diff --git a/sections/structure.tex b/sections/structure.tex
@@ -11,7 +11,7 @@ Given an interval \(I\), recall that a continuous curve \(\gamma : I \to
continuous, \(\dot \gamma(t)\) exists for almost all \(t \in I\) and
\(\dot\gamma \in L^2(I, \RR^n)\). It is a well known fact that the so called
\emph{Sobolev space \(H^1(I, \RR^n)\)} of all class \(H^1\) curves in \(\RR^n\)
-is a Hilbert space under the inner product given by
+is a Hilbert space under the inner product given by
\[
\langle \gamma, \eta \rangle_1
= \int_0^1 \gamma(t) \cdot \eta(t) + \dot\gamma(t) \cdot \dot\eta(t) \; \dt
@@ -68,11 +68,11 @@ approach \(\ell(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
\centering
\begin{tikzpicture}
\draw (4, 1) -- (1, 4);
- \draw (4, 1) -- (4, 2)
- -- (3, 2)
- -- (3, 3) node[right]{$\gamma_n$}
- -- (2, 3) node[left]{$\gamma$}
- -- (2, 4)
+ \draw (4, 1) -- (4, 2)
+ -- (3, 2)
+ -- (3, 3) node[right]{$\gamma_n$}
+ -- (2, 3) node[left]{$\gamma$}
+ -- (2, 4)
-- (1, 4);
\draw[dotted] (4.5, .5) -- (4, 1);
\draw[dotted] (.5, 4.5) -- (1, 4);
@@ -90,7 +90,7 @@ 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
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
+natural candidate for a norm in \({C'}^\infty(I, \RR^n)\) is
\[
\norm{\gamma}_1^2 = \norm{\gamma}_0^2 + \norm{\dot\gamma}_0^2,
\]
@@ -104,7 +104,7 @@ it is natural to expect \({C'}^\infty(I, \RR^n)\) to be Banach space. In
particular, \({C'}^\infty(I, \RR^n)\) must be a normed Banach space. This is
unfortunately not the case for \({C'}^\infty(I, \RR^n)\) the norm
\(\norm\cdot_1\), but we can consider its completion, which, by a classical
-result by Lebesgue, just so happens to coincide with \(H^1(I, M)\).
+result by Lebesgue, just so happens to coincide with \(H^1(I, M)\).
It's also interesting to note that the completion of \({C'}^\infty(I, \RR^n)\)
with respect to the norm \(\norm\cdot_\infty\) is the space \(C^0(I, \RR^n)\)
@@ -126,7 +126,7 @@ find\dots
of \(E\) is the completion of the space \({C'}^\infty(E)\) of peace-wise
smooth sections of \(E\) under the inner product given by
\[
- \langle \xi, \eta \rangle_1
+ \langle \xi, \eta \rangle_1
= \int_0^1 \langle \xi_t, \eta_t \rangle +
\langle \nabla \xi_t, \nabla \eta_t \rangle \; \dt
\]
@@ -166,13 +166,13 @@ find\dots
\[
\begin{split}
\norm{\xi}_\infty^2
- & = \norm{\xi_{t_0}}^2
+ & = \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
+ & = \norm{\xi_{t_0}}^2
+ 2 \int_{t_0}^{t_1} \langle \xi_s, \nabla \xi_s \rangle \; \dd s \\
\text{(Cauchy-Schwarz)}
- & \le \norm{\xi_{t_0}}^2
+ & \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 2 \norm{\xi}_1^2
@@ -222,7 +222,7 @@ We begin with a technical lemma.
Let \(C^0(W_\gamma) = \{ X \in C^0(\gamma^* TM) : X_t \in W_{\gamma, t} \;
\forall t \}\). We claim \(C^0(W_\gamma)\) is open in \(C^0(\gamma^* TM)\).
Indeed, given \(X \in C^0(W_\gamma)\) there exists \(\delta > 0\) such
- that
+ that
\[
\begin{split}
\norm{X - Y}_\infty < \delta
@@ -240,7 +240,7 @@ We begin with a technical lemma.
Let \(W \subset TM\) be an open neighborhood of the zero section in \(TM\) such
that \(\exp\!\restriction_W : W \to \exp(W)\) is invertible -- whose existence
follows from the fact that the injectivity radius depends continuously on \(p
-\in M\).
