diff --git a/sections/structure.tex b/sections/structure.tex
@@ -121,18 +121,6 @@ of sections of vector bundles over the unit interval \(I\). Explicitly, we
find\dots
\begin{proposition}
- Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a
- Riemannian metric -- the space \(H^1(E)\) of all class class \(H^1\) sections
- of \(E\) is the completion of the space \({C'}^\infty(E)\) of peace-wise
- smooth sections of \(E\) under the inner product given by
- \[
- \langle \xi, \eta \rangle_1
- = \int_0^1 \langle \xi_t, \eta_t \rangle +
- \langle \nabla \xi_t, \nabla \eta_t \rangle \; \dt
- \]
-\end{proposition}
-
-\begin{proposition}
Given an Euclidean bundle \(E \to I\), the space \(C^0(E)\) of all continuous
sections of \(E\) is is the completion of \({C'}^\infty(E)\) under the norm
given by
@@ -150,6 +138,20 @@ find\dots
\]
\end{proposition}
+\begin{proposition}
+ Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a
+ Riemannian metric -- the space \(H^1(E)\) of all class class \(H^1\) sections
+ of \(E\) is the completion of the space \({C'}^\infty(E)\) of peace-wise
+ smooth sections of \(E\) under the inner product given by
+ \[
+ \langle \xi, \eta \rangle_1
+ = \langle \xi, \eta \rangle_0 +
+ \left\langle
+ \nabla_{\frac\dd\dt} \xi, \nabla_{\frac\dd\dt} \eta
+ \right\rangle_0
+ \]
+\end{proposition}
+
\begin{proposition}\label{thm:continuous-inclusions-sections}
Given an Euclidean bundle \(E \to I\), the inclusions
\[
@@ -169,12 +171,14 @@ find\dots
& = \norm{\xi_{t_0}}^2
+ \int_{t_0}^{t_1} \frac{\dd}{\dd s} \norm{\xi_s}^2 \; \dd s \\
\text{(\(\nabla\) is compatible with the metric)}
- & = \norm{\xi_{t_0}}^2
- + 2 \int_{t_0}^{t_1} \langle \xi_s, \nabla \xi_s \rangle \; \dd s \\
+ & = \norm{\xi_{t_0}}^2 + 2 \int_{t_0}^{t_1}
+ \left\langle \xi_s, \nabla_{\frac\dd{\dd s}} \xi_s \right\rangle
+ \; \dd s \\
\text{(Cauchy-Schwarz)}
- & \le \norm{\xi_{t_0}}^2
- + 2 \int_0^1 \norm{\xi_s} \cdot \norm{\nabla \xi_s} \; \dd s \\
- & \le \norm{\xi}_0^2 + \norm{\xi}_0^2 + \norm{\nabla \xi}_0^2 \\
+ & \le \norm{\xi_{t_0}}^2 + 2 \int_0^1
+ \norm{\xi_s} \cdot \norm{\nabla_{\frac\dd{\dd s}} \xi_s} \; \dd s \\
+ & \le \norm{\xi}_0^2
+ + \norm{\xi}_0^2 + \norm{\nabla_{\frac\dd\dt} \xi}_0^2 \\
& \le 2 \norm{\xi}_1^2
\end{split}
\]
@@ -189,8 +193,8 @@ find\dots
\end{proof}
\begin{note}
- Apply proposition~\ref{thm:continuous-inclusions-sections} to the bundle \(I
- \times \RR^n \to I\) to get the continuity of the maps in
+ Apply proposition~\ref{thm:continuous-inclusions-sections} to the trivial
+ bundle \(I \times \RR^n \to I\) to get the continuity of the maps in
(\ref{eq:continuous-inclusions-rn-curves}).
\end{note}
@@ -393,6 +397,6 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
\end{proof}
This last result will be the basis for our analysis of the Riemannian structure
-of \(H^1(I, M)\). We may now describe the metric of \(H^1(I, M)\) in terms of
+of \(H^1(I, M)\): we may now describe the metric of \(H^1(I, M)\) in terms of
the fibers \(T_\gamma H^1(I, M) \cong H^1(\gamma^* TM)\).