Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

We consider the family ℰ (s, r, d) of all singular complex analytic vector fields X ( z ) = Q ( z ) P ( z ) e E ( z ) ∂ ∂ z $X(z)=\frac{Q(z)}{P(z)}{{e}^{E(z)}}\frac{\partial }{\partial z}$ on the Riemann sphere where Q, P, ℰ are polynomials with deg Q = s, deg P = r and deg ℰ = d ≥ 1. Using the pullback action of the affine group Aut(ℂ) and the divisors for X, we calculate the isotropy groups Aut(ℂ) X of discrete symmetries for X ∈ ℰ (s, r, d). The subfamily ℰ (s, r, d)id of those X with trivial isotropy group in Aut(ℂ) is endowed with a holomorphic trivial principal Aut(ℂ)-bundle structure. A necessary and sufficient arithmetic condition on s, r, d ensuring the equality ℰ (s, r, d) = ℰ (s, r, d)id is presented. Moreover, those X ∈ ℰ (s, r, d) \ ℰ (s, r, d)id with non-trivial isotropy are realized. This yields explicit global normal forms for all X ∈ ℰ (s, r, d). A natural dictionary between analytic tensors, vector fields, 1-forms, orientable quadratic differentials and functions on Riemann surfaces M is extended as follows. In the presence of nontrivial discrete symmetries Γ < Aut(M), the dictionary describes the correspondence between Γ-invariant tensors on M and tensors on M /Γ.

A KAHLER MANIFOLD ADMITTING SEMI-SYMMETRIC METRIC CONNECTION A.

In present paper we study the properties of Kahler manifold satisfying the semi - symmetric metric connection. Symmetric and skew-symmetric conditions for Nijenhuis tensor of the connection in Kahler manifold has been discussed. The paper also includes some properties of contravariant almost analytic vector field in a Kahler manifold.

Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables

In this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-valued functions analytic in the unit ball $\begin{array}{} \mathbb{B}^2\! = \!\{z\!\in\!\mathbb{C}^2: |z|\! = \!\small\sqrt{|z_1|^2+|z_2|^2}\! \lt \! 1\}, \end{array} $ where L = (l1, l2): 𝔹2 → $\begin{array}{} \mathbb{R}^2_+ \end{array} $ is a positive continuous vector-valued function.Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L-index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of (p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L(z).