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