diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -129,13 +129,13 @@ manifolds to spaces modeled after Banach spaces. We begin by the former.
is continuous. Since \(\mathcal{L}(V, W)\) is a Banach space under the
operator norm, we may recursively define functions of class \(C^n\) for \(n >
1\): a function \(f : U \to W\) of class \(C^{n - 1}\) is called
- \emph{differentiable of class \(C^n\)} if the map
+ \emph{differentiable of class \(C^n\)} if the map\footnote{Here we consider
+ the \emph{projective tensor product} of Banach spaces. See
+ \cite[ch.~1]{klingenberg}.}
\[
d^{n - 1} f :
U \to \mathcal{L}(V, \mathcal{L}(V, \cdots \mathcal{L}(V, W)))
\cong \mathcal{L}(V^{\otimes n}, W)
- \footnote{Here we consider the \emph{projective tensor product} of Banach
- spaces. See \cite[ch.~1]{klingenberg}.}
\]
is of class \(C^1\). Finally, a map \(f : U \to W\) is called
\emph{differentiable of class \(C^\infty\)} or \emph{smooth} if \(f\) is of
@@ -151,7 +151,7 @@ chain rule}.
: U_2 \to V_3\), the composition map \(g \circ f : U_1 \to V_3\) is smooth
and its derivative is given by
\[
- (d g \circ f)_p = d g_{f(p)} \circ d f_p
+ d (g \circ f)_p = d g_{f(p)} \circ d f_p
\]
\end{lemma}
@@ -223,13 +223,13 @@ in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
(\phi_{p, i} \circ \phi_{p, j}^{-1}) v
& = (\varphi_i \circ \varphi_j^{-1} \circ \gamma_v)'(0) \\
\text{(chain rule)}
- & = (d \varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} \dot\gamma_v(0) \\
- & = (d \varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} v
+ & = d (\varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} \dot\gamma_v(0) \\
+ & = d (\varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} v
\end{split}
\]
where \(v \in V_j\) and \(\gamma_v : (-\epsilon, \epsilon) \to V_j\) is any
smooth curve with \(\gamma_v(0) = \varphi_j(p)\) and \(\dot\gamma_v(0) = v\):
- \(\phi_{p, i} \circ \phi_{p, j}^{-1} = (d \varphi_i \circ
+ \(\phi_{p, i} \circ \phi_{p, j}^{-1} = d (\varphi_i \circ
\varphi_j^{-1})_{\varphi_j(p)}\) is continuous by definition.
\end{proof}
diff --git a/sections/structure.tex b/sections/structure.tex
@@ -48,10 +48,10 @@ Denote by \({C'}^\infty(I, \mathbb{R}^n)\) the space of piece-wise curves in
energy functional and the length functional are smooth maps. As most function
spaces, \({C'}^\infty(I, \mathbb{R}^n)\) admits several natural topologies.
Some of the most obvious candidates are the uniform topology and the topology
-of the \(\norm\cdot_0\) norm, which are the topologies induces by the norms
+of the \(\norm\cdot_0\) norm, which are the topologies induced by the norms
\begin{align*}
- \norm{\gamma}_\infty & = \sup_t \norm{\gamma(t)} \\
- \norm{\gamma}_0 & = \int_0^1 \norm{\gamma(t)}^2 \; \dt
+ \norm{\gamma}_\infty & = \sup_t \norm{\gamma(t)} \\
+ \norm{\gamma}_0 & = \sqrt{\int_0^1 \norm{\gamma(t)}^2 \; \dt}
\end{align*}
respectively.
@@ -99,7 +99,7 @@ natural candidate for a norm in \({C'}^\infty(I, \mathbb{R}^n)\) 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 \(H^0(I,
+\rangle_1\) -- here \(\norm{\cdot}_0\) denotes the norm of \(H^0(I,
\mathbb{R}^n) = L^2(I, \mathbb{R}^n)\).
