Bumpy Metrics Theorem for Geodesic Nets

TL;DR

This paper proves that for a generic class of Riemannian metrics, all connected embedded stationary geodesic nets on a manifold are non-degenerate, extending the understanding of geodesic structures in differential geometry.

Contribution

It establishes a Bumpy Metrics Theorem for geodesic nets, showing generic non-degeneracy of stationary geodesic networks on smooth manifolds.

Findings

All connected embedded stationary geodesic nets are non-degenerate for generic metrics.

The result extends classical bumpy metric theorems to geodesic nets.

Provides a new framework for studying geodesic networks in Riemannian geometry.

Abstract

Stationary geodesic networks are the analogs of closed geodesics whose domain is a graph instead of a circle. We prove that for a Baire-generic Riemannian metric on a smooth manifold $M$, all connected embedded stationary geodesic nets are non-degenerate.