# Topological Sound and Flocking on Curved Surfaces

## TL;DR

This paper explores how active polar flocks behave on curved surfaces, revealing that curvature induces topological modes that enable robust, unidirectional information transport along specific geodesics, akin to quantum Hall edge states.

## Contribution

It analytically demonstrates the emergence of topologically protected sound modes in active flocks on curved surfaces, linking geometry, activity, and topological phenomena.

## Key findings

- Curvature induces inhomogeneous steady states with topological defects.
- Active flow and curvature create localized, symmetry-protected topological modes.
- These modes act as robust, unidirectional channels for information transport.

## Abstract

Active systems on curved geometries are ubiquitous in the living world. In the presence of curvature orientationally ordered polar flocks are forced to be inhomogeneous, often requiring the presence of topological defects even in the steady state due to the constraints imposed by the topology of the underlying surface. In the presence of spontaneous flow the system additionally supports long-wavelength propagating sound modes which get gapped by the curvature of the underlying substrate. We analytically compute the steady state profile of an active polar flock on a two-sphere and a catenoid, and show that curvature and active flow together result in symmetry protected topological modes that get localized to special geodesics on the surface (the equator or the neck respectively). These modes are the analogue of edge states in electronic quantum Hall systems and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering.