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},