Smooth maps between manifolds?

[benjamin-macadam]

Hi there,

This is a very interesting project! I was wondering if there is support for smooth maps between manifolds, and if so, do you have any examples?

-Ben

[mateuszbaran]

Hi!

Thanks! What would you like to do with these smooth maps between manifolds? There is not much in that area yet but I'm working on adding Jacobians of smooth functions.

[mateuszbaran]

I know very little category theory and it looks quite abstract. Maybe something like https://github.com/AlgebraicJulia/Catlab.jl would help you express the category theory part? By the way, are you looking more for some CAS-type support or number crunching?

[benjamin-macadam]

Well to be more concrete, I would really like to manipulate second order ODEs in a slightly CAS manner (e.g. vector fields on TM that are stable under the canonical flip c:TTM —> TTM), but still be able to handle some number crunching. For everything else, projections and maps into Euclidean space should suffice.

[kellertuer]

For differential equations, we started in ManifoldsDiffEq.jl to collect algorithms for ODEs on Manifolds and Lie groups, maybe that is a good starting point then? It is not yet feature complete, so maybe the algorithms you need might be missing, but it might give a good impression on how to combined manifolds and ODEs, so that you can use that to implement (and maybe contribute) one or the other algorithm?

From and to Euclidean spaces we have either the idea of coordinates (that is basically local charts based on a tangent space) or project and embed (the map into Euclidean space) within the EmbeddedManifold regime.

[mateuszbaran]

To add a bit to @kellertuer 's answer, the CAS part doesn't currently exist in Manifolds.jl. I think at some point we would need to add differential-geometric abstractions to https://github.com/JuliaSymbolics/Symbolics.jl and then support lowering that to ManifoldsDiffEq.jl, similar to what they currently do for ODEs and DifferentialEquations.jl. I don't have any plans to tackle symbolics on manifolds myself in the near future but I could help a bit if you are interested in doing that.

[kellertuer]

I second that. I currently have no own plans to tackle CAS but I am happy to provide help.