diff --git a/sections/structure.tex b/sections/structure.tex
@@ -10,7 +10,7 @@ Given an interval \(I\), recall that a continuous curve \(\gamma : I \to
\RR^n\) is called \emph{a class \(H^1\)} curve if \(\gamma\) is absolutely
continuous, \(\dot \gamma(t)\) exists for almost all \(t \in I\) and
\(\dot\gamma \in H^0(I, \RR^n) = L^2(I, \RR^n)\). It is a well known fact that
-the so called \emph{Sobolev space \(H^1(I, \RR^n)\)} of all class \(H^1\)
+the so called \emph{Sobolev space \(H^1([0, 1], \RR^n)\)} of all class \(H^1\)
curves in \(\RR^n\) is a Hilbert space under the inner product given by
\[
\langle \gamma, \eta \rangle_1
@@ -47,8 +47,8 @@ described in section~\ref{sec:introduction}, we would like \({C'}^\infty(I,
\RR^n)\) to be a Banach manifold under which both the energy functional and the
length functional are smooth maps. As most function spaces, \({C'}^\infty(I,
\RR^n)\) admits several natural topologies. Some of the most obvious candidates
-are the uniform topology and the topology of \(H^0\) norm, which are the
-topologies induces by the norms
+are the uniform topology and the topology of the \(\norm\cdot_0\) norm, which
+are the topologies induces by the norms
\begin{align*}
\norm{\gamma}_\infty & = \sup_t \norm{\gamma(t)} \\
\norm{\gamma}_0 & = \int_0^1 \norm{\gamma(t)}^2 \; \dt
@@ -91,20 +91,21 @@ approach \(\ell(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
The issue with this particular example is that while \(\gamma_n \to \gamma\)
uniformly, \(\dot\gamma_n\) does not converge to \(\dot\gamma\) in the uniform
-topology. This hints at the fact that in order to \(E\) and \(\ell\) to be
+topology. This hints at the fact that in order for \(E\) and \(\ell\) to be
continuous maps we need to control both \(\gamma\) and \(\dot\gamma\). Hence a
natural candidate for a norm in \({C'}^\infty(I, \RR^n)\) is
\[
\norm{\gamma}_1^2 = \norm{\gamma}_0^2 + \norm{\dot\gamma}_0^2,
\]
which is, of course, the norm induced by the inner product \(\langle \, ,
-\rangle_1\) -- here \(\norm{\cdot}_0\) denote the norm of \(H^0(I, \RR^n)\).
+\rangle_1\) -- here \(\norm{\cdot}_0\) denote the norm of \(H^0(I, \RR^n) =
+L^2(I, \RR^n)\).
The other issue we face is one of completeness. Since \(\RR^n\) has a global
chart, we expect \({C'}^\infty(I, \RR^n)\) to be affine too. In other words, it
is natural to expect \({C'}^\infty(I, \RR^n)\) to be Banach space. In
particular, \({C'}^\infty(I, \RR^n)\) must be complete. This is unfortunately
-not the case for \({C'}^\infty(I, \RR^n)\) the norm \(\norm\cdot_1\), but we
+not the case for \({C'}^\infty(I, \RR^n)\) in the \(\norm\cdot_1\) norm, but we
can consider its completion. Lo and behold, a classical result by Lebesgue
establishes that this completion just so happens to coincide with \(H^1(I,
\RR^n)\).
@@ -125,9 +126,9 @@ of sections of vector bundles over the unit interval \(I\). Explicitly, we
find\dots
\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
+ Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a
+ Riemannian metric -- the space \(C^0(E)\) of all continuous sections of \(E\)
+ is the completion of \({C'}^\infty(E)\) under the norm given by
\[
\norm{\xi}_\infty = \sup_t \norm{\xi_t}
\]
@@ -135,7 +136,7 @@ find\dots
\begin{proposition}\label{thm:h0-bundle-is-complete}
Given an Euclidean bundle \(E \to I\), the space \(H^0(E)\) of all square
- integrable sections of \(E\) is is the completion of \({C'}^\infty(E)\) under
+ integrable sections of \(E\) is the completion of \({C'}^\infty(E)\) under
the inner product given by
\[
\langle \xi, \eta \rangle_0 = \int_0^1 \langle \xi_t, \eta_t \rangle \; \dt
@@ -143,10 +144,9 @@ 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 piece-wise
- smooth sections of \(E\) under the inner product given by
+ Given an Euclidean bundle \(E \to I\), the space \(H^1(E)\) of all class
+ \(H^1\) sections of \(E\) is the completion of the space \({C'}^\infty(E)\)
+ of piece-wise smooth sections of \(E\) under the inner product given by
\[
\langle \xi, \eta \rangle_1
= \langle \xi, \eta \rangle_0 +
@@ -166,9 +166,16 @@ find\dots
\end{proposition}
\begin{proof}
- Fix \(\xi \in H^1(E)\) and \(t_0, t_1 \in I\) with \(\norm{\xi}_0 \ge
- \norm{\xi_{t_0}}\) and \(\norm{\xi}_\infty = \norm{\xi_{t_1}}\).
