# Difference between revisions of "Tensorial map"

From Diffgeom

(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...) |
m (2 revisions) |
||

(One intermediate revision by the same user not shown) | |||

Line 7: | Line 7: | ||

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 is a differential manifold and are vector bundles over . A -linear map:

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

- There exists a linear map from to for every point , which gives rise to , in the sense that for any and :
- is linear with respect to the algebra of
*all*real-valued functions on . In other words, for any real-valued function we have: - is linear with respect to the algebra of all continuous real-valued functions on . In other words, for any continuous real-valued function we have the above condition.
- is linear with respect to the algebra of all smooth () functions. In other words, for any smooth real-valued function we have the above condition