# Green's functions, Biot-Savart Operators and Linking Numbers on Negatively Curved Symmetric Spaces

## TL;DR

This paper constructs fundamental solutions for the Laplacian on negatively curved symmetric spaces, enabling the definition of Biot-Savart operators and linking integrals in this geometric setting.

## Contribution

It introduces radial Green's functions for the form Laplacian on symmetric spaces and derives Biot-Savart operators and linking formulas from these solutions.

## Key findings

- Constructed explicit Green's functions for the form Laplacian.
- Established the existence of Biot-Savart operators in this setting.
- Derived linking integrals analogous to Gauss linking numbers.

## Abstract

We construct radial fundamental solutions for the differential form Laplacian on negatively curved symmetric spaces. At least one of these Green's functions also yields a Biot-Savart Opearator, i.e. a right inverse of the exterior differential on closed forms with image in the kernel of the codifferential. Any Biot-Savart operator gives rise to a Gauss linking integral.