$($a$)$ If E$1,$ E$2$ is a tangent frame field on M with connection form w$12,$

show that

K$=$E$[$w $($E$)]-$E$[$w $($E$)]-$w $($E$)2-$w $($E$)2.21211122121122$

$($Hint: Write w$12=$ f$1$q$1+$ f$2$q$2,$ where fi $=$ w$12($Ei$),$ and use Cor. $2\mathrm{.}3.)($b$)$ Check this formula on the sphere in Example $1\mathrm{.}6$