Infinitesimal conformal transformations of locally conformal Kähler manifolds

Background. Motivations for investigation of infinitesimal transformations are the development of physics, particularly mechanics and the probability theory, and the reached results have applications in many branches of technical sciences, especially in modelling of dynamical processes. Attention is paid also to special classes of Hermitian manifolds that are distinguished by some differential conditions on the complex structure. The manifolds can be mapped conformally on Kähler manifolds, therefore they are called conformally Kähler mansfolds. Materials and methods. The author uses local coordinates, assumes that all functions under consideration are sufficiently differentiable and applies tensor methods. Results. 1. Infinitesimal transformations relative to a covariant almost analitic field preserve a Nijenhuis' tensor, i. e. its Lie derivative is identically equal to zero: $L_\xi N^k_{ij}=0$. 2.The researcher has found an expression for a Lie derivative of a Lee form relative to a covariant almost analitic field for locally conformal Kähler manifolds: $L_\xi \omega_i=-\varphi_i$. 3. Also article considers compact orientable locally conformal Kähler manifolds and reveals the identity $\int_{M_n} \omega_\alpha J^\alpha_i \xi^i d\sigma =\frac{2}{n-2} \int_{M_n} J^\alpha_i \xi^i_\alpha d\sigma $. That is condition on the complex structure $J^\alpha_i$, the vector field $\xi^i$, and its derivatives $\xi^i_\alpha$.