Tensor field

Definition

Let $M$ be a smooth manifold of dimension $n$, let $T_pM$ be the tangent space at $p \in M$, and let $T_p^*M$ be its dual space. Let $k, l \ge 0$ be integers. A tensor of type $(k, l)$ at $p$ is an element of the tensor product $$T^k_l(T_pM) = \underbrace{T_pM \otimes \cdots \otimes T_pM}_{k} \otimes \underbrace{T_p^*M \otimes \cdots \otimes T_p^*M}_{l}.$$ These spaces form a smooth vector bundle $T^k_l M \to M$, built from the tangent bundle as $TM$ is. A tensor field of type $(k, l)$ on $M$ is a smooth section of $T^k_l M$. In a chart with coordinates $x^1, \dots, x^n$, a tensor field $T$ is written as follows, with summation over repeated indices. $$T = T^{i_1 \cdots i_k}_{j_1 \cdots j_l} \, \partial_{i_1} \otimes \cdots \otimes \partial_{i_k} \otimes dx^{j_1} \otimes \cdots \otimes dx^{j_l}.$$ Smoothness means that the component functions $T^{i_1 \cdots i_k}_{j_1 \cdots j_l}$ are smooth in every chart.