Tensorial map: Difference between revisions

Revision as of 21:24, 2 April 2008

Definition

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

$f : Γ (E) \to Γ (F)$

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

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 $s \in Γ (E)$ : $f_{m} (s (m)) = (f (s)) (m)$
$f$ 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: $g (m) (f (s)) (m) = f (g (m) s) (m)$
$f$ 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.
$f$ 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

@@ Line 7: / Line 7: @@
 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 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