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, 17 insertions, 7 deletions

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -65,13 +65,23 @@ explored in detail in \cite[sec.~6]{eells}. The 11th chapter of \cite{palais}
 is also a great reference for the general theory of spaces of sections of fiber
 bundles.
 
-% TODO: Finilize this
-We should point out that due to limitations of space we will primarily focus on
-the broad strokes of the theory ahead. 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 theory.
+We should point out that we will primarily focus on
+the broad strokes of the theory ahead and that we will leave many results
+unproved. The reasoning behind this is twofold. First, we don't want to bore
+the reader with the numerous technical details of some of the constructions
+we'll discuss in the following. Secondly, and this is more important, this
+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.
 
 Before moving to the next section we would like to review the basics of the
 theory of real Banach manifolds.