Infinitesimal Transformations of Locally Conformal Kähler Manifolds

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.


Locally conformal Kähler manifolds
A Hermitian manifold (M 2m , J, g) is called a locally conformal Kähler manifold (LCK -manifold) if there is an open cover U = U α α∈A of M 2m and a family {σ α } α∈A of C ∞ functions σ α : U α → R so that each local metricĝ α = e −2σ α g| U α is Kählerian. An LCK -manifold is endowed with some form ω, so called a Lee form which can be calculated as [3] The form should be closed: dω = 0.
One can compute covariant derivative an almost complex structure with respect to the Levi-Civita connection of (M 2m , J, g) using the formula

Infinitesimal transformations of manifolds
34 Definition 1. Transformation of a manifold M n is called infinitesimal transformation of a manifold M n . Vector ξ(x 1 , x 2 , . . . , x n ) is often refered as a generator 35 of transformation. An arbitrary small parameter is independent on x i . 36 Lie derivative of a tensor of type (p, q) T i 1 ...i p j 1 ...j q with respect to a vector field ξ may be calculated by te equation [1, p. 196 In particular, for a metric tensor g we get If a manifold M n was transformed then its metric tensor g of the transformed M n is where h ij = L ξ g ij = ξ i,j + ξ j,i [7, p. 275]. For the Christoffel symbols we have also[17, p. 8]: Transvecting (7) with g hi we get: The item g hi L ξ Γ h jk depends on transformation type. We are interested primarily in the case when a vector field ξ(x 1 , x 2 , . . . , x n ) generates a transformation preserving the complex structure [16]: The field is called a contravariant analytic vector field, and the infinitesimal transformation is refered as a holomorphic one. It is worth to note that since exterior differentiation and the Lie derivation with respect to ξ are commutative dL ξ ω = L ξ dω (10) hence any infinitesimal transformation preserves the closeness property of Lee form. If a transformation (3) does not change geodesics of a manifold, it is called a projective transformation. Mikeš an Radulovich in [4] proved that LCK-manifolds (n > 2) do not admit nontrivial finite geodesic mappings onto Hermitian manifolds if a preserving complex structure is requared. 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 ψ is a scalar whose gradient ψ i = ∂ i ψ and a vector ξ generate the transformation. Then combining (8) and its conditions of integrability, we obtain: is satisfied [7, p. 275]. Since the metric g ij is Hermitian, we get: Also, since deformed metric g ij is Hermitian and the complex structure is preserved, hence on the deformed manifold M n , 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 g ij , we get: Then we use (2) and (11): Then, let us regroup the items: Transvecting (17) with η j produces: It follows from (18) that χ k = αθ k . Hence, It follows from (19) that one of the equations holds, namely θ i = 0, or αg jk + h jk = 0. In the former 39 case we have that the manifold M n is Kählerian since ω i = 0 and the transformation is trivial because 40 ψ i = 0. In the latter case the equation h jk = −αg jk means that the transformations is a conformal one.

41
But one knows that if a transformation is simultaneously conformal and projective then it is a trivial 42 one. Hence we obtain the theorem.
It is known, if a vector field ξ generates conformal infinitesimal transformations, the field and invariant ϕ satisfy the system [6], [5]: 3.3. Nijenhuis tensor and Lee form under conformal infinitesimal transformations 48 Taking into account (9) and (10), we have that a necessary and sufficient condition that under conformal infinitesimal transformation an LCK-manifold remains also locally conformal Kählerian is that the Lie derivative of Nijenhuis tensor must be equal to zero: The Lie derivative of a Nijenhuis tensor is because of(9).

