The covariant derivative on the dual bundle is defined as follows:
$\nabla^{*}: \Gamma(TM) \times \Gamma(E^*) \ni (X, t) \mapsto \nabla_X^{*} t \in \Gamma(E^*)$
, where for any section $s \in \Gamma(E)$,
$(\nabla_X^{*} t)(s) = L_X(t(s)) – t(\nabla_X s)$.
Remark: $\Gamma(E^*)$ is the set of sections of the dual bundle and $\Gamma(TM)$ is the set of vector fields.
I would like to check whether this is indeed a covariant derivative. I have already proved that it satisfies function-linearity, but now I have difficulties to show that it satisfies Leibniz rule, i.e. $(\nabla_X^* ft) = (L_X f)t + f\nabla^*_X t$.
I need to verify that
$(\nabla_X^* ft)(s) = L_X(ft(s)) – ft(\nabla_X s) = \ldots = (L_X f)t(s) + f(\nabla^*_X t)(s). $ Can someone help me to find the intermediate steps ?
Thanks.
Best Answer
As written in the post, $$\nabla^*_Xft(s)=X(ft(s))-ft\nabla_Xs.$$ Now, using the Leibniz rule for the first summand on the right, $$=X(f)t(s)+fX(t(s))-ft\nabla_Xs=X(f)t(s)+f\nabla^*_Xt(s).$$