Tensorial map

From Diffgeom
Jump to: navigation, search

Definition

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

f: \Gamma(E) \to \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 \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 \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 \R we have the above condition.
  4. f is linear with respect to the algebra of all smooth (C^\infty) functions. In other words, for any smooth real-valued function g: M \to \R we have the above condition