Offenes Intervall/Trivialer Zusammenhang/Basisschnitte/Christoffelsymbole/1/Aufgabe/Lösung

1. Es ist

   ${}\det {\begin{pmatrix}1+t^{2}&t^{3}\\t&1+t^{2}\end{pmatrix}}=1+2t^{2}+t^{4}-t^{4}=1+2t^{2}>0\,$

   stets positiv, daher sind die beiden Vektoren ${}f_{1}(t)$ und ${}f_{2}(t)$ zu jedem ${}t\in \mathbb {R}$ eine Basis.

2. Aus der Beziehung

   ${}{\begin{pmatrix}f_{1}\\f_{2}\end{pmatrix}}={\begin{pmatrix}1+t^{2}&t^{3}\\t&1+t^{2}\end{pmatrix}}{\begin{pmatrix}e_{1}\\e_{2}\end{pmatrix}}\,$

   folgt

   ${}{\begin{pmatrix}e_{1}\\e_{2}\end{pmatrix}}={\frac {1}{1+2t^{2}}}{\begin{pmatrix}1+t^{2}&-t^{3}\\-t&1+t^{2}\end{pmatrix}}{\begin{pmatrix}f_{1}\\f_{2}\end{pmatrix}}\,,$

   also

   ${}e_{1}={\frac {1+t^{2}}{1+2t^{2}}}f_{1}-{\frac {t^{3}}{1+2t^{2}}}f_{2}\,$

   und

   ${}e_{2}=-{\frac {t}{1+2t^{2}}}f_{1}+{\frac {1+t^{2}}{1+2t^{2}}}f_{2}\,.$

3. Für den trivialen Zusammenhang ist

   ${}\nabla _{\partial }(f)=\partial f\,.$

   Somit ist

   ${}{\begin{aligned}\nabla _{\partial }(f_{1})&=\nabla _{\partial }{\left({\left(1+t^{2}\right)}e_{1}+t^{3}e_{2}\right)}\\&=2te_{1}+3t^{2}e_{2}\\&=2t{\left({\frac {1+t^{2}}{1+2t^{2}}}f_{1}-{\frac {t^{3}}{1+2t^{2}}}f_{2}\right)}+3t^{2}{\left(-{\frac {t}{1+2t^{2}}}f_{1}+{\frac {1+t^{2}}{1+2t^{2}}}f_{2}\right)}\\&={\left({\frac {2t+2t^{3}}{1+2t^{2}}}-{\frac {3t^{3}}{1+2t^{2}}}\right)}f_{1}+{\left(-{\frac {t^{3}}{1+2t^{2}}}+{\frac {3t^{2}+3t^{4}}{1+2t^{2}}}\right)}f_{2}\\&={\frac {2t-t^{3}}{1+2t^{2}}}f_{1}+{\frac {3t^{2}-t^{3}+3t^{4}}{1+2t^{2}}}f_{2}\end{aligned}}$

   und

   ${}{\begin{aligned}\nabla _{\partial }(f_{2})&=\nabla _{\partial }{\left(te_{1}+{\left(1+t^{2}\right)}e_{2}\right)}\\&=e_{1}+2te_{2}\\&={\frac {1+t^{2}}{1+2t^{2}}}f_{1}-{\frac {t^{3}}{1+2t^{2}}}f_{2}+2t{\left(-{\frac {t}{1+2t^{2}}}f_{1}+{\frac {1+t^{2}}{1+2t^{2}}}f_{2}\right)}\\&={\left({\frac {1+t^{2}}{1+2t^{2}}}-{\frac {2t^{2}}{1+2t^{2}}}\right)}f_{1}+{\left(-{\frac {t^{3}}{1+2t^{2}}}+{\frac {2t+2t^{3}}{1+2t^{2}}}\right)}f_{2}\\&={\frac {1-t^{2}}{1+2t^{2}}}f_{1}+{\frac {2t+t^{3}}{1+2t^{2}}}f_{2}.\end{aligned}}$

   Die Koeffizientenfunktionen sind die Christoffelsymbole.

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