On Geometry of Vector Fields

Principal cycles of one dimensional foliations associated to a plane field in $\mathbb{E}^3$

“…This work was motivated by the extrinsic geometry of vector fields and plane fields ∆η as developed by Y. Aminov [1]. This approach came back to the classical works, G. Rogers [21], A. Voss [27] and others.…”

“…Following the classical approach of principal curvature lines (see [1], [23], [24], [26]) the extremal values of k η restricted to the plane field ∆ η are called η−principal curvatures (denoted by k 1 ≤ k 2 ) and the associated directions are called η−principal directions (denoted by e 1 and e 2 ). The integral curves of e 1 and e 2 are called η−principal lines, defining the two η−principal foliations F i (η), i = 1, 2.…”

“…They are associated to the normal curvature of a plane distribution defined by an unit vector field η. See [1]. After that, it is obtained the differential equation of the η−principal line field directions.…”

“…The extrinsic geometry of vector fields and flows can be applied in the dynamic of mechanical systems, control systems, theoretical elasticity and fluid flows, see [1] and [4]. Also, in foliation theory the extrinsic geometry of leaves and flows is present, see [22] for a recent exposition on this subject.…”

Command of Collective Dynamics by Topological Defects in Spherical Crystals

“…Prior to examining the dynamical modes on the sphere, we first present general discussions about the topology of a vector field on the sphere. As a classical problem of differential topology, an even-dimensional sphere admits no regular tangent vector field, and singularities are inevitable as a topological constraint [35]. A singularity at point p in the vector field V is characterized by its index Ind p V :…”