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

Diffstat

1 file changed, 5 insertions, 5 deletions

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -202,11 +202,11 @@ find\dots
   \end{equation}
 
   Similarly, if \(t_0 > t_1\) then by inverting the orientation of the curve
-  \(\gamma\) we can see that \(\norm{\xi}_\infty < \sqrt{2} \norm{\xi}_0\).
-  More precisely, if we set \(\eta(t) = \gamma(1 - t)\) and \(\xi' \in
-  H^1(\eta^* TM)\) with \(\xi'_t = \xi_{1 - t}\) then \(\norm{\xi}_\infty =
-  \norm{\xi'}_\infty \le \sqrt{2} \norm{\xi'}_1 = \sqrt{2} \norm{\xi}_1\)
-  because of equation (\ref{eq:one-norm-le-sqrt-two-infty-norm}).
+  \(\gamma\) we can see that \(\norm{\xi}_\infty \le \sqrt{2} \norm{\xi}_1\).
+  More precisely, if we set \(\eta(t) = \gamma(1 - t)\) and \(\zeta \in
+  H^1(\eta^* TM)\) with \(\zeta_t = \xi_{1 - t}\) then \(\norm{\xi}_\infty =
+  \norm{\zeta}_\infty \le \sqrt{2} \norm{\zeta}_1 = \sqrt{2} \norm{\xi}_1\)
+  because of the inequality (\ref{eq:one-norm-le-sqrt-two-infty-norm}).
 \end{proof}
 
 \begin{note}