Let \(M\) be a smooth manifold. We define the bundle of covariant \(k\)-tensors on M by
\begin{equation*} T^kT^*M=\coprod_{p\in M} T^k(T^*_pM), \end{equation*}the bundle of contravariant \(k\)-tensors on \(M\) by
\begin{equation*} T^kTM=\coprod_{p\in M} T^k(T_pM) \end{equation*}and the bundle of mixed tensors of type \((k,l)\) on \(M\) by
\begin{equation*} T^{(k,l)}TM=\coprod{p\in M} T^{(k,l)}(T_pM). \end{equation*}
Remarks
- \(T^{(0,1)}TM=T^*M\)
- \(T^{(1,0)}TM=TM\)
- \(T^{(0,k)}TM=T^kT^*M\)
- \(T^{(k,0)}TM=T^kTM\)