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, 5 insertions, 1 deletion

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -308,7 +308,11 @@ precisely\dots
                    \gamma & \mapsto f \circ \gamma
   \end{align*}
   is smooth. In addition, \(H^1(I, f \circ g) = H^1(I, f) \circ H^1(I, g)\) and
-  \(H^1(I, \id) = \id\) for any composable smooth maps \(f\) and \(g\).
+  \(H^1(I, \id) = \id\) for any composable smooth maps \(f\) and \(g\). We thus
+  have a functor \(H^1(I, -) : \categoryname{Rnn} \to \categoryname{BMnd}\)
+  from the category \(\categoryname{Rnn}\) of finite-dimensional Riemannian
+  manifolds and smooth maps onto the category \(\categoryname{BMnd}\) of Banach
+  manifolds and smooth maps.
 \end{theorem}
 
 We would also like to point out that this is a particular case of a more