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, 8 insertions, 8 deletions

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -74,14 +74,14 @@ 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 --
-a generalization of the notion of local compactness \cite[ch.~2]{klingenberg}
--- and its applications to the study of closed geodesics out of this 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' discussion of condition (C)
+-- wich 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 this 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.