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.