Speaker
Description
The linking number of two disjoint oriented null-homologous submanifolds K,L in an oriented manfold M is the primary obstruction to disentangling them. By means of the Thom isomorphism in deRham theory, the linking number can be rewritten as an integral over the product of the two submanfolds, analogous to Gauss' famous linking integral for disjoint loops in euclidean space.
The integrand of such linking integrals is closely related to the fundamental solution of the Cartan differential, i.e. to the integral kernel of a right inverse. In R^n such a fundamental solution is provided by the Riesz potential, which coincidentally also solves the adjoint equation. In magnetostatics, this is known as the Biot-Savart law which computes the magnetic field caused by a stationary current. In a series of papers, DeTurck, Gluck and Cantarella have constructed such a "Biot-Savart Operator" for space forms. We extend part of this to negatively curved symmetric spaces exploiting their harmonicity.
For the linking number however, one may ignore the adjoint equation, only a right inverse of the Cartan differential is needed. On the rank-1 symmetric spaces we apply Cartan's homotopy ("magic") formula to the gradient flows of the Busemann functions to construct this.
