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, 4 insertions, 2 deletions

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -466,8 +466,10 @@ We are now ready to state Morse's index theorem.
 
 \begin{theorem}[Morse]
   Let \(\gamma \in \Omega_{p q} M\) be a critical point of \(E\). Then the
-  index of \(\gamma\) is given of the sum of the multiplicities of the
-  proper conjugate points of \(\gamma\) in the interior of \(I\).
+  index of \(\gamma\) is given of the sum of the multiplicities of the proper
+  conjugate points of \(\gamma\)\footnote{By ``conjugate points of $\gamma$''
+  we of course mean points conjugate to $\gamma(0) = p$ along $\gamma$.} in the
+  interior of \(I\).
 \end{theorem}
 
 Unfortunately we do not have the space to include the proof of Morse's theorem