Tensorial map

Definition

Suppose $M$ is a differential manifold and $E,F$ are vector bundles over $M$. A $\mathbb{R}$-linear map:

$f:\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(F)$

between the spaces of sections is termed tensorial or pointwise if it satisfies the following equivalent conditions:

1. There exists a linear map $f_m$ from $E_m$ to $F_m$ for every point $m$, which gives rise to $f$, in the sense that for any $m\in M$ and $s\in \mathrm{\Gamma }(E)$: $f_m(s(m))=(f(s))(m)$
2. $f$ is linear with respect to the algebra of all real-valued functions on $M$. In other words, for any real-valued function $g:M\to \mathbb{R}$ we have: $g(m)(f(s))(m)=f(g(m)s)(m)$
3. $f$ is linear with respect to the algebra of all continuous real-valued functions on $M$. In other words, for any continuous real-valued function $g:M\to \mathbb{R}$ we have the above condition.
4. $f$ is linear with respect to the algebra of all smooth ($C^{\mathrm{\infty }}$) functions. In other words, for any smooth real-valued function $g:M\to \mathbb{R}$ we have the above condition