Daniel Peralta-Salas
On the electric field of point charges in 2-dimensional Riemannian manifolds
In this work we will study the topological and
geometrical properties of the electric fields created by point
charges on 2-dimensional Riemannian manifolds. The problems that
we will rise and solve include
- Existence on compact manifolds
- Geodesic character of
the orbits
- Basin boundaries of the field and topological
invariants
- Kupka--Smale generic behavior
- Isometries of
the space and symmetries of the field
We will illustrate our claims with examples in constant curvature
spaces ($S^1\times\mathbb R$, $\mathbb H^2$, $S^2$, $\ldots$ ), rotationally
symmetric spaces, $\ldots$
All these results will show the deep relationship between
electrostatics of point charges and the geometry--topology of the
underlying manifold. This is a very poorly studied topic in the
literature and as far as we know most of the results that we will
report are new. This is a joint work with A. Enciso.