- Then
+ Given \(\xi \in H^0(E)\) we have
+ \[
+ \norm{\xi}_0^2
+ = \int_0^1 \norm{\xi_t}^2 \; \dt
+ \le \int_0^1 \norm{\xi}_\infty \; \dt
+ = \norm{\xi}_\infty
+ \]
+
+ Now given \(\xi \in H^1(E)\) fix \(t_0, t_1 \in I\) with \(\norm{\xi}_\infty
+ = \norm{\xi_{t_1}}\). Then
\[
\begin{split}
\norm{\xi}_\infty^2
@@ -181,19 +188,11 @@ find\dots
\text{(Cauchy-Schwarz)}
& \le \norm{\xi_{t_0}}^2 + 2 \int_0^1
\norm{\xi_s} \cdot \norm{\nabla_{\frac\dd{\dd s}} \xi_s} \; \dd s \\
- & \le \norm{\xi}_0^2
+ & \le \norm{\xi}_\infty^2
+ \norm{\xi}_0^2 + \norm{\nabla_{\frac\dd\dt} \xi}_0^2 \\
& \le 2 \norm{\xi}_1^2
\end{split}
\]
-
- Finally, given \(\xi \in H^0(E)\) we have
- \[
- \norm{\xi}_0^2
- = \int_0^1 \norm{\xi_t}^2 \; \dt
- \le \int_0^1 \norm{\xi}_\infty \; \dt
- = \norm{\xi}_\infty
- \]
\end{proof}
\begin{note}
@@ -318,8 +317,8 @@ precisely\dots
We would also like to point out that this is a particular case of a more
general construction: that of the Banach manifold \(H^1(E)\) of class \(H^1\)
sections of a smooth fiber bundle \(E \to I\) -- not necessarily a vector
-bundle. Our construction of \(H^1(I, M)\) is equivalent to of of the manifold
-\(H^1(I \times M)\) in the sense that the canonical map
+bundle. Our construction of \(H^1(I, M)\) is equivalent to that of the manifold
+\(H^1(I \times M)\), in the sense that the canonical map
\[
\arraycolsep=1pt
\begin{array}{rl}
@@ -339,7 +338,9 @@ of a vector bundle -- the so called \emph{vector bundle neighborhoods of
\(E\)}. This construction is highlighted in great detail and generality in the
first section of \cite[ch.~11]{palais}, but unfortunately we cannot afford such
a diversion in this short notes. Having said that, we are now finally ready to
-discuss the Riemannian structure of \(H^1(I, M)\).
+layout the Riemannian structure of \(H^1(I, M)\).
+
+We are finally ready to discuss some applications.
\subsection{The Metric of \(H^1(I, M)\)}
@@ -352,13 +353,13 @@ proposition~\ref{thm:tanget-space-topology}. In fact, this isomorphisms may be
extended to a canonical isomorphism of vector bundles, as seen in\dots
\begin{lemma}\label{thm:alpha-fiber-bundles-definition}
- Given \(i = 0, 1\), the collection \(\{(\psi_{i, \gamma}(H^1(U_\gamma) \times
+ Given \(i = 0, 1\), the collection \(\{(\psi_{i, \gamma}(H^1(W_\gamma) \times
H^i(\gamma^* TM)), \psi_{i, \gamma}^{-1})\}_{\gamma \in {C'}^\infty(I,
M)}\) with
\[
\arraycolsep=1pt
\begin{array}{rl}
- \psi_{i, \gamma} : H^1(U_\gamma) \times H^i(\gamma^* TM) \to
+ \psi_{i, \gamma} : H^1(W_\gamma) \times H^i(\gamma^* TM) \to
& \coprod_{\eta \in H^1(I, M)} H^i(\eta^* TM) \\
(X, Y) \mapsto &
\begin{array}[t]{rl}
@@ -384,20 +385,20 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
\end{proposition}
\begin{proof}
- Note that the sets \(H^1(U_\gamma) \times T_\gamma H^1(I, M)\) are precisely
+ Note that the sets \(H^1(W_\gamma) \times T_\gamma H^1(I, M)\) are precisely
the images of the canonical charts
\[
\varphi_\gamma :
- \varphi_\gamma^{-1}(H^1(U_\gamma) \times T_\gamma H^1(I, M))
+ \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
\subset T H^1(I, M)
- \to H^1(U_\gamma) \times T_\gamma H^1(I, M)
+ \to H^1(W_\gamma) \times T_\gamma H^1(I, M)
\]
of \(T H^1(I, M)\).
By composing charts we get local vector bundle isomorphism \(\psi_{1, \gamma}
\circ (\id, \varphi_\gamma) \circ \varphi_\gamma :
- \varphi_\gamma^{-1}(H^1(U_\gamma) \times T_\gamma H^1(I, M)) \isoto \psi_{1,
- \gamma}(H^1(U_\gamma) \times H^1(\gamma^* TM))\). Because of the fact that
+ \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M)) \isoto \psi_{1,
+ \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM))\). Because of the fact that
\(\varphi_\gamma\) and \(\psi_{1, \gamma}^{-1}\) are charts, this
isomorphisms agree in the intersections, so they may be glued together into a
global vector bundle isomorphism. Furthermore, by construction the
@@ -467,7 +468,7 @@ words, we'll show\dots
\begin{proposition}\label{thm:partial-is-smooth-sec}
The map
\begin{align*}
- \partial : H^1(I, M) & \to \coprod_{\gamma} H^0\gamma^* TM \\
+ \partial : H^1(I, M) & \to \coprod_{\gamma} H^0(\gamma^* TM) \\
\gamma & \mapsto \dot\gamma
\end{align*}
is a smooth section of \(\coprod_{\gamma \in H^1(I, M)} H^0(\gamma^* TM) \to
@@ -482,8 +483,8 @@ words, we'll show\dots
\begin{align*}
\mathfrak{X}(H^1(I, M))
& \to \Gamma\left(\coprod_\gamma H^0(\gamma^* TM)\right) \\
- \xi
- & \mapsto \nabla_\xi^0 \partial
+ \tilde X
+ & \mapsto \nabla_{\tilde X}^0 \partial
\end{align*}
is such that
\[