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

Commit
376233b254b35a85281bd0c16f2ec0a6e838fe5d
Parent
1dfc5bb9948bd9e62f74b8305b0756a4a442c006
Author
Pablo <pablo-escobar@riseup.net>
Date

Fixed some typos

Diffstat

1 file changed, 5 insertions, 4 deletions

Status File Name N° Changes Insertions Deletions
Modified sections/structure.tex 9 5 4
diff --git a/sections/structure.tex b/sections/structure.tex
@@ -1,6 +1,6 @@
 \section{The Structure of \(H^1(I, M)\)}\label{sec:structure}
 
-Throughout this sections let \(M\) be a finite-dimensional Riemannian manifold.
+Throughout this section let \(M\) be a finite-dimensional Riemannian manifold.
 As promised, in this section we will highlight the differential and Riemannian
 structures of the space \(H^1(I, M)\) of class \(H^1\) curves in a \(M\). The
 first question we should ask ourselves is an obvious one: what is \(H^1(I,
@@ -411,16 +411,17 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
     (X, Y) & \mapsto (d \exp_\gamma)_X \phi_\gamma(Y)
   \end{align*}
 
-  By composing charts we get fiber-preserving, fiber-wise linear diffeomorphism
+  By composing charts we get a fiber-preserving, fiber-wise linear
+  diffeomorphism
   \[
     \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
     \subset T H^1(I, M)
     \isoto
     \psi_{1, \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM)),
   \]
-  which takes \(\varphi^{-1}(X, Y) \in T_{exp_\gamma(X)} H^1(I, M)\) to
+  which takes \(\varphi^{-1}(X, Y) \in T_{\exp_\gamma(X)} H^1(I, M)\) to
   \(\psi_{1, \gamma}(X, \phi_\gamma(Y)) \in H^1(\exp_\gamma(X)^* TM)\). With
-  enought patience, one can deduce from the fact that \(\varphi_\gamma\) and
+  enough patience, one can deduce from the fact that \(\varphi_\gamma\) and
   \(\psi_{1, \gamma}^{-1}\) are charts that these maps agree in the
   intersections of the open subsets \(\varphi_\gamma^{-1}(H^1(W_\gamma) \times
   T_\gamma H^1(I, M))\), so that they may be glued together into a global