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: M→TM (or ξ: M→T1M 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.