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, 9 insertions, 8 deletions

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -75,14 +75,15 @@ notes are meant to be concise. Hence we do not have the necessary space to
 discuss neither technicalities nor more involved applications of the theory we
 will develop.
 
-In particular, we leave the intricacies of Palais' discussion of condition (C)
--- which can be seen as a substitute for the failure of a proper Hilbert space
-to be locally compact \cite[ch.~2]{klingenberg} -- and its applications to the
-study of closed geodesics out of these notes. As previously stated, many results
-are left unproved, but we will include references to other materials containing
-proofs. We'll assume basic knowledge of differential and Riemannian geometry,
-as well as some familiarity with the classical theory of the calculus of
-variations -- see \cite[ch.~5]{gorodski} for the classical approach.
+In particular, we leave the intricacies of Palais' and Smale's discussion of
+condition (C) -- which can be seen as a substitute for the failure of a proper
+Hilbert space to be locally compact \cite[ch.~2]{klingenberg} -- and its
+applications to the study of closed geodesics out of these notes. As previously
+stated, many results are left unproved, but we will include references to other
+materials containing proofs. We'll assume basic knowledge of differential and
+Riemannian geometry, as well as some familiarity with the classical theory of
+the calculus of variations -- see \cite[ch.~5]{gorodski} for the classical
+approach.
 
 Before moving to the next section we would like to review the basics of the
 theory of real Banach manifolds.