I am trying to calculate the derivative of the exponential map, and I am getting stuck on a few points.
Given a Lie group $G$, the Lie algebra $\mathfrak{g}$ can either be thought of as left-invariant vector fields of $G$ (which I will denote $\mathcal{Lie}(G)$) or the tangent space at the identity (denoted $T_eG$). Given a left-invariant vector field, there exists a flow $\phi_X(t)$ such that $\phi_X(0)=e$ and $\dot{\phi}_X(t)=X_{\phi(t)}$. The exponential map
$$\text{exp}:g \rightarrow G \\
X \mapsto \phi_X(1)$$
then satisfies certain properties, and in particular
$$\text{exp}(tX) = \phi_X(t).
$$
I want to show that the derivative of exp at the identity is given by the identity. It seems to follow immediately from looking at
$$
D_0\text{exp}(X) = \left.\frac{d}{dt}\right|_{t=0} \text{exp}(tX) = \left.\frac{d}{dt}\right|_{t=0}\phi_X(t) = X_e,
$$
where I understand the first equality to hold since we are looking at the derivative at the identity, i.e. with $X=0$ (is this correct?).
However, I want to understand the spaces in which everything is defined a bit better. It seems be the case that everything is defined 'up to isomorphism' in a sense, but I would like to clarify is this is correct.
To begin, the Lie algebra is a vector space, and so there is clearly an isomorphism between the Lie algebra $\mathfrak{g}$ and the tangent space at the identity of the Lie algebra $T_0\mathfrak{g}$. Thus, the above map begins by using this isomorphism so that $D_0\text{exp}$ is a map from the Lie algebra $\mathcal{Lie}(G)$ (which we consider in terms of vector fields). The map is then to the Lie algebra $T_eG$, now in terms of the tangent space. Hence, the result is the identity up to the isomorphism between them.
So to summarise:
Is the first step in my proof correct?
Is my understanding of the map correct?
Best Answer
It is helpful to first understand the definition of the derivatives in the abstract.
I will use the definition that for a smooth manifold $M$, $T_pM$ is the set of equivalence classes of curves $\gamma: (-1,1)\to M$ with $\gamma(0)=p$, where two curves are equivalent if their derivative in any chart is equal, i.e. $$\gamma\sim\gamma' \iff \frac{d}{dt}\bigg\rvert_{t=0}\phi\circ \gamma=\frac{d}{dt}\bigg\rvert_{t=0}\phi\circ \gamma'$$ for a chart $\phi: U\to \mathbb{R}^n$ a chart containing $p$. We say that the derivative of a curve $\gamma$ at time $\tau$ is the equivalence class of $\gamma$, i.e. \begin{equation}\frac{d}{dt}\bigg\rvert_{t=\tau}\gamma:= [\gamma(t-\tau)]\quad (\ast)\end{equation}
The defining equation for the flow of a vector field is as follows:Given a time dependent diffeomorphism $\phi: \mathbb{R}\times M\to M$ and a point $p\in M$, we can associate a curve $\gamma_p: I\to M$ by $\gamma_p(t)=\phi_t(p)$. A time dependent diffeomorphism is said to be the flow for a vector field $X\in \mathfrak{X}(M)$ if for each $p\in M$ $[\gamma_p]=X(p)$ or more suggestively $$\frac{d}{dt}\bigg\rvert_{t=0}\phi_{t}(p)=X(p)$$
We then can easily define the derivative of a map $f: M\to N$ as $D_pf: T_pM\to T_{f(p)}N$ by $D_pf[\gamma]=[f\circ \gamma]$
Now we will look at the specific case of $f=\exp: \mathfrak{g}\to G$. If we define $\exp(tX)$ to be the flow of the left invariant vector field corresponding to $X$, $\phi_X(t)$ must satisfy $\frac{d}{dt}\bigg\rvert_{t=0}\phi_X(t)(g)=X(g)$ for all $g\in G$. Recall that $D_0\exp: T_0 \mathfrak{g}\to T_{\exp(0X)}G=T_eG$ and since $\mathfrak{g}$ is a vector space we have a canonical isomorphism $\mathfrak{g}\cong T_0\mathfrak{g}$. This means that the map $D_0\exp$ is really an endomorphism $\mathfrak{g}\to \mathfrak{g}$.
$D_0\exp(X):=\frac{d}{dt}\bigg\rvert_{t=0} \exp(\gamma(t))$ where $\gamma(t)$ is a curve with $\frac{d}{dt}\bigg\rvert_{t=0} \gamma(t)=X$, so in this case we can take $\gamma(t)=tX$ (under the canonical isomorphism $T_0\mathfrak{g}\cong \mathfrak{g}$.)
EDIT: The definition of $(D_0\exp)(X)$ is as follows, if $\gamma: I\to \mathfrak{g}$ with $[\gamma]=X$ then $D_0\exp(X):=[\exp\circ \gamma]$. Since $\frac{d}{dt}\bigg\rvert_{t=0} (tX)=\left(\frac{d}{dt}\rvert_{t=0} t\right)X=X$ then $[tX]=X$ and $D_0\exp(X)=[\exp(tX)]=\frac{d}{dt}\bigg\rvert_{t=0} \exp(tX)$ (this last equation comes from the notation given in equation $(\ast)$).