49
It is known the identity [16, p. 159]: where Γ k ji are components of a symmetric affine connection which is compatible with a metric g ij . Because of (9), from (23) we get Let us calculate the Lie derivative of a Nijenhuis tensor with respect to the vector field ξ, taking into account (24) Removing the parentheses and collecting similar terms in (25) we obtain that the Lie derivative of a Nijenhuis tensor is equal to zero L ξ N k ij = 0.
Taking into account that any infinitesimal transformation preserves the closeness property of its Lee 50 form we obtain the theorem. Proof. Let us calculate a Lie derivative of a Lee form. Because of (9), from (1) we have On other hand, since a Lie derivative and contraction are commutative hence contracting for k and j from (24) we obtain Substituting (27) into (26) we find that Theorem 3. 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) Proof. It is worth to note that according to (4) On other hand, Since the Lee form is closed then ω i,j = ω j,i , and hence from (4) it follows that Hence the scalar ϕ mentioned in may be expressed by the equation where C is an arbitrary constant. Hence taking into account the conditions (9) the PDE system (21) becomes Let us find the conditions of integrability of (30). According to [17, p. 17] for the Levi-Civita connection the conditions are For the present case we have Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 June 2019 doi:10.20944/preprints201906.0232.v1 Since for the conformal transformations the equations are satisfied hence (32) can be presented in the form where Also there is identity [17, p. 16] that for the present case is Taking account of (32), (33), (32), from (31) we obtain Finally we have where Q h ijk is defined as Differentiating several times (35) we get a system of differential prolongations. For convenience we use the identity for Lie derivative of tensor covariant derivative [17, p.16] and we obtain first differential prolongation for (35) where L ξ Γ h jk and Q h ijk are defined (32) and (36) respectively. We can continue the process until it turn 56 out that the new equations are satisfied identically or the system have became inconsistent.

57
The equations (30 1 ) are solvable for n = 2m unknown functions, and the equations (30 3 ) are solvable for n 2 = 4m 2 unknown functions. The equations (30 2 ) include n(n+1) 2 = 2m(2m+1) 2 restrictions. It is easy to see that (30 4 ) determines 2m 2 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 (z α , zα) the conditions (30 4 ) are presented in the form Hence we have Lowering the indices we obtain Hence we find that the equations (30 2 ) include m(m + 1) restrictions which involve (30 4 ). It follows that solution of the system (30) involves not more then constants.

58
Theorem 4. In order that an LCK-manifold (M n , J, g) admits a group of conformal transformations, it is necessary and sufficient that the equations then the LCK-manifold admits a r = (m + 1) 2 − k parameter group of conformal transformations.

59
Considering the system (30) we can find that if ω α ξ α = 0, then the system may also be written in the form   Let Kählerian metricĝ be locally conformal to the metric of an LCK-manifold (M n , J, g). According to the definition g ij = g ij e −2σ , ω i = 2σ ,i . Then is the Levi-Civita connection which is compatible with the metricĝ. Let us define a contravariant vector field ξ i on (M n , J, g). Let us denote Then we differentiate covariantlyξ i with respect to the Levi-Civita connection which is compatible with the metricĝ. Covariant derivative with respect to the connectionΓ k ij is denoted as | . Covariant derivative with respect to the connection Γ k ij is denoted as usual by comma. We get Suppose that a field ξ i generates a homothetic group of the metricĝ. Then it must satisfy equationŝ Substituting (39) into (40) we obtain e −2σ ξ i,j + ξ j,i = Cĝ ij + ω α ξ αĝij ; e −2σ ξ i,j + ξ j,i = e −2σ ω α ξ α g ij + Cg ij ; Since e −2σ = 0 holds, (30 2 ) are necessarily satisfied Let us differentiate covariantlyξ i|j with respect to the connectionΓ k ij . Since (38) holds, we obtain Since according to (21 2 ) in the case of conformal transformations we have L ξ g jk = ϕg jk , hence ω i L ξ g jk − ω α (L ξ g iα )g jk = 0, and (41) can be written aŝ where ||ω|| 2 = ω i ω j g ij . On other hand, it follows from (38) that the curvature tensorR of a Kähler metricĝ and the curvature tensor R of an LCK-metric are related by the following expression It is known that if a field ξ i generates homothetic transformation of metricĝ then the field satisfies also the equation Substituting (42) and (43) into (44), taking into account thatξ i = ξ i e −2σ , we get Again, it follows from e −2σ = 0 that (30 3 ) is satisfied The condition that for a Kähler metricĝ a vector field ξ i satisfies if and only if the similar conditions(9) is satisfied. Hence if a vector field ξ i satisfies the system (30), then it satisfies the system 1)ξ i,j =ξ ij ; 2)ξ i,j +ξ j,i = Cĝ ij ; We obtain the theorem 67 Theorem 6. If an LCK-manifold (M n , J, g), n = 2m admits a group G r of infinitesimal conformal 68 transformations preserving the complex structure, then the group G r is isomorphic to the group of homothetic 69 transformations of the Kähler metricĝ conformally corresponding to the LCK-metric.

