global-analysis-and-the-banach-manifold-of-class-h1-curvers

Riemannian Geometry course project on the manifold H¹(I, M) of class H¹ curves on a Riemannian manifold M and its applications to the geodesics problem

Commit
7515687c9555b84811248641d85d25e4eebafe10
Parent
83cacbeeac86bbb8a771b9929db3c5a152014bba
Author
Pablo <pablo-escobar@riseup.net>
Date

Further clarifications in notation

Diffstat

1 file changed, 24 insertions, 20 deletions

Status File Name N° Changes Insertions Deletions
Modified sections/structure.tex 44 24 20
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 \(T_\gamma H^1(I, M) \cong H^1(\gamma^* TM)\).