Let \(F\colon M\to N\) be smooth. Then the pullback \(F^*\) of a smooth covariant \(k\)-tensor field \(A\) is defined by
\begin{equation*} (F^*A)_p(v_1, \ldots, v_k)=A_{F(p)}(dF_p(v_1),\ldots,dF_p(v_k)). \end{equation*}
Remark
- \(F^*A\) is smooth if \(A\) also is.