Alexander Yampolsky, V.N. Karazin National University, Kharkiv, Ukraine
A vector field $\xi$ on a Riemannian manifold $(M,g)$ defines a mapping $\xi:M\to TM$ ( or $\xi:M\to T_1M$ in case of $|\xi|=1)$. Endowing $TM$ with the Sasaki metric gives rise to the Riemannian metric on $\xi(M)\subset TM$ or $\xi(M)\subset T_1M$, respectively. This idea allows to assign geometric properties from the geometry submanifold to the vector field. So, one can talk about 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 up-to-date achievements in geometry of unit vector fields focused on minimal, harmonic and totally geodesic properties.
Also, the generalization to the Riemannian bundles will be discussed.
MC 5413