diff --git a/sections/structure.tex b/sections/structure.tex
@@ -392,19 +392,19 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
\begin{proof}
Note that the sets \(H^1(W_\gamma) \times T_\gamma H^1(I, M)\) are precisely
- the images of the canonical charts
+ the images of the charts
\[
- \varphi_\gamma :
- \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
+ \varphi_\gamma^{-1} :
+ \varphi_\gamma(H^1(W_\gamma) \times T_\gamma H^1(I, M))
\subset T H^1(I, M)
\to H^1(W_\gamma) \times T_\gamma H^1(I, M)
\]
- of \(T H^1(I, M)\) -- i.e. the charts given by\footnote{Once more, we use the
+ of \(T H^1(I, M)\) given by\footnote{Once more, we use the
canonical identification $T_X H^1(W_\gamma) \cong H^1(\gamma^* TM)$ to apply
the vector $\phi_\gamma(Y) \in H^1(\gamma^* TM)$ to $(d \exp_\gamma)_X : T_X
H^1(W_\gamma) \to T_{\exp_\gamma(X)} H^1(I, M)$.}
\begin{align*}
- \varphi_\gamma^{-1} : H^1(W_\gamma) \times T_\gamma H^1(I, M)
+ \varphi_\gamma : H^1(W_\gamma) \times T_\gamma H^1(I, M)
& \to T H^1(I, M) \\
(X, Y) & \mapsto (d \exp_\gamma)_X \phi_\gamma(Y)
\end{align*}
@@ -412,16 +412,16 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
By composing charts we get a fiber-preserving, fiber-wise linear
diffeomorphism
\[
- \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
+ \varphi_\gamma(H^1(W_\gamma) \times T_\gamma H^1(I, M))
\subset T H^1(I, M)
\isoto
\psi_{1, \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM)),
\]
- which takes \(\varphi^{-1}(X, Y) \in T_{\exp_\gamma(X)} H^1(I, M)\) to
- \(\psi_{1, \gamma}(X, \phi_\gamma(Y)) \in H^1(\exp_\gamma(X)^* TM)\). With
- enough patience, one can deduce from the fact that \(\varphi_\gamma\) and
+ which takes \(\varphi(X, Y) \in T_{\exp_\gamma(X)} H^1(I, M)\) to \(\psi_{1,
+ \gamma}(X, \phi_\gamma(Y)) \in H^1(\exp_\gamma(X)^* TM)\). With enough
+ patience, one can deduce from the fact that \(\varphi_\gamma^{-1}\) and
\(\psi_{1, \gamma}^{-1}\) are charts that these maps agree in the
- intersections of the open subsets \(\varphi_\gamma^{-1}(H^1(W_\gamma) \times
+ intersections of the open subsets \(\varphi_\gamma(H^1(W_\gamma) \times
T_\gamma H^1(I, M))\), so that they may be glued together into a global
smooth map \(\Phi : T H^1(I, M) \to \coprod_{\eta \in H^1(I, M)} H^1(\eta^*
TM)\).