An important class of problems in Riemannian geometry can be stated as follows: given a smooth and orientable Riemanninan manifold M, find a hypersphere U in M having prescribed mean curvature K at each point.
We will be mainly focused on the case when the target M is the Euclidean plane, and the unknown U is a planar loop. Besides its geometrical interpretation, this (apparently) simple problem naturally arises in the study the planar motion of an electrified particle that experiences a Lorentz force produced by a magnetostatic field. It can be regarded as a model for a more general question raised by V.I. Arnold in [Uspekhi Mat. Nauk 1986] within Newtonian setting.
In the last part of the course we will be focused on related questions in the setting of special relativity.