- Commit
- 7515687c9555b84811248641d85d25e4eebafe10
- Parent
- 83cacbeeac86bbb8a771b9929db3c5a152014bba
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Further clarifications in notation
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
Further clarifications in notation
1 file changed, 24 insertions, 20 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/structure.tex | 44 | 24 | 20 |
diff --git a/sections/structure.tex b/sections/structure.tex @@ -121,18 +121,6 @@ of sections of vector bundles over the unit interval \(I\). Explicitly, we find\dots \begin{proposition} - Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a - Riemannian metric -- the space \(H^1(E)\) of all class class \(H^1\) sections - of \(E\) is the completion of the space \({C'}^\infty(E)\) of peace-wise - smooth sections of \(E\) under the inner product given by - \[ - \langle \xi, \eta \rangle_1 - = \int_0^1 \langle \xi_t, \eta_t \rangle + - \langle \nabla \xi_t, \nabla \eta_t \rangle \; \dt - \] -\end{proposition} - -\begin{proposition} Given an Euclidean bundle \(E \to I\), the space \(C^0(E)\) of all continuous sections of \(E\) is is the completion of \({C'}^\infty(E)\) under the norm given by @@ -150,6 +138,20 @@ find\dots \] \end{proposition} +\begin{proposition} + Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a + Riemannian metric -- the space \(H^1(E)\) of all class class \(H^1\) sections + of \(E\) is the completion of the space \({C'}^\infty(E)\) of peace-wise + smooth sections of \(E\) under the inner product given by + \[ + \langle \xi, \eta \rangle_1 + = \langle \xi, \eta \rangle_0 + + \left\langle + \nabla_{\frac\dd\dt} \xi, \nabla_{\frac\dd\dt} \eta + \right\rangle_0 + \] +\end{proposition} + \begin{proposition}\label{thm:continuous-inclusions-sections} Given an Euclidean bundle \(E \to I\), the inclusions \[ @@ -169,12 +171,14 @@ find\dots & = \norm{\xi_{t_0}}^2 + \int_{t_0}^{t_1} \frac{\dd}{\dd s} \norm{\xi_s}^2 \; \dd s \\ \text{(\(\nabla\) is compatible with the metric)} - & = \norm{\xi_{t_0}}^2 - + 2 \int_{t_0}^{t_1} \langle \xi_s, \nabla \xi_s \rangle \; \dd s \\ + & = \norm{\xi_{t_0}}^2 + 2 \int_{t_0}^{t_1} + \left\langle \xi_s, \nabla_{\frac\dd{\dd s}} \xi_s \right\rangle + \; \dd s \\ \text{(Cauchy-Schwarz)} - & \le \norm{\xi_{t_0}}^2 - + 2 \int_0^1 \norm{\xi_s} \cdot \norm{\nabla \xi_s} \; \dd s \\ - & \le \norm{\xi}_0^2 + \norm{\xi}_0^2 + \norm{\nabla \xi}_0^2 \\ + & \le \norm{\xi_{t_0}}^2 + 2 \int_0^1 + \norm{\xi_s} \cdot \norm{\nabla_{\frac\dd{\dd s}} \xi_s} \; \dd s \\ + & \le \norm{\xi}_0^2 + + \norm{\xi}_0^2 + \norm{\nabla_{\frac\dd\dt} \xi}_0^2 \\ & \le 2 \norm{\xi}_1^2 \end{split} \] @@ -189,8 +193,8 @@ find\dots \end{proof} \begin{note} - Apply proposition~\ref{thm:continuous-inclusions-sections} to the bundle \(I - \times \RR^n \to I\) to get the continuity of the maps in + Apply proposition~\ref{thm:continuous-inclusions-sections} to the trivial + bundle \(I \times \RR^n \to I\) to get the continuity of the maps in (\ref{eq:continuous-inclusions-rn-curves}). \end{note} @@ -393,6 +397,6 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots \end{proof} This last result will be the basis for our analysis of the Riemannian structure -of \(H^1(I, M)\). We may now describe the metric of \(H^1(I, M)\) in terms of +of \(H^1(I, M)\): we may now describe the metric of \(H^1(I, M)\) in terms of the fibers \(T_\gamma H^1(I, M) \cong H^1(\gamma^* TM)\).