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In this paper, we study the existence of torqued and anti-torqued vector fields on the hyperbolic ambient space $\mathbb{H}^n$. Although there are examples of proper torqued vector fields on open subsets of $\mathbb{H}^n$, we prove that…
We consider the standard Darboux space equipped with the radial symmetric contact form. We study co-orientation preserving contactomorphisms between relatively compact domains up to the boundary. We determine the contactomorphism classes…
We study ordinary differential equations in the complex domain given by meromorphic vector fields on K\"ahler compact complex surfaces. We prove that if such an equation has a maximal single valued solution with Zariski-dense image (in…
For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree d, it is proved that any bifurcation which preserves the multiplicity of equilibrium points can be realized as a composition of a…
We determine the shapes of pure cubic fields and show that they fall into two families based on whether the field is wildly or tamely ramified (of Type I or Type II in the sense of Dedekind). We show that the shapes of Type I fields are…
Torse-forming vector fields are generalizations of some important vector fields. In this paper, we present some techniques to transform a proper torse-forming vector field into its special cases. Concrete examples are given.
Eberhard-type theorems are statements about the realizability of a polytope (or more general polyhedral maps) given the valency of its vertices and sizes of its polygonal faces up to a linear linear degree of freedom. We present new…
We give the classification of all complete algebraic vector fields on Danielewski surfaces (smooth surfaces given by $xy=p(z)$). We use the fact that for each such vector field there exists a certain fibration that is preserved under its…
We introduce and study $H$-paracontact metric manifolds, that is, paracontact metric manifolds whose Reeb vector field $\xi$ is harmonic. We prove that they are characterized by the condition that $\xi$ is a Ricci eigenvector. We then…
After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective action of a torus whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we…
We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset…
We have been interested in graphs and realizing them as Reeb graphs of explicit real algebraic functions. The Reeb graph of a differentiable function is the quotient space of the manifold of the domain, regarded as the space consisting of…
In this article, we present a complete study of two disjoint classes of conformal vector fields on doubly warped product manifolds as well as on doubly warped space-times. Then we study Ricci solitons on doubly warped product manifollds…
Let $F$ be a smooth vector field defined in a neighborhood of the origin in $\mathbb{R}^n$, $F(O)=0$, and let $F_t$ be its local flow. Denote by $E$ the set of germs of diffeomorphisms $h:\mathbb{R}^n \to \mathbb{R}^n$ preserving orbits of…
New splitting theorems in a semi-Riemannian manifold which admits an irrotational vector field (not necessarily a gradient) with some suitable properties are obtained. According to the extras hypothesis assumed on the vector field, we can…
We obtain several results for (iterated) planar contact manifolds in higher dimensions: (1) Iterated planar contact manifolds are not weakly symplectically semi-fillable. This generalizes a 3-dimensional result of Etnyre to a…
A contact manifold admittting a supporting contact form without contractible Reeb orbits is called hypertight. In this paper we construct a Rabinowitz Floer homology associated to an arbitrary supporting contact form for a hypertight…
In this paper, we characterize conformal vector fields of any (regular or singular) $(\alpha,\beta)$-space with some PDEs. Further, we show some properties of conformal vector fields of a class of singular $(\alpha,\beta)$-spaces satisfying…
In this paper, we give some properties of biharmonic hypersurface in Riemannian manifold has a torse-forming vector field.
In this paper, we prove that there exists a residual subset of contact forms $\lambda$ (if any) on a compact connected orientable manifold $M$ for which the foliation de Rham cohomology of the associated Reeb foliation $F_\lambda$ is…