Improve exp and log documentation strings

[kellertuer]

Currently it is not mentioned that

• X is from the Lie algebra, as we use that as the default representation of tangent vectors
• we have the left-invariant metric in mind

(cc @olivierverdier )

[olivierverdier]

Just being pedantic: this has nothing to do with a metric, it is rather the exponential associated to the canonical group connection (which need not be metric). There are always two exactly equivalent ways to compute the exponential (here $\exp_1$ is the Lie exponential):

• $\exp_p(X) = p \exp_{1}(p^{-1}X)$
• $\exp_p(X) = \exp_{1}(Xp^{-1}) p$

With the convention of storing a vector field $X$ at $p$ as $X = pY$, the first formulation is much better, since in terms of $Y$ you have (just caring about the storage): $\exp_p(Y) \equiv \exp_1(Y)$.

[kellertuer]

Ah yes, I sometimes am not exact enough with metric and connection. Yes our default convention would be pY and just store Y; but we could also reflect that in tangent-vector-types as also discussed yesterday.

[olivierverdier]

Same issue with the log docstring. If $\log_1$ is the Lie logarithm, then the group log fulfills either of those

• $\log_p(q) = p\log_1(p^{-1}q)$
• $\log_p(q) = \log_1(qp^{-1})p$

And again, with the left-invariant storage convention, the first formulation is better, since in the end you just compute $\log_p(Y) \equiv \log_1(p^{-1}q)$.

[kellertuer]

Sure. I'll add the title here right away.