What are some state of the art methods/algorithms to find geodesics on arbitrary manifold meshes?
There can be much dispute on which work is considered best or state-of-the-art. It will depend on your application. However, I'd like to recommend to give a look at this paper by Keenan Crane et al. It is impressive in terms of computational speed. Also, It is simple to implement.
Crane et al. Geodesics in heat: A new approach to computing distance based on heat flow. ACM TOG. 2013. (https://dl.acm.org/doi/10.1145/2516971.2516977)