# Impossible configurations for geodesics on negatively-curved surfaces

## TL;DR

This paper generalizes a known example of a graph on a punctured sphere that cannot be realized by geodesics in negatively curved metrics to more complex surfaces with higher genus and punctures.

## Contribution

It extends the concept of impossible geodesic configurations from a specific case to all surfaces of genus n with k punctures, broadening understanding of geometric constraints.

## Key findings

- Identifies new impossible configurations on complex surfaces.
- Generalizes previous specific examples to all genus and puncture cases.
- Provides theoretical framework for understanding geodesic realization limitations.

## Abstract

Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus $n$ with $k$ punctures for any $n$ and $k$.