Hyperbolische Halbebene/Kurve/Tangentiale Beschleunigung/2/Aufgabe/Lösung

Wir arbeiten auf dem zurückgezogenen Vektorbündel ${}\mathbb {R} \times \mathbb {R} ^{2}$ über ${}\mathbb {R}$ mit dem zurückgezogenen Zusammenhang, dessen Christoffelsymbole ${}{\tilde {\Gamma }}_{\ell }^{k}$ in Aufgabe bestimmt wurden. Es seien ${}e_{1},e_{2}$ die Basisschnitte. Dann ist nach Fakt

${}{\begin{aligned}\nabla _{\partial }{\left(\psi _{1}'e_{1}+\psi _{2}'e_{2}\right)}&=D_{\partial }(\psi _{1}')e_{1}+\psi _{1}'\nabla _{\partial }e_{1}+D_{\partial }(\psi _{2}')e_{2}+\psi _{2}'\nabla _{\partial }e_{2}\\&=\psi _{1}^{\prime \prime }e_{1}+\psi _{1}'\nabla _{\partial }e_{1}+\gamma _{2}^{\prime \prime }e_{2}+\psi _{2}'\nabla _{\partial }e_{2}\\&=\psi _{1}^{\prime \prime }e_{1}+\psi _{1}'{\left({\tilde {\Gamma }}_{1}^{1}e_{1}+{\tilde {\Gamma }}_{1}^{2}e_{2}\right)}+\psi _{2}^{\prime \prime }e_{2}+\psi _{2}'{\left({\tilde {\Gamma }}_{2}^{1}e_{1}+{\tilde {\Gamma }}_{2}^{2}e_{2}\right)}\\&={\left(\psi _{1}^{\prime \prime }+\psi _{1}'{\tilde {\Gamma }}_{1}^{1}+\psi _{2}'{\tilde {\Gamma }}_{2}^{1}\right)}e_{1}+{\left(\psi _{2}^{\prime \prime }+\psi _{1}'{\tilde {\Gamma }}_{1}^{2}+\psi _{2}'{\tilde {\Gamma }}_{2}^{2}\right)}e_{2}.\end{aligned}}$

Dies ergibt

${}{\begin{aligned}\nabla _{\partial }{\left(\psi _{1}'e_{1}+\psi _{2}'e_{2}\right)}(t)&={\left({\tilde {\Gamma }}_{1}^{1}(t)+e^{t}{\tilde {\Gamma }}_{2}^{1}(t)\right)}e_{1}+{\left(e^{t}+{\tilde {\Gamma }}_{1}^{2}(t)+e^{t}{\tilde {\Gamma }}_{2}^{2}(t)\right)}e_{2}\\&={\left(-1-e^{t}e^{-t}\right)}e_{1}+{\left(e^{t}+e^{-t}-e^{t}\right)}e_{2}\\&=-2e_{1}+e^{-t}e_{2}.\end{aligned}}$

Zur gelösten Aufgabe