Difference between revisions of "Tensorial map"

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(New page: ==Definition== Suppose <math>M</math> is a differential manifold and <math>E,F</math> are vector bundles over <math>M</math>. A <math>\R</math>-linear map: <math>f: \Gamma(E) \to...)
 
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between the spaces of sections is termed '''tensorial''' or '''pointwise''' if it satisfies the following equivalent conditions:
 
between the spaces of sections is termed '''tensorial''' or '''pointwise''' if it satisfies the following equivalent conditions:
  
# There exists a linear map <math>f_m</math> from <math>E_m</math> to <math>F_m</math> for every point <math>m</math>, which gives rise to <math>f</math>, in the sense that for any <math>m \in M</math> and <math>s \in \Gamma(E)</math>:
+
# There exists a linear map <math>f_m</math> from <math>E_m</math> to <math>F_m</math> for every point <math>m</math>, which gives rise to <math>f</math>, in the sense that for any <math>m \in M</math> and <math>s \in \Gamma(E)</math>: <math>f_m(s(m)) = (f(s))(m)</math>
<math>f_m(s(m)) = (f(s))(m)</math>
+
# <math>f</math> is linear with respect to the algebra of ''all'' real-valued functions on <math>M</math>. In other words, for any real-valued function <math>g: M \to \R</math> we have: <math>g(m)(f(s))(m) = f(g(m)s)(m)</math>
# <math>f</math> is linear with respect to the algebra of ''all'' real-valued functions on <math>M</math>. In other words, for any real-valued function <math>g: M \to \R</math> we have:
 
<math>g(m)(f(s))(m) = f(g(m)s)(m)</math>
 
 
# <math>f</math> is linear with respect to the algebra of all continuous real-valued functions on <math>M</math>. In other words, for any continuous real-valued function <math>g: M \to \R</math> we have the above condition.
 
# <math>f</math> is linear with respect to the algebra of all continuous real-valued functions on <math>M</math>. In other words, for any continuous real-valued function <math>g: M \to \R</math> we have the above condition.
 
# <math>f</math> is linear with respect to the algebra of all smooth (<math>C^\infty</math>) functions. In other words, for any smooth real-valued function <math>g: M \to \R</math> we have the above condition
 
# <math>f</math> is linear with respect to the algebra of all smooth (<math>C^\infty</math>) functions. In other words, for any smooth real-valued function <math>g: M \to \R</math> we have the above condition

Latest revision as of 20:10, 18 May 2008

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