70
It is worth to note that the obtained theorem is very similar to the results obtained by R. F. Bilyalov Then we raise the index i in (45) On other hand, it'known [18], that a necessary and sufficient condition for a vector field ξ 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. Let a vector field ξ generate one-parameter continuous group of isometries of an LCK-manifold. Then the vector field ξ satisfies Killing equtions.
Note, that we denote by comma covariant differentiation with respect to the Levi-Civita connection of (M n , J, g). Taking account of (39), expressing (49) with respect to the Levi-Civita connection which is compatible with the Kählerian metricĝ, we obtain But it follows from the Theorem 3 that Kählerian metric does not admit nontrivial conformal 85 transformations. Hence ξ α ω α = const, and we obtain the theorem. Let us consider a pseudo-Vaisman manifold [8] i. e. the LCK-manifold whose Lee form satisfies the equation where Φ 4 is the fourth Obata projector. It follows from (51) that, Lie derivative with respect to the vector field B = ω # satisfies the equations Let us find a Lie derivative of a fundamental form Ω ij = J s i g sj . According to [10, p. 4] on an LCK-manifold, covariant derivative of the complex structure in the directions of B or A is equal to zero Here A = −JB is so called anti-Lee fields. Hence Since (51) is equivalent to Let us find a Lie derivative of the fundamental form with respect to the anti-Lee field A = −JB = θ # . Since (52) holds, we have Removing the parentheses in (55), and taking into account that Lee form is closed, we have We obtain the theorem.

91
Theorem 9. On a pseudo-Vaisman manifold i. e. on an LCK-manifold whose Lee form satisfies the condition Lie derivatives of the fundamental form with respect to the Lee field B = ω # and anti-Lee field A = −JB = θ # satisfy the equations 1)L B Ω ij = ||ω|| 2 Ω ij , 2)L A Ω ij = 0.
Let us find a Lie derivative of the complex structure with respect to the Lee field B and the anti-Lee field A taking account of (52).
Removing the parentheses in (57) and collecting similar terms, we obtain that Let us find a Lie derivative of the LCK-metric with respect to the anti-Lee field A Finally, we get Now let us consider the case when the Lee form satisfies strong pseudo-Vaisman condition ∇ω(X, Y) = ||ω|| 2 2 g(X, Y) Hence the Lee field satisfies the equations ω i,j + ω j,i = ||ω|| 2 g ij Comparing the equations with (30 2 ) ξ i,j + ξ j,i = ω α ξ α + C g ij , Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 June 2019 doi:10.20944/preprints201906.0232.v1 we obtain, that the Lee field ω # generates on the LCK-manifold one-parameter conformal group for which in (30 2 ) the condition C = 0 holds. We get ω i,j + ω j,i = ω α ω α g ij .
Taking account of (39) we obtain that for the connection which is compatible with the Kählerian metriĉ g ij = e −2σ g ij the equationsω i|j +ω j|i = 0, are satisfied. Here we noteω i = ω t g stĝ ti = e −2σ ω i . It follows from (60) that the vector field ω # generates one-parameter isometry group of the Kählerian metricĝ ij . Also it follows from (56) that if the Lee form satisfies strong pseudo-Vaisman condition, the we have Hence The Lee 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 L A J k i = 0, It means that anti-Lee field is also contravariant analytic. Hence we write (59) in the form L A g ij = 0., That mean also that anti-Lee field θ # is a Killing field. Taking into account Theorem 8 we make the 92 following deductions.

98
The manifolds under consideration are LCK-manifolds. The investigations use local coordinates. 99 We assume that all functions under consideration are sufficiently differentiable, and use tensor methods

101
Complex geometry deals primarily with Kählerian manifolds i.e. manifolds carrying some