Match the Manifolds.jl notation in doc

[Affie]

Nice!

I have two short comments after a short first look: In the intro you introduce Lie groups – at the end you link to JuliaManifolds/Manifolds.jl, but you could actually link to the documentation about GroupManifolds? https://juliamanifolds.github.io/Manifolds.jl/latest/manifolds/group.html

Second, more e technical remark: We redesigned ManifoldsBase to have a neat trait system, so AbstractGroupManifold will vanish in the next (breaking) release of Manifolds.jl (based on the already released ManifoldsBase.jl 0.13.0 with its own documentation now) and there will just be a IsGroupManifold trait.

Oh I just saw two further things down the road:

• You write $T_M(p)$ for the tangent space, we actually write $T_p\mathcal M$ (manifolds are calligraphic). This is just a remark, if you have a different notation that is fine of course, if you just use this notation in this one place, maybe it might be helpful to the reader to use only one notation.
• you write $vee$. and $Euclidean$ and such in Math mode. To me that reads as variables multiplied (so it would acutally be $e^2v$), since they are in italics, the function would be $\mathrm{vee}$ (but that's also just me, a math and notation nerd).

Originally posted by @kellertuer in https://github.com/JuliaManifolds/Manifolds.jl/issues/355#issuecomment-1088476581

[dehann]

Hi @kellertuer, great, thanks for the notes and pointers on ManifoldsBase.jl v0.13.0. Will do the fixes.

I caught the wave to hold back with the updates until Manifolds.jl is tagged but have not yet been able to dig through all the code to learn what the changes were. I saw some of those PRs were pretty large.

So the few comments here help us a lot, thanks!

Oh yeah, I strongly second the notation nerd comment, I'm biased to following the notation you have in JuliaManifolds. For us working the navigation problem, the number one issue is ALWAYS figuring out what all conventions are, as used by different groups. So notation is super super important and my sense is there should be zero friction when looking at the factor graph solver here or the JuliaManifolds ecosystem. If we stray but a little, we are going to get these impossible error stacks and funky numerics and then trying to debug will be a nightmare.

[kellertuer]

For the notation – sure I am not saying you must change it, just if you could mention that the notation is different in Manifolds.jl or something.

An also for the update – suer, we still need a few more iterations to finish the Manifolds.jl release – Manopt is already done for example – but Manifolds.jl has just a few small errors to fix left.