SOME VECTORS FIELDS ON THE TANGENT BUNDLE WITH A SEMI-SYMMETRIC METRIC CONNECTION

Let $M$ is a (pseudo-)Riemannian manifold and $TM$ be its tangent bundlewith the semi-symmetric metric connection $\overline{\nabla }$. In thispaper, we examine some special vector fields, such as incompressible vectorfields, harmonic vector fields, concurrent vector fields, conformal vectorfields and projective vector fields on $TM$ with respect to thesemi-symmetric metric connection $\overline{\nabla }$ and obtain someproperties related to them.

Analogs of S.N. Bernstein and V.I. Smirnov inequalities for harmonic polynomials

Гармонические отображения и, в частности, гармонические полиномы находят приложения во многих задачах прикладной математики, математической физики, механики и электротехники. Это связано с ключевой ролью, которую гармонические функции играют в краевых задачах математической физики. Гармонические полиномы используются при описании плоских гармонических векторных полей в гидродинамике, в теории жидких кристаллов, в теории плоского потенциала. Оценки гармонических полиномов и их производных применяются при разработке неравномерных сеток и триангуляций во многих вычислительных схемах и математическом моделировании. В середине двадцатого столетия советскими математиками С.Н. Бернштейном и В.И. Смирновым были доказаны результаты из области дифференциальных неравенств, связывающих многочлены $P(z)=a_n z^n+ a_{n-1} z^{n-1}+ \dots a_1 z+ a_0$ в комплексной плоскости $\mathbb{C}$ и их производные $P'(z)$. Данная тематика сохраняет актуальность, о чем свидетельствует большое число посвященных ей новых публикаций российских и зарубежных математиков. В настоящей работе доказаны результаты, обобщающие неравенства С.Н. Бернштейна и В.И. Смирнова на случай гармонических многочленов $F=H+\overline G,$ где $H, G$ - аналитические многочлены. В частности получены условия типа мажорирующих неравенств на единичной окружности, позволяющие связать производные аналитических и антианалитических частей гармонических многочленов, все нули которых расположены в единичном круге. Доказательства основных результатов получены с помощью топологического аналога известного в теории функций принципа аргумента, позволяющего свести некоторые задачи теории гармонических многочленов к аналитическому случаю. Из полученных результатов следуют классические неравенства Смирнова и Бернштейна в случае аналитических многочленов. Доказанные теоремы проиллюстрированы примером, демонстрирующим точность сформулированных нами условий и оценок. Harmonic mapings and, in particular, harmonic polynomials find applications in many problems of mathematics, mathematical physics, mechanics and electrical engineering. This is due to the key role that harmonic functions play in boundary value problems of mathematical physics. Harmonic polynomials are used to describe plane harmonic vector fields in hydrodynamics, in the theory of liquid crystals, in the theory of plane potential. Estimates of harmonic polynomials and their derivatives are used in the development of non-uniform grids and triangulations in many computational schemes. In the middle of the twentieth century, Soviet mathematicians S.N. Bernstein and V.I. Smirnov proved results several differential inequalities connecting the polynomials $P(z) = a_n z^n + a_{n-1} z^{n-1} + \dots a_1 z + a_0$ in the complex plane $\mathbb{C}$ and their derivatives. This topic remains important, as evidenced by the large number of new publications of Russian and foreign mathematicians. In this paper, we proved results that generalize the inequalities of S.N. Bernstein and V.I. Smirnov for the case of harmonic polynomials $F = H + \overline G,$ where $H, G$ are analytic polynomials. In particular, conditions of the type of majorizing inequalities on the unit circle are obtained, which make it possible to estimate the derivatives of the analytic and antianalytic parts of harmonic polynomials, all of whose zeros are located in the unit disk. The proofs of the main results are obtained using a topological analogue of the principle of the argument known in the theory of functions, which makes it possible to reduce some problems of the theory of harmonic polynomials to the analytic case. The classical inequalities of Smirnov and Bernstein in the case of analytic polynomials follow from the results of current paper. The proved theorems are illustrated by an example that demonstrates the accuracy of the conditions and estimates formulated by us.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

Harmonic vector fields on a weighted Riemannian manifold arising from a Finsler structure

In the present work, the harmonic vector field is defined on closed Finsler measure spaces through different approaches. At first, the weighted harmonic vector field is obtained as the solution space of a PDE system. Then a suitable Dirichlet energy functional is introduced. A σ-harmonic vector field is considered as the critical point of related action. It is proved that a σ-harmonic vector field on a closed Finsler space with an extra unit norm condition is an eigenvector of the defined Laplacian operator on vector fields. Moreover, we prove that a unit weighted harmonic vector field on a closed generalized Einstein manifold is a σ-harmonic vector field.