Update notation

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

Author: mseri

7-tensors.tex

@@ -463,7 +463,7 @@ \section{Tensor bundles}
 	We know that the pullback induces on the fibres a diffeomorphism of cotangent bundles, obtained by collecting the pullbacks on each fibre into a single map:
 	\begin{equation}
 		T^*N \to T^* M, \quad
-		(q,\omega) \mapsto (\varphi^*\omega)_q = \left(\varphi^{-1}(q), d\varphi^*\omega|_{\varphi^{-1}(q)}\right).
+		(q,\omega) \mapsto (\varphi^*\omega)_q = \left(\varphi^{-1}(q), d\varphi^*_{\varphi^{-1}(q)}\omega\right).
 	\end{equation}
 	Since $\phi$ is a differential, this can be inverted giving rise to the so-called \emph{cotangent lift}\footnote{This is often denoted also $\varphi_\sharp$.} of $\varphi$:
 	\begin{equation}
@@ -480,9 +480,9 @@ \section{Tensor bundles}
 	Note that if $p\in M$, for any $\omega\in T_p^* M$ and any $v\in T_pM$, we have
 	\begin{align}
 		(d\varphi^\dagger_p \omega \;\mid\; d\varphi_p v)
-		& =  d(\varphi^{-1})^*(\omega|_{\varphi^{-1}\circ\varphi(p)} (d\varphi_p v) ) \\
-		& = \omega_p(d\varphi^{-1}_{\varphi(p)} \circ d\varphi_p v ) \\
-		& = \omega_p(v) = (\omega_p \mid v).
+		& =  d(\varphi^{-1})^*_{\varphi^{-1}\circ\varphi(p)}(\omega (d\varphi_p v) ) \\
+		& = \omega(d\varphi^{-1}_{\varphi(p)} \circ d\varphi_p v ) \\
+		& = \omega(v) = (\omega \mid v).
 	\end{align}
 
 	\newthought{Step II}.