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The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. We have also calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric.
Kählerian manifolds, because of their properties, have been used for modeling of physical processes for a long time, for instance in supersymmetric theories , in string theories (e.g., , p. 411). A manifold is called locally conformal Kähler manifold (for brevity, LCK-manifolds) if its metric is conformal to some local Kählerian metric in the neighborhood of each point of the manifold. On the other hand, one knows that conformal mappings preserve the Petrov type of a manifold . The LCK-manifolds are also used for physical modeling. For instance, in  authors offered a‘Kaluza-Klein model with spontaneous compactification, using a generalized Hopf manifold. Also, researchers use locally conformally Calabi-Yau manifolds to build M-theory models. According to  a locally conformally Calabi-Yau manifold is an LCK-manifold with a Ricci-flat metric. For example an eight-dimensional Hopf manifold admits a Ricci-flat metric, hence it may be used in a model of eleven-dimensional Supergravity.
The objects under consideration in the article are the LCK-manifolds for which . LCK-manifolds were explored by [6,7,8]. The book  is also worth noting as one of the most distinguished in this area. Infinitesimal conformal transformations were explored in [10,11]. Infinitesimal conformal transformations of complex manifolds were studied by Yano . Transformations of LCK-manifolds were explored in . The main goal of the article is also to explore transformations of LCK-manifolds.
2. Locally Conformal Kähler Manifolds
A Hermitian manifold is called a locally conformal Kähler manifold (LCK-manifold) if there is an open cover of and a family of functions so that each local metric
is Kählerian. An LCK-manifold is endowed with some form , called the Lee form, which can be calculated as 
The form should be closed:
One can compute covariant derivative of an almost complex structure with respect to the Levi-Civita connection of using the formula
Here and below, we denote by comma covariant differentiation with respect to the Levi-Civita connection of .
3. Infinitesimal Transformations of Manifolds
Transformation of a manifold
is called infinitesimal transformation of a manifold . Vector is often referred to as a generator of transformation. An arbitrary small parameter ϵ is independent on .
The Lie derivative of a tensor of type with respect to a vector field may be calculated by using the formula (, p. 196):
In particular, for a metric tensor we get
If a manifold is transformed then indices of the metric tensor of the transformed is
where , and is the arbitrary small parameter mentioned in the Definition 1 (, p. 275). For the Christoffel symbols we have also (, p. 8):
The item depends on transformation type. We are interested primarily in the case when a vector field generates a transformation preserving the complex structure :
The field is called a contravariant analytic vector field, and the infinitesimal transformation is referred to as a holomorphic one. It is worth noting that since exterior differentiation and the Lie derivation with respect to are commutative
hence any infinitesimal transformation preserves the closeness property of a Lee form.
3.1. Projective Transformations and LCK-Manifolds
If a transformation (3) does not change geodesics of a manifold, it is called a projective transformation. Mikeš and Radulovich in  proved that LCK-manifolds () do not admit nontrivial finite geodesic mappings onto Hermitian manifolds if a preserving complex structure is required. We have to explore whether nontrivial projective transformations preserving a complex structure are admitted on LCK-manifolds. Hence let us suppose that such transformation is admitted. Then
where is a scalar whose gradient and a vector generates the transformation. Then combining (8) and its conditions of integrability, we obtain:
Also the equation
is satisfied (, p. 275). Since the metric is Hermitian, we get:
Also, since deformed metric is Hermitian and the complex structure is preserved, hence on the deformed manifold , the identity
is satisfied. Taking into account (6) and (12), from (13) we obtain:
Differentiating covariantly (14) with respect to the Levi-Civita connection which is compatible with a metric , we get:
It follows from (19) that one of the equations holds, namely , or , where is a certain function of the variables . In the former case we have that the manifold is Kählerian since and the transformation is trivial because . In the latter case the equation means that the transformation is a conformal one. But one knows that if a transformation is simultaneously conformal and projective then it is a trivial one. Hence we obtain the theorem.
An LCK-manifold , does not admit nontrivial projective transformations with respect to the Levi-Civita connection preserving its complex structure.
Note that the technique we use proving the theorem is very similar to the one offered in .