+\in M\).
\begin{definition}
Given \(\gamma \in {C'}^\infty(I, M)\) let \(W_\gamma, W_{\gamma, t} \subset
@@ -249,7 +249,7 @@ follows from the fact that the injectivity radius depends continuously on \(p
\arraycolsep=1pt
\begin{array}{rl}
\exp_\gamma : H^1(W_\gamma) & \to H^1(I, M) \\
- X &
+ X &
\begin{array}[t]{rl}
\mapsto \exp \circ X : I & \to M \\
t & \mapsto \exp_{\gamma(t)}(X_t)
@@ -278,7 +278,7 @@ M)\). Moreover, since \({C'}^\infty(I, M)\) is dense, \(\{U_\gamma\}_{\gamma
\in {C'}^\infty(I, M)}\) is an open cover of \(H^1(I, M)\). The real difficulty
of this proof is showing that the transition maps
\[
- \exp_\eta^{-1} \circ \exp_\gamma :
+ \exp_\eta^{-1} \circ \exp_\gamma :
\exp_\gamma^{-1}(U_\gamma \cap U_\eta) \subset H^1(\gamma^* TM)
\to H^1(\eta^* TM)
\]
@@ -287,9 +287,9 @@ leave this details we leave as an exercise to the reader -- see theorem 2.3.12
of \cite{klingenberg} for a full proof.
The charts \(\exp_\gamma^{-1}\) are modeled after separable Hilbert spaces,
-with tipical representatives \(H^1(\gamma^* TM) \cong H^1(I,
+with typical representatives \(H^1(\gamma^* TM) \cong H^1(I,
\RR^n)\)\footnote{Any trivialization of $\gamma^* TM$ induces an isomorphism
-$H^1(\gamma^* TM) \isoto H^1(I, \RR^n)$.}.
+$H^1(\gamma^* TM) \isoto H^1(I, \RR^n)$.}.
It's interesting to note that this construction is functorial. More
precisely\dots
@@ -307,14 +307,14 @@ 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 necessarity a vector
+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
\[
\arraycolsep=1pt
\begin{array}{rl}
\tilde{\cdot} : H^1(I, M) \to & H^1(I \times M) \\
- \gamma \mapsto &
+ \gamma \mapsto &
\begin{array}[t]{rl}
\tilde\gamma : I & \to I \times M \\
t & \mapsto (t, \gamma(t))
@@ -328,7 +328,7 @@ The space \(H^1(E)\) is modeled after the Hilbert spaces \(H^1(F)\) of class
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 divergion in this short notes. Having saied that, we are now finally ready to
+a diversion in this short notes. Having said that, we are now finally ready to
discuss the Riemannian structure of \(H^1(I, M)\).
\subsection{The Metric of \(H^1(I, M)\)}
@@ -337,8 +337,8 @@ We begin our discussion of the Riemannian structure of \(H^1(I, M)\) by looking
at its tangent bundle. Notice that for each \(\gamma \in {C'}^\infty(I, M)\)
the chart \(\exp_\gamma^{-1} : U_\gamma \to H^1(\gamma^* TM)\) induces a
canonical isomorphism \(\phi_\gamma : T_\gamma H^1(I, M) \isoto H^1(\gamma^*
-TM)\), as described in propostion~\ref{thm:tanget-space-topology}. In fact,
-this isomorphisms may be extended to a cononical isomorphism of vector bundles,
+TM)\), as described in 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}
@@ -348,7 +348,7 @@ as seen in\dots
\[
\arraycolsep=1pt
\begin{array}{rl}
- \psi_{i, \gamma} : H^1(U_\gamma) \times H^i(\gamma^* TM) \to
+ \psi_{i, \gamma} : H^1(U_\gamma) \times H^i(\gamma^* TM) \to
& \prod_{\eta \in H^1(I, M)} H^i(\eta^* TM) \\
(X, Y) \mapsto &
\begin{array}[t]{rl}
@@ -372,12 +372,12 @@ as seen in\dots
\begin{proof}
Note that the sets \(H^1(U_\gamma) \times T_\gamma H^1(I, M)\) are precisely
- the images of the canonical charts
+ the images of the canonical charts
\[
- \varphi_\gamma :
+ \varphi_\gamma :
\varphi_\gamma^{-1}(H^1(U_\gamma) \times T_\gamma H^1(I, M))
\subset T H^1(I, M)
- \to H^1(U_\gamma) \times T_\gamma H^1(I, M)
+ \to H^1(U_\gamma) \times T_\gamma H^1(I, M)
\]
of \(T H^1(I, M)\).
@@ -392,7 +392,7 @@ as seen in\dots
{C'}^\infty(I, M)\) is given by \(\phi_\gamma\).
\end{proof}
-This las result will be the basis for our analysis of the Riemannian structure
+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
the fibers \(T_\gamma H^1(I, M) \cong H^1(\gamma^* TM)\).