (see for example [18]).
Since then, to our knowledge, no systematic study of locally conformal, near-Kenmotsu manifolds has been undertaken. Therefore, we consider the class of near-contact metric manifolds called locally conformal, near-Kenmotsu manifolds. We wondered whether a locally conformal, nearly Kenmotsu manifold can admit a 1-form ω proportional to η.
The thesis is organized as follows: In Chapter 2 we recall some preliminary denitions about almost Kenmotsu, Kenmotsu and locally conformal (l.c.) almost Kenmotsu structures. We prove that, under a certain condition, the class of locally conformal almost Kenmotsu structures belongs to the class of β-Kenmotsu structures.
Almost Kenmotsu and Kenmotsu structures
In this chapter we recall some general denitions and basic properties of contact metric structures and (almost) Kenmotsu manifolds with particular attention to locally consistent almost Kenmotsu manifolds. We assume (unless otherwise stated) that all manifolds in this thesis are smooth and paracompact. If nowJ is integrable, we say that the almost contact structure (φ, ξ, η) is normal (Sasaki and Hatakeyama [32]).
Since the vanishing of the Nijenhuis rotation of J is a necessary and sufficient condition for integrability, we seek to express the normality condition in terms of the Nijenhuis rotation of φ. However, the vanishing of N(1) implies the vanishing of N(2), N(3), and N(4), so the normality condition is straightforward.
Locally conformal almost Kenmotsu
Furthermore, Kenmotsu proved that such a manifoldM2n+1 is locally a skew product (−, )×f N2n, N2n is a Kähler manifold and f2 =ce2t, approximately positive onstant. Kenmotsu manifold, then the structures (φt, ξt, ηt, gt) are Kenmotsu, that is, they are normal almost Kenmotsu.
Example of l.c. Kenmotsu manifolds
So for e3 = ξ, given the definition of a nearly contact-metric manifold, (φ, ξ, η, g) yields a nearly contact-metric structure on M.
Classes of l.c. almost Kenmotsu structures
Let D := kerη be the contact distribution and D⊥ the spread distribution in the vector structure ξ. The metric is said to be bundle-like for the leaves Fif the induced metric on the transverse distribution D⊥ is parallel with respect to the inner bound of D⊥. If for a given foliationF, the Riemannian metric on M is bundle-like for F, then we say that F is a Riemannian foliation on (M, φ, ξ, η, g).
Now we provide necessary and sufficient conditions for the metric on an l.c. almost Kenmotsu manifold to be bundle-like for foliationsF and F⊥. Therefore, we have the following results. almost Kenmotsu manifold and let F be a foliation on M of codimension 1. Then the following assertions are equivalent: i). The metric g on M is bundle-like for the foliation F. ii) The double vector field B of ω has no components along D. We therefore have the following. almost Kenmotsu manifold and let F be a foliation on M of codimension 1. Then the following assertions are equivalent: a).
When the dierential 1-form ω is reduced to ω =f η, where f is a function such that df ∧η= 0, then M becomes the β-Kenmotsu manifold with β =α+ exp(−σt) and the relation (2.81) for some leaves ofM to be Kählerian becomes. If the metric g of M is bundle-like for the leaves of F, then the leaves of F are Kähler and completely umbilical. Since B is normal to Mf, we compare the tangential part and the normal part in the above equation.
Contact CR -submanifolds
Similarly, for all vector fields X we have tangents to Find any vector field V normal to Mf.
Integrability of distributions in CR -submanifolds
The distribution De⊥ is completely integrable and its largest integral submanifold is a finite-dimensional anti-invariant submanifold of M normal to ξ. The relation (3.65) in Theorem 3.3.2 is also equivalent to the integrability of the De distribution. Assume that the distribution De is integrable with a non-zero normal component of the vector field B (or a non-zero function ω(ξ)), then the integral manifolds of the distribution De cannot be minimal.
We proved that a locally conformal nearly Kenmotsu manifold with the glat1 form, ω, that is proportional to the contact structure, η, i.e. In addition, the integrability of distributions was investigated and the results have shown that the contact distribution D= kerη admits leaves whose leaves are l.c. We also proved that there exist classes of near-contact structures that allow leaves whose leaves are Kähler and umbilical leaves.
We investigated CR submanifolds of l.c. almost Kenmotsu manifold and was aware of the 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 nid-dimensional anti-invariant submanifold of M normal to ξ. The latter can then be extended to the foliation on the surrounding 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 if the leaves of F are (almost) Kähler or allow some other geometric structure. However, studies have indicated that we are only at the beginning of the first study of locally conformal (almost) Kenmotsu manifolds. There are known to be more than four thousand near-contact structures, so we plan to pay attention to some of them and see how we can extract their even-dimensional structure to see the light of the proof of the Golberg conjecture which says that: Any Einstein symplectic structure is Kähler.
Banaru, A new characterization of the Gray:Hervella classes of nearly Hermitian varieties, in: 8th International Conference on Animal Geometry and Its Applications, Opava, Czech Republic.
