diff --git a/sections/structure.tex b/sections/structure.tex
@@ -1,6 +1,6 @@
\section{The Structure of \(H^1(I, M)\)}\label{sec:structure}
-Throughout this sections let \(M\) be a finite-dimensional Riemannian manifold.
+Throughout this section let \(M\) be a finite-dimensional Riemannian manifold.
As promised, in this section we will highlight the differential and Riemannian
structures of the space \(H^1(I, M)\) of class \(H^1\) curves in a \(M\). The
first question we should ask ourselves is an obvious one: what is \(H^1(I,
@@ -411,16 +411,17 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
(X, Y) & \mapsto (d \exp_\gamma)_X \phi_\gamma(Y)
\end{align*}
- By composing charts we get fiber-preserving, fiber-wise linear diffeomorphism
+ By composing charts we get a fiber-preserving, fiber-wise linear
+ diffeomorphism
\[
\varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
\subset T H^1(I, M)
\isoto
\psi_{1, \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM)),
\]
- which takes \(\varphi^{-1}(X, Y) \in T_{exp_\gamma(X)} H^1(I, M)\) to
+ 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
- enought patience, one can deduce from the fact that \(\varphi_\gamma\) and
+ enough patience, one can deduce from the fact that \(\varphi_\gamma\) and
\(\psi_{1, \gamma}^{-1}\) are charts that these maps agree in the
intersections of the open subsets \(\varphi_\gamma^{-1}(H^1(W_\gamma) \times
T_\gamma H^1(I, M))\), so that they may be glued together into a global