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, 15 insertions, 12 deletions

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -542,15 +542,18 @@ We should point out that part \textbf{(i)} of theorem~\ref{thm:jacobi-darboux}
 is weaker than the classical formulation of the Jacobi-Darboux theorem -- such
 as in theorem 5.5.3 of \cite{gorodski} for example -- in two aspects. First, we
 do not compare the length of curves \(\gamma\) and \(\eta \in U\). This could
-be amended by showing that the length functional \(L : H^1(I, M) \to \RR\)
-is smooth and that its Hessian \(d^2 L_\gamma\) is given by \(C \cdot d^2
-E_\gamma\) for some \(C > 0\).
-
-Secondly, unlike the classical formulation we only consider curves in an
-\(H^1\)-neighborhood of \(\gamma\) -- instead of a neighborhood of \(\gamma\)
-in \(\Omega_{p q} M\) in the uniform topology. On the other hand, part
-\textbf{(ii)} is definitively an improvement of the classical formulation: we
-can find curves \(\eta = i(v)\) shorter than \(\gamma\) already in an
-\(H^1\)-neighborhood of \(\gamma\).
-
-We hope that short notes could provide the reader
+be amended by showing that the length functional \(L : H^1(I, M) \to \RR\) is
+smooth and that its Hessian \(d^2 L_\gamma\) is given by \(C \cdot d^2
+E_\gamma\) for some \(C > 0\). Secondly, unlike the classical formulation we
+only consider curves in an \(H^1\)-neighborhood of \(\gamma\) -- instead of a
+neighborhood of \(\gamma\) in \(\Omega_{p q} M\) in the uniform topology. On
+the other hand, part \textbf{(ii)} is definitively an improvement of the
+classical formulation: we can find curves \(\eta = i(v)\) shorter than
+\(\gamma\) already in an \(H^1\)-neighborhood of \(\gamma\).
+
+This concludes our discussion of the applications of our theory to the
+geodesics problem. We hope that this short notes could provide the reader with
+a glimpse of the rich theory of the calculus of variations and global anylisis
+at large. We once again refer the reader to \cite[ch.~2]{klingenberg},
+\cite[ch.~11]{palais} and \cite[sec.~6]{eells} for further insight on modern
+variational methods.