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.

Zero sets of Lie algebras of analytic vector fields on real and complex two-dimensional manifolds

Let$M$be an analytic connected 2-manifold with empty boundary, over the ground field$\mathbb{F}=\mathbb{R}$or$\mathbb{C}$. Let$Y$and$X$denote differentiable vector fields on$M$. We say that$Y$tracks$X$if$[Y,X]=fX$for some continuous function$f:\,M\rightarrow \mathbb{F}$. A subset$K$of the zero set$\mathsf{Z}(X)$is an essential block for$X$if it is non-empty, compact and open in$\mathsf{Z}(X)$, and the Poincaré–Hopf index$\mathsf{i}_{K}(X)$is non-zero. Let${\mathcal{G}}$be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field$X$. Let$K\subset \mathsf{Z}(X)$be an essential block. Assume that if$M$is complex and$\mathsf{i}_{K}(X)$is a positive even integer, no quotient of${\mathcal{G}}$is isomorphic to$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then${\mathcal{G}}$has a zero in$K$(main result). As a consequence, if$X$and$Y$are analytic,$X$is non-trivial, and$Y$tracks$X$, then every essential component of$\mathsf{Z}(X)$meets$\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

Infinitesimal Hartman-Grobman Theorem in Dimension Three

ABSTRACTIn this paper we give the main ideas to show that a real analytic vector field in R3 with a singular point at the origin is locally topologically equivalent to its principal part defined through Newton polyhedra under non-degeneracy conditions.

On a Hsu-Unified Structure Manifold with a Quarter-Symmetric Non-Metric Connection

In the present paper, we have defined a Hsu-unified structure manifold and a Hsu-Kahler manifold and studied some properties of the quarter-symmetric non-metric connection. Certain interesting results on such manifolds have been obtained. We have also studied the properties of the contravariant almost analytic vector field on these manifolds equipped with the quarter-symmetric non-metric connection.

Cyclicity of period annuli and principalization of Bautin ideals

Let Π be an open period annulus of a plane analytic vector field X0. We prove that the maximal number of limit cycles which bifurcate from Π under a given multi-parameter analytic deformation Xλ of X0 is the same as in an appropriate one-parameter analytic deformation Xλ(ε), provided that this cyclicity is finite. Along the same lines, we also give a bound for the cyclicity of homoclinic saddle loops.

Topological Transitivity on the Torus

T. Ding has shown that a topologically transitive flow on the torus given by a real analytic vector field is orbitally equivalent to a Kronecker flow on the torus, modified so as to have a finite number of fixed points, provided the original flow had only a finite number of fixed points. In this paper it is shown that the assumption that there are only finitely many fixed points is unnecessary.