3.2. Conformal Infinitesimal Transformations of Locally Conformal Kähler Manifolds
Infinitesimal transformations are called conformal if the equations hold (, p. 275):
where is some function of the variables .
It is well known that, if a vector field generates conformal infinitesimal transformations, then the field and the invariant satisfy the system [3,11]:
3.3. Nijenhuis Tensor and Lee form under Conformal Infinitesimal Transformations
A necessary and sufficient condition for an almost Hermitian manifold to be Hermitian is
If a vector field ξ generates a conformal infinitesimal transformation of an LCK-manifold, then components of Lie derivatives of the Lee form are equal to the partial derivatives of the invariant φ defined by the system (21)
Differentiating several times (35) we get a system of differential prolongations. For convenience we use the identity for Lie derivative of tensor covariant derivative (, p.16) and we obtain first differential prolongation for (35)
where and are defined by (32) and (36) respectively. We can continue the process until it turns out that the new equations are satisfied identically or the system has become inconsistent.
The Equation (30a) is solvable for unknown functions, and the Equation (30c) is solvable for unknown functions. The Equation (30b) includes restrictions. It is easy to see that (30d) determines independent restrictions. Since an LCK-manifold is a Hermitian one, then it follows from integrability of its almost complex structure that there exists a system of complex coordinate neighbourhoods. In the complex coordinate system the condition (30d) is presented in the form
Hence we have
Lowering the indices we obtain
Hence we find that the Equation (30b) includes restrictions which involve (30d). It follows that solution of the system (30) involves not more then
In order for an LCK-manifold to admit a group of conformal transformations, it is necessary and sufficient that the equations
the conditions of integrability (35), their differential prolongations (37), ... etc, be algebraically consistent with respect to and . If there are, among the Equations (35) and (37), ..., exactly k equations which are linearly independent among themselves and of
then the LCK-manifold admits a parameter group of conformal transformations.
Considering the system (30) we can find that if , then the system may also be written in the form
Thus we have the following theorem.
If on an LCK-manifold the Lie algebra of conformal vector fields includes such subalgebra that everywhere on holds, then the subalgebra generates a group of homothetic transformations.
The Theorem follows immediately from the Frobenius Theorem (, p. 201). □
3.4. Local Isomorphism between Conformal Group of an LCK-Manifold and Homothetic Group of the Corresponding Kählerian Metric
Let the Kählerian metric be locally conformal to the metric of an LCK-manifold . According to the definition , . Then
is the Levi-Civita connection which is compatible with the metric . Let us define a contravariant vector field on . Let us denote
Then we differentiate covariantly with respect to the Levi-Civita connection which is compatible with the metric . Covariant derivative with respect to the connection is denoted as “|”. Covariant derivative with respect to the connection is denoted as usual by comma. We get
Suppose that a field generates a homothetic group of the metric . Then it must satisfy equations
The condition that for the Kähler metric a vector field satisfies
if and only if the similar conditions (9) is satisfied. Hence if a vector field satisfies the system (30), then it satisfies the system
We obtain the theorem.
If an LCK-manifold , admits a group of infinitesimal conformal transformations preserving the complex structure, then the group is isomorphic to the group of homothetic transformations of the Kähler metric conformally corresponding to the LCK-metric.
It is worth noting that the obtained theorem is very similar to the results obtained by R. F. Bilyalov (, p. 274) for real Lorenzian manifolds. Namely, let be a group of conformal transformations of a Lorenzian manifold which is not conformally flat. Then we can find a manifold , conformally corresponding to whose homothetic group is isomorphic to the group of conformal transformations of the . But our result does not require that the manifold needs not to be conformally flat.
Applying the Theorems 6 and 4 to conformally flat manifolds, in particularly to a Hopf manifold, equipped by the Boothby metric, we obtain that conformal groups of the manifolds depend on parameters, where .
3.5. Conformal Infinitesimal Transformations on Compact LCK-Manifolds
Let be a compact LCK-manifold, vector field generates conformal transformations (30b). Contracting (30c) with we have
On the other hand, it is known , that a necessary and sufficient condition for a vector field in a compact almost Hermitian space to be contravariant almost analytic is
For LCK-manifolds, taking account of (2) and (1), we have
Comparing (46) and (47), taking account of (48) we obtain the theorem.
