global-analysis-and-the-banach-manifold-of-class-h1-curvers

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -429,8 +429,8 @@ words, we'll show\dots
   Given \(\gamma \in {C'}^\infty(I, M)\) and \(X \in H^1(\gamma^*)\), let
   \begin{align*}
     g_X^\gamma : H^0(\gamma^* TM) \times H^0(\gamma^* TM) & \to \RR \\
-    (Y, Z) & 
-    \mapsto \int_0^1 
+    (Y, Z) &
+    \mapsto \int_0^1
     \langle (d\exp)_{X_t} Y_t, (d\exp)_{X_t} Z_t \rangle \; \dt
   \end{align*}
 
@@ -447,16 +447,16 @@ words, we'll show\dots
   Furthermore, given \(\gamma \in {C'}^\infty(I, M)\) and \(X, Y \in
   H^0(\gamma^* TM)\) by construction we have
   \[
-    g_\gamma(X, Y) 
-    = g_0^\gamma(X, Y) 
-    = \int_0^1 \langle (d \exp)_0 X_t, (d \exp)_0 Y_t \rangle \; \dt 
-    = \int_0^1 \langle X_t, Y_t \rangle \; \dt 
+    g_\gamma(X, Y)
+    = g_0^\gamma(X, Y)
+    = \int_0^1 \langle (d \exp)_0 X_t, (d \exp)_0 Y_t \rangle \; \dt
+    = \int_0^1 \langle X_t, Y_t \rangle \; \dt
     = \langle X, Y \rangle_0
   \]
 \end{proof}
 
 \begin{proposition}\label{thm:partial-is-smooth-sec}
-  The map 
+  The map
   \begin{align*}
     \partial : H^1(I, M) & \to     \coprod_{\gamma} H^0\gamma^* TM \\
                   \gamma & \mapsto \dot\gamma
@@ -471,14 +471,14 @@ words, we'll show\dots
   \Gamma\left(\coprod_{\gamma} H^0(\gamma^* TM)\right)\) the Levi-Civita
   connection of \(\coprod_\gamma H^0(\gamma^* TM)\). The map
   \begin{align*}
-    \mathfrak{X}(H^1(I, M)) 
+    \mathfrak{X}(H^1(I, M))
     & \to \Gamma\left(\coprod_\gamma H^0(\gamma^* TM)\right) \\
-    \xi 
+    \xi
     & \mapsto \nabla_\xi^0 \partial
   \end{align*}
   is such that
   \[
-    (\nabla_X^0 \partial)_\gamma 
+    (\nabla_X^0 \partial)_\gamma
     = \nabla_{\frac\dd\dt} X
     = \frac\nabla\dt X
   \]
@@ -495,8 +495,8 @@ now finally describe the canonical Riemannian metric of \(H^1(I, M)\).
 \begin{definition}\label{def:h1-metric}
   Given \(\gamma \in H^1(I, M)\) and \(X, Y \in H^1(\gamma^* TM)\), let
   \[
-    \langle X, Y \rangle_1 
-    = \langle X, Y \rangle_0 
+    \langle X, Y \rangle_1
+    = \langle X, Y \rangle_0
     + \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle_0
   \]
 \end{definition}
@@ -504,7 +504,7 @@ now finally describe the canonical Riemannian metric of \(H^1(I, M)\).
 At this point it should be obvious that definition~\ref{def:h1-metric} does
 indeed endow \(H^1(I, M)\) with the structure of a Riemannian manifold: the
 inner products \(\langle \, , \rangle_1 : H^1(\gamma^* TM) \times H^1(\gamma^*
-TM) \to \RR\) may be glued together into a single positive-definite 
+TM) \to \RR\) may be glued together into a single positive-definite
 section \(\langle \, , \rangle_1 \in \Gamma( \Sym^2 \coprod_\gamma H^1(\gamma^*
 TM))\) -- whose smoothness follows from
 theorem~\ref{thm:h0-has-metric-extension},