Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$, and $exp:\mathfrak{g}\to G$ be its exponential map. The group $G$ could be finite or infinite dimensional. Let $G$ have the property that 


$\bullet$ 
For each smooth curve 
$X\in C^{\infty}(\mathbb R,\mathfrak g)$ there exists a curve 
$g\in C^{\infty}(\mathbb R,G)$ whose right logarithmic derivative is $X$, i.e.,
$$
g(0) = e, \qquad
\partial\_t g(t) = T\_e(\mu^{g(t)})X(t) = X(t).g(t),\quad\text{where } \mu(a,b)=\mu\_a(b)=\mu^b(a) = a.b
$$
The curve $g$ is uniquely determined by its initial value $g(0)$, if it
exists.


$\bullet$
Put $\operatorname{evol}^r\_G(X)=g(1)$ where $g$ is the unique solution required above. 
Then $\operatorname{evol}^r\_G: C^{\infty}(\mathbb R,\mathfrak g)\to G$ is required to be
$C^{\infty}$ also. 


Note that for $X$ constant in time, $\operatorname{evol}^r\_G(X)=\exp(X)$. So each regular Lie group admits an exponential mapping.


My questions are: 


1) Is there any expression for the derivative of $\operatorname{evol}^r\_G$. It's motivated by the fact that that there's an expression of $(Dexp)\_{\mathfrak g}$ using the adjoint action on $\mathfrak g$ and derivative of the left translation in $G$.


2) Can we construct a new Lie group $H\subset = C^{\infty}(\mathbb R,G)$ with Lie algebra $C^{\infty}(\mathbb R,\mathfrak g)$ so that $\operatorname{evol}^r$ becomes the exponential map $C^{\infty}(\mathbb R,\mathfrak g)\to H$?


I must admit that I got the idea of the question from Peter Michor's answer to this question:


[Exponential map](https://mathoverflow.net/questions/121579/exponential-map/121611#121611)



Generalization of the Lie group exponential map and its derivative
Answer to 1:


$TG$ is again a Lie group, semidirect product go $G$ over the normal $\mathfrak g$. See 6.7 of [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf). $TC^\infty(\mathbb R, \mathfrak g) = C^\infty(\mathbb R, \mathfrak g\times \mathfrak g)$. Then 
$$
T(evol\_G^r) = evol^r\_{TG}. 
$$
Be careful with the identifications. This is treated in detail in 38.10 of [this book](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf).


Answer to 2:


See 38.11 and 42.21 of [this same second book](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf).


