Higher genus manifolds

[FedeClaudi]

Hi all,

Following a conversation on Slack with @mateuszbaran and @kellertuer I wanted to open this space to start talking about ways to implement higher genus manifolds.
I'm happy to do to the work, but some guidance from folks with more experience with Manifolds.jl would be very super helpful.

@mateuszbaran on Slack you mentioned that the best way might be to construct a higher genus manifolds from charts, I tend to agree as there doesn't seem to be a nice parametrization for them. Could you point me in the direction of an example of how this approach is used to construct other manifolds with Manifolds.jl? Thank you

[kellertuer]

Hi,
In general the way would be to implement a smooth atlas, see https://juliamanifolds.github.io/Manifolds.jl/latest/features/atlases.html, which we only use for one example for now, the stereographic atlas on the Sphere, see https://juliamanifolds.github.io/Manifolds.jl/latest/manifolds/sphere.html#Manifolds.StereographicAtlas

[mateuszbaran]

BTW, here is a link I've posted on Slack that may be helpful for defining charts on higher genus manifolds: https://math.stackexchange.com/questions/3427382/how-to-construct-a-g-genus-torus-by-identifying-sides-of-a-4g-gon . I think mathematically what we need at this point is an atlas covering such a higher genus manifold with a metric tensor and Christoffel symbols and we could work from there in Manifolds.jl.

[FedeClaudi]

Allright, thanks both of you for the help.
I'll have a look during this week for ways to construct an atlas for such manifolds. I might just start with the 2-torus and we can see if there's a way to generalize it to other manifolds later.

[mateuszbaran]

Cool, I'm going to sketch a code for calculating geodesics on a 1-torus in my proposed approach.

[kellertuer]

Cool, also an atlas for the Torus would be great, since having a few atlases would be great to be sure out interface idea works :)

[FedeClaudi]

that's fair. Perhaps that's a good place to start for me to familiarize myself with the interface with a simpler problem.