In a compact LCK-manifold any vector field ξ which generates nontrivial conformal transformations is contravariant almost analytic.
3.6. Isometries of LCK-Manifolds
Let a vector field generates one-parameter continuous group of isometries of an LCK-manifold. Then the vector field satisfies the Killing equations.
Taking account of (39), expressing (49) with respect to the Levi-Civita connection which is compatible with the Kählerian metric , we obtain
But it follows from the Theorem 3 that Kählerian metric does not admit nontrivial conformal transformations. Hence , and we obtain the theorem.
Isometric Group of an LCK-manifold is isomorphic to some subgroup of homothetic group of the corresponding local Kählerian metric. In particular, if a vector field orthogonal to the Lee vector field is Killing with respect to the LCK-metric g then the field is also Killing with respect to the local Kählerian metric .
3.7. Transformations Generated by the Lee Fields and Anti-Lee Vector Fields on Pseudo-Vaisman Manifolds
Let us consider a pseudo-Vaisman manifold , i.e., the LCK-manifold whose Lee form satisfies the equation
where is the fourth Obata projector. It follows from (51) that, Lie derivative with respect to the vector field satisfies the equations
Let us find a Lie derivative of a fundamental form . According to (, p. 4) on an LCK-manifold, covariant derivatives of the complex structure in the directions of B or A are equal to zero:
Here is so called the anti-Lee vector field, the symbol denotes the covariant derivative of the Riemannian connection defined by the LCK-metric g with respect to B, etc. Hence
we obtain, that the Lee vector field generates on the LCK-manifold one-parameter conformal group for which in (30b) the condition holds. We get
Taking account of (39) we obtain that for the connection which is compatible with the Kählerian metric the equations
are satisfied. Here we note . It follows from (60) that the vector field generates one-parameter isometry group of the Kählerian metric . Also it follows from (56) that if the Lee form satisfies the strong pseudo-Vaisman condition, then we have
Hence the Lee vector field is contravariant analytic, i.e., a transformation generated by the field preserves the complex structure. Also, substituting the strong pseudo-Vaisman condition into (57), we obtain
It means that the anti-Lee field is also contravariant analytic. Hence we write (59) in the form
That means also that the anti-Lee vector field is a Killing vector field. Taking into account Theorem 8 we make the following deductions.
Let be an LCK-manifold, and its Lee form satisfies the strong pseudo-Vaisman condition
The Lee and anti-Lee vector fields (respectively and ) are contravariant analytic.
On the manifold the Lie field generates one-parameter conformal group, and anti-Lee field generates one-parameter group of isometry.
The Lee and anti-Lee vector fields generate one-parameter isometric groups of the Kählerian metric . The Kählerian metric is conformally corresponding to the LCK-metric g.
The manifolds under consideration are LCK-manifolds. The investigations use local coordinates. We assume that all functions under consideration are sufficiently differentiable, and use tensor methods (c.f. ).
Complex geometry deals primarily with Kählerian manifolds, i.e., manifolds carrying some Kählerian metric. Although, some complex manifolds, such as complex Hopf manifolds, admit no global Kählerian metrics at all. However, we can often find for every map of atlas a multiplyer which transforms a metric into a Kählerian one. One can say that a metric g is a locally conformal Kähler (LCK) metric if g is conformal to some local Kählerian metric in the neighborhood of each point of a manifold. In fact, the locally conformal Kähler manifolds were introduced by W. Westlake in 1954, some publications were soon made by P. Libermann, but mainly through the works of Vaisman since the 1970s has the geometry of LCK-manifolds been developed. Mappings and transformations of LCK-manifolds were explored by V. F. Kirichenko, K. Matsumoto, J. Mikeš, A. Moroianu, L. Ornea. The presented paper is devoted to infinitesimal transformations. We have obtained that LCK-manifolds does not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Finally, we have got the necessary and sufficient conditions in order that the an LCK-manifold admit a group of conformal motions. In addition, we have calculated the number of parameters which the group depends on. We have proved that the group of conformal motions admitted by an LCK-manifold is isomorphic to the homothetic group admitted by the corresponding Kählerian metric.
All authors contributed equally to this research. The research was carried out by all authors. The manuscript was prepared together and they all read and approved the final version.
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