The other issue we face is one of completeness. Since \(\mathbb{R}^n\) has a
@@ -168,12 +168,12 @@ find\dots
\begin{proof}
Given \(\xi \in H^0(E)\) we have
- \[
+ \begin{equation}\label{eq:zero-norm-le-infty-norm}
\norm{\xi}_0^2
= \int_0^1 \norm{\xi_t}^2 \; \dt
- \le \int_0^1 \norm{\xi}_\infty \; \dt
- = \norm{\xi}_\infty
- \]
+ \le \int_0^1 \norm{\xi}_\infty^2 \; \dt
+ = \norm{\xi}_\infty^2
+ \end{equation}
Now given \(\xi \in H^1(E)\) fix \(t_0, t_1 \in I\) with \(\norm{\xi}_\infty
= \norm{\xi_{t_1}}\). Then
@@ -183,14 +183,21 @@ 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}
- \left\langle \xi_s, \nabla_{\frac\dd{\dd s}} \xi_s \right\rangle
+ & = \norm{\xi_{t_0}}^2 + \int_{t_0}^{t_1}
+ 2 \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_{\frac\dd{\dd s}} \xi_s} \; \dd s \\
+ & \le \norm{\xi_{t_0}}^2 + \int_0^1
+ 2 \norm{\xi_s} \cdot \norm{\nabla_{\frac\dd{\dd s}} \xi_s} \; \dd s \\
+ & \le \norm{\xi}_\infty^2
+ + \int_0^1 \norm{\xi_s}^2 + \norm{\nabla_{\frac\dd\dt} \xi_s}^2
+ \; \dd s \\
& \le \norm{\xi}_\infty^2
+ \norm{\xi}_0^2 + \norm{\nabla_{\frac\dd\dt} \xi}_0^2 \\
+ % TODO: This is actually wrong. Fix this.
+ \text{(because of equation (\ref{eq:zero-norm-le-infty-norm}))}
+ & \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}
\]
@@ -276,10 +283,10 @@ Finally, we find\dots
for \(H^1(I, M)\) under the final topology of the maps \(\exp_\gamma\) --
i.e. the coarsest topology such that such maps are continuous. This atlas
gives \(H^1(I, M)\) the structure of a \emph{separable} Banach manifold
- modeled after separable Hilbert spaces, with typical representatives
- \(H^1(\gamma^* TM) \cong H^1(I, \mathbb{R}^n)\)\footnote{Any trivialization
- of $\gamma^* TM$ induces an isomorphism $H^1(\gamma^* TM) \isoto H^1(I,
- \mathbb{R}^n)$.}.
+ modeled after separable Hilbert spaces, with typical
+ representatives\footnote{Any trivialization of $\gamma^* TM$ induces an
+ isomorphism $H^1(\gamma^* TM) \isoto H^1(I, \mathbb{R}^n)$.} \(H^1(\gamma^*
+ TM) \cong H^1(I, \mathbb{R}^n)\).
\end{theorem}
The fact that \(\exp_\gamma\) is bijective should be clear from the definition
@@ -371,9 +378,9 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
\]
gives \(\coprod_{\gamma \in {C'}^\infty(I, M)} H^i(\gamma^* TM) \to H^1(I,
M)\) the structure of a smooth vector bundle\footnote{Here we use the
- canonical identification $T_{\gamma(t)} M \cong T_{X_t}^\vee TM$ to apply
- the vector $Y_t \in T_{\gamma(t)} M$ to the map $(d \exp)_{X_t} : T_{X_t} TM
- \to T_{\exp_{\gamma(t)}(X_t)} M$.}.
+ canonical identification $T_{\gamma(t)} M \cong T_{X_t} TM$ to apply the
+ vector $Y_t \in T_{\gamma(t)} M$ to the map $(d \exp)_{X_t} : T_{X_t} TM \to
+ T_{\exp_{\gamma(t)}(X_t)} M$.}.
\end{lemma}
\begin{proposition}
@@ -394,17 +401,43 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
\subset T 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 \(\psi_{1, \gamma}
- \circ (\operatorname{id}, \varphi_\gamma) \circ \varphi_\gamma :
- \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M)) \isoto \psi_{1,
- \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM))\). 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
- restriction of this isomorphism to \(T_\gamma H^1(I, M)\) with \(\gamma \in
- {C'}^\infty(I, M)\) is given by \(\phi_\gamma\).
+ of \(T H^1(I, M)\) -- i.e. the charts given by\footnote{Once more, we use the
+ canonical identification $T_X H^1(W_\gamma) \cong H^1(\gamma^* TM)$ to apply
+ the vector $\phi_\gamma(Y) \in H^1(\gamma^* TM)$ to $(d \exp_\gamma)_X : T_X
+ H^1(W_\gamma) \to T_{\exp_\gamma(X)} H^1(I, M)$.}
+ \begin{align*}
+ \varphi_\gamma^{-1} : H^1(W_\gamma) \times T_\gamma H^1(I, M)
+ & \to T H^1(I, M) \\
+ (X, Y) & \mapsto (d \exp_\gamma)_X \phi_\gamma(Y)
+ \end{align*}
+
+ By composing charts we get fiber-preserving, fiber-wise linear diffeomorphism
+ \[
+ \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
+ \subset T H^1(I, M)
+ \isoto
+ \psi_{1, \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM)),
+ \]
+ which takes \(\varphi^{-1}(X, Y) \in T_{exp_\gamma(X)} H^1(I, M)\) to
+ \(\psi_{1, \gamma}(X, \phi_\gamma(Y)) \in H^1(\exp_\gamma(X)^* TM)\). With
+ enought patience, one can deduce from the fact that \(\varphi_\gamma\) and
+ \(\psi_{1, \gamma}^{-1}\) are charts that this maps agree in the intersection
+ of the open subsets \(\varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma
+ H^1(I, M))\), so that they may be glued together into a global smooth map
+ \(\Phi : T H^1(I, M) \to \coprod_{\eta \in H^1(I, M)} H^1(\eta^* TM)\).
+
+ Since this map is a fiber-preserving, fiber-wise linear local diffeomorphism,
+ this is an isomorphism of vector bundles.
+ Furthermore, by construction
+ \[
+ \Phi(X)_t
+ = \psi_{1, \gamma}(0, \phi_\gamma(X))_t
+ = (d \exp)_{0_{\gamma(t)}} \phi_\gamma(X)_t
+ = \phi_\gamma(X)_t
+ \]
+ for each \(\gamma \in {C'}^\infty(I, M)\) and \(X \in T_\gamma H^1(I, M)\).
+ In other words, \(\Phi\!\restriction_{T_\gamma H^1(I, M)} = \phi_\gamma\) as
+ required.
\end{proof}
At this point it may be tempting to think that we could now define the metric
@@ -516,11 +549,14 @@ 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 \(\langle \, , \rangle_1 : H^1(\gamma^* TM) \times H^1(\gamma^*
TM) \to \mathbb{R}\) may be glued together into a single positive-definite
-section \(\langle \, , \rangle_1 \in \Gamma( \operatorname{Sym}^2
-\coprod_\gamma H^1(\gamma^* TM))\) -- whose smoothness follows from
+section \(\langle \, , \rangle_1 \in \Gamma\left(\operatorname{Sym}^2
+\coprod_\gamma H^1(\gamma^* TM)\right)\) -- whose smoothness follows from
theorem~\ref{thm:h0-has-metric-extension},
proposition~\ref{thm:partial-is-smooth-sec} and
proposition~\ref{thm:covariant-derivative-h0} -- which is then mapped to a
positive-definite section of \(\operatorname{Sym}^2 T H^1(I, M)\) by the
-induced isomorphism \(\operatorname{Sym}^2 \coprod_\gamma H^1(\gamma^* TM)
-\isoto \operatorname{Sym}^2 T H^1(I, M)\).
+induced isomorphism
+\[
+ \Gamma\left(\operatorname{Sym}^2 \coprod_\gamma H^1(\gamma^* TM)\right)
+ \isoto \Gamma(\operatorname{Sym}^2 T H^1(I, M))
+\]