- Commit
- d49430bf20e2e9d2ebe2611d6de516013acacf44
- Parent
- e9d4b817491e56e3d1817041eee5e08000363084
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Minor correction to the section on the Morse index theorem
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, 3 insertions, 3 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/applications.tex | 6 | 3 | 3 |
diff --git a/sections/applications.tex b/sections/applications.tex @@ -324,7 +324,7 @@ see lemma 2.4.6 of \cite{klingenberg}. As a first consequence, we prove\dots -\begin{proposition}\label{thm:energy-is-morse-function} +\begin{proposition}\label{thm:morse-index-e-is-finite} Given a critical point \(\gamma\) of \(E\) in \(\Omega_{p q} M\), the self-adjoint operator \(A_\gamma : T_\gamma \Omega_{p q} M \to T_\gamma \Omega_{p q} M\) given by @@ -461,9 +461,9 @@ the index of a geodesic \(\gamma\) can be defined without the aid of the tools developed in here, by using of the Hessian form \(d^2 E_\gamma\) we can place definition~\ref{def:morse-index} in the broader context of Morse theory. In fact, the geodesics problems and the energy functional where among Morse's -original proposed applications. Proposition~\ref{thm:energy-is-morse-function} +original proposed applications. Proposition~\ref{thm:morse-index-e-is-finite} and definition~\ref{def:morse-index} amount to a proof that the Morse index of -\(E\) at a critical point \(\gamma\) is well defined. +\(E\) at a critical point \(\gamma\) is finite. We are now ready to state Morse's index theorem.