Geometry & Topology Seminar

Thursday, March 23, 2023 2:30 pm - 2:30 pm EDT

Alexander Yampolsky, V.N. Karazin National University, Kharkiv, Ukraine

"Some aspects of geometry of unit vector fields"

A vector field ξ on a Riemannian manifold (M,g) defines a mapping ξ:  MTM (or ξ: MT1M in case of |ξ|= 1). Endowing TM with the Sasaki metric gives rise to the Riemannian metric on ξ(M) ⊂ TM or ξ(M) ⊂ T1M, respectively. This idea allows to assign the geometric properties from the geometry of submanifolds to the vector field. So, one can talk about the intrinsic or extrinsic geometry of vector fields.

The most developed idea in a given setting is the idea of harmonic and minimal unit vector fields. The report outlines the geometry of unit vector fields focused on examples of minimal, harmonic and totally geodesic properties. Mean curvature of the Reeb vector field on (α,β) - trans-Sasakian manifold will be presented as well as its minimality and total geodesity conditions. The properties of invariant unit vector fields on the oscillator Lie group will be considered in more detail.

MC 5417