I just started studying differentiable manifolds, and I've encountered a few allegedly simple questions that aren't really simple to me.
If possible, I'd be very happy to see full proofs (without leaving logical gaps left out for adept readers, since I'm just getting used to these formalisms) of following identities:
- $L_X (\omega \wedge \eta) = (L_X \omega ) \wedge \eta + (-1)^p \omega \wedge L_X \eta $ where $\omega$ is a p-form.
- $L_X \omega = di_X \omega + i_Xd \omega$ where $d$ is exterior derivative and $i_X$ is interior derivative.
- $ (L_{Y_0}\omega)(Y_1 \cdots Y_p) = L_{Y_0} (\omega(Y_1 \cdots Y_p )) - \sum_{k=1}^p \omega(Y_1 \cdots Y_{k-1} (L_{Y_0}Y_k) , \cdots Y_p)$
- $d\omega (Y_0 \cdots Y_p) = \sum_{i=0}^p (-1)^i Y_i (\omega(Y_0 \cdots \hat Y_i \cdots Y_p)) + \sum_{i<j} (-1)^{i+j} \omega([Y_i,Y_j], Y_0 , \cdots \hat Y_i , \cdots \hat Y_j \cdots Y_p)$ where $\hat\cdot$ represents omission of that symbol.
Thank you so much in advance.