Riemannsche Mannigfaltigkeit/Zweidimensional/Levi-Civita-Zusammenhang/Schnittkrümmung/Berechnung/Fakt/Beweis

Wir können direkt die Situation auf einer Karte betrachten. Nach Fakt kann man im zweidimensionalen Fall die Schnittkrümmung mit einer beliebigen Basis ausrechnen, wir nehmen ${\displaystyle {}\partial _{1}}$ und ${\displaystyle {}\partial _{2}}$. Für die Schnittkrümmung ist unter Verwendung von Fakt  (5)

${\displaystyle {}{\begin{aligned}K(\partial _{1},\partial _{2})&={\frac {\left\langle R(\partial _{1},\partial _{2})\partial _{2},\partial _{1}\right\rangle }{\left\langle \partial _{1},\partial _{1}\right\rangle \left\langle \partial _{2},\partial _{2}\right\rangle -\left\langle \partial _{1},\partial _{2}\right\rangle ^{2}}}\\&={\frac {\left\langle R(\partial _{1},\partial _{2})\partial _{2},\partial _{1}\right\rangle }{g_{11}g_{22}-g_{12}^{2}}}\\&=-{\frac {\left\langle R(\partial _{1},\partial _{2})\partial _{1},\partial _{2}\right\rangle }{g_{11}g_{22}-g_{12}^{2}}}\end{aligned}}}$

Dabei ist gemäß Fakt

${\displaystyle {}R(\partial _{1},\partial _{2})(\partial _{1})=R_{121}^{1}\partial _{1}+R_{121}^{2}\partial _{2}\,.}$

Dies ergibt

${\displaystyle {}{\begin{aligned}K(\partial _{1},\partial _{2})&=-{\frac {\left\langle R(\partial _{1},\partial _{2})\partial _{1},\partial _{2}\right\rangle }{g_{11}g_{22}-g_{12}^{2}}}\\&=-{\frac {R_{121}^{1}g_{12}+R_{121}^{2}g_{22}}{g_{11}g_{22}-g_{12}^{2}}}.\end{aligned}}}$

Nach Fakt  (4) ist

${\displaystyle {}\left\langle R(\partial _{1},\partial _{1})\partial _{1},\partial _{2}\right\rangle =R_{121}^{1}g_{11}+R_{121}^{2}g_{12}=0\,.}$

Wir multiplizieren den Zähler der Schnittkrümmung mit ${\displaystyle {}g_{11}}$ und erhalten

${\displaystyle {}{\begin{aligned}g_{11}{\left(R_{121}^{1}g_{12}+R_{121}^{2}g_{22}\right)}&=g_{11}{\left(R_{121}^{1}g_{12}+R_{121}^{2}g_{22}\right)}-g_{12}{\left(R_{121}^{1}g_{11}+R_{121}^{2}g_{12}\right)}\\&=g_{11}g_{22}R_{121}^{2}-g_{12}^{2}R_{121}^{2}\\&={\left(g_{11}g_{22}-g_{12}^{2}\right)}R_{121}^{2}.\end{aligned}}}$

Also ist

${\displaystyle {}K(\partial _{1},\partial _{2})=-{\frac {R_{121}^{2}}{g_{11}}